

MATHEMATICS AND 
THE IMAGINATION 


MATHEMATICS and 
the IMAGINATION 

Edward Kasner and James Newman 


With Drawings and Diagrams by Rufus Isaacs 
LONDON; G. BELL AND SONS, LTD 


ALL RIGHTS RESERVED 
INCLUDING THE RIGHT OF REPRODUCTION 
IN WHOLE OR IN PART IN ANY FORM 


33247 


British Edition first published 194^ 
ReprinUd /p 5 o, 1952, 1956, 19S9 


PRINTED IN GREAT BRITAIN 


BY JARROLD AND 


SONS LIMITED, NORWICH 


Xo R.G. 

without whose selfless help 
and understanding there would 
have been no book. 


ACKNOWLEDGMENT 


We are indebted to many books, too many to enumerate. 
Some of them are listed in the selected bibliography. 

And we wish to acknowledge particularly the services of Mr. 
Don Mittleman of Columbia University, whose help in prepa¬ 
ration of the manuscript has been generous and invaluable. 


Table of Contents 


Introduction 

I. NEW NAMES FOR OLD 

Easy words Jot hard ideas . . . Transcendental . . . Non- 
simple curve . . . Simple curve . . . Simple group . . . Bolshe¬ 
viks and giraffes . . . Turbines. . . Turns and slides . . . 
Circles and cycles . . . Patho-circles . . . Clocks . . . Hexagons 
and parhexagons . . . Radicals, hyperradicals, and ultrarad¬ 
icals {nonpoliticaf) . . . New numbers for the nursery . . . 
Googol and googolplex . . . Miracle of the rising book . . . The 
mathescope. 

II. BEYOND THE GOOGOL 

Counting—the language of number . . . Counting, matching, 
and '‘"‘Going to Jerusalem'" . . . Cardinal numbers . . . Cosmic 
chess and googols . . . The sand reckoner . . . Mathematical 
induction . . . The infinite and its progeny . . . ^eno . . . Puz¬ 
zles and quarrels . . . Bolzano . . . Galileo''s puzzle . . . 
Cantor . . . Measuring the measuring rod. . . The whole is 
no greater than some of its parts ... The first transfinite— 
Alepho. . . Arithmetic Jot morons . . . Common sense hits a 
snag . . . Cardinality oj the continuum , . . Extravagances oj 
a mathematical madman . . . The tortoise unmasked . . . 
Motionless motion ... Private life oJ a number ... The 
house that Cantor built. 


III. TT, i, e (pie) 

Chinamen and chandeliers . .. Twilight oJ common sense . . . 
TT, i, e . . . Squaring the circle and its cousins . . . Mathe¬ 
matical impossibility . . . Silk purse, sow''s ear, ruler and 
compass . . , Rigor mortis . . . Algebraic equations and tran- 

vii 


Vlll 


Contents 


scendental numbers . . . Galois and Greek epidemics . . . Cube 
duplicators and angle trisectors . . . Biography of tt . . . 
Infancy: Archimedes^ the Bible^ the Egyptians . . . Adoles¬ 
cence: Vieta, Van Ceulen . . . Maturity: Wallis^ Newton, 
Leibniz • • • Old Age: Base, Richter, Shanks . . . Victim of 
schizophrenia . . . Boon to insurance companies . . . (e) . . . 
Logarithms or tricks of the trade . . . Mr. Briggs is surprised 
. . . Mr. Napier explains . . . Biography oft; or e, the bank¬ 
er's boon . . . Pituitary gland of mathematics: the exponential 
function . . . (i) . . . Humpty Dumpty, Doctor of Semantics 
. . . Imaginary numbers . . . The y/- 1, or Where am /?” 
• • • Biography of i, the self-made amphibian . . . Omar 
Khayyam, Cardan, Bombelli, and Gauss . . . i and Soviet 
Russia . . . Program music of mathematics . . . Breakfast in 
bed; or, How to become a great mathematician . . . Analytic 
geometry . . . Geometric representation of \. . . Complex plane 
. . . A famous formula, faith, and humility. 


IV. 


ASSORTED GEOMETRIES—PLANE AND FANCY 

The talking fish and St. Augustine ... A new alphabet . . . 
High priests and mumbo jumbo . . . Pure and applied mathe¬ 
matics . . . Euclid and Texas . . . Mathematical tailors . . . 
Geometry a game . . . Ghosts, table-tipping, and the land of 
the dead . . . Fourth-dimension flounders . . . Henry More to 
the rescue . . . Fourth dimension—a new gusher . . . A cure 
for arthritis . . . Syntax suffers a setback . . . The physicisCs 
delight ... Dimensions and manifolds . . . Distance formulae 

■ - . Scaling blank walls . . . Four-dimensional geometry de¬ 
fined , . . Moles and tesseracts . . . A four-dimensional fancy 

. Romance of flatland . . . Three-dimensional cats and two- 
dimensional kings . . . Gallant Gulliver and the gloves . . . 
Beguiling voices and strange footprints . . . Non-Euclidean 
geometry . . . Space credos and millinery . . . Private and 
public space . . . Rewriting our textbooks ...The prince and 
the Bofthiam ...The flexible flfth ... The mathematicians 
unite nothing to lose but their chains . . . Lobachevsky breaks 
u link . . . Riemann breaks another . . . Checks and double 
checks in nrathematics ...The tractnx and the psendosphere 

■ . . Great circles and bears ...The skeptic persists-and ts 


I 


Contents 


IX 


stepped on . . . Geodesics. . . Seventh Day Adventists . . . 
Curvature . . . Lobachevskian Eiffel Towers and Riemannian 
Holland Tunnels. 

V. PASTIMES OF PAST AND PRESENT TIMES 1 56 

Puzzle acorns and mathematical oaks . . . Charlemagne and 
crossword puzzles . . . Mark Twain and the “farmer's daugh¬ 
ter" . . . The syntax of puzzles . . . Carolyn Flaubert and the 
cabin boy . . . A wolf^ a goat, and a head of cabbage . . . 
Brides and cuckolds . . . Til be switched . . . Poisson, the mis¬ 
fit.. . High finance; or. The international beer wolf . . . 
Lions and poker players . . . The decimal system . . . Casting 
out nines . . . Buddha, God, and the binary scale . . . The 
march of culture; or, Russia, the home of the binary system . . . 

The Chinese rings . . . The tower of Hanoi. . . The ritual of 
Benares: or, Charley horse in the Orient. . . Nim, Sissa Ben 
Dahir, and Josephus . . . Bismarck plays the boss . . . The 15 
puzzle plague . . . The spider and the fly ... A nightmare of 
relatives . . . The magic square . . . Take a number from 1 to 
10 . . . Fermat's last theorem . . . Mathematics' lost legacy. 

VI. PARADOX LOST AND PARADOX REGAINED 193 

Great paradoxes and distant relatives . . . Three species of par¬ 
adox . . . Paradoxes strange but true . . . Wheels that move 
faster on top than on bottom . . . The cycloid family . . . The 
curse of transportation; or. How locomotives can't make up 
their minds . . . Reformation of geometry . . . Ensuing troubles 
. . . Point sets—the Arabian Nights of mathematics . . . 
Hausdorff spins a tall tale . . . Messrs. Banach and Tarski 
rub the magic lamp . . . Baron Munchhausen is stymied by a 
pea . . . Mathematical fallacies . . . Trouble from a bubble; 
or, Dividing by zero . . . The infinite—troublemaker par ex¬ 
cellence . . . Geometrical fallacies . . . Logical paradoxes—the 
folk tales of mathematics . . . Deluding dialectics of the poacher 
and the prince; of the introspective barber; of the number 
111777; of this book and Confucius; of the Hon. Bertrand 
Russell. . . Scylla and Charybdis; or, What shall poor mathe- 
mathics do? 


X 


Contents 


VII. CHANCE AND CHANCEABILITY 223 

The clue of the billiard cue ... A little chalk, a lot of talk . . . 
Watson gets his leg pulled by probable inference . . . Finds it 
all absurdly simple . . . Passionate oysters, waltzing ducks, 
and the syllogism . . . The twilight of probability . . . Inter^ 
esting behavior of a modest coin . . . Biological necessity and a 
pair of dice . . . What is probability? ... A poll of views: a 
meteorologist, a bootlegger, a bridge player . . . The subjective 
view—based on insufficient reason, contains an element of truth 
. . . The jackasses on Afars . . . The statistical view . . . 

What happens will probably happen . . . Experimental euryth- 
mics; or, Pitching pennies . . . Relative frequencies . . . The 
adventure of the dancing men . . . Scheherezode and John 
Wilkes Booth—a challenge to statistics . . . The red and the 
black . . . Charles Peirce predicts the weather . . . How far is 
^''away'’'’? . . . Herodotus explains . . . The calculus of chance 
... The benefits of gambling . . . De Mere and Pascal . . . 

Mr. Jevons omits an acknowledgment . . . The study of craps 
the very guide of life . . . Dice, pennies, permutations, and 
combinations . . . Measuring probabilities . . . D'Alembert 
drops the ball . . . Count Buffon plays with a needle . . . The 
point ... A black ball and a white ball ... The binomial 
theorem . . . The calculus of probability re-examined . . . 
Found to rest on hypothesis . . . Laplace needs no hypothesis 
. . . Twits Napoleon, who does . . . The Marquis de Condorcet 
has high hopes . . . M.le Marquis omits a factor and loses his 
head . . . Fourier of the Old Guard ... Dr. Darwin of the 
New . . . The syllogism scraps a standby . . . Mr. Socrates 
may not die .. . Ring out the old logic, ring in the new. 


VIII. RUBBER-SHEET GEOMETRY 265 

Seven bridges over a stein of beer .. . Euler shivers . . . Is 
warmed by news from home . . . Invents topology . . . Dis¬ 
solves the dilemma of Sunday strollers . . . Babies' cribs and 
Pythagoreans . . . Taltsmen and queer figures . . . Position is 
everything m topology ...Da Vinci' and Dali . . . Invariants 
. ..Transformations ...The immutable derby . . . Com¬ 
petition for the cahpk's cup; or. Sifting out the suitors by 


Contents 


XI 


science . . . Mr. Jordan's theorem . . . Only seems idiotic . . . 
Dejormed circles . . . Odd facts concerning Times Square and 
a balloonist's head . . . Eccentric deportment of several dis¬ 
tinguished gentlemen at Princeton . . . Their passion for pret¬ 
zels . . . Their delving in doughnuts . . . Enforced modesty of 
readers and authors . . . The ring . . . Lachrymose recital 
around a Pans pissoir . . . “ Who staggered how many times 
around the walls of what?" . . . In and out the doughnut . . . 
Gastric surgery—from doughnut to sausage in a single cut . . . 
^/-dimensional pretzels . . . The Mobius strip . . . Just as 
black as it is painted . . . Foments industrial discontent . . . 
Never takes sides . . . Bane of painter and paintpot alike . . . 

The iron rings . . . Mathematical cotillion; or., How on earth 
do I get rid of my partner? . . . Topology—the pinnacle of 
perversity; or, Removingyour vest without your coat . . . Down 
to earth—map coloring . . . Four-color problem . . . Euler's 
theorem . . . The simplest universal law . . . Brouwer's puz¬ 
zle .. . The search for invariants. 

IX. CHANGE AND CHANGEABILITY 299 

The calculus and cement . . . Meaning of change and rate of 
change . . . ^eno and the movies . . . '‘‘‘Flying Arrow" local — 
stops at all points . . . Geometry and genetics . . . The arith¬ 
metic men dig pits . . . Lamentable analogue of the boomerang 
. . . History of the calculus . . . Kepler . . . Fermat . . . 

Story of the great rectangle . . . Newton and Leibniz • • • 
Archimedes and the limit . . . Shrinking and swelling; or, 

“ Will the circle go the limit?" . . . Brief dictionary of mathe¬ 
matics and physics . . . Military idyll; or. The speed of the 
falling bomb . . . The calculus at work . . . The derivative 
. . . Higher derivatives and radius of curvature . . . Laudable 
scholarship of automobile engineers . . . The third derivative 
as a shock absorber . . . The derivative finds its male . . . 
Integration . . . Kepler and the bungholes . . . Measuring 
lengths; or. The yawning regress . . . Methods of approx¬ 
imation . . . Measuring areas under curves . . . Method of 
rectangular strips . . . The definite S • • • Indefinite J' . . . 

One the inverse of the other . . . The outline of history and the 
descent of man: or, y = e^ . . . Sickly curves and orchidaceous 


XU 


Contents 


ones . . . The snowflake . . . Inflnite perimeters and postage 
stamps . . . Anti-snowflake . . . Super-colossal pathological 
specimen—the curve that fills space . . . The unbelieveable 
crisscross. 


EPILOGUE. MATHEMATICS AND THE IMAGINATION 


357 


Introduction 


The fashion in books in the last decade or so has turned increas¬ 
ingly to popular science. Even newspapers^ Sunday supplements 
and magazit^s have given space to relativity^ atomic physics^ 
and the newest marvels of astronomy and chemistry. Symptomatic 
as this is of the increasing desire to know what happens in 
laboratories and observatories, as well as in the awe-inspiring 
conclaves of scientists and mathematicians, a large part of 
modern science remains obscured by an apparently impenetrable 
veil of mystery. The feeling is widely prevalent that science, 
like magic and alchemy in the Aiiddle Ages, is practiced and 
can be understood only by a small esoteric group. The mathema¬ 
tician is still regarded as the hermit who knows little of the ways 
of life outside his cell, who spends his time compounding incredible 
and incomprehensible theories in a strange, clipped, unintelligible 
jargon. 

Nevertheless, intelligent people, weary of the nervous pace of 
their own existence—the sharp impact of the happenings of the 
day—are hungry to learn of the accomplishments of more leisurely, 
contemplative lives, timed by a slower, more deliberate clock 
than their own. Science, particularly mathematics, though it 
seems less practical and less real than the news contained in the 
latest radio dispatches, appears to be building the one permanent 
and stable edifice in an age where all others are either crumbling 
or being blown to bits. This is not to say that science has not 
also undergone revolutionary changes. But it has happened 
quietly and honorably. That which is no longer useful has been 
rejected only after mature deliberation, and the building has been 

reared steadily on the creative achievements of the past. 

• • • 
xm 


xiv Introduction' 

ThuSy in a certain sense, the popularization oj science is a 
duty to be performed, a duty to give courage and comfort to the 
men and women of good will everywhere who are gradually losing 
their faith in the life of reason. For most of the sciences the veil 
of mystery is gradually being torn asunder. Mathematics, in 
large measure, remains unrevealed. What most popular books on 
mathematics have tried to do is either to discuss it philosophically, 
or to make clear the stuff once learned and already forgotten. 
In this respect our purpose in writing has been somewhat 
different. ^‘Haute vulgarisation*' is the term applied by the 
French to that happy result which neither offends by its condescen¬ 
sion nor leaves obscure in a mass of technical verbiage. It has 
been our aim to extend the process of '■'■haute vulgarisation" to 
those outposts of mathematics which ate mentioned, if at all, 
only in a whisper; which are referred to, if at all, ordy by name; 
to show by its very diversity something of the character of 
mathematics, of its bold, untrammeled spirit, of how, as both 
an art and a science, it has continued to lead the creative faculties 
beyond even imagination and intuition. In the compass of so 
brief a volume there can only be snapshots, not portraits. Yet, it 
is hoped that even in this kaleidoscope there may be a stimulus 
to further interest in and greater recognition of the proudest queen 
of the intellectual world. 


MATHEMATICS AND 
THE IMAGINATION 


I tuill not go so Jar as to say that to construct a history 
of thought without profound study of the mathematical 
ideas of successive epochs is like omitting Hamlet from the 
play which is named after him. That would be claiming 
too much. But it is certainly analogous to cutting out the 
part of Ophelia. This simile is singularly exact. For 
Ophelia is quite essential to the play, she is very charm¬ 
ing,—and a little mad. Let us grant that the pursuit of 
mathematics is a divine madness of the hutnan spirit, a 
refuge from the goading urgency of contingent happenings. 

-ALFRED NORTH WHITEHEAD, 

Science and the Modern World. 


New Names for Old 


For out of aide feldeSy as men seithy 
Cometh al this newe corn fro yeer to yere; 

And out of olde bakes, in good feith, 

Cometh al this newe science that men lere. 

—CHAUCER 


Every once in a while there is house cleaning in mathe¬ 
matics. Some old names are discarded, some dusted off 
and refurbished; new theories, new additions to the 
household are assigned a place and name. So what our 
title really means is new words in mathematics; not new 
names, but new words, new terms which have in part 
come to represent new concepts and a reappraisal of old 
ones in more or less recent mathematics. There are surely 
plenty of words already in mathematics as well as in other 
subjects. Indeed, there are so many words that it is even 
easier than it used to be to speak a great deal and say 
nothing. It is mostly through words strung together like 
beads in a necklace that half the population of the world 
has been induced to believe mad things and to sanctify 
mad deeds. Frank Vizetelly, the great lexicographer, 
estimated that there are 800,000 words in use in the 
English language. But mathematicians, generally quite 
modest, are not satisfied with these 800,000; let us give 
them a few more. 

We can get along without new names until, as we ad¬ 
vance in science, we acquire new ideas and new forms. 

3 


2 


4 Mathematics and the Imagination 

A peculiar thing about mathematics is that it does not 
use so many long and hard names as the other sciences. 
Besides, it is more conservative than the other sciences 
in that it clings tenaciously to old words. The terms 
used by Euclid in his Elements are current in geometry 
today. But an Ionian physicist would find the terminol¬ 
ogy of modern physics, to put it colloquially, pure Greek. 
In chemistry, substances no more complicated than 
sugar, starch, or alcohol have names like these: Meth- 

ylpropenylenedihydroxycinnamenylacrylic acid, or, 0- 
anhydrosulfaminobenzoine, or, protocatcchuicaldehyde- 
methylene. It would be inconvenient if we had to use such 
terms in every'day conversation. Who could imagine 
even the aristocrat of science at the breakfast table asking, 
Please pass the O-anhydrosulfaminobenzoic acid,” 
when all he wanted was sugar for his coffee? Biology also 
has some tantalizing tongue twisters. The purpose of 
these long words is not to frighten the exoteric, but to 
describe with scientific curtness what the literary man 
would take half a page to express. 

In mathematics there are many easy words like 
group, “family,” “ring,” “simple curve,” “limit,” etc. 
But these ordinary words are sometimes given a very 
peculiar and technical meaning. In fact, here is a booby- 
prize definition of mathematics: Mathematics is the science 
which uses easy words for hard ideas. In this it differs from 
any other science. There are 500,000 known species of 
insect and every one has a long Latin name. In math¬ 
ematics we are more modest. We talk about “fields,” 
“groups,” “families,” “spaces,” although much more 
meaning is attached to these words than ordinary con¬ 
versation implies. As its use becomes more and more 
technical, nobody can guess the mathematical meaning 


New Names for Old ^ 

of a word any more than one could guess that a “drug 
store ’ is a place where they sell ice-cream sodas and 
umbrellas. No one could guess the meaning of the word 
“group” as it is used in mathematics. Yet it is so impor¬ 
tant that whole courses are given on the theory of 
‘ groups,” and hundreds of books are written about it. 

Because mathematicians get along with common words, 
many amusing ambiguities arise. For instance, the word 
“function” probably expresses the most important idea 
in the whole history of mathematics. Yet, most people 
hearing it would think of a “function” as meaning an 
evening social affair, while others, less socially minded, 
would think of their livers. The word “function” has 
at least a dozen meanings, but few people suspect the 
mathematical one. The mathematical meaning (which 
we shall elaborate upon later) is expressed most simply 
by a table. Such a table gives the relation between two 
variable quantities when the value of one variable quan¬ 
tity is determined by the value of the other. Thus, one 
variable quantity may express the years from 1800 to 
1938, and the other, the number of men in the United 
States wearing handle-bar mustaches; or one variable 
may express in decibels the amount of noise made by a 
political speaker, and the other, the blood pressure units 
of his listeners. You could probably never guess the mean¬ 
ing of the word “ring” as it has been used in mathematics. 
It was introduced into the newer algebra within the last 
twenty years. The theory of rings is much more recent 
than the theory of groups. It is now found in most of the 
new books on algebra, and has nothing to do with cither 
matrimony or bells. 

Other ordinary words used in mathematics in a pe¬ 
culiar sense are “domain,” “integration,” “differentia- 


6 


Mathematics and the Imagination 

tion.” The uninitiated would not be able to guess what 
they represent; only mathematicians would know about 
them. The word “transcendental” in mathematics has 
not the meaning it has in philosophy. A mathemati¬ 
cian would say: The number tt, equal to 3.14159 . . . , 
is transcendental, because it is not the root of any alge¬ 
braic equation with integer coefficients. 

Transcendental is a very exalted name for a small 
number, but it was coined when it was thought that 
transcendental numbers were as rare as quintuplets. 
The work of Georg Cantor in the realm of the infinite 
has since proved that of all the numbers in mathematics, 
the transcendental ones are the most common, or, to 
use the word in a slighdy different sense, the least tran¬ 
scendental. We shall talk of this later when we speak of 
another famous transcendental number, e, the base of 
the natural logarithms. Immanuel Kant’s “transcen¬ 
dental epistemology” is what most educated people 
might think of when the word transcendental is used, 
but in that sense it has nothing to do with mathematics. 
Again, take the word “evolution,” used in mathematics 
to denote the process most of us learned in elementary 
school, and promptly forgot, of extracting square roots, 
cube roots, etc. Spencer, in his philosophy, defines 
evolution as ‘ an integration of matter, and a dissipation 
of motion from an indefinite, incoherent homogeneity 
to a definite, coherent heterogeneity,” etc. But that, 
formnately, has nothing to do with mathematical evo¬ 
lution either. Even in Tennessee, one may extract square 
roots without running afoul of the law. 

As \'. e see, mathematics uses simple words for com¬ 
plicated ideas. An example of a simple word used in a 
complicated way is the word “simple.” “Simple curve” 


New Names jor Old 7 

and “simple group” represent important ideas in higher 
mathematics. 


The above is not a simple curve. A simple curve is a 
closed curve which does not cross itself and may look like 
Fig. 2. There are many important theorems about such 
figures that make the word worth while. Later, we are 


FIG. 2 


going to talk about a queer kind of mathematics called 
“ntbber-sheet geometry,” and will have much more to 
say about simple curves and nonsimple ones. A French 
mathematician^ Tordan. gave the fundamental theorem: 
every simple curve has one inside and one outside. That 
is, every simple curve divides the plane into two regions, 
one inside the curve, and one outside. 

There are some groups in mathematics that arc 
“simple” groups. The definition of “simple group” is 
really so hard that it cannot be given here. If we wanted 
to get a clear idea of what a simple group was, we should 


8 Mathematics and the Imagination 

probably have to spend a long time looking into a great 
many books, and then, without an extensive mathemat¬ 
ical background, we should probably miss the point. 
First of all, we should have to define the concept “group.” 
Then we should have to give a definition of subgroups, 
and then of self-conjugate subgroups, and then we should 
be able to tell what a simple group is. A simple group 
is simply a group without any self-conjugate subgroups— 
simple, is it not? 

Mathematics is often erroneously referred to as the 
science of common sense. Actually, it may transcend 
common sense and go beyond either imagination or 
intuition. It has become a very strange and perhaps 
frightening subject from the ordinary point of view, but 
anyone who penetrates into it will find a veritable fairy¬ 
land, a fairyland which is strange, but makes sense, if 
not common sense. From the ordinary point of view 
mathematics deals with strange things. We shall show 
you that occasionally it does deal with strange things, 
but mostly it deals with familiar things in a strange way. 
If you look at yourself in an ordinary mirror, regardless 
of your physical attributes, you may find yourself amus- 
ing, but not strange; a subway ride to Coney Island, and 
a glance at yourself in one of the distorting mirrors will 
convince you that from another point of view you may be 
strange as well as amusing. It is largely a matter of what 
you arc accustomed to. A Russian peasant came to Mos¬ 
cow for the first time and went to see the sights. He went 
to the zoo and saw the giraffes. You may find a moral in 
his reaction as plainly as in the fables of La Fontaine. 
“Look/’ he said, “at what the Bolsheviks have done to 
our horses.” That is what modern mathematics has done 
to simple geometry and to simple arithmetic.^ 


New Names for Old 9 

There are other words and expressions, not so familiar, 
which have been invented even more recently. Take, 
for instance, the word “turbine.’* Of course, that is 
already used in engineering, but it is an entirely new 
word in geometry. The mathematical name applies to 
a certain diagram. (Geometry, whatever others may 
think, is the study of different shapes, many of them very 
beautiful, having harmony, grace and symmetry. Of 
course, there are also fat books written on abstract geom¬ 
etry, and abstract space in which neither a diagram nor 
a shape appears. This is a very important branch of 
mathematics, but it is not the geometry studied by the 
Egyptians and the Greeks. Most of us, if we can play 
chess at all, are content to play it on a board with wooden 


\ U / / 


/ / M \ 


FIG. 3.—Turbines. 


chess pieces; but there are some who play the game 
blindfolded and without touching the board. It might 
be a fair analogy to say that abstract geometry is like 
blindfold chess—it is a game played without concrete 
objects.) Above you see a picture of a turbine, in fact, two 
of them. 

A turbine consists of an infinite number of “elements” 
filled in continuously. An element is not merely a point; 


I o Mathematics and the Imagination 

it is a point with an associated direction—like an iron 
filing. A turbine is composed of an infinite number of 
these elements, arranged in a peculiar way: the points 
must be arranged on a perfect circle, and the inclination 
of the iron filings must be at the same angle to the circle 
throughout. There are thus, an infinite number of ele¬ 
ments of equal inclination to the various tangents of 
the circle. In the special case where the angle between 
the direction of the element and the direction of the 
circle is zero, what would happen? The turbine would 
be a circle. In other words, the theory of turbines is a 
generalization of the theory of the circle. If the angle 
is ninety degrees, the elements point toward the center 
of the circle. In that special case we hav^ a normal 
turbine (see left-hand diagram). 

There is a geometry of turbines, instead of a geometry 

of circles. It is a rather technical branch of mathematics 

which concerns itself with working out continuous groups 

of transformations connected with differential equations 

and differential geometry. The group connected with 

the turbine bears the rather odd name of “turns and 
slides.” 

♦ 

The circle is one of the oldest figures in mathematics. 

The straight line is the simplest line, but the circle is 

the simplest nonstraight line. It is often regarded as the 

limit of a polygon with an infinite number of sides. You 

can see for yourself that as a series of polygons is inscribed 

in a circle with each polygon having more sides than its 

predecessor, each polygon gets to look more and more 
like a circle.^ 

The Greeks were already familiar with the idea that 
as a regular polygon increases in the number of its sides, 


New Names for Old 11 

it differs less and less from the circle in which it is in¬ 
scribed. Indeed, it may well be that in the eyes of an 
omniscient creature, the circle would look like a polygon 
with an infinite number of straight sides. ^ However, in 
the absence oi complete omniscience, we shall continue 


FIG. 4.—The circle as the limit of inscribed polygons. 


to regard a circle as being a nonstraight line. There are 
some interesting generalizations of the circle when it 
is viewed in this way. There is, for example, the concept 
denoted by the word “cycle,” which was introduced by 
a French mathematician, Laguerre. A cycle is a circle 
with an arrow on it, like this: 


If you took the same circle and put an arrow on it in 
the opposite direction, it would become a different cycle. 
The Greeks were specialists in the art of posing prob- 


12 Mathematics and the Imagination 

lems which neither they nor succeeding generations of 
mathematicians have ever been able to solve. The three 
most famous of these problems—the squaring of the 
circle, the duplication of the cube, and the trisection of 
an angle—we shall discuss later. Many well-meaning, 
self-appointed, and self-anointed mathematicians, and 
a motley assortment of lunatics and cranks, knowing 
neither history nor mathematics, supply an abundant 
crop of “solutions” of these insoluble problems each year. 
However, some of the classical problems of antiquity 
have been solved. For example, the theory of cycles was 
used by Laguerre in solving the problem of Apollonius: 
given three fixed circles, to find a circle that touches 
them all. It turns out to be a matter of elementary high 


New Names Jor Old 13 

school geometry, although it involves ingenuity, and 
any brilliant high school student could work it out. It 
has eight answers, as shown in Fig. 6(a). 

They can all be constructed with ruler and compass, 
and many methods of solution have been found. Given 
three circles^ there will be eight circles touching all of them. 
Given three cycles^ however, there will be only one cycle 
that touches them all. (Two cycles are said to touch each 
other only if their arrows agree in direction at the point 
of contact.) Thus, by using the idea of cycles, we have 
one definite answer instead of eight. Laguerre made 
the idea of cycles the basis of an elegant theory. 


FIG. 6(b). —The eight solutions of Appolonius 
merged into one diagram. 

Another variation of the circle introduced by the emi¬ 
nent American mathematician, C. J. Keyser, is obtained 
by taking a circle and removing one point.^ This creates 
a serious change in conception. Keyser calls it “a patho- 
circle,” (from pathological circle). He has used it in 
discussing the logic of axioms. 


14 Mathematics and the Imagination 

We have made yet another change in the concept of 
circle, which introduces another word and a new di¬ 
agram. Take a circle and instead of leaving one point 
out, simply emphasize one point as the initial point. 
This is to be called a “clock.’’ It has been used in the 
theory of polygenic functions. “Pplygeiiic” is a word 
recently introduced into the theory of complex functions 
—about 1927. There was an important word, “mono¬ 
genic,” introduced in the nineteenth century by the 
famous French mathematician, Augustin Cauchy, and 
used in the classical theory of functions. It is used to 
denote functions that have a single derivative at a point, 
as in the differential calculus. But most functions, in the 
complex domain, have an infinite number of derivatives 
at a point. If a function is not monogenic, it can never 
be bigenic, or trigenic. Either the derivative has one 
value or an infinite number of values—either monogenic 
or polygenic, nothing intermediate. Monogenic means 
one rate of growth. Polygenic means many rates of 
growth. The complete derivative of a polygenic function 
is represented by a congruence (a double infinity) of 
clocks, all with different starting points, but with the 


same uniform rate of rotation. It would be useless to 
attempt to give a simplified explanation of these con¬ 
cepts. (The neophyte will have to bear with us over a 

few intervals like this for the sake of the more experienced 
mathematical reader.) 


New Names jor Old 


The going has been rather hard in the last paragraph, 
and if a few of the polygenic seas have swept you over¬ 
board, we shall throw you a hexagonal life preserver. 
We may consider a very simple word that has been intro¬ 
duced in elementary geometry to indicate a certain kind 
of hexagon. The word on which to fix your attention is 
‘"parhexagon.” An ordinary hexagon has six arbitrary 
sides. A parhexagon is that kind of hexagon in which 


any side is both equal and parallel to the side opposite 
to it (as in Fig, 7). 

If the opposite sides of a quadrilateral are equal and 
parallel, it is called a parallelogram. By the same rea¬ 
soning that we use for the word parhexagon, a parallelo¬ 
gram might have been called a parquadrilateral. 

Here is an example of a theorem about the parhex¬ 
agon: take any irregular hexagon, not necessarily a 
parhexagon, ABCDEF. Draw the diagonals AC, BD, 
CE, DF, EA, and FB, forming the six triangles, ABC, 
BCD, CDE, DEF, EFA, and FAB. Find the six centers 
of gravity, A', B', C', D', E', and F' of these triangles. 
(The center of gravity of a triangle is the point at which 
the triangle would balance if it were cut out of cardboard 
and supported only at that point; it coincides with the 


D 


FIG. -ABCDEF \s an irregular hexagon. A'B' 
C'D'E'F' is a parhexagon. 


16 Mathematics and the Imagination 

point of intersection of the medians.) Draw A'B', B'C', 
C'D', D'E', E'F', and F'A'. Then the new inner hex¬ 
agon A'B'C'D'ET' will always be a parhexagon. 

The word radical, favorite call to arms among Repub¬ 
licans, Democrats, Communists, Socialists, Nazis, Fas¬ 
cists, Trotskyites, etc., has a less hortatory and bellicose 
character in mathematics. For one thing, everybody 
knows its meaning: i.e., square root, cube root, fourth 
root, fifth root, etc. Combining a word previously de¬ 
fined with this one, we might say that the extraction of a 
root is the evolution of a radical. The square root of 9 is 
3; the square root of 10 is greater than 3, and the most 
famous and the simplest of all square roots, the first in¬ 
commensurable number discovered by the Greeks, the 
square root of 2, is 1.414. . . There are also composite 

radicals—expressions like \/7 + "V^IO. The symbol for a 
radical is not the hammer and sickle, but a sign three or 
four centuries old, and the idea of the mathematical 
radical is even older than that. The concept of the 
‘‘hypcrradical,” or “ultraradical,” which means some¬ 
thing liigher than a radical, but lower than a transcen¬ 
dental, is of recent origin. It has a symbol which we shall 
see in a moment. First, we must say a few words about 
radicals in general. There are certain numbers and 
functions in mathematics which are not expressible in 
the language of radicals and which are generally not 
well understood. Many ideas for which there are no 
concrete or diagrammatic representations are difficult to 
explain. Most people find it impossible to think without 
words; it is necessary to give them a word and a symbol 
to pin their attention. Hyperradical or ultraradical, for 
which hitherto there have been neither words, nor sym¬ 
bols, fall into this category. 


New Names Jor Old 


We first meet these ultraradicals, not in Mexico City, 
but in trying to solve equations of the fifth degree. The 
Egyptians solved equations of the first degree perhaps 
4000 years ago. That is, they found that the solution 
of the equation ax b = 0^ which is represented in 


geometry by a straight line, is x = —. The quadratic 
equation ax^ + ix- + c = 0 was solved by the Hindus and 


V 


the Arabs with the formula x 


— ^ ± \/b^ — 4ac 


2a 

The various conic sections, the circle, the ellipse, the 
parabola, and the hyperbola, are the geometric pictures 
of quadratic equations in two variables. 

Then in the sixteenth century the Italians solved the 
equations of third and fourth degree, obtaining long 
formulas involving cube roots and square roots. So that 
by the year 1550, a few years before Shakespeare was 
born, the equation of the first, second, third, and fourth 


degrees had been solved. Then there was a delay of 250 
years, because mathematicians were struggling with the 
equation of the fifth degree—the general quintic. Finally, 
at the beginning of the nineteenth century, Ruffi ni and 
Abel showed that equations of the fifth degree couTd not 
be solved with radicals. The general quintic is thus not 
like the general quadratic, cubic or biquadratic. Never¬ 
theless, it presents a problem in algebra which theoret¬ 
ically can be solved by algebraic functions. Only, these 
functions are so hard that they cannot be expressed by 
the symbols for radicals. TThese new higher things are 


FIG. 9.—A portrait of two radicals. 


18 Mathematics and the Imagination 

named “ultraradicals,” and they too have their special 
symbols (shown in Fig. 9). 

With such symbols combined with radicals, we can 
solve equations of the fifth degree. For example, the 
solution of -j- X = a may be written x = 

X = Jo". The usefulness of the special symbol and 
name is apparent. Without them the solution of the 
quintic equation could not be compactly expressed. 

+ 

We may now give a few ideas somewhat easier than 
those with which we have thus far occupied ourselves. 
These ideas were presented some time ago to a number 
of children in kindergarten. It was amazing how well 
they understood everything that was said to them. In¬ 
deed, it is a fair inference that kindergarten children 
can enjoy lectures on graduate mathematics as long as 
the mathematical concepts are clearly presented. 

It was raining and the children were asked how many 
raindrops would fall on New York. The highest answer 
was 100. They had never counted higher than 100 and 
what they meant to imply when they used that number 
was merely something very, very big—as big as they 
could imagine. They were asked how many raindrops 
hit the roof, and how many hit New York, and how many 
single raindrops hit all of New York in 24 hours. They 
soon got a notion of the bigness of these numbers even 
though they did not know the symbols for them. They 
were certain in a little while that the number of rciindrops 
was a great deal bigger than a hundred. They were asked 
to think of the number of grains of sand on the beach at 
Coney Island and decided that the number of grains of 
sand and the number of raindrops were about the same. 
But the important thing is that they realized that the 


New Names for Old i g 

number \wa.s finite^ not infinite. In this respect they showed 
their distinct superiority over many scientists who to 
this day use the word infinite when they mean some big 
number, like a billion billion. 

Counting, something such scientists evidently do not 
realize, is a precise operation.* It may be wonderful 
but there is nothing vague or mysterious about it. If 
you count something, the answer you get is either per¬ 
fect or all wrong; there is no half way. It is very much like 
catching a train. You either catch it or vou miss it, and 
if you miss it by a split second you might as well have 

come a week late. There is a famous quotation which 
illustrates this: 

“Oh, the little more, and how much it is! 

And the little less, and what worlds away!” 

A big number is big, but it is definite and it is finite. 

Of course in poetry, the finite ends with about three 

thousand; any greater number is infinite. In many poems, 

the poet will talk to you about the infinite number of 

stars. But, if ever there was a hyperbole, this is it, for 

nobody, not even the poet, has ever seen more than three 

thousand stars on a clear night, without the aid of a 
telescope. 

With the Hottentots, infinity begins at three.t Ask 
a Hottentot how many cows he owns, and if he has more 
than three he’ll say “many.” The number of raindrops 

♦ No one would say that 1 + 1 is "about equal to 2.” It is just as 
silly to say that a billion billion is not a finite number, simply because 
It IS big. Any number which may be named, or conceived of in icrins 

ol (he integers is finite. Infinite means something quite different, as we shall 
see in ihe chapter on the 

^ in all fairness, it must be pointed out that some of the 

tribes of the Belgian Congo can count to a million and beyond. 


3 


20 Mathematics and the Imagination 

falling on New York is also “many.” It is a large finite 
number, but nowhere near infinity. 

Now here is the name of a very large number; “Goo- 
gol.”* Most people would say, “A googol is so large 
that you cannot name it or talk about it; it is so large 
that it is infinite.” Therefore, we shall talk about it, 
explain exactly what it is, and show that it belongs to 
the very same family as the number 1. 

A googol is this number which one of the children in 
the kindergarten wrote on the blackboard: 
100000000000000000000000000000000000000000000000 
000000000000000000000000000000000000000000000000 
00000 

The definition of a googol is: 1 followed by a hundred 
zeros. It was decided, after careful mathematical re¬ 
searches in the kindergarten, that the number of rain¬ 
drops falling on New York in 24 hours, or even in a year - 
or in a century, is much less than a googol. Indeed, the 
googol is a number just larger than the largest numbers 
that are used in physics or astronomy. All those numbers 
require less than a hundred zeros. This information is, 
of course, available to everyone, but seems to be a great 
secret in many scientific quarters. 

A very distinguished scientific publication recently 
came forth with the revelation that the number of snow 
crystals necessary to form the ice age wais a billion to the 
billionth power. This is very startling and also very silly. 

A billion to the billionth power looks like this: 

lOOOOOOOOO^ooo^ooooo^ 

A more reasonable estimate and a somewhat smaller 
number would be 10^®. As a matter of fact, it has been 
estimated that if the entire universe, which you will con- 

• Not even approximately a Russian author. 


ft 


Lihtmn 


21 


New Names for Old 

cede is a trifle larger than the earth, were filled with 
protons and electrons, so that no vacant space remained, 
the total number of protons and electrons would be 
10 (i*c*) 10 with 110 zeros after it). Unfortunately, 

as soon as people talk about large numbers, they run 
amuck. They seem to be under the impression that since 
zero equals nothing, they can add as many zeros to a 
number as they please with practically no serious con¬ 
sequences. We shall have to be a little more careful than 
that in talking about big numbers. 

To return to Coney Island, the number of grains of 
sand on the beach is about lO^o, or more descriptively, 

100000000000000000000. That is a large number, but 

not as large as the number mentioned by the divorcee 
in a recent divorce suit who had telephoned that she 
loved the man “a million billion billion times and eight 
times around the world.” It was the largest number that 
she could conceive of, and shows the kind of thing that 
may be hatched in a love nest. 

Though people do a great deal of talking, the total 
output since the beginning of gabble to the present day, 
including all baby talk, love songs, and Congressional 
debates, totals about 10*®. This is ten million billion. Con¬ 
trary to popular belief, this is a larger number of words 
than is spoken at the average afternoon bridge. 

A great deal of the veneration for the authority of 
the printed word would vanish if one were to calculate 
the number of words which have been printed since the 
Gutenberg Bible appeared. It is a number somewhat 
larger than 10*®. A recent popular historical novel alone 
accounts for the printing of several hundred billion words. 

The largest number seen in finance (though new 
records are in the making) represents the amount of 


22 


Mathematics and the Imagination 

money in circulation in Germany at the peak of the 
inflation. It was less than a googol—merely 

496,585,346,000,000,000,000. 

A distinguished economist vouches for the accuracy of 
this figure. The number of marks in circulation was very 
nearly equal to the number of grains of sand on Coney 
Island beach. 

The number of atoms of oxygen in the average thimble 
is a good deal larger. It would be represented by perhaps 

1000000000000000000000000000. The number of elec¬ 
trons, in size exceedingly smaller than the atoms, is much 
more enormous. The number of electrons which pass 
through the filament of an ordinary fifty-watt electric 
lamp in a minute equals the number of drops of water 
that flow over Niagara Falls in a century. 

One may also calculate the number of electrons, not 
only in the average room, but over the whole earth, and 
out through the stars, the Milky Way, and all the neb¬ 
ulae. The reason for giving all these examples of very 
large numbers is to emphasize the fact that no matter 
how large the collection to be counted, a finite number 
will do the trick. We will have occasion later on to speak 
of infinite collections, but those encountered in nature, 
though sometimes very large, are all definitely finite. 
A celebrated scientist recently stated in all seriousness 
that he believed that the number of pores (through which 
lca\ es breathe) of all the leaves, of all the trees in all the 
woild, would certainly be infinite. Needless to say, he 
was not a niathematician. The number of electrons in a 
single leaf is much bigger than the number of pores of 
all the leaves of all the trees of all the world. And still the 
num er of all the electrons in the entiie universe can be 
found by means of the physics of Einstein. It is a good 


New Names for Old 23 

deal less than a googol—perhaps ten with seventy-nine 
zeros, 10^®, as estimated by Eddington. 

Words of wisdom are spoken by children at least as 
often as by scientists. The name “googol” was invented 
by a child (Dr. Kasner’s nine-year-old nephew) who was 
asked to think up a name for a very big number, namely, 
1 with a hundred zeros after it. He was very certain that 
this number was not infinite, and therefore equally 
certain that it had to have a name. At the same time that 
he suggested “googol” he gave a name for a still larger 
number: “Googolplex.” A googolplex is much larger 
than a googol, but is still finite, as the inventor of the 
name was quick to point out. It was first suggested that a 
googolplex should be 1, followed by writing zeros until 
you got tired. This is a description of what would happen 
if one actually tried to write a googolplex, but different 
people get tired at different times and it would never do 
to have Camera a better mathematician than Dr. Ein¬ 
stein, simply because he had more endurance. The goo¬ 
golplex then, is a specific finite number, with so many 
zeros after the 1 that the number of zeros is a googol. A 
googolplex is much bigger than a googol, much bigger 
even than a googol limes a googol. A googol times a 
googol would be 1 with 200 zeros, whereas a googolplex 
is 1 with a googol of zeros. You will get some idea of the 
size of this very large but finite number from the fact 
that there would not be enough room to write it, if you 
went to the farthest star, touring all the nebulae and put¬ 
ting down zeros every inch of the way. 

One might not believe that such a large number would 
ever really have any application; but one who felt that 
way would not be a mathematician. A number as large 
as the googolplex might be of real use in problems of 


24 Mathematics and the Imagination 

combination. This would be the type of problem in 
which it might come up scientifically: 

Consider this book which is made up of carbon and 
nitrogen and of other elements. The answer to the ques¬ 
tion, “How many atoms are there in this book?” would 
certainly be a finite number, even less than a googol. 
Now imagine that the book is held suspended by a string, 
the end of which you are holding. How long will it 
be necessary to wait before the book will jump up into 
your hand? Could it conceivably ever happen? One 
answer might be “No, it will never happen without 
some external force causing it to do so.” But that is not 
correct. The right answer is that it will almost certainly 

happen sometime in less than a googolplex of years—per¬ 
haps tomorrow. 

The explanation of this answer can be found in physical 
chemistry, statistical mechanics, the kinetic theory of 
gases, and the theory of probability. We cannot dispose 
of all these subjects in a few lines, but we will try. 
Molecules are always moving. Absolute rest of molecules | 
would mean absolute zero degrees of temperature, and I 
absolute zero degrees of temperature is not only non- I 
existent, but impossible to obtain. All the molecules of 1 
the surrounding air bombard the book. At present the 
bombardment from above and below is nearly the same 
and gravity keeps the book down. It is necessary to wait 
or the favorable moment when there happens to be an 
enormous numb- • of molecules bombarding the book 
from below and x. ry few from above. Then gravity will 
be overcome and the book will rise. It would be some¬ 
what hke the effect known in physics as the Brownian 
movement, which describes the behavior of small par- 

tic es m a liquid as they dance about under the impact 


New Names Jot Old 25 

of molecules. It would be analogous to the Brownian 
movement on a vast scale. 

But the probability that this will happen in the near 
future or, for that matter, on any specific occasion that 

we might mention, is between ^—- and -5-. 

googol googolplex 

To be reasonably sure that the book will rise, we should 

have to wait between a googol and a googolplex of years. 

When working with electrons or with problems of 
combination like the one of the book, we need larger 
numbers than are usually talked about. It is for that 
reason that names like googol and googolplex, though 
they may appear to be mere jokes, have a real value. 
The names help to fix in our minds the fact that we are 
still dealing with finite numbers. To repeat, a googol is 
a googolplex is 10 to the googol power, which may 

be written 

We have seen that the number of years that one would 
have to wait to see the miracle of the rising book would 
be less than a googolplex. In that number of years the 
earth may well have become a frozen planet as dead as 
the moon, or perhaps splintered to a number of meteors 
and comets. The real miracle is not that the book will 
rise, but that with the aid of mathematics, we can 
project ourselves into the future and predict with accu¬ 
racy when it will probably rise, i.c., some time between 
today and the year googolplex. 

♦ 

We have mentioned quite a few new names in mathe¬ 
matics new names for old and new ideas. There is one 
more new name which it is proper to mention in con¬ 
clusion. Watson Davis, the popular science reporter, has 
given us the name “mathescope.” With the aid of the 


26 


Mathematics and the Imagination 

magnificent new microscopes and telescopes, man, mid¬ 
way between the stars and the atoms, has come a little 
closer to both. The mathoscope is not a physical instru¬ 
ment; it is a purely intellectual instrument, the ever- 
increasing insight which mathematics gives into the fairy¬ 
land which lies beyond intuition and beyond imagina¬ 
tion. Mathematicians, unlike philosophers, say nothing 
about ultimate truth, but patiently, like the makers of 
the great microscopes, and the great telescopes, they 
grind their lenses. In this book, w'e shall let you see 
through the newer and greater lenses which the mathe¬ 
maticians have ground. Be prepared for strange sights 
through the mathescope! 

FOOTNOTES 

1. Sec the Chapter on pie. —P. 10. 

2. See the Chapter on Change and Changeability—Section on Path¬ 
ological Curves.—P.ll. 

3. NMi. iliis is a diagram which the reader will have to imagine, 
lor It IS beyond the capacity of any printer to make a circle with 
one point omitted. A point, having no dimensions, will, like 
many of the persons on the Lord High Executioner’s list, never 

c inissc . o the circle with one point missing is purely con- 
cepiual, not an idea wliich can be pictured.—P.13. 


Beyond the Googol 


If you do not expect the unexpected^ you will not find it; 
for It IS hard to be sought out, and difficult. 

—HERACLITUS 


Mathematics may well be a science of austere logical 
propositions in precise eanonical form, but in its count¬ 
less applications it serves as a tool and a language, the 
language of description, of number and size. It describes 
with economy and elegance the elliptic orbits of the plan¬ 
ets as readily as the shape and dimensions of this page 
or a corn field. The whirling dance of the electron can 
be seen by no one; the most powerful telescopes can re¬ 
veal only a meager bit of the distant stars and nebulae 
and the cold far corners of space. But with th(' aid of 
mathematics and the imagination the very small, the 
very large—all things may be brought within man’s 
domain. 

To count is to talk the language of number. To count 
to a googol, or to count to ten is part of the same ijroeess; 
the googol is simply harder to pronounce. The essential 
thing to realize is that the googol and ten arc kin, like 
the giant stars and the electron. Arithmetic—this count¬ 
ing language—makes the whole v/orld kin, both in 
space and in time. 

To grasp the meaning and importance of mathematics, 
to appreciate its beauty and its value, arithmetic must 
first be understood, for mostly, since its beginning, mathc- 

27 


28 Mathematics and the Imagination 

matics has been arithmetic in simple or elaborate attire. 
Arithmetic has been the queen and the handmaiden of 
the sciences from the days of the astrologers of Chaldea 
and the high priests of Egypt to the present days of 
relativity, quanta, and the adding machine. Historians 
may dispute the meaning of ancient papyri, theologians 
may wrangle over the exegesis of Scripture, philosophers 
may debate over Pythagorean doctrine, but all will con¬ 
cede that the numbers in the papyri, in the Scriptures and 
in the writings of Pythagoras are the same as the num- 
bers of today. As arithmetic, mathematics has helped 
man to cast horoscopes, to make calendars, to predict 
the risings of the Nile, to measure fields and the height 
of the Pyramids, to measure the speed of a stone as it fell 
from a tower in Pisa, the speed of an apple as it fell 
from a tree in Woolsthorpe, to weigh the stars and the 
atoms, to mark the passage of time, to find the curvature 
of space. And although mathematics is also the calculus, 
the theory of probability, the matrix algebra, the science 
of the infinite, it is still the art of counting. 

* 

Everyone who will read this book can count, and yet, 
what is counting? The dictionary definitions are about 
as helpful as Johnson’s definition of a net: “A series of 
reticulated interstices.’* Learning to compare is learning to 
count. Numbers come much later; they are an artificiality, 
an abstraction. Counting, matching, comparing are al¬ 
most as indigenous to man as his fingers. Without the 
faculty of comparing, and without his fingers, it is un¬ 
likely that he would have arrived at numbers. 

One who knows nothing of the formal processes of 
counting is still able to compare two classes of objects, 
to determine which is the greater, which the less. With- 


Beyond the Googol 29 

out knowing anything about numbers, one may ascertain 
whether two classes have the same number of elements; 
for example, barring prior mishaps, it is easy to show 
that we have the same number of fingers on both hands 
by simply matching finger with finger on each hand. 

To describe the process of matching, which underlies 
counting, mathematicians use a picturesque name. They 
call it putting classes into a “one-to-one reciprocal cor¬ 
respondence” with each other. Indeed, that is all there 
is to the art of counting as practiced by primitive peoples, 

by us, or by Einstein. A few examples may serve to make 
this clear. 

In a monogamous country it is unnecessary to count 
both the husbands and the wives in order to ascertain 
the number of married people. If allowances are made 
for the few gay Lotharios who do not conform to either 
custom or statute, it is sufficient to count either the 
husbands or the wives. There are just as many in one 
class as in the other. The correspondence between the 
two classes is one-to-one. 

There are more useful illustrations. Many people are 
gathered in a large hall where seats are to be provided. 
The question is, are there enough chairs to go around? 
It would be quite a job to count both the people and the 
chairs, and in this case unnecessary. In kindergarten 
children play a game called “Going to Jerusalem”; in a 
room full of children and chairs there is alwa>a one less 
chair than the number of children. At a signal, each 
child runs for a chair. The child left standing is “out.” 
A chair is removed and the game continues. Here is 
the solution to our problem. It is only necessary to ask 
everyone in the hall to be seated. If everyone sits down 
and no chairs are left vacant, it is evident that there 


30 Alatfumatics and the Imagination 

are as many chairs as people. In other words, without 
actually knowing the number of chairs or people, one 
does know that the number is the same. The two classes— 
chairs and people—have been shown to be equal in 
number by a one-to-one correspondence. To each person 
corresponds a chair, to each chair, a person. 

In counting any class of objects, it is this method alone 
which is employed. One class contains the things to be 
counted; the other class is always at hand. It is the class 
of integers, or “natural numbers,” which for convenience 
we regard as being given in serial order: 1, 2, 3, 4, 5, 6, 
7 . . . Matching in one-to-one correspondence the ele¬ 
ments of the first class with the integers, we experience a 
common, but none the less wonderful phenomenon—the 
last integer necessary to complete the pairings denotes 
how many elements there are. 

* 

In clarifying the idea of counting, we made the un¬ 
warranted assumption that the concept of number was 

everyone. The number concept may seem 
intuitively clear, but a precise definition is required. 
While the definition may seem worse than the disease, 
it is not as difficult as appears at first glance. Read it 

t you will find that it is both explicit and 

economical. 

Given a class C containing certain elements, it is 
possible to find other classes, such that the elements of 
each may be matched one to one with the elements of 
C. (Each of these classes is thus called “equivalent to C.”) 
All such classes, including C, whatever the character of 
their elements, share one property in common: all of 
them have the same cardinal number^ which is called the 
cardinal number of the class Cd 


Beyond the Googol 21 

The cardinal number of the class C is thus seen to be 
the symbol representing the set of all classes that can be 
put into one-to-one correspondence with C. For example, 
the number 5 is simply the name, or symbol, attached 
to the set of all the classes, each of which can be put into 
one-to-one correspondence with the fingers of one hand. 

Hereafter we may refer without ambiguity to the 
number of elements in a class as the cardinal number of 
that class or, briefly, as “its cardinality.’* The question, 
“How many letters are there in the word mathematics?'^ 
is the same as the question, “What is the cardinality of 
the class whose elements are the letters in the word 
mathematics?" Employing the method of one-to-one cor¬ 
respondence, the following graphic device answers the 


question, and illustrates the method: 


M A 

T 

H 

E 

M 

A 

T 

I 

C S 

1 I 

t 

t 

i 

t 

t 


k 

t 

t t 

4- 1 


X 

X 

X 

X 


X 

X X 

1 2 

3 

4 

5 

6 

1 

8 

9 

10 11 

It must 

now 

be 

evident that 

this 

method 

is neither 


strange nor esoteric; it was not invented by mathema¬ 
ticians to make something natural and easy seem un¬ 
natural and hard. It is the method employed when we 
count our change or our chickens; it is the proper 
method for counting any class, no matter how large, 
from ten to a googolplex—and beyond. 

Soon we shall speak of the “beyond” when we turn to 
classes which are not finite. Indeed, we shall try to measure 
our measuring class —the integers. One-to-one correspond¬ 
ence should, therefore, be thoroughly understood, for 
an amazing revelation awaits us: Infinite classes can 
also be counted, and by the very same means. But before 


32 Mathematics and the Imagination 

we try to count them, let us practice on some very big 
numbers—big, but not infinite. 

* 

“Googol” is already in our vocabulary: It is a big 
number one, with a hundred zeros after it. Even bigger 
is the googolplex: 1 with a googol zeros after it. Most 
numbers encountered in the description of nature are 
much smaller, though a few are larger. 

Enormous numbers occur frequently in modern sci¬ 
ence. Sir Arthur Eddington claims that there are, not 
approximately, but exactly 136-2256 protons,* and an 
equal number of electrons, in the universe. Though 
not easy to visualize, this number, as a symbol on paper, 
takes up little room. Not quite as large as the googol, 
it is completely dwarfed by the googolplex. None the 

less, Eddington’s number, the googol, and the googolplex 
are finite. 

A veritable giant is Skewes* number, even bigger than 

^ googolplex. It gives information about the distribution 
of primes^ and looks like this: 


10 


10 


ID 


Or, for example, the total possible number of moves in 
a game of chess is: 


10 


10 


oO 


And speaking of chess, as the eminent English mathe¬ 
matician, G. H. Hardy, pointed out—if we imagine the 

• Let no one suppose that Sir Arthur has counted them. But he 

if'n ^ ^ justify his claim. Anyone with a better theory 

may challenge Sir Arthur, for who can be referee? Here is his number 

653,951,181,555,468,- 

he says,’ to t’he" Hst ,425,076,185,631,031,276-accurate, 


Beyond the Googol 33 

entire universe as a chessboard, and the protons in it 
as chessmen, and if we agree to call any interchange in 
the position of two protons a “move” in this cosmic game, 
then the total number of possible moves, of all odd coin¬ 
cidences, would be Skewes’ number: 


No doubt most people believe that such numbers are 
part of the marvelous advance of science, and that a few 
generations ago, to say nothing of centuries back, no one 
in dream or fancy could have conceived of them. 

There is some truth in that idea. For one thing, the 
ancient cumbersome methods of mathematical notation 
made the writing of big numbers difhcult, if not actually 
impossible. For another, the average citizen of today en¬ 
counters such huge sums, representing armament ex¬ 
penditures and stellar distances, that he is quite conver¬ 
sant with, and immune to, big numbers. 

But there were clever people in ancient times. Poets 
in every age may have sung of the stars as infinite in 
number, when all they saw was, perhaps, three thousand. 
But to Archimedes, a number as large as a googol, or 
even larger, was not disconcerting. He says as much in 
an introductory passage in The Sand Reckoner, realizing 
that a number is not infinite merely because it is enor¬ 
mous. 

There are some, King Gelon, who think that the number of 
the sand is infinite in multitude; and I mean by the sand, not 
only that which exists about Syracuse and the rest of Sicily, 
but also that which is found in every region whether inhabited 
or uninhabited. Again there are some who, without regarding 


34 Mathematics and the Imagination 

it as infinite, yet think that no number has been named which 
is great enough to exceed its multitude. And it is clear that 
they who hold this view, if they imagined a mass made up of 
sand in other respects as large as the mass of the earth, in¬ 
cluding in it all the seas and the hollows of the earth filled up 
to a height equal to that of the highest of the mountains, would 
be many times further still from recognizing that any number 
could be expressed which exceeded the multitude of the sand 
so taken. But I will try to show you by means of geometrical 
proofs, which you will be able to follow, that, of the numbers 
named by me and given in the work which I sent to Zeuxippus, 
some exceed not only the number of the mass of sand equal in 
magnitude to the earth filled up in the way described, but 
also that .of a mass equal in magnitude to the universe. 

The Greeks had verv definite ideas about the infinite. 
Just as we are indebted to them for much of our wit and 
our learning, so are we indebted to them for much of 
our sophistication about the infinite. Indeed, had we 
always retained their clear-sightedness, many of the prob¬ 
lems and paradoxes connected with the infinite would 
never have arisen. 

Above everything, we must realize that “very big” and 
“infinite’' arc entirely difTejrent.* By using the method 
of one-to-one correspondence, the protons and electrons 
in the universe may theoretically be counted as easily 
as the buttons on a vest. Sufficient and more than 
sufficient lor that task, or for the task of counting any 
finite collection, are the integers. But measuring the 


* There is no point where the very big starts to merge into the 
infinite. \ou may write a number as big as you please; it will be no 
nearer the infinite than the number 1 or the number 7. Make sure 
that you keep this distinction very clear and you will have mastered 
many of the subtleties of the transfinitc. 


Beyond the Googol 3 ^ 

totality of integers is another problem. To measure such 
a class demands a lofty viewpoint. Besides being, as the 
German mathematician Kronecker thought, the work of 
God, which requires courage to appraise, the class of 
integers is infinite—which is a great deal more in¬ 
convenient. It is worse than heresy to measure our own 
endless measuring rod! 

♦ 

. The problems of the infinite have challenged man’s 
I mind and have fired his imagination as no other single 
problem in the history of thought. The infinite appears 
both strange and familiar, at times beyond our grasp, at 
times natural and easy to understand. In conquering 
it, man broke the fetters that bound him to earth. All 
his faculties were required for this conquest—his rea.son- 
ing powers, his poetic fancy, his desire to know. 

To establish the science of the infinite involves the 

principle of mathematical imita tion. This principle affirms 

the power of reasoning by recurrence. It typifies almost 

all mathematical thinking, all that we do when we 

construct complex aggregates out of simple elements. 

It is, as Poincare remarked, “at once necessary to the 

mathematician and irreducible to logic.” His statement 

of the principle is: “If a property be true of the number 

one, and if we establish that it is true of h -f 1,* provided 

it be of «, it will be true of all the whole numbers.” 

Mathematical induction is not derived from experience, 

rather is it an inherent, intuitive, almost instinctive 

property of the mind, “ What we have once done we can do 
again.’’’* 

If we can construct numbers to ten, to a million, to a 
googol, we are led to believe that there is no stopping, 

* Where n is any integer. 


4 


36 Mathematics and the Imagination 

no end. Convinced of this, we need not go on forever; 
the mind grasps that which it has never experienced— 
the infinite itself. Without any sense of discontinuity, 
without transgressing the canons of logic, the mathema¬ 
tician and philosopher have bridged in one stroke the 
gulf between the finite and the infinite. The mathematics 
ol the infinite is a sheer affirmation of the inherent power 
of reasoning by recurrence. 

In the sense that “infinite” means “without end, with¬ 
out bound,” simply “not finite,” probably everyone un¬ 
derstands its meaning. No difficulty arises where no 
precise definition is required. Nevertheless, in spite of 
the famous epigram that mathematics is the science in 
which we do not know what we are talking about, at 
least we shall have to agree to talk about the same thing. 
Apparently, even those of scientific temper can argue 
bitterly to the point of mutual vilification on subjects 
ranging from Marxism and dialectical materialism to 
group theory and the uncertainty principle, only to find, 
on the verge of exhaustion and collapse, that they are on 
the same side of the fence. Such arguments are generally 
the results of vague terminology; to assume that everyone 
is familiar with the precise mathematical definition of 
infinite is to build a new Tower of Babel. 

Before undertaking a definition, we might do well to 
glance backwards to see how mathematicians and philos¬ 
ophers of other times dealt with the problem. 

The infinite has a double aspect—the infinitely large, 
and the infinitely small. Repeated arguments and demon- 
stiations, of apparently apodictic force, were advanced, 
overwhelmed, and once more resuscitated to prove or 
disprove its existence. Few of the arguments were ever 


Beyond the Googol 37 

refuted—each was buried under an avalanche of others. 
The happy result was that the problem never became 
any clearer. * 

The warfare began in antiquity with the paradoxes 
of Zeno; it has never ceased. Fine points were debated 
with a fervor worthy of the earliest Christian martyrs, 
but without a tenth part of the acumen of medieval 
theologians. Today, some mathematicians think the 
infinite has been reduced to a state of vassalage. Others 
are still wondering what it is. 

Zeno’s puzzles may help to bring the problem into 
sharper focus. Zeno of Elea, it will be recalled, said some 
disquieting things about motion, with reference to an 
arrow, Achilles, and a tortoise. This strange company 
was employed on behalf of the tenet of Eleatic philosophy 
—that all motion is an illusion. It has been suggested, 
probably by “baffled critics,” that “Zeno had his longue 
in cheek when he made his puzzles.” Regardless of mo¬ 
tive, they arc immeasurably subtle, and perhaps still 
defy solution.! 

One paradox—the Dichotomy—states that it is im¬ 
possible to cover any given distance. The argument: 
First, half the distance must be trav'ersed, then half of 
the remaining distance, then again half of what remains, 

* No one has written more brilliantly or more wittily on this subject 
than Bertrand Russell. Sec particularly his essays in the volume Mys¬ 
ticism and Lof^ic. 

tTo be sure, a variety of explanations have been t^iven for the 
paradoxes. In the last analysis, the explanations for the riddles rest 
upon the interpretation of the foundations of mathematics. Mathe¬ 
maticians like Brouwer, who reject the infinite, would probably not 
accept any of the solutions given. 


38 Mathematics and the Imagination 

and so on. It follows that some portion of the distance 
to be covered always remains, and therefore motion is 
impossible! A solution of this paradox reads: 


The successive distances to be covered form an infinite 
geometric series: 

i + i + i + l + _L+ 3 

2 ^ 4 ^ 816 ^ 32 ^ *" 

each term of which is half of the one before. Although 
this series has an infinite number of terms, its sum is 
finite and equals 1. Herein, it is said, lies the flaw of the 
Dichotomy. Zeno assumed that any totality composed 
of an infinite number of parts must, itself, be infinite, 
whereas we have just seen an infinite number of elements 
which make up the finite totality—1. 

The paradox of the tortoise states that Achilles, running 
to overtake the tortoise, must first reach the place where 
it started: but the tortoise has already departed. This 
comedy, however, is repeated indefinitely. As Achilles 
arrives at each new point in the race, the tortoise having 
been there, has already left. Achilles is as unlikely to 
catch him as a rider on a carrousel the rider ahead. 

Finally. the arrow in flight must be moving every 
instant of time. But at ever)’ instant it must be somewhere 
in space. However, if the arrow must always be in some 


Beyond the Googol 3g 

one place, it cannot at every instant also be in transit, 
for to be in transit is to be nowhere. 

Aristotle and lesser saints in almost every age tried 
to demolish these paradoxes, but not very creditably. 
Three German professors succeeded where the saints 
had failed. At the end of the nineteenth century, it 
seemed that Bolzano, Weierstrass and Cantor had laid 
the infinite to rest, and Zeno’s paradoxes as well. 

The modern method of disposing of the paradoxes is 
not to dismiss them as mere sophisms unworthy of serious 
attention. The history of mathematics, in fact, recounts a 
poetic vindication of Zeno’s stand. Zeno was, at one lime, 
as Bertrand Russell has said, “A notable victim of pos¬ 
terity’s lack of judgement.” That wrong has been righted. 
In disposing of the infinitely small, Weierstrass showed 
that the moving arrow is really always at rest, and that 
we live in Zeno’s changeless world. The work of Georg 
Cantor, which we shall soon encounter, showed that 
if we are to believe that Achilles can catch the tortoise, 
we shall have to be prepared to swallow a bigger paradox 
than any Zeno ever conceived of: the whole is no 

GREATER THAN MANY OF ITS PARTS* 

The infinitely small had been a nuisance for more than 
two thousand years. At best, the innumerable opinions 
it evoked deserved the laconic verdict of Scotch Juries: 
“Not proven.” Until Weierstrass appeared, the total 
advance was a confirmation of Zeno’s argument against 
motion. Even the jokes were better. Leibniz, according 
to Carlyle, made the mistake of trying to explain the 
infinitesimal to a Queen—Sophie Charlotte of Prussia. 
She informed him that the behavior of her courtiers 
made her so familiar with the infinitely small, that she 
needed no mathematical tutor to explain it. But philos- 


40 Mathematics and the Imagination 

ophers and mathematicians, according to Russell, ‘‘hav¬ 
ing less acquaintance with the courts, continued to dis¬ 
cuss this topic, though without making any advance.’’ 

Berkeley, with the subtlety and humor necessary for an 
Irish bishop, made some pointed attacks on the infini¬ 
tesimal, during the adolescent period of the calculus, 
that had the very best, sharp-witted, scholastic sting. 
One could perhaps speak, if only with poetic fervor, of the 
infinitely large, but what, pray, was the infinitely small? 
The Greeks, with less than their customary sagacity, 
introduced it in regarding a circle as differing infini¬ 
tesimally from a polygon with a large number of equal 
sides. Leibniz used it as the bricks for the infinitesimal 
calculus. Still, no one knew what it was. The infinitesimal 
had wondrous properties. It was not zero, yet smaller 
than any quantity. It could be assigned no quantity or 
size, yet a sizable number of infinitesimals made a very 
definite quantity. Unable to discover its nature, happily 
able to dispense with it, Weierstrass interred it alongside 
of the phlogiston and other once-cherished errors. 

* 

The infinitely large offered more stubborn resistance. 
Whatever it is, it is a doughty weed. The subject of reams 
of nonsense, sacred and profane, it was first discussed 
fully, logically, and without benefit of clergy-like prej¬ 
udices by Bernhard Bolzano. Die Paradoxien des ZJnendlichen, 
a remarkable little volume, appeared posthumously in 
1851. Like the work of another Austrian priest, Gregor 
Mendel, whose distinguished treatise on the principles 
of heredity escaped oblivion only by chajr^e, this im¬ 
portant book, charmingly written, made no great im¬ 
pression on Bolzano’s contemporaries. It is the creation 
of a clear, forceful, penetrating intelligence. For the 


Beyond the Googol 41 

first time in twenty centuries the infinite was treated as a 
problem in science, and not as a problem in theology. 

Both Cantor and Dedekind are indebted to Bolzano 
for the foundations of the mathematical treatment of the 
infinite. Among the many paradoxes he gathered and 
explained, one, dating from Galileo, illustrates a typical 
source of confusion: 

Construct a square— ABCD. About the point .1 as cen¬ 
ter, with one side as radius, describe a quarter-circle, in¬ 
tersecting the square at B and Z). Draw PR parallel to 
dZ), cutting AB at P, CD at /?, the diagonal AC at N, and 
the quarter-circle at A/. 


A 


FIG. 11.—Extract triangle APM from the figure. It is 
not hard to see that its three sides equal respectively the 
radii of the three circles. 

Thus 

RP - R^i = R,2 
or, 

wRi^ - = irRs^ 

or, the two shaded areas are equal. 


42 


Mathematics and the Imagination 

By a well-known geometrical theorem, it can be shown 
that if PjV, PM and PR are radii, the following relation¬ 
ship exists: 

irPN = itTR - wPM^ (1) 

Permit PR to approach AD. Then circle with PH as 
radius becomes smaller, and the ring between the circles 
with PiM and PR as radii becomes correspondingly 
smaller. Finally, when PR becomes identical with AD, the 
radius PH vanishes, leaving the point A, while the ring 
between the two circles PM and PR contracts into one 
periphery with AD as radius. From equation (1) it may 
be concluded that the point A takes up as much area as 
the circumference of the circle with AD as radius. 

Bolzano realized that there is only an appearance of a 

paradox. The two classes of points, one composed of a 

single member, the point A^ the other of the points in 

the circumference of the circle with AB as radius, take 

up exactly the same amount of area. The area of both is 

zero! The paradox springs from the erroneous conception 

that the number of points in a given configuration is an 

indication of the area which it occupies. Points, finite or 

infinite in number, have no dimensions and can therefore 
occupy no area. 

Through the centuries such paradoxes had piled up. 
Born of the union of vague ideas and vague philosophical 
reflections, they were nurtured on sloppy thinking. Bol¬ 
zano cleared away most of the muddle, preparing the way 
for Cantor. It is to Cantor that the mathematics of the in¬ 
finitely large owes its coming of age. 

* 

Georg Cantor was born in St. Petersburg in 1845, 
SIX years before Bolzano’s book appeared. Though born 
m Russia, he lived the greater part of his life in Germany, 


Beyond the Googol 43 

where he taught at the University of Halle. While Weier- 
strass was busy disposing of the infinitesimal, Cantor 
set himself the apparently more formidable task at the 
other pole. The infinitely small might be laughed out of 
existence, but who dared laugh at the infinitely large? 
Certainly not Cantor! Theological curiosity prompted 
his task, but the mathematical interest came to subsume 
every other. 

In dealing with the science of the infinite, Cantor 
realized that the first requisite was to define terms. His 
definition of “infinite class” which we shall paraphrase, 
rests upon a paradox, an infinite class has the unique 

PROPERTY THAT THE WHOLE IS NO GREATER THAN SOME 
OF ITS PARTS. That statement is as essential for the mathe¬ 
matics of the infinite as the whole is greater than any 
OF ITS PARTS is for finite arithmetic. When we recall that 
two classes are equal if their elements can be put into 
one-to-one correspondence, the latter statement be¬ 
comes obvious. Zeno would not have challenged it, in 
spite of his scepticism about the obvious. But what is 
obvious for the finite is false for the infinite; our extensive 
experience with finite classes is misleading. Since, for 
example, the class of men and the class of mathemati¬ 
cians are both finite, anyone realizing that some men 
arc not mathematicians would correctly conclude that 
the class of men is the larger of the two. He might also 
conclude that the number of integers, even and odd, is 
greater than the number of even integers. But we see 
from the following pairing that he would be mistaken: 


1 


1 

2 


2 

T 

i 

4 


3 

t 

6 


4 5 6 1 ... 

t t t t 

i 4 ' i f 

8 10 12 14 ... 


44 Mathematics and the Imagination 

Under every integer, odd or even, we may write its 
double—an even integer. That is, we place each of the 
elements of the class of all the integers, odd and even, into 
a one-to-one correspondence with the elements of the 
class composed solely of even integers. This process may 
be continued to the googolplex and beyond. 

Now, the class of integers is infinite. No integer, no 
matter how great, can describe its cardinality (or numer- 
osity). Yet, since it is possible to establish a one-to-one 
correspondence between the class of even numbers and 
the class of integers, we have succeeded in counting the 
class of even numbers just as we count a finite collection. 
The two classes being perfectly matched, we must con¬ 
clude that they have the same cardinality. That their 
cardinality is the same we know, just as we knew that the 
chairs and the people in the hall were equal in number 
when every chair was occupied and no one was left 
standing. Thus, we arrive at the fundamental paradox of 
all infinite classes:—There exist component parts of an 
infinite class which are just as great as the class itself. 

THE WHOLE IS NO GREATER THAN SOME OF ITS PARTS! 

The class composed of the even integers is thinned out 
as compared with the class of all integers, but evidently 
thinning out” has not the slightest effect on its .cardi¬ 
nality. Moreover, there is almost no limit to the number 

• 

of times this process can be repeated. For instance, there 
are as many square numbers and cube numbers as there 
are integers. The appropriate pairings are: 

] 23456... 1 2 3 4 5 6... 

I I I I I I I I I I I I 

1 4 9 16 25 36... 1 g 27 64 125 216.. . 

U 2“ 3^ 4^ 5^ 6^ p 2^ 3^ 4^ 53 


Beyond the Googol 45 

Indeed, from any denumerable class there can always 
be removed a denumerably infinite number of denumer- 
ably infinite classes without affecting the cardinality of 
the original class. 

* 

Infinite classes which can be put into one-to-one cor¬ 
respondence with the integers, and thus “counted,’’ 
Cantor called countable^ or denumerably infinite. Since all 
finite sets are countable, and we can assign to each one 
a number, it is natural to try to extend this notion and 
assign to the class of all integers a number representing 
its cardinality. Yet, it is obvious from our description of 
“infinite class” that no ordinary integer would be ade¬ 
quate to describe the cardinality of the whole class of in¬ 
tegers. In effect, it would be asking a snake to swallow 
itself entirely. Thus, the first of the transfinite numbers 
was created to describe the cardinality of countable 
infinite classes. Etymologically old, mathematically new, 
(aleph), the first letter of the Hebrew alphabet, was 
suggested. However, Cantor finally decided to use the 
compound symbol (Aleph-Null). If asked, “How 
many integers are there?” it would be correct to reply, 
“There are integers.” 

Because he suspected that there were other transfinite 
numbers, in fact an infinite number of transfinites, and 
the cardinality of the integers the smallest. Cantor affixed 
to the first N a small zero as subscript. The cardinality of 
a denumerably infinite class is therefore referred to as Xo 
(Aleph-Null). The anticipated transfinite numbers form a 
hierarchy of alephs; ^<3 . . . 

All this may seem very strange, and it is quite excus¬ 
able for the reader by now to be thoroughly bewildered. 
Yet, if you have followed the previous reasoning step 


4 ^ Mathematics and the Imagination 

by step, and will go to the trouble of rereading, you will 
see that nothing which has been said is repugnant to 
straight thinking. Having established what is meant by 
counting in the finite domain, and what is meant by 
number, we decided to extend the counting process to 
infinite classes. As for our right to follow such a pro¬ 
cedure, we have the same right, for example, as those who 
decided that man had crawled on the surface of the earth 
long enough and that it was about time for him to fly. It 
is our right to venture forth in the world of ideas as it is 
our right to extend our horizons in the physical univer.se. 
One restraint alone is laid upon us in these adventures of 
ideas: that we abide by the rules of logic. 

Upon extending the counting process it was evident 
at once that no finite number could adequately describe 
an infinite class. If any number of ordinary arithmetic 
describes the cardinality of a class, that class must be 
finite, even though there were not enough ink or enough 
space or enough time to write the number out. We shall 
then require an entirely new kind of number, nowhere 
to be found in finite arithmetic, to describe the cardi¬ 
nality of an infinite class. Accordingly, the totality of inte¬ 
gers was assigned the cardinality “aleph.” Suspecting that 
there were other infinite classes with a cardinality greater 
than that of the totality of integers, we supposed a whole 
hierarchy of alephs, of which the cardinal number of the 
totality of integers was named Aleph-Null to indicate it 
was the smallest of the transfinites. 

Having had an interlude in the form of a summary, 

let us turn once more to scrutinize the alephs, to find if, 

upon closer acquaintance, they may not become easier 
to understand. 

The arithmetic of the alephs bears little resemblance 


Beyond the Googol 47 

to that of the finite integers. The immodest behavior of 
No is typical. 

A simple problem in addition looks like this: 

No + 1 = No 

No + googol = No 
No + No = No 

The multiplication table would be easy to teach, easier 

to learn: 

.1 X No = No 

2 X No = No 

3 X No = No 
« X No = No 

where n represents any finite number. 

Also, 

(No) 2 = No X No 
= No 

And thus, 

(No)" = No 


when is a finite integer. 

There seems to be no variation of the theme; the 
monotony appears inescapable. But it is all very deceptive 
and treacherous. We go along obtaining the same result, 
no matter what we do to No, when suddenly we try: 

(No)''^ 

This operation, at last, creates a new transfinite. But 
before considering it, there is more to be said about 
countable classes. 

* 

Common sense says that there are many more fractions 
than integers, for between any two integers there is an in- 


48 Mathematics and the Imagination 

finite number of fractions. Alas—common sense is amidst 
alien corn in the land of the infinite. Cantor discovered a 
simple but elegant proof that the rational fractions form a 
denumerably infinite sequence equivalent to the class of 
integers. Whence, this sequence must have the same car¬ 
dinality.* 

The set of all rationed fractions is arranged, not in 
order of increasing magnitude, but in order of ascending 
numerators and denominators in an array: 


FIG. 12.—Cantor's diagonal array. 

Since each fraction may be written as a pair of integers, 

i.e., f as (3,4). the familiar one-to-one correspondence 

It has been suggested that at this point the tired reader puts the 
book down with a sigh—and goes to the movies. VVe can only offer 


Beyond the Googol 


with the integers may be effected. This is illustrated in 
the above array by the arrows. 


1 2 3 4 5 6 7 8 

I I I I I I I I 

(1,1) (2,1) (1,2) (1,3) (2,2) (3,1) (4.1) (3,2) (2,3) 

Cantor also found, by means of a proof (too technical 
to concern us here) based on the “height" of algebraic 
equations, that the class of all algebraic numbers, num¬ 
bers which are the solutions of algebraic equations with 
integer coefficients, of the form: 


^ + . . . + = 0 

is denumerablv infinite. 

But Cantor felt that there were other transfinites, that 
there were classes which were not countable, which 
could not be put into one-to-one correspondence with 
the integers. And one of his greatest triumphs came when 
he succeeded in showing that there are classes with a 
cardinality greater than No. 

The class of real numbers composed of the rational 
and irrational numbers! is such a class. It contains those 
irrationals which are algebraic as well as those which 
are not. The latter are called transcendental numbers^ 


in mitigation that this proof, like the one which follows on the non¬ 
countability of the real numbers, is tough and no bones about it. 
You may grit your teeth and try to get what you can out of them, or 
conveniently omit them. The essential thing to come away with 
is that Cantor found that the rational fractions are countable but that 
the set of real numbers is not. Thus, in spite of what common sense; 
tells you, there are no more fractions than there are integers and 
there are more real numbers between 0 and 1 than there are elements 
in the whole class of integers. 

^ Irrational numbers are numbers which cannot be expressed as 
rational fractions. For example, \/2, VX e, tt. The class of real 
numbers is made up of rationals like 1, 2, 3, i, ‘ .J, and irrationals as 
above. 


Mathematics and the Imagination 

Two important transcendental numbers were known 
to exist in Cantor’s time: tt, the ratio of the circumference 
of a circle to its diameter, and the base of the natural 
logarithms. Little more was known about the class of 
transcendcntals: it was an enigma. What Cantor had 
to prove, in order to show that the class of real numbers 
was nondenumerable (i.e., too big to be counted by the 
class of integers), was the unlikely fact that the class of 
transcendcntals was nondenumerable. Since the rational 
and the algebraic numbers were known to be denumer¬ 
able, and the sum of any denumerable number of de¬ 
numerable classes is also a denumerable class, the sole 
remaining class which could make the totality of real 
numbers nondenumerable was the class of transcendcntals. 

He was able to devise such a proof. If it can be shown 
that the class of real numbers between 0 and 1 is non- 
dcnumerable, it will follow a Jortiori that all the real 
numbers arc nondenumerable. Employing a device often 
used in advanced mathematics, the reductio ad absurdum^ 
Cantor assumed that to be true which he suspected was 
false, and then showed that this assumption led to a 
contradiction. He assumed that the real numbers be¬ 
tween 0 and 1 were countable and could, therefore, be 
paired with the integers. Having proved that this as¬ 
sumption led to a contradiction, it followed that its 
opposite, namely, that the real numbers could not be 
paired with the integers (and were therefore not count¬ 
able), was true. 

To count the real numbers between 0 and 1, it is 
required that they all be expressed in a uniform way 
and a method of writing them down in order be devised 
so that they can be paired one to one with the integers. 
The first requirement can be fulfilled, for it is possible 


Beyond the Googol 


to express every real number as a nonterminating dec¬ 
imal. Thus, for example; ® 


1 

3 

1 

9 


. 3333 ... 

. 1111111 ... 


. 2142857121428571 ... 


1 . 414 . . . 


- = .707 


Now, the second requirement confronts us. How shail 
we make the pairings? What system ma>- be devised to en¬ 
sure the appearance of every decimal? We did find a 
method for ensuring the appearance of every rational 
fraction. Of course, we could not actually write them all, 
any more than we could actually write all the integers; 
but the method of increasing numerators and denomina¬ 
tors was so explicit that, if we had had an infinite time 
in which to do it, we could actually have set down all 
the fractions and have been certain that we had not 
omitted any. Or, to put it another way: It was always 
certain and determinate after a fraction had been paired 
with an integer, what the next fraction would be, and 
the next, and the next, and so on. 

On the other hand, when a real number, expressed 
as a nonterminating decimal, is paired with an integer, 
what method is there for determining what the next 
decimal in order should be? You have only to ask your¬ 
self, which shall be the first of the nonterminating dec¬ 
imals to pair with the integer 1, and you have an inkling 
of the difficulty of the problem. Cantor however assumed 
that such a pairing does exist, without attempting to 
give its explicit form. His scheme was: With the integer 
1 pair the decimal .aia 2 a 3 . . . , with the integer 2, 
.bib 2 b 3 . . . , etc. Each of the letters represents a digit 
of the nonterminating decimal in which it appears. The 


5 


52 Mathematics and the Imagination 

determinate array of pairing between the decimals and 
the integers would then be: 

—^ 0 . a\ a<i Os 04 as , . » 

2 ^ * 0 . bi 62 b 4 bs ■ • ‘ 

Cl C2 C3 C4 Cs • • • 

4* *^ 0 . di 1.2 dz d4 ds • • • 


That was Cantor’s array. But at once it was evident 
that it glaringly exhibited the very contradiction for 
which he had been seeking. And in this defeat lay his 
triumph. For no matter how the decimals are arranged, 
by whatever system, by whatever scheme, it is always 
possible to construct an infinity of others which are not 
present in the array. The point is worth repeating: 
having contrived a general form for an array which we 
believed would include every decimal, we find, in spite 
of all our efforts, that some decimals are bound to be 
omitted. This, Cantor showed by his famous “diagonal 
proof.” The conditions for determining a decimal omitted 
from the array are simple. It must differ from the first 
decimal in the array in its first place, from the second 
decimal in the array in its second place, from the third 
decimal in its third place, and so on. But then, it must 

every decimal in the entire array in at least one 
place. If (as illustrated in the figure) we draw a diagonal 
line through our model array and write a new decimal, 
each digit of which shall differ from every digit inter¬ 


cepted by the diagonal, this 
found in the array. 


decimal cannot be 


new 


53 


Beyond the Googol 

1 * * 0. Q \ 02 ^3 O5 . 

2 * -► 0. />2 63 ^4 ^5 . 

3 <—> 0. iTi C2 Cz Cs . 

4 ^ * 0 . (/2 di d\ di, . 

5 <—► 0. ^2 ^3 ^4 ^6 . 


The new decimal may be written:— 

0. ai 02 03 04 as . . .; 

where oi differs from oi, 03 differs from /> 2 , 0.3 from ^3 
04 from d^y 03 from ^ 5 , etc. Accordingly, it will differ 
from each decimal in at least one place, from the «th 
decimal in at least its nth place. This proves conclusively 
that there is no way of including all the decimals in any 
possible array, no way of pairing them off with the inte¬ 
gers. Therefore, as Cantor set out to prove: 

1. The class of transcendental numbers is not only infinite, 
but also not countable, i.e., nondcnumerably infinite. 

2. The real numbers between 0 and 1 are infinite and not 
countable. 

3. A fortiori, the class of all real numbers is nondenumerable. 

♦ 

To the noncountable class of real numbers, Cantor as¬ 
signed a new transfinite cardinal. It was one of the alephs, 
but which one remains unsolved to this day. It is sus¬ 
pected that this transfinite, called the “cardinal of the 
continuum,” which is represented by c or C, is identical 
with Ki. But a proof acceptable to most mathematicians 
has yet to be devised. 


54 


Mathematics and the Imagination 

The arithmetic of C is much the same as that of Xo* 
The multiplication table has the same dependable mon¬ 
otone quality. But when C is combined with Xo> it swal¬ 
lows it completely. Thus: 

C + Xo = C <7- Xo = C 

C X X 0 = C and even C X C = C 

Again, we hope for a variation of the theme when we 
come to the process of involution. Yet, for the moment, 
we are disappointed, for C^o = C. But just as (Xo)^® 
does not equal Xo, so does not equal C. 

We are now in a position to solve our earlier problem 
in involution, for actually Cantor found that (Xo)^° = C. 
Likewise gives rise to a new transfinite, greater than 
C. This transfinite represents the cardinality of the class 
of all one-valued functions. It is also one of the X’s, but 
again, which one is unknown. It is often designated by the 
letter F.® In general, the process of involution, when re¬ 
peated, continues to generate higher transfinites. 

Just as the integers served as a measuring rod for 
classes with the cardinality Xo, the class of real numbers 
serves as a measuring rod for classes with the cardinality 
C. Indeed, there are classes of geometric elements which 
can be measured in no other way except by the class of 
real numbers. 

From the geometric notion of a point, the idea is 
evolved that on any given line segment there are an 
infinite number of points. The points on a line segment 
are also, as mathematicians say, “everywhere dense.” 
This means that between any two points there is an 
infinitude of others. The concept of two immediately 
adjoining points is, therefore, meaningless. This property 
of being “everywhere dense,” constitutes one of the es- 


Beyond the Googol 55 

sential characteristics of a continuum. Cantor, in referring 
to the “cardinality of the continuum,” recognized that it 
applies alike to the class of real numbers and the class 
of points on a line segment. Both are everywhere dense, 
and both have the same cardinality, C. In other words, 
it is possible to pair the points on a line segment with 
the real numbers. 

Classes with the cardinality C possess a property similar 
to classes with the cardinadity No: they may be thinned 
out without in any way affecting their cardinality. In 
this connection, we see in very striking fashion another 
illustration of the principle of transfinite arithmetic, 
that the whole is no greater than many of its parts. For 
instance, it can be proved that there are as many points 
on a line one foot long as there are on a line one yard 
long. The line segment AB in Fig. 13 is three times 
as long as the line A*B‘. Nevertheless, it is possible to 
put the class of all points on the segment AB into a one- 
to-one correspondence with the class of points on the 
segment A'B\ 


L 


FIG. 13. 

Let L be the intersection of the lines AA' and BB'. 
If then to any point Ad of AB^ there corresponds a point 


5 ^ Mathematics and the Imagination 

M' oi A'B\ which is on the line LM, we have established 
the desired correspondence between the class of points 
on A'B' and those on AB. It is easy to see intuitively 
and to prove geometrically that this is always possible, 
and that, therefore, the cardinality of the two classes 
of points is the same. Thus, since A*B^ is smaller than 
AB^ it may be considered a proper part of AB^ and we 
have again established that an infinite class may contain 
as proper parts, subclasses equivalent to it. 

There are more startling examples in geometry which 
illustrate the power of the continuum. Although the 
statement that a line one inch in length contains as many 
points as a line stretching around the equator, or as a 
line stretching from the earth to the most distant stars, 
is startling enough, it is fantastic to think that a line 
segment one-millionth of an inch long has as many points 
as there are in all three-dimensional space in the entire 
universe. Nevertheless, this is true. Once the principles of 
Cantor s theory of transfinites is understood, such state¬ 
ments cease to sound like the extravagances of a mathe¬ 
matical madman. The oddities, as Russell has said, “then 
become no odder than the people at the antipodes who 
used to be thought impossible because they would find it 
so inconvenient to stand on their heads.*’ Even conceding 
that the treatment of the infinite is a form of mathemati¬ 
cal madness, one is forced to admit, as does the Duke in 
Measure for Measure: 

If she be mad,—as I believe no other,— 

Her madness hath the oddest frame of sense, 

Such a dependency of thing on thing. 

As e’er I heard in madness.” 

♦ 

Until now we have deliberately avoided a definition 


Beyond the Googol 57 

of “infinite class.” But at last our equipment makes it 
possible to do so. We have seen that an infinite class, 
whether its cardinality is C, or greater, may be 
thinned out in a countless variety of ways, without 
affecting its cardinality. In short, the whole is no greater 
than many of its parts. Now, this property does not 
belong to finite classes at all; it belongs only to infinite 
classes. Hence, it is a unique method of determining 
whether a class is finite or infinite. Thus, our definition 
reads: An infinite class is one which can be put into one-to-one 
reciprocal correspondence with a proper subset of itself. 

Equipped with this definition and the few ideas we 
have gleaned we may re-examine some of the paradoxes 
of Zeno. That of Achilles and the tortoise may be ex¬ 
pressed as follows: Achilles and the tortoise, running 
the same course, must each occupy the same number of 
distinct positions during their race. However, if Achilles 
is to catch his more leisurely and determined opponent, 
he will have to occupy more positions than the tortoise, 
in the same elapsed period of time. Since this is man¬ 
ifestly impossible, you may put your money on the 
tortoise. 

But don’t be too hasty. There are better ways of saving 
money than merely counting change. In fact, you had 
best bet on Achilles after all, for he is likely to win the 
race. Even though we may not have realized it, we have 
just finished proving that he could overtake the tortoise 
by showing that a line a millionth of an inch long has 
just as many points as a line stretching from the earth 
to the furthest star. In other words, the points on the 
tiny line segment can be placed into one-to-one corre¬ 
spondence with the points on the great line, for there 
is no relation between the number of points on a line 


58 Mathematics and the Imagination 

and its length. But this reveals the error in thinking that 
Achilles cannot catch the tortoise. The statement that 
Achilles must occupy as many distinct positions as the 
tortoise is correct. So is the statement that he must travel 
a greater distance than the tortoise in the same time. 
The only incorrect statement is the inference that since 
he must occupy the same number of positions as the 
tortoise he cannot travel further while doing so. Even 
though the classes of points on each line, which cor¬ 
respond to the several positions of both Achilles and the 
tortoise are equivalent, the line representing the path of 
Achilles is much longer than that representing the path 
of the tortoise. Achilles may travel much further than 
the tortoise without successively touching more points. 

The solution of the paradox involving the arrow in 
flight requires a word about another type of continuum. 
It is convenient and certainly familiar to regard time as a 
continuum. The time continuum has the same properties 
as the space continuum: the successive instants in any 
elapsed portion of time, just as the points on a line, may 
be put into one-to-one correspondence with the class of 
real numbers; between any two instants of time an 
infinity of others may be interpolated; time also has the 
mathematical property mentioned before—it is every¬ 
where dense. 

Zeno’s argument stated that at every instant of time 
the arrow was somewhere, in some place or position, 
and therefore, could not at any instant be in motion. 
Although the statement that the arrow had at every 
moment to be in some place is true, the conclusion that, 
therefore, it could not be moving is absurd. Our natural 
tendency to accept this absurdity as true springs from our 
firm conviction that motion is entirely different from rest. 


Beyond the Googol 

We are not confused about the position of a body when 

it is at rest we feel there is no mystery about the state 

of rest. We should feel the same when we consider a 
body in motion. 

When a body is at rest, it is in one position at one 
instant of time and at a later instant it is still in the same 
position. When a body is in motion, there is a one-to-one 
correspondence between every instant of time and every 
new position. To make this clear we may construct two 
tables: One will describe a body at rest, the other, a 
body in motion. The “rest” table will tell the life history 


MOTION 


On Bedloe’s Island 9 a.m. In the city 


On Bedloe’s Island 11 a.m. Over the river 


On Bedloe’s Island 3 p.m. In the mountains. 

FIG. 14.—At the times shown, the Statue of 
Liberty is at the point shown, while the taxi’s 
passengers see the different scenes shown at the 
right. 


6o Mathematics and the Imagination 

and the life geography of the Statue of Liberty, while 
the “motion” table will describe the Odyssey of an auto¬ 
mobile. 

The tables indicate that to every instant of time there 
corresponds a position of the Statue of Liberty and of the 
taxi. There is a one-to-one space-time correspondence for 
rest as well as for motion. 

No paradox is concealed in the puzzle of the arrow 
when we look at our table. Indeed, it would be strange if 
there were gaps in the table; if it were impossible, at any 
instant, to determine exactly what the position of the 
arrow is. 

Most of us would swear by the existence of motion, 
but we are not accustomed to think of it as something 
which makes an object occupy different positions at 
different instants of time. We are apt to think that motion 
endows an object with the strange property of being 
continually nowhere. Impeded by the limitations of our 
senses which prevent us from perceiving that an object in 
motion simply occupies one position after another and 
does so rather quickly, we foster an illusion about the 
nature of motion and weave it into a fairy tale. Mathe¬ 
matics helps us to analyze and clarify what we perceive, 
to a point where we are forced to acknowledge, if we no 
longer wish to be guided by fairy tales, that we live either 
in Mr. Russell’s changeless world or in a world where 
motion is but a form of rest. The story of motion is the 
same as the story of rest. It is the same story told at a 
quicker tempo. The story of rest is: “It is here.” The story 
of motion is; “It is here, it is there.” Because, in this re¬ 
spect, it resembles Hamlet’s father’s ghost is no reason to 
doubt its existence. Most of our beliefs are chained to less 
substantial phantoms. Motion is perhaps not easy for our 


Beyond the Googol 61 

senses to grasp, but with the aid of mathematics, its 
essence may first be properly understood. 

* 

At the beginning of the twentieth century it was 
generally conceded that Cantor’s work had clarified the 
concept of the infinite so that it could be talked of and 
treated like any other respectable mathematical concept. 
The controversy which arises wherever mathematical 
philosophers meet, on paper, or in person, shows that 
this was a mistaken view. In its simplest terms this con¬ 
troversy, so far as it concerns the infinite, centers about 
the questions: Does the infinite exist? Is there such a 
thing as an infinite class? Such questions can have little 
meaning unless the term mathematical “existence” is first 
explained. 

In his famous “Agony in Eight Fits,” Lewis Carroll 
hunted the snark. Nobody was acquainted with the 
snark or knew much about it except that it existed and 
that it was best to keep away from a boojum. The 
infinite may be a boojum, too, but its existence in any 
form is a matter of considerable doubt. Boojum or garden 
variety, the infinite certainly does not exist in the same 
sense that we say, “There are fish in the sea.” For that 
matter, the statement “There is a number called 7” 
refers to something which has a different existence from 
the fish in the sea. “Existence” in the mathematical sense 
is wholly different from the existence of objects in the 
physical world. A billiard ball may have as one of its 
properties, in addition to whiteness, roundness, hardness, 
etc., a relation of circumference to diameter involving 
the number tt. We may agree that the billiard ball and 
TT both exist; we must also agree that the billiard ball 
and TT lead different kinds of lives. 


62 


Mathematics and the Imagination 

There have been as many views on the problem of 
existence since Euclid and Aristode as there have been 
philosophers. In modern times, the various schools of 
mathematical philosophy, the Logistic school. Formalists, 
and Intuitionists, have all disputed the somewhat less 
than glassy essence of mathematical being. All these dis¬ 
putes are beyond our ken, our scope, or our intention. A 
stranger company even than the tortoise, Achilles, and 
the arrow, have defended the existence of infinite classes 
—defended it in the same sense that they would defend 
the existence of the number 7. The Formalists, who think 
mathematics is a meaningless game, but play it with no 
less gusto, and the Logistic school, which considers that 
mathematics is a branch of logic—both have taken 
Cantor’s part and have defended the alephs. The defense 
rests on the notion of self-consistency. “Existence” is a 
metaphysical expression tied up with notions of being and 
other bugaboos worse even than boojums. But the ex¬ 
pression, “self-consistent proposition” sounds like the 
language of logic and has its odor of sanctity. A propo¬ 
sition which is not self-contradictory is, according to the 
Logistic school, a true existence statement. From this 
standpoint the greater part of Cantor’s mathematics of 
the infinite is unassailable. 

New problems and new paradoxes, however, have 
been discovered arising out of parts of Cantor’s structure 
because of certain difficulties already inherent in class¬ 
ical logic. They center about the use of the word “all.” 
The paradoxes encountered in ordinary parlance, such 
as “All generalities are false including this one,” con¬ 
stitute a real problem in the foundations of logic, just as 
did the Epimenides paradox whence they sprang. In the 
Epimenides, a Cretan is made to say that all Cretans are 


Beyond the Googol 63 

liars, which, if true, makes the speaker a liar for telling 
the truth. To dispose of this type of paradox the Logistic 
school invented a “Theory of Types.” The theory of 
types and the axiom of reducibility on which it is based 
must be accepted as axioms to avoid paradoxes of this 
kind. In order to accomplish this a reform of classical 
logic is required which has already been undertaken. 
Like most reforms it is not wholly satisfactory—even to 
the reformers—but by means of their theory of types the 
last vestige of inconsistency has been driven out of the 
house that Cantor built. The theory of transfinites may 
still be so much nonsense to many mathematicians, but 
it is certainly consistent. The serious charge Henri Poin¬ 
care expressed in his aphorism, “La logistique n’est plus 
sterile; elle engendre la contradiction,” has been success¬ 
fully rebutted by the logistic doctrine so far as the infinite 
is concerned. 

To Cantor’s alephs then, we may ascribe the same 
existence as to the number 7. An existence statement free 
from self-contradiction may be made relative to either. 
For that matter, there is no valid reason to trust in the 
finite any more than in the infinite. It is as permissible 
to discard the infinite as it is to reject the impressions of 
one’s senses. It is neither more, nor less scientific to do so. 
In the final analysis, this is a matter of faith and taste, 
but not on a par with rejecting the belief in Santa Claus. 
Infinite classes, judged by finite standards, generate para¬ 
doxes much more absurd and a great deal less pleasing 
than the belief in Santa Claus; but when they are judged 
by the appropriate standards, they lose their odd appear¬ 
ance, behave as predictably as any finite integer. 

At last in its proper setting, the infinite has assumed a 
respectable place next to the finite, just as real and just as 


Mathematics and the Imagination 


dependable, even though wholly different in character. 
Whatever the infinite may be, it is no longer a purple 
cow. 


FOOTNOTES 

1. We distinguish cardinal from ordinal numberSy which denote the re¬ 
lation of an element in a class to the others, with reference to some 
system of order. Thus, we speak of the first Pharaoh of Egypt, 
or of the fourth integer, in their customary order, or of the third 
day of the week, etc. These are examples of ordinals. P. 30. 

2. For the definition of primes, see the Chapter on pie. —P. 32. 

3. This series is said to converge to a limit —1. Discussion of this 
concept must be postponed to the chapters on pie and the cal¬ 
culus.—P. 38. 

4. A transcendental number is one which is not the root of an 
algebraic equation with integer coefficients. See pie. —P. 49. 

5. Any terminating decimal, such as .4, has a nonterminating form 
.3999. . . —P. 51. 

6. A simple geometric interpretation of the class of all one-valued 
functions F is the following: With each point of a line segment, 
associate a color of the spectrum. The class F is then composed 
of all possible combinations of colors and points that can be 
conceived.—P. 54. 


PIE (TrJ.e) 

Transcendental and Imaginary 


In order to reach the Truth, it is necessary, once in one's 
life, to put everything in doubt—so far as possible. 

—DESCARTES 


Perhaps pure science begins where common sense ends; 
perhaps, as Bergson says, “Intelligence is characterized 
by a natural lack of comprehension of life.’' ' But we have 
no paradoxes to preach, no epigrams to sell. It is only 
that the study of science, particularly mathematics, often 
leads to the conclusion that one need only say that a thing 
is unbelievable, impossible, and science will prove him 
wrong. Good common sense makes it plain that the earth 
is flat and stands still, that the Chinese and the Antipo- 
deans walk about suspended by their feet like chandeliers, 
that parallel lines never meet, that space is infinite, that 
negative numbers are as real as negative cows, that -1 
has no square root, that an infinite series must have an 
infinite sum, or that it must be possible with ruler and 
compass alone to construct a square exactly equal in area 
to a given circle. 

Just how far have we been carried by common sense 
in arriving at these conclusions? Not very far! Yet some of 
the statements seem quite plausible, even inescapable. 
It would be wrong to say that science has proved that all 
are false. We may still cling to the Euclidean hypothesis 
that parallel lines never meet and remain always equi- 


66 Mathematics and the Imagination 

distant, as long as we remember it is merely a hypothesis, 
but the statements about the squaring of the circle, the 
square root of —1, and about infinite series belong in a 
different category. 

The circle can not be squared with ruler and compass, 
— 1 has a square root. An infinite series can have a finite 
sum. Three symbols, tt, e, have enabled mathemati¬ 
cians to prove these statements, three symbols which rep¬ 
resent the fruits of centuries of mathematical research. 
How do they stand up to common sense? 

* 

The most famous problem in the entire history of math¬ 
ematics is the “squaring of the circle.’^ Two other prob¬ 
lems which challenged Greek geometers, the “duplication 
of the cube” and the “trisection of an angle,” may, as a 
matter of interest, be briefly considered with the first, 
even though squaring the circle alone involves tt. 

In the infancy of geometry, it was discovered that it 
was possible to measure the area of a figure bounded by 
straight lines. Indeed, geometry was devised for that very 
purpose—to measure the fields in the valley of the Nile, 
where each year the floods from the rising river obliter¬ 
ated every mark made by the farmer to indicate which 
fields were his and which his neighbors. Measuring areas 
bounded by curved lines presented greater difficulties, 
and an effort was made to reduce every problem of this 
type to one of measuring areas with straight boundaries. 
Clearly, if a square can be constructed with the area 
of a given circle, by measuring the area of the square, 
that of the circle is determined. The expression “squar¬ 
ing the circle” derives its name from this approach. 

The number tt is the ratio of the circumference of a 
circle to its diameter. The area of a circle of radius r is 


PIE (tt, i, e)—Transcendental and Imaginary 67 

given by the formula Now the area of a square with 
side of length A is A^. Thus, the algebraic statement: 
A^ = irr^ expresses the equivalence in area between a 
given square and a circle. Taking square roots of both 
sides of this equation yields A = As r is a known 

quantity, the problem of squaring the circle is, in effect, 
the computation ^ of the value of tt. 

Since mathematicians have succeeded in computing 
TT with extraordinary exactitude, what then is meant by 
the statement, “It is impossible to square the circle?’* 
Unfortunately, this question is still shrouded in many 
misapprehensions. But these would vanish if the problem 
were understood. 

* 

Squaring the circle is proclaimed impossibUy but what 
does “imfxjssible” mean in mathematics? The first steam 
vessel to cross the Atlantic carried, as part of its cargo, a 
book that “proved” it was impossible for a steam vessel 
to cross anything, much less the Atlantic. Most of the 
savants of two generations ago “proved” that it would be 
forever impossible to invent a practical heavier-than-air 
flying machine. The French philosopher, Auguste Comte, 
demonstrated that it would always be impossible for the 
human mind to discover the chemical constitution of the 
stars. Yet, not long after this statement was made the 
spectroscope was applied to the light of the stars, and we 
now know more about their chemical constitution, in¬ 
cluding those of the distant nebulae, than we know about 
the contents of our medicine chest. As just one illustra¬ 
tion, helium was discovered in the sun before it was found 
in the earth. 

Museums and patent offices are filled with cannons, 
clocks, and cotton gins, already obsolete, each of which 


6 


68 Mathematics and the Imagination 

confounded predictions that their invention would be im¬ 
possible. A scientist who says that a machine or a project 
is impossible only reveals the limitations of his day. What¬ 
ever the intentions of the prophet, the prediction has none 
of the qualities of prophecy. “It is impossible to fly to the 
moon” is meaningless, whereas “We have not yet devised 
a means of flying to the moon” is not. 

Statements about impossibility in mathematics are of 
a wholly different character. A problem in mathematics 
which may not be solved for centuries to come is not 
always impossible. “Impossible” in mathematics means 
theoretically impossible, and has nothing to do with the 
present state of our knowledge. “Impossible” in mathe¬ 
matics does not characterize the process of making a silk 
purse out of a sow’s ear, or a sow’s ear out of a silk purse; 
it does characterize an attempt to prove that 7 dmes 6 is 
43 (in spite of the fact that people not good at arithmetic 
often achieve the impossible). By the rules of arithmetic 
7 times 6 is 42, just as by the rules of chess, a pawn must 
make at least 5 moves before it can be queened. 

Where theoretical proof that a problem cannot be 
solved is lacking, it is legitimate to attempt a solution, no 
matter how improbable the hope of success. For centuries 
the construction of a regular polygon of 17 sides was 
rightly considered difficult, but falsely considered im¬ 
possible, for the nineteen-year-old Gauss in 1796 suc¬ 
ceeded in finding an elementary construction.® On the 
other hand, many famous problems, such as Fermat’s 
Last Theorem,^ have defied solution to this day in spite 
of heroic researches. To determine whether we have the 
right to say that squaring the circle, trisecting the angle, 
or duplicating the cube is impossible^ we must find logical 
proofs, involving purely mathematical reasoning. Once 


PIE (tt, z, e)—Transcendental and Imaginary 69 

such proofs have been adduced, to continue the search 
for a solution is to hunt for a three-legged biped.^ 

* 

Having determined what mathematicians mean by 
impossible, the bare statement, “It is impossible to square 
the circle” still remains meaningless. To give it meaning 
we must specify how the circle is to be squared. When 
Archimedes said, “Give me a place to stand and I will 
move the earth,” he was not boasting of his physical 
powers but was extolling the principle of the lever. When 
it is said that the circle cannot be squared, all that is 
meant is that this cannot be done with ruler and compass alone^ 
although with the aid of the integraph or higher curves 
the operation does become possible. 

Let us repeat the problem; It is required to construct 
a square equal in area to a given circle, by means of an 
exact theoretical plan, using only two instruments: the 
ruler and compass. By a ruler is meant a straightedge, 
that is, an instrument for drawing a straight line, not 
for measuring lengths. By a compass is meant an instru¬ 
ment with which a circle with any center and any radius 
can be drawn. These instruments are to be used a finite 
number of times, so that limits or converging processes 
with an infinite number of steps may not be employed.® 
The construction, by purely logical reasoning, depending 
only on Euclid’s axioms and theorems, is to be absolutely 
exact. 

The concepts of “limit” and “convergence” are more 
fully explained elsewhere,’ but a word about them here 
is in place. 

Consider the familiar series 1 + ^ + l + 

3 T "t" • - • The sum of the first 5 terms of this series is 
1.9375; the sum of the first 10 terms is 1.9980 . . . ; the 


70 Mathematics and the Imagination 

sum of the first 15 is 1.999781 . . . What is readily appar¬ 
ent is that this series tends to choke off, i.e., the additional 
terms which are added become so small that even a vast 
number will not cause the series to grow beyond a finite 
bound. In this instance the bound, or limit, is 2. Such a 
series which chokes off is said to ^^convergd"^^ to a ^^limitJ*^ 


FIG. 15,—An infinite number of terms with a finite sum. 

If the width of the first block is one foot, the width of 
the second foot, of the third \ foot, of the fourth J foot, 
and so one, then an infinite number of blocks rests on the 
2-foot bar, that is: 

The geometric analogues of the concepts of limit and 
convergence are equally fruitful. A circle may be re¬ 
garded as the limit of the polygons with increasing num¬ 
ber of sides which may be successively inscribed in it, or 
circumscribed about it, and its area as the common limit 
of both of these sets of polygons. 

This is not a rigorous definition of limit and conver¬ 
gence, but too often mathematical rigor serves only to 
bring about another kind of rigor —rigor mortis of math¬ 
ematical creativencss. 


PIE (tt, i, e)—Transcendental and Imaginary 71 

To return to squaring the circle: the Greeks, and later 
mathematicians, sought an exact construction with ruler 
and compass, but always failed. As we shall see later, all 
ruler and compass constructions are geometric equiva¬ 
lents of first- and second-degree algebraic equations and 
combinations of such equations. But the German mathe¬ 
matician Lindemann, in 1882, published a proof that tt 
is a transcendental number and thus any equation which 
satisfies it cannot be algebraic and surely not algebraic of 
first or second degree. It follows that the statement, “The 
squaring of the circle is impossible with ruler and compass 
alone,” is meaningful. 

So far as the other two problems are concerned, thanks 
in part to the work of “the marvelous boy . . . who 
perished in his prime,” the sixteen-year-old Galois, it was 
established about one hundred years ago that the dupli¬ 
cation of the cube and the trisection of an angle are also 
impossible with ruler and compass. We may allude to 
them briefly. 

There is a story among the Greeks that the problem of 
duplicating the cube originated in a visit to the Delphic 
oracle. There was an epidemic raging at the time, and 
the oracle said the epidemic would cease only if a cubical 
altar to Apollo were doubled in size. The masons and 
architects made the mistake of doubling the side of the 
cube, but that made the volume eight times as great. Of 
course the oracle was not satisfied, and the Greek math¬ 
ematicians, on re-examining the problem began to see 
that the right answer involved, not doubling the side, 
but multiplying it by the cube root of 2. This could not 
be done geometrically with ruler and compass. They 
finally succeeded by using other instruments and higher 
curves. The oracle was appeased and the epidemic 


72 Mathematics and the Imagination 

ceased. You may believe the story or not, much as you 
choose, but you cannot “duplicate the cube.”® 

The trisection of an angle has received a good deal of 
attention in the newspapers during the past few years 
because monographs continue to crop up which claim to 
solve the problem completely. The fallacies contained in 
these “solutions” are of four kinds: they are sometimes 
merely approximate and not exact; instruments other 
than the ruler and compass are occasionally used, either 
wittingly or unwittingly; at times there is a logical fallacy 
in the intended proof; and often only special and not 
general angles are considered. An angle can be bisected 
but not trisected by elementary geometry, since the first 
problem involves merely square roots, while the second 
involves cube roots, which, as we have stated, cannot be 
constructed with ruler and compass. 

★ 

The difficulty in squaring the circle, as stated at the 
outset, lies in the nature of the number tt. This remark¬ 
able number, as Lindemann proved, cannot be the root 
of an algebraic equation with integer coefficients.^® It is 
therefore not expressible by rational operations, or by the 
extraction of square roots, and as only such operations 
can be translated into an equivalent ruler and compass 
construction, it is impossible to square the circle. The pa¬ 
rabola is a more complicated curve than a circle, but 
nevertheless, as Archimedes knew, any area bounded by a 
parabola and a straight- line can be determined by ra¬ 
tional operations, and hence the “parabola can be 
squared.” 

Lindemann’s proof is too technical to concern us here. 
If, however, we consider the history and development of 
TT, we shall be in a better position to understand its 


PIE (x, z, e)—Transcendental and Imaginary 73 

purpose without being compelled to master its difficulties. 
If a triangle is inscribed in a circle (fig. 16), the area of 

the inscribed triangle will be less than the area of the 
circle: 


no. 16.—^The circle as the limit of inscribed and 
circumscribed polygons. 


The difference between the area of the circle and the 
triangle are the three shaded portions of the circle. Now 
consider the same circle with a triangle circumscribed about 
it (Fig. 16). The area of the circumscribed triangle will 
be greater than the area of the circle. The three shaded 
portions of the triangle again represent the difference in 
area. It may readily be seen that if the number of sides of 
the inscribed figure is doubled, the area of the resulting 
hexagon will be less than the area of the circle, but closer 
to it than the area of the inscribed triangle. Similarly, if 
the number of sides of the circumscribed triangle is 
doubled, the area of the circumscribed hexagon will still 
be greater than the area of the circle but, again, closer to 


*74 Mathematics and the Imagination 

it than the area of the circumscribed triangle. By well- 
known, simple, geometric methods, employing only ruler 
and compass, the number of sides of the inscribed and 
circumscribed polygons may be doubled as many times as 
desired. The area of the successively inscribed polygons 
will approach that of the circle, but will always remain 
slightly less; the area of the circumscribed polygons will 
also approach that of the circle but their area will always 
remain slightly greater. The common value approached 
by both is the area of the circle. In other words, the circle 
is the limit of these two series of polygons. If the radius of 
the circle is equal to 1, its area, which equals ttt^, is 
simply TT. 

This method of increasing and decreasing polygons for 
computing the value of tt was known to Archimedes, who, 
employing polygons of 96 sides, showed that tt is less than 

and greater than • Somewhere in between lies the 
area of the circle. 

Archimedes’ approximation for tt is considerably closer 
than that given in the Bible. In the Book of Kings, and in 
Chronicles, tt is given as 3. Egyptian mathematicians 
gave a somewhat more accurate value—3.16. The fa¬ 
miliar decimal—3.1416, used in our schoolbooks, was 
already known at the time of Ptolemy in 150 a.d. 

Theoretically, Archimedes’ method for computing tt 
by increasing the number of sides of the polygons may 
be extended indefinitely, but the requisite calculations 
soon become very cumbersome. None the less, during 
the Middle Ages such calculations were zealously carried 
out. 

Francisco Vieta, the most eminent mathematician of 
the sixteenth century, though not a professional, made a 
great advance in the calculation of tt in determining its 


PIE (tt, 2 , e)—Transcendental and Imaginary 75 

value to ten decimal places. In addition to giving the 
formula: 


a nonterminating product, and making many other im¬ 
portant mathematical discoveries, Vieta rendered service 
to King Henry IV of France, in the war against Spain, 
by deciphering intercepted letters addressed by the Span¬ 
ish Crown to its governors of the Netherlands, The Span¬ 
iards were so impressed that they attributed his discovery 
of the cipher key to magic. It was neither the first nor the 
last time that the efforts of mathematicians were branded 
as necromancy. 

In 1596 Ludolph van Ceulen, the German mathe¬ 
matician, long a resident in Holland, calculated 35 
decimal places for tt. Instead of the epitaph, “died at 40, 
buried at 60,” appropriate where cerebration ceases 
just when life is supposed to begin, van Ceulen, who 
worked on tt almost to the day of his death at the age of 
70, requested that the 35 digits of tt which he had com¬ 
puted be inscribed as a fitting epitaph on his tombstone. 
This was actually done. The value he gave for tt is, in 
part, 3.14159 26535 89793 23846... In memory of his 
achievement the Germans still call this number the Lu- 

dolphian number. We propose to call tt the Archimedean 
number. 

* 

The number tt reached maturity with the invention 
of the calculus by Newton and Leibniz. The Greek 
method was abandoned and the purely algebraic device 
of convergent infinite series, products, and continued 
fractions came into vogue. John Wallis (1616-1703), the 


76 Mathematics and the Imagination 

Englishman, contributed one of the most famous prod 
ucts: 


Leibniz’ infinite series, unlike Wallis’ product for tt, 
is a sum: 


The successive products and sums of the terms of these 
series yield values of tt as accurate as desired. These proc¬ 
esses, typical of the powerful methods of approximation 
used not only in mathematics but in the other sciences, 
although much less cumbersome than the method em¬ 
ployed by the Greeks, still entail a great deal of calcula¬ 
tion. The products of Wallis’ series are: 


1 ’13 3’1^3^3 9’1^3 3 5 45’ 


etc. 


2 


FIG. 17.—Wallis’ product. 


PIE (tt, i, e)—Transcendental and Imaginary 77 

Taking the successive sums of Leibniz’ series, we obtain; 

1 , = 1.1 1 76 

’ 3 3 ’ ^ 3 ^ 5 “ 15 ’ 


1 — i -|- i _ 1 = 

3 ^ 5 7 105 ’ 


etc. 


1 


•’A 


♦ Vs -Vt 


y I _ I 


+ 


...)- 


FIG. 18. —Leibniz’ series. 

^ = 0.795 . . . 

5 = i -1 + 1_1 + !_± + 

After taking the first 50 terms of these series, the next 
50 will not yield an appreciably more accurate value 
of TT, for the series converge rather slowly. The rapidly 
convergent series 

= J- +J_^ 

4 \5 3-53 ^5-5^ 7-5^ 

U _^ + _^ 

\239 3-2393 ^ 5.2393 7.2397 -r . . -y 

is much more useful, and is frequently employed in mod¬ 
ern mathematics. Its relation to tt was established by 
Machin (1680-1752). Using even more rapidly converg¬ 
ing series, Abraham Sharp, in 1699, calculated tt to 71 
decimal places. Dase, a lightning calculator employed by 
Gauss, worked out 200 places in 1824. In 1854, Richter 
computed 500 places, and finally, in 1873, Shanks, an 
English mathematician, achieved a curious kind of im¬ 
mortality by determining tt to 707 decimal places. Even 


78 Mathematics and the Imagination 

today it would require 10 years of calculation to deter¬ 
mine TT to 1000 places. Yet that does not seem like a waste 
of time as compared with the billions of hours spent by 
millions of people on crossword puzzles and contract 
bridge, to say nothing of political debates. 

Of course Shank’s result has no conceivable use in ap¬ 
plied science. No more than 10 decimal places for tt are 
ever needed in the most precise work. The famous Amer¬ 
ican astonomer and mathematician, Simon Newcomb, 
once remarked, “Ten decimal places are sufficient to give 
the circumference of the earth to the fraction of an inch, 
and thirty decimals would give the circumference of the 
whole visible universe to a quantity imperceptible with 
the most powerful telescope.” 

Why, then, has so much time zind effort been devoted 
to the calculation of tt? The reason is twofold. First, by 
studying infinite series mathematicians hoped they 
might find some clue to its transcendental nature. 
Second, the fact that tt, a purely geometric ratio, could 
be evolved out of so many arithmetic relationships—out 
of infinite series, with apparendy little or no relation to 
geometry—was a never-ending source of wonder and a 
never-ending stimulus to mathematical acdvity. 

Who would imagine—that is, who but a mathemati¬ 
cian—that the number expressing a fundamental relation 
between a circle and its diameter could grow out of the 
curious fraction communicated by Lord Brouncker 
(1620-1684) to John Wallis? 

4 

1 + P 

2 + 32 
2 + 52 

2 + 72 . . . 


TT 


PIE (tt, z, e)—Transcendental and Imaginary 79 

But just such relations between infinite series and tt 
illustrate the profound connection between most mathe¬ 
matical forms, geometric or algebraic. It is mere coinci¬ 
dence, a mere accident that tt is defined as the ratio of 
the circumference of a circle to its diameter. No matter 
how mathematics is approached, tt forms an integral 
part.*^ In his Budget of Paradoxesy Augustus De Morgan 
illustrated how little the usual definition of tt suggests its 
origin. He was explaining to an actuary what the chances 
were that, at the end of a given time a certain proportion 
of a group of people would be alive, and quoted the for¬ 
mula employed by actuaries which involves tt. On ex¬ 
plaining the geometric meaning of tt, the actuary, who 
had been listening with interest, interrupted and ex¬ 
claimed, “My dear friend, that must be a delusion. 
What can a circle have to do with the number of people 
alive at the end of a given time?” 

To recapitulate briefly, the problem of squaring the 
circle turns out to be an impossible construction with 
ruler and compass alone. The only constructions possible 
with these instruments correspond to first- and second- 
degree algebraic equations. Lindcmann proved that tt is 
not only not the root of a first- or second-degree algebraic 
equation, but is not the root of any algebraic equation 
(with integer coefficients), no matter how great the de¬ 
gree; therefore tt is transcendental. Here, then, is the end 
of every hope of proving this classical problem in the in¬ 
tended way. Here is mathematical impossibility. 

* 

When the Greek philosophers found that the square 

root of 2 is not a rational number, t^ey celebrated the 
discovery by sacrificing 100 oxen. The much more pro¬ 
found discovery that tt is a transcendental number de- 


8o 


Mathematics and the Imagination 

serves a greater sacrifice. Again mathematics triumphed 
over common sense, tt, a finite number the ratio of the 
circumference of a circle to its diameter is accurately 
expressible only as the sum or product of an infinite series 
of wholly different and apparently unrelated numbers. 
The area of the simplest of all geometric figures, the 
circle, Ccinnot be determined by finite (Euclidean) means. 

e 

In the seventeenth century, perhaps the greatest of all 
for the development of mathematics, there appeared a 
work which in the history of British science can be placed 
second only to Sir Isaac Newton’s monumental Principia. 
In 1614, John Napier of Merchiston issued his Mirifici 
Logariihmorum Canonis Descriptio, (“A Description of the 
Admirable Table of Logarithms”), the first treatise on 
logarithms.*® To Napier, who also invented the decimal 
point, we are indebted for an invention which is as im¬ 
portant to mathematics as Arabic numerals, the concept 
of zero, and the principle of positional notation.*'* With¬ 
out these, mathematics would probably not have ad¬ 
vanced much beyond the stage to which it had been 
brought 2000 years ago. Without logarithms the com¬ 
putations accomplished daily with ease by every math¬ 
ematical tyro would tax the energies of the greatest math¬ 
ematicians. 

Since e and logarithms have the Scime genealogical tree 
and were brought up together, we may for the moment 
turn our attention to logarithms to ascertain something 
of the nature of the number e. 

Stupendous calculations being required to construct 
trigonometric tables for navigation and astronomy, Na¬ 
pier was prompted to invent some device to facilitate 
these computations. Although contemporaries like Vieta 


PIE (tt, z, e)—Transcendental and Imaginary 8i 

and Ceulen vied with each other in performing almost 
unbelievably difficult feats of arithmetic, it was at best 
a labor of love, an exalted drudgery and self-immolation, 
with love’s labor often lost as the result of one small 
slip. 


Napier succeeded in achieving his purpose, in abbre¬ 
viating the operations of multiplication and division, op¬ 
erations “so fundamental in their nature that to shorten 
them seems impossible.” Nevertheless, by means of loga¬ 
rithms, every problem in multiplication and division, no 


matter how elaborate, reduces to a relatively easy one 
in addition and subtraction. Multiplying and dividing 
googols and googolplexes becomes as easy as adding a 
simple column of figures. 


Like many another of the profound and fecund inven¬ 


tions of mathematics, the underlying idea was so simple 
that one wonders why it had not been thought of earlier. 
Cajori recounts that Henry Briggs (1556-1631), professor 
of geometry at Oxford, “was so struck with admiration 


of Napier’s book, that he left his studies in London to 
do homage to the Scottish philosopher. Briggs was de¬ 
layed in his journey, and Napier complained to a com¬ 
mon friend, ‘Ah, John, Mr. Briggs will not come.’ At 
that very moment knocks were heard at the gate, and 
Briggs was brought into the lord’s chamber. Almost one 
quarter of an hour was spent, each beholding the other 
without speaking a word. At last Briggs began: ‘My lord, 
I have undertaken this long journey purposely to see your 
person, and to know by what engine of wit or ingenuity 
you came first to think of this most excellent help in as¬ 
tronomy, viz. the logarithms; but, my lord, being by you 

found out, I wonder nobody found it out before, when 
now known it is so easy.* ” 


32 Mathematics and the Imagination 


Napier’s conception of logarithms was based on an 
ingenious and well-known idea: a comparison between 
2 moving points, one of which generates an arithmetical, 

the other a geometric progression. 

The two progressions: 

Arithmetical— 0 1 23 4567 8... 

Geometric— 1 2 4 8 16 32 64 128 256 ... 


bear to each other this interesting relationship: If the 
terms of the arithmetical progression are regarded as 
exponents (powers) of 2, the corresponding terms of the 
geometric progression represent the quantity resulting 
from the indicated operation. Thus,'^ 2° = 1, 2' == 2, 


22 ^ 4 23 = 8, 2^ = 16, 2^ = 32, etc. Furthermore, to 
determine the value of the product 2^ X 2^, it is 
necessary to add the exponents, obtaining 2^^ = 2®, 
which is the desired product. Calling 2 the base, each term 
in the arithmetical progression is the logarithm oJ the corre¬ 
sponding term of the geometric progression. 

Napier explained this notion geometrically as follows: 
A point S moves along a straight line, AB, with a velocity 
at each point Si proportional to the remaining distance 
SiB. Another point R moves along an unlimited line, CD, 
with a uniform velocity equal to the initial velocity of S. 
If both points start from A and C at the same time, then 
the logarithm of the number measured by the distance 
S\B is measured by the distance CR\. 


FIG. 19.—Napier’s d*;Tiamic interpretation of 
logarithms. 


PIE (ir, iy e)—Transcendental and Imaginary 83 

By this method, as S^B decreases, its logarithm CR^ 
increases. But it soon became apparent that it was advan¬ 
tageous to define the logarithm of 1 as zero, and to have 
the logarithm grow with the number. Napier changed his 
system accordingly. 

One of the fruits of the higher education is the illumi¬ 
nating view that a logcu-ithm is merely a number that 
is found in a table. We shall have to widen the curric¬ 
ulum. If a, by and c are three numbers related by the 
equation ~ c, then by the exponent of a, is the loga¬ 
rithm of c to the base a. in other words, the logarithm of a 
number to the base a is the power to which a must be 
raised to obtain that number. In the example, 2^ = 8, 
the logarithm of 8 to the base 2 is 3. Or 10^ = 100, and 
the logarithm of 100 to the base 10 equals 2. The concise 
way of expressing this is: 3 = log 2 8, 2 = log 10 100. The 
simple table below gives all the essential properties of 
logarithms: 


(1) loga {b X c) 

(2) log. 0 = 


= loga b -f loga 

loga b — loga C, 


(3) loga b‘ = C X loga b. 

(4) logaV^ b = (7) 


Equations (1) and (2) indicate how to multiply or 
divide two numbers; nothing more is required than to 
add or subtract their respective logarithms. The result 
obtained is the logarithm of the product, or quotient. 
Equations (3) and (4) show that with the aid of loga¬ 
rithms the operations of raising to powers and extracting 
roots may be replaced by the much simpler ones of 
multiplication and division. 


7 


84 Mathematics and the Imagination 

Extensive tables of logarithms were soon constructed 
to the base 10 and to the Napierian or natural base e. So 
widely were these tables distributed that mathematicians 
all over Europe were able to avail themselves of the use of 
logarithms within a very short time of their invention. 
Kepler was one who not only saw the tables of Napier but 
himself advanced their development; he was thus one of 
the first of the legion of scientists whose contributions to 
knowledge were greatly facilitated by logarithms. 

The two systems of logs to the two bases, 10 and e 
(the Briggs and the natural base respectively), are the 
principal ones still in use, with e predominating.^® Like 
TT, the number e is transcendental and like tt it is what 
P. W. Bridgman names a “progrcim of procedure,” 
rather than a number, since it can never be completely 
expressed ( 1 ) in a finite number of digits, ( 2 ) as the root 
of an algebraic equation with integer coefficients, (3) ^ a 
nonterminating but repeating decimal. It can only be 
expressed with accuracy as the limit of a convergent 
infinite series or of a continued fraction. The simplest 
and most familiar infinite series giving the value of e is: 


Accordingly, its value may be approximated as closely 
as we please by taking additional terms of the series. 
To the tenth decimal places = 2.718281285. A glance at 
the table below will indicate how an infinite convergent 
series behaves as more and more of its terms are summed. 


( 1 ) 1 + 1 ; = 2 - 
( 2 ) 1 + ^ + ^ 


= 2.5 


PIE (tt, i, e)—Transcendental and Imaginary 85 

( 3 ) 1 + ^ ^ ^ = 2.6666666 ... 

(4) l+^ + ^ + l + l = 2.7083334 ... 

(5) l+i + ^ + i^ + l + ± = 2.7166666 ... 

W 1 + ^ + . . . + i = 2.7180555 ... 

(■ 7 ) 1 + + • . . + = 2.7182539 ... 

(8) 1 + ^ + . . . + ^ = 2.7182787 ... 

(9) 1 + ^ + . . . + ^ = 2.7182818 . .. 

Upon taking a few more terms, e looks like this: 

2.7182818284590452353602874 . . . 

Euler, who undoubtedly had the Midas touch in math¬ 
ematics, not only invented the symbol e and calculated its 
value to 23 places, but gave several very interesting ex¬ 
pressions for it, of which these two are the most important: 

(1) e = 2+ 1 

1 -hi _ 

2 + 2 

3 + 3 
4 + 4 

5 + 5... 


86 Mathematics and the Imagination 


( 2 ) = 1+1 

T +j_ 

1 + 1 _ 

1 + 1 _ 

5 + ^ 

1 

1 +J_ 

1 +2 

9... 


The need for navigational tables was not alone re¬ 
sponsible for the development of logarithms. Big business, 
particularly banking, played its part as well. A remark¬ 
able series, the limiting value of which is e, arises in the 


preparation of tables of compound interest. This series 
is obtainable from the expansion of ^1 


comes infinite. The origin of this important expression 
is interesting. 

Suppose your bank pays 3 per cent interest yearly on 
deposits. If this interest is added at the end of each year, 
for a period ot three years, the total amount to your 
credit, assuming an original capital of $1.00, is given 
by the formula; (1 + .03)®. If the interest is compounded 
semiannually, after the three-year period the total of 


principal plus interest would be 


.03\ 2X3 

t) ■ 


Imagine however that you are fortunate enough to 
find a philanthropic bank which decides to pay 100 per 
cent interest a year. Then the amount to your credit at 
the end of the year will be(l + 1)^ = $2.00. If the inter¬ 
est is compounded semiannually, the amount will be 


PIE (tt, i, e)—Transcendental and Imaginary 87 


~1” 2 )^^ “ or $2.25. If it is compounded quarterly, 
it will be (1 + = $2.43. It seems clear that the more 

often the interest is compounded, the more money you 
will have in the bank. By a further stretch of the imagina¬ 
tion, you may conceive of the possibility that the philan¬ 
thropic bank decides to compound the interest continuously, 
that is to say at every instant throughout the year. How 
much money will you then have at the end of the year? 
No doubt a fortune. At least, that is what you would sus¬ 
pect, even allowing for what you know about banks. In¬ 
deed you might become, not a millionaire, not a billion¬ 
aire, but more nearly what could be described as an “in- 
finitaire.” Alas, banish all delusions of grandeur, for the 
process of compounding interest continuously, at every 
instant, generates an infinite series which converges to the 
limit e. The sum on deposit after this hectic year, with its 
apparent promise of untold riches, would be not quite 


$2.72. For, if one takes the trouble to expand 


as n becomes very large, the successive values thus ob¬ 
tained approximate to the value of e, and where n be¬ 


comes infinite, 


actually yields the infinite series 


for e: 


Besides serving as the base for the natural logarithms, 
f is a number useful everywhere in mathematics and 
applied science. No other mathematiccil constant, not 
even tt, is more closely connected with human affairs. 
In economics, in statistics, in the theory of probability, 
and in the exponential function, e has helped to do one 


88 Mathematics and the Imagination 

thing and to do that better than any number yet dis¬ 
covered. It has played an integral part in helping mathe¬ 
maticians describe and predict what is for man the most 
important of all natural phenomena—that of growth. 

The exponential function, y = is the instrument 
used, in one form or another, to describe the behavior 
of growing things. For this it is uniquely suited: it is the 
only function of x with a rate of change with respect to x equal 
to the function itself.'^ A function, it will be remembered, 
is a table giving the relation between two variable quan¬ 
tities, where a change in one implies some change in the 
other. The cost of a quantity of meat is a function of its 
weight; the speed of a train, a function of the quantity of 
coal consumed; the amount of perspiration given off, a 
function of the temperature. In each of these illustrations, 
a change in the second variable; weight, quantity of coal 
consumed, and temperature, is correlated with a change 
in the first variable: cost, speed, and volume of perspira¬ 
tion. The symbolism of mathematics permits functional 
relationships to be simply and concisely expressed. Thus, 
y = x^ y = x"^, y — sin x, y = csch x^ y = e^ are ex¬ 
amples of functions. 

A function is not only adequate to describe the behav¬ 
ior- of a projectile in flight, a volume of gas under changes 
of pressure, an electric current flowing through a wire, 
but also of other processes which entail change, such as 
growth of population, growth of a tree, growth of an 
amoeba, or as we have just seen, growth of capital and 
interest. What is peculiar to every organic process is that 
the rate of growth is proportional to the state of growth. 
The bigger something is, the faster it grows. Under ideal 
conditions, the larger the population of a country be¬ 
comes, the faster it increases. The rate of speed of many 


PIE (ttj 2, e)—Transcendental and Imaginary 89 

chemical reactions is proportional to the quantity of the 
reacting substances which are present. Or, the amount 
of heat given off by a hot body to the surrounding me¬ 
dium is proportional to the temperature. The rate at 
which the total quantity of a radioactive substance dimin¬ 
ishes at any instant, owing to emanations, is proportional 
to the total quantity present at that instant. All these phe¬ 
nomena, which either are, or resemble, organic processes, 
may be accurately described by a form of the exponential 
function (the simplest beings = for this has the prop¬ 
erty that its rate of change is proportional to the rate 
of change of its variable. 

♦ 

A universe in which e and tt were lacking, would not, 
as some anthropomorphic soul has said, be inconceivable. 
One could hardly imagine that the sun would fail to 
rise, or the tides cease to flow for lack of tt and But 
without these mathematical artifacts, what we know 
about the sun and the tides, indeed our ability to describe 
all natural phenomena, physical, biological, chemical 
or statistical, would be reduced to primitive dimensions. 

Alice was criticizing Humpty Dumpty for the liberties 
he took with words: “When I use a word,” Humpty 
replied, in a scornful tone, “it means just what I choose 
it to mean—neither more nor less.” “The question is,” 
said Alice, whether you can make a word mean so 
mai\y different things.” “The question is,” said Humpty, 
“which is to be master, that’s all.” 

Those who are troubled (and there are many) by the 
word imaginary” as it is used in mathematics, should 
hearken unto the words of H. Dumpty. At most, of 
course, it is a small matter. In mathematics familiar 


go Mathematics and the Imagination 

words are repeatedly given technical meanings. But as 
Whitehead has so aptly said, this is confusing only to 
minor intellects. When a word is precisely defined, and 
signifies only one thing, there is no more reason to 
criticize its use than to criticize the use of a proper name. 
Our Christian names may not suit us, may not suit our 
friends, but they occasion litde misunderstanding. Con¬ 
fusion arises only when the same word packs several 
meanings and is what Humpty D. calls a “portmanteau.** 

Semantics, a rather fashionable science nowadays, is 
devoted to the study of the proper use of words. Yet there 
is much more need for semantics in other branches of 
knowledge than in mathematics. Indeed, the larger part 
of the world’s troubles today arise from the fact that 
some of its more voluble magnificoes are definitely anti- 
semantic. 

An imaginary number is a precise mathematical idea. 
It forced itself into algebra much in the same way as 
did the negative numbers. We shall see more clearly how 
imaginary numbers came into use if we consider the 
development of their progenitors—the negatives. 

Negative numbers appeared as roots of equations as 
soon as there were equations, or rather, as soon as mathe¬ 
maticians busied themselves with algebra. Every equa¬ 
tion of the form ax -\- b — 0, where a and b are greater 
than zero, has a negative root. 

The Greeks, for whom geometry was a joy and algebra 
a necessary evil, rejected negative numbers. Unable to 
fit them into their geometry, unable to represent them 
by pictures, the Greeks considered negative numbers no 
numbers at all. But algebra needed them if it were to 
grow up. Wiser than the Greeks, wiser than Omar 
Khayyam,the Chinese and the Hindus recognized 
negative numbers even before the Christian era. Not as 


PIE (ttj 2, e) ’Transcendental and Imaginary gi 

learned in geometry, they had no qualms about numbers 
of which they could draw no pictures. There is a repeti¬ 
tion of that indifference to the desire for concrete repre¬ 
sentation of abstract ideas in the contemporary theories 
of mathematical physics, (relativity, the mechanics of 
quanta, etc.) which, although understandable as symbols 
on paper, defy diagrams, pictures, or adequate metaphors 
to explain them in terms of common experience. 

Cardan, eminent mathematician of the sixteenth cen¬ 
tury, gambler, and occasional scoundrel, to whom al¬ 
gebra is vasdy indebted, first recognized the true im¬ 
portance of negative roots. But his scientific conscience 
twitted him to the point of calling them “fictitious.” 
Raphael Bombelli of Bologna carried on from where 
Cardan left off. Cardan had talked about the square roots 
of negative numbers, but he failed to understand the con¬ 
cept of imaginaries. In a work published in 1572, Bom¬ 
belli pointed out that imaginary quantities were essential 
to the solution of many algebraic equations. He saw that 
equations of the form -\- a = Oy where a is any num- 
ber g^reater than 0, could not be solved except with the 
aid of imaginaries. In trying to solve a simple equation 

+ 1 = 0, there are two alternatives. Either the equa¬ 
tion is meaningless, which is absurd, or is the square 
root of —1, which is equally absurd. But mathematics 
thrives on absurdities, and Bombelli helped it along by 
accepting the second alternative. 

♦ 

Three hundred and fifty years have gone by since 
Bombelli made his choice. Philosophers, sciendsts, and 
those with that minor-key quality of mind known as 
plain common sense have criticized, in ever-increasing 
diminuendo, the concept of the imaginary. All of these 
worthies are dead, most of them forgotten, while imagi- 


92 


Mathematics and the Imagination 

nary numbers flourish wickedly and wantonly over the 

whole field of mathematics. 

Occasionally, even the. masters snickered. Leibniz 
thought: “Imaginary numbers are a fine and wonderful 
refuge of the Holy Spirit, a sort of amphibian between 
being and not being.’* Even the mighty Euler said that 
numbers like the square root of minus one “are neither 
nothing, nor less than nothing, which necessarily con¬ 
stitutes them imaginary, or impossible.” He was quite 
right, but what he omitted to say was that unaginanes 
were useful and essential to the development of mathe¬ 
matics. And so they were allotted a place in the number 
domain with all the rights, privileges, and immunities 
thereunto appertaining. In time, the fears and queasiness 
about their essence all but vanished, so that the judgment 
of Gauss is the judgment of today: 

Our general arithmetic, so far surpassing in extent the 
geometry of the ancients, is entirely the creation of modem 
times. Starting originally from the notion of absolute integers, 
it has gradually enlarged its domain. To integers have been 
added fractions, to rational quantities, the irrational, to pos¬ 
itive, the negative, and to the real, the imaginary. This ad¬ 
vance, however, had always been made at first with timorous 
and hesitating steps. The early algebraists called the negative 
roots of equations false roots, and this is indeed the case when 
the problem to which they relate has been stated in such a 
form that the character of the quantity sought allows of no 
opposite. But just as in general arithmetic no one would 
hesitate to admit fractions, although there are so many count¬ 
able things where a fraction has no meaning, so we would not 
deny to negative numbers the rights accorded to positives, 
simply because innumerable things admit of no opposite. The 
reality of negative numbers is sufficiently justified since m 
innumerable other cases they find an adequate interpretauon. 


PIE (tt, z, e)—Transcendental and Imaginary 93 

This has long been admitted, but the imaginary quantities, 
formerly, and occasionally now, improperly called impossible, 
as opposed to real quantities—are still rather tolerated than 
fully naturalized; they appear more like an empty play upon 
symbols, to which a thinkable substratum is unhesitatingly 
denied, even by those who would not depreciate the rich 
contribution which this play upon symbols has made to the 
treasure of the relations of real quantities. 

* 

Imaginary numbers, like four-dimensional geometry, 
developed from the logical extension of certain proc¬ 
esses. The process of extracting roots is called evolution. 
It is an apt name, for imaginary numbers were literally 
evolved out of the extension of the process of extracting 

ro ots. I f 's/a, y/l, 's/ll had meaning, why not \/ — 4 , 

a/ ~ 7, \/ — 11? If — 1 =0 had a solution, why not 

+ I =0? The recognition of imaginaries was much 
like the United States recognizing Soviet Russia—the 
existence was undeniable, all that was required was for¬ 
mal sanction and approval. 

is the best-known imaginary. Euler represented 
It by the symbol “z"” which is still in use.^^ It is idle to 
be concerned with the question, “What number when 
multiplied by itself equals —1?’’ Like all other numbers, 
i is a symbol which represents an abstract but very 
precise idea. It obeys all the rules of arithmetic with the 
added convention that iXi = —1. Its obedience to these 
rules and its manifold uses and applications justify its 
existence regardless of the fact that it may be an anomaly. 

The formal laws of operation for i are easy: 

Since the rule of signs provides: 

( + 1) X (+1) = + 1 I /(-I) X ( + 1) = -1 
( + 1) X (-1) = - 1 / \(-l) X (-1) = +1 


Mathematics and the Imagination 
Accordingly: 

i X ( + 1 ) = 
i X ( - 1 ) = - V - 1 

- t X ( - 1 ) = + ^ 

= 

i X ! = 

= - 1 

i X i y. i = _ 

= (V“) 

= - 1 ) 

= - 

ixiyiyi = ,_ 

= ( - 1 ) X ( - 1) 

= + 1 

!' X z' X i X i X I = ^ 

= (V^)* (V -1 )Mv' - 1) 

= ( - 1) X ( - 1) X V~^ 
= ( + 1) X V - 1 
= — 1, etc.* 


• From which we may construct a convenient table: 


11 

< 

11 

. 


= V -1V -1 = -1 


i* 

= (V~1)’-(V -D’ = +1 


= +iV -1 = >■ 


= +i(V^)* = -1 

p 


= -i (V- i)'= +1 


_ 


The table shows that odd powers of i are equal to — *, or + *, and even 
powers of i are equal to — 1 or +1. 


PIE (tt, iy e') TTanscendental and Imaginary 

Extension of the use of imaginaries has led to complex 
numbers of the form a + iby where a and b are real 
numbers (as distinguished from imaginaries). Thus 3 + 
4i, 1 — 2 4- 3/ are examples of complex numbers. 

The enormously fruitful field of function theory is a 
direct consequence of the development of complex num¬ 
bers. While this is a subject too technical and specialized, 
we shall have occasion to mention complex numbers 
again when we explain the geometric representation of 
imaginaries. To that end, we must turn for a moment to 
that mathematical idea which, as Boltzmann once said 
seems almost cleverer than the man who invented it— 
the science of Analytical Geometry. 

♦ 

Program music is distinguished from absolute music, 
which owes its coherence to structure, in that the purposes 
of the former is to teU a story. In a certain sense, analyti¬ 
cal geometry can be distinguished from the geometry of 
the Greeks as program music from absolute music. Geom- 
etry, practical in its origin, was cultivated and developed 
for its own sake both as a logical discipline and as a study 
of form. Geometry was a manifestation of a striving for 
the ideal. Shapes and forms that were beautiful, har¬ 
monious, and symmetric were appreciated and eagerly 
studied. But the Greeks cultivated the practical only as 
long as it had a beautiful side; beyond that, their math¬ 
ematics was hampered by their aesthetics. 

There was left to Descartes the task of writing the 
program music of mathematics, of devising a geometry 
which tells a story. When it is said that every algebraic 
equation has a picture, we are describing the relation 
between analytical geometry and algebra. And just as 
program music is as important and significant in itself as 


g6 Mathematics and the Imagination 

the stories it illustrates, so analytical geometry has its 
own dignity and importance—is an autonomous math¬ 
ematical discipline. 

* 

The Jesuit Fathers were often very wise: at their school 
at La Fleche, young Rene Descartes was permitted, be¬ 
cause of his delicate health, to remain in bed each day 
until noon. What McGufFey would have prophesied 
about the future of such a child is not difficult to imagine. 
But Descartes did not turn out a complete profligate. 
Indeed, his delightful habit of staying in bed until noon 
bore at least one remarkable fruit. Analytical geometry 
came to him one morning as he lay pleasantly in bed. 

It is powerful, this idea of a co-ordinate geometry, yet 
easy to understand. Consider two lines (axes) in a plane: 
intersecting at right angles at a point R: 


Y 

FIG. 20.—The point p has the co-ordinates {rriy in'). 


Any point in the entire plane may then be unique y 
determined by its perpendicular distance from the lines 
xx' Sind yy'. The point P, for example, by the distances 
m and Thus, a pair of numbers representing scalar 


PIE (tt, 2 , e) Transcendental and Imaginary gy 

distances along jirV and^' will determine every point in 

the plane, and conversely, every point in the plane 

determines a pair of numbers. These numbers are called 
the co-ordinates of the point. 

All distances on ata:' measured to the right of R are 
called positive, to the left of /?, negative. Similarly, all 
distances measured on yy' above R are positive, all 
distances below, negative. The point of intersection, the 
origin, is designated by the co-ordinates (0, 0). The con- 


y' 


FIG. 21.—The co-ordinate axes in the real plane. 


vention for writing co-ordinates is to put down the 
distance/rom they/ axis (i.e. the distance along the A.r' 
axis) first, the distance/rom the xx' axis, along the yy' axis 
second; thus: (0, 0), (4, 3), ( - 1, 5), (6, 0),' (0, 6), 


Mathematics and the Imagination 

( _ 6, - 5), (3, - 3), (- 8, 0), (0, - 8) are the co¬ 
ordinates of the points in Fig. 21. 


Y' 


Y 

FIG. 22(a).—Graphic represenution of the equa¬ 
tion y = Jc®. 


FIG. 22(b).—Graphic representation of the equation 
y — sin X. This is the famous wave curve used to represent 
many regular and periodic phenomena, i.e., electrical 
current, the motion of a pendulum, radio trans^ssion, 
sound and light waves, etc. (For the meaning of sin x, see 
note in the chapter on the calculus.) 

Coupling this notion with that of a function, it is not 
difficult to see how an equation may be pictured in the 


PIE (tt, e, e)—Transcendental and Imaginary 99 

plane of analytic geometry. When a- and^ are functionally 
related, to each value of x there corresponds a value of 
which two values determine a point in the plane. The 
totality of such number pairs, that is, all the values of 
y corresponding to all the values of ;r, when joined by a 
smooth curve as in Figs. 22(a,b,c), make up the geometri¬ 
cal portrait of an equation. 


FIG. 22(c).—Graphic representation of the equa¬ 
tion^ = e*. This curve illustrates the property com¬ 
mon to all phenomena of growth: rate of growth is O 
proportional to state of growth. 

Employing co-ordinate geometry, how shall we rep¬ 
resent an imaginary number like A theorem in 


A 


8 


100 Mathematics and the Imagination 

elementary geometry, relating to the geometric mean, 
furnishes the clue (see Fig.- 23). 

In the right triangle ABC, the perpendicular AD divides 
BC into two portions: BD^ DC. The length of the perpen¬ 
dicular AD equcils \/BD X DC, and is called the geo¬ 
metric mean of BD and DC. (Fig. 23.) 

A Norwegian surveyor, Wessel, and a Parisian book¬ 
keeper, Argand, at the close of the eighteenth and begin¬ 
ning of the nineteenth centuries, independendy found 
that imaginary numbers could be represented by the 
application of this theorem. In Fig. 24: 


Y' 


Y 


FIG. 24.—Geometric interpretation of i. 

the distance S, from the origin to +1, is the geometric 
mean of the triangle, bounded by the sides L and L , and 
the base formed by that portion of the xx' axis from — 1 
to 1. 

Then 6' = -\/—1+1 — — \ — i 


PIE (tt, 2 , e)—Transcendental and Imaginary \ o i 

Here, then, is a geometric representation of an imaginary 
number. 

Extending this idea, Gauss built up the entire complex 
plane. In the complex plane every point represented by a 
complex number of the form x iy corresponds to the 
point in the plane fixed by the co-ordinates a: and y. In 
other words, a complex number may be regarded as a 
pair of real numbers with the addition of the number i. 
The use of i appears only on performing the operations of 
multiplication and division. Conceive of a line joining 
the point {a -j- ib) to the origin R. Then the operation 
of multiplying by — 1 is equivalent to rotating that line 
about the origin through 180° and shifting the point 
from (+(2 -\-ib) to ( a — ib). The effect of multiplying a 
number by i is such that when performed twice, P is 
obtained, which is equivalent to multiplication by —1. 


FIG. 25.—Multiplication by i is a rotation throueh 90°.- 

Let /* = (a -j- ib'). 

Then, P X i = (a ib) X i 

= (a X i) + (A X I X i) 

= ta + 6 • -1 
= —b ia 

= d- 


102 Mathematics and the Imagination 

Therefore, multiplication by i is a rotation through only 
90°. 

Complex numbers may be added, subtracted, multi¬ 
plied, and divided, just as though they were real numbers. 
The formal rules of these operations (the most interesting 
being the substitution of — 1 for i^) are illustrated in the 
examples below. 

(1) X iy = x' iy' if, and only \S x = x' znd y = y' 

(2) {x + iy) + ix' + iy') = {x + x') + i(y + /) 

(3) (x -I- iy) - (x' + ty') = (x - x') -h i{y - /) 

(4) (x -h iy) (x' -h iy') = (xx' - y/) -h Hxy' -f yx') 


FIG. 26.—The complex plane. 


Figure 26 shows the same points in the plane given in 
Fig. 21, except that for the co-ordinates of x and ^ of 
each point we have substituted the corresponding com¬ 
plex number x -|- iy. 

By virtue of the peculiar properties of i, complex num- 


PIE (tt, 2 , e)—Transcendental and Imaginary 103 

bers may be used to represent both magnitude and direc¬ 
tion. With their aid some of the most essential notions in 
physics such as velocity, force, acceleration, etc., are con¬ 
veniently represented. 

Enough has now been said to indicate the general 
nature of i, its purpose and importance in mathematics, 
its challenge to and final victory over the cherished tenets 
of common sense. Undaunted by its paradoxical appear¬ 
ance, mathematicians used it as they used tt and e. The 
result has been to make possible almost the entire edifice 

of modern physical science.* 

* 

One thing remains. There is a famous formula—per¬ 
haps the most compact and famous of all formulas—de¬ 
veloped by Euler from a discovery of the French mathe¬ 
matician, De Moivre: + 1 =0. Elegant, concise and 

full of meaning, we can only reproduce it and not stop 
to inquire into its implications. It appeals equally to the 
mystic, the scientist, the philosopher, the mathematician. 
For each it has its own meaning. Though known for over 
a century, De Moivre’s formula came to Benjamin Peirce, 
one of Harvard's leading mathematicians in the nine¬ 
teenth century, as something of a revelation. Having 
discovered it one day, he turned to his students and made 
a remark which supplies in dramatic quality and ap¬ 
preciation what it may lack in learning and sophistica- 

* Let us have this much balm for the reader who has bravely gone 
through the pages on analytical geometry and complex numbers. The 
average college course on analytic geometry (not including complex 
numbers) takes six months. It is, therefore, a little too much to expect 
that it can be learned in about five pages. On the other hand, if the 
basic idea has been put over, that every number, every equation of 
algebra, can be graphically represented, the harrowing details may 
be left to more intrepid adventurers. 


104 Mathematics and the Imagination 

tion: “Gentlemen,” he said, “that is surely true, it is 
absolutely paradoxical; we cannot understand it, and 
we don’t know what it means, but we have proved it, 
and therefore, we know it must be the truth.” 

When there is so much humility and so much vision 
everywhere, society will be governed by science and not 
by its clever people. 

APPENDIX 
BIRTH OF A CURVE 

(1) Let US consider the equation^ = x^. Take a few 
sample values of x and find the corresponding values of 
arranging the results in a table: 


X 

y 

0 

0 

1 

1 

2 

4 

3 

9 

4 

16 


That is 2^ = 4, 3^ = 9, etc. Plotting these points on 
the co-ordinate plane, we obtain Fig. A. 

(2) Now, what about the negative values of xl We see, 
for example, ( — 2)^ = —2X —2 = 4. This is evidently 
true for all values of x; thus there corresponds to every 
point plotted in Fig. A another point which is its mirror 
image, the axis OT being the mirror. Adding these gives 
the second figure (Fig. B). 

(3) The arrangement of the points suggests that we 
draw a smooth curve through them. (Fig. C.) 


PIE (tt, iy e) 'Transcendental and Imaginary 105 


Y 


FIO. B. 


But does this curve embrace other points which arise 

m our functional table. Let us test this, tabulating some 
fractional values of x. 


io6 Mathematics and the Imagination 


FIG. C, 


If we plot these new points, it may be seen that they all 
lie on the curve (Fig. D). Indeed, if we continue further, 
we would find that every point which might arise in the 
table will lie on the curve; the totality of such points will 
form the curve known as the parabola. 


PIE (ttj e)—Transcendental and Imaginary 107 


The parabola is formed by the section of a cone 
cut by a plane parallel to the opposite edge. 


You can make a parabola for A jet of water forms a parabola, 
yourself with the help of a flash- So does the path of a projectile, 

light, holding it so that the upper But the curve formed by a loop of 

boundary of the beam will be string held at the ends, hangin 
parallel to the floor. freely, is not a parabola, but 

catenary. 


io8 Mathematics and the Imagination 

FOOTNOTES 

1. Henri Bergson, Creative Evolution. —P. 65. 

2. It is a simple matter geometrically to determine the square root 
of a given length.—P. 67. 


FIG. 27.—Let AB be the given length. Extend it 
to C so that BC = 1. Draw a semicircle having AC 
as diameter. Erect a perpendicular at B meeting 
semicircle at D. BD is the required square root ofL. 

3. Gauss made an exhaustive study to determine what other poly¬ 
gons could be constructed with ruler and compass. The Greeks 
had been able to construct regular polygons of 3 and 5 sides, but 
not those with 7, 11, or 13 sides. Gauss, with marvelous precocity, 
gave the formula which showed what polygons were constructible 
in the classical way. It had been thought that only regular poly¬ 
gons, the number of whose sides could be expressed by the forms: 
2", 2” X 3, 2" X 5, 2" X 15 (where n is an integer), could be so 
constructed. Gauss’ formula proves that polygons with a prime 
number of sides may be constructed as follows; Let P be the num¬ 
ber of sides and n any integer up to 4, then P =* 2*" -|- 1. If « = 0, 
1, 2, 3, 4, P = 3, 5, 17, 257, 65537. Where n is greater than 4, 
there are no known primes of the form 2^" + 1. 

(A prime number is one which is not evenly divisible by any 
number other than 1 or itself. Thus, 2, 3, 5, 7, 11, 13, 17 are 
examples of primes. A famous proof of Euclid, which appears in 
his Elements^ shows that the number of primes is infinite. See 
p. 192.) 

It is an amazing fact that of all the possible polygons, the 
number of whose sides is prime, only the five given above are 
known to be constructible with ruler and compass.—P. 68. 

4. See Chap. 5.—P. 68. 

5. As long ago as 1775, the Paris Academy was so overwhelmed 


PIE (ttj z, e )— Transcendental and Imaginary 109 

with pretended solutions from circle squarers, angle trisectors, ■ 
and cube duplicators, that a resolution was passed that no more 
would be accepted. But at that time the impossibility of these 
solutions was only suspected and not yet mathematically demon¬ 
strated; thus the arbitrary action of the academy can only be 
explained on the grounds of self-preser\'ation.—P. 69. 

6 . Limits and converging processes with an infinite number of 
steps, as we shall soon see, were used in computing tt.—P. 69. 

7. See the chapter on the calculus.—P. 69. 

8 . Most infinite series are divergenty that is, the sum of the series 
exceeds any assignable integer. A typical divergent series is 
1 + ^ + ^+ ^ + !+ . . . This series seems to differ very 
little from the convergent series given in the text, and only the 
most subtle mathematical operations reveal whether a series is 
convergent or divergent. —P. 70. 

9. A square can be duplicated by dra^ving a square on the diagonal 
of the given square, but a cube cannot be duplicated because the 
cube root of 2 is involved, and this, like tt, is not the root of an 
algebraic equation of first or second degree, and therefore cannot 
be constructed with ruler and compass. In four-dimensional 
space, the figure which corresponds to the cube, called a “tes- 
saract” (see the chapter on assorted geometries) can be duplicated 
by ruler and compass, because the fourth root of 2 , which is 
what is required, can be written as the square root of the square 
root of 2.—P. 70. 

10. What is meant by “the root of an algebraic equation with 
integer coefficients”? A word may suffice to jog the memory of 
those who have had a course in elementary algebra. The root 
of an equation is the value that must be substituted for the 
unknown quantity in the equation in order to satisfy it. Thus, 
in the equation a: — 9 = 0, 9 is the root, since if you substitute 
9 for X, the equation is satisfied. Similarly —4 and 4 are the 
roots of the equation x* — 16 = 0 , because when either value is 
substituted for x, the equation balances. “Algebraic” equations 
arc the kind of equations we have just been talking about. 
There are also trigonometric equations, differential equations 
and others, and the term “algebraic” is intended to distinguish 
equations of the form 

flox" + <2ix"~‘ + + . . . + an-\x + = 0. 

The coefficients of an equation are the numbers which appear 
before the unknown quantity or quantities. In the equation 

3x* + 17x3 -f >/2 x3 - /x + ttx = 0 


I 10 


Mathematics and the Imaginatibn 

3, 17, \/2, z, and x are the coefficients. This is an example of an 
algebraic equation with queer coefficients. In defining an algebraic 
number (see page 49), we demand that n be a positive integer and 
that the a’s be integers.—P. 72. 

11. See Buffon’s Needle Problem in the chapter on Chance and 
Chanceability.—P. 79. 

12. The y/l when written as a decimal is just as complicated as x, 
for it never repeats, never ends, and there is no known law 
giving the succession of its digits; yet this complicated decimal is 
easily obtained with exactitude by a ruler and compass con¬ 
struction. It is the diagonal of a square whose side is equal to 1. 
—P. 79. 

13. Jobst Biirgi of Prague had prepared tables of logarithms before 
Napier’s Descriptio appeared. Biirgi however failed to publish 
his tables until 1620 because, as he explained, he was busy on 
some other problem.—P. 80. 

14. According to the principle of positional notation, the value of a 
digit depends on its position in relation to the other digits in 
the number in which they appear.—P. 80. 

15. The rules for operating with exponents in multiplication and 
division are: 

A) Multiplication 

a”* 'K — 42 "*+"; thus, 

X = d®; or, 

d® X d* = (d d d) X (d d) 

B) DiiAsion 


= d® 


^8 


= a 


d®-* = d 


But, if wj b equal to n, 

d"* „ ^ 

d" 

d® 

— = /, 3-3 = ^0 = ? 


= 1 


d® 4X^X4 

TTierefore we agree upon 

a® = 1 _p. 82. 


16 


17 


18 


19 


20 , 


21 


PIE (tt, iy e) Ttanscendental and Imaginary 


111 


22 . 

23. 


Because e possesses certain unique properties valuable in many 
branches of mathematics, particularly the calculus, because of 
the relation between logarithmic and exponential functions, e is 
the natural” base for the logarithmic system.—P. 84. 

The first proof that e is transcendental (i.c., not the root of an 
algebraic equation with integer coefficients), was given by Her- 
mite, the distinguished French mathematician, in 1873, nine 
years before Lindemann’s proof of the transcendental character of 
TT a^eared. Since that time several others succeeded in simplify¬ 
ing Hermite’s proof. The general method is to “assume e to be the 
root of an algebraic equation,/(e) = 0, and show that a multiplier 
iW can be chosen such that when each side of the equation is mul- 
Uphed by Af, (the value of) Mj{e) is reduced to the sum of an 
integer ny zero and a number between 1 and 0, showing that the 
assumption that e can be the root of an algebraic equation is un¬ 
tenable.” See U. G. Mitchell and M. Strain, in Osiris, Studies in 
History of Science, Vol. I.—P. 84. 

The symbol ! as used in mathematics does not indicate surprise 
or excitement, although in this case it might not be amiss, since 
the simplicity and beauty of this series is amazing. ! means 
take the factorial of the number after which ! appears.” The 
factorial of a number is the product of its components; thus 

11 = 1,2! = 1X2, 3! = IX2X3, 41 = 1X2X3X4 

5! = 1X2X3X4X5.-P.84. X ^ X 3 X 4, 

Actually « need only be equal to 1000 (i.e., the interest computed 
thnee daily) to give S2.72.—P. 87. 

The derivative of _>- = e* h equal to the function itself. For a 
further discussion of the derivative and of problems involving 
rate of change, see the chapter on the calculus.—P. 88. 

being the author of the well-worn 
Rubaiyat, ’ was also a mathematician of distinction, but one 
whose prophetic vision failed for negative numbers.— P 90 

Translated in Dantzig, Mumb.r, the Language ojScience (New York; 
Macmillan), 1933, p. 190.—P. 93. 

It was once suggested that appropriate symbols for the two 
constants, r and should be 0 for r. and (p for f in order to avoid 

confusion. But printers balked at making new type and the old 
•symbol^s remained. More often than is realized, such considera¬ 
tions determined the character of mathematical notation.- 


Assorted Geometries—Plane and Fancy 


They say that habit is second nature, 
nature is only first habit? 


Who knows but 

—PASCAL 


Among our most cherished convictions, none is more 
precious than our beliefs about space and time, yet none 
is more difficult to explain. The talking fish of Grimm’s 
fairy tale would have had great difficulty in explaining 
how it felt to be always wet, never having tasted the 
pleasure of being dry. We have similar difficulties in 
talking about space, knowing neither what it is, nor what 
it would be like not to be in it. Space and time are “too 
much with us late and soon” for us to detach ourselves 
and describe them objectively. 

“For what is time?” asked Saint Augustine. “Who can 
easily and briefly explain it? Who even in thought can 
comprehend it, even to the pronouncing of a word con¬ 
cerning it? But what in speaking do we refer to more 
familiarly and knowingly than time? And certainly we 
understand when we speak of it; we understand also 
when we hear it spoken of by another. What, then, is 
time? If no one ask of me, I know; if I wish to explain to 
him who asks, I know not.”^ 

And this could as well be said of space. Though space 
cannot be defined, there is little difficulty in measuring 
distances and areas, in moving about, in charting vast 


I 12 


Assorted Geometries—Plane and Fancy \ 13 

courses, or in seeing through millions of light years. 
Everywhere there is overwhelming evidence that space 
is our natural medium and confronts us with no in¬ 
superable problems. 

But this professes to be no philosophical treatise and 
no German Handbook on an Introduction to the Theory of 
Space in 14 volumes. Our intention is to explain in the 
simplest, most general manner, not the physical space 
of sense perception, but the space of the mathematician. 
To that end, all preconceived notions must be cast aside 
and the alphabet learned anew. 

In this chapter we propose to discuss two kinds of 
geometry—four-dimensional and non-Euclidean. Neither 
of these subjects is beyond the comprehension of the non¬ 
mathematician prepared to do a little straight thinking. 
To be sure, they have both been described, like the theory 
of relativity (to which they are in some ways related) in 
high and mighty mumbo jumbo. High priests in every 
profession devise elaborate rituals and obscure language 
as much to conceal their own ineptness as to awe the 
uninitiate. But the corruptness of the clergy should not 
deter us. The basic ideas underlying four-dimensional 
and non-Euclidean geometry are simple, and this we 
aim to prove. 

♦ 

Euclid, in writing the Elements^ recognized no great 
obstacles. Starting with certain fundamental ideas (pre¬ 
sumably understood by everyone) expressed as postu¬ 
lates and axioms, he built upon these as foundations. 
This ideal method for developing a logical system has 
never been improved upon, although occasionally it has 
been neglected or forgotten with sad results. 

Although Euclid’s Elements constitute an imposing in- 


114 Mathematics and the Imagination 

tellectual achievement, they fail to make an important 
distinction between two types of mathematics —fure and 
applied —a distinction which has only come to light in 
modern theoretical developments in mathematics, logic 
and physics. 

A geometry which treats of the space of experience, 
is applied mathematics. If it says nothing about that space 
—if, in other words, it is a system composed of abstract 
notions, elements, and classes, with rules of combination 
obeying the laws of formed logic, it is pure mathematics. 
Its propositions are of the form: If A is true, then B is 
true, regardless of what A and B may possibly be.^ Should 
a system of pure mathematics be applicable to the physi¬ 
cal world, its fruitfulness may be regarded either as mere 
chance, or as further evidence of the profound connection 
between the forms of nature and those of mathematics. 
Yet, in either case, this essential fact must be borne in 
mind—the fruitfulness of a logical system neither dimin¬ 
ishes nor augments its validity. 

As applied mathematics, Euclid’s geometry is a good 
approximation within a restricted field. Good enough to 
help draw a map of Rhode Island, it is not good enough 
for a map of Texas or the United States, or for the meas¬ 
urement of either atomic or stellar distances. As a system 
of pure mathematics, its propositions are true in a most 
general way. That is to say, they have validity only as 
propositions of logic, only if they have been correctly 
deduced from the axioms. Other geometries with dif¬ 
ferent postulates are therefore possible—indeed, as many 
others as the mathematician chooses to devise. All that 
is necessary is to assemble certain fundamental ideas 
(classes, elements, rules of combination), declare these to 
be undefinable, make certain that they are not self-con- 


Assorted Geometries—Plane and Fancy 115 

tradictory, and the groundwork has been laid for a new 
edifice, a new geometry. Whether this new geometry 
will be fruitful, whether it will prove as useful in survey¬ 
ing or navigation as Euclidean geometry, whether its 
fundamental ideas measure up to any standard of truth 
other than self-consistency, doesn’t concern the mathe¬ 
matician a jot. The mathematician is the tailor to the 
pntry of science. He makes the suits, anyone who fits 
into them can wear them. To put it another way, the 
mathematician makes the rules of the game; anyone 
who wishes may play, so long as he observes them. There 
is no sense in complaining afterwards that the game was 
without profit. 

* 

If we wish to pay a mathematical system the highest 
possible compliment, to indicate that it partakes of the 
same generality and has the same validity as logic, we 
may call it a game. A four-dimensional geometry is a 
game: so is the geometry of Euclid. To object to four¬ 
dimensional geometry on the grounds that there are 
only three dimensions is absurd. Chess can be played as 
well by those who believe in comrades or dictators as by 
those who cling to the vanishing glory of kings and 
queens. What sense is there in objecting to chess on the 
grounds that kings and queens belong to a past age, and 
that, in any case, they never did behave like chess pieces 
—no, not even bishops. What merit is there to the con¬ 
tention that chess is an illogical game because it is im¬ 
possible to conceive that a private citizen may be crowned 
queen merely by moving forward five steps. 

Perhaps these are ridiculous examples, but they are 
no more so than the complaints of the faint of heart who 
say that three dimensions make space and space makes 


9 


116 Mathematics and the Imagination 

three dimensions, “that is all ye know on earth and all 
ye need to know.” If we can rake the doubters fore, we 
can rake them aft—indeed, from stem to stern. For there 
is no proof, in the scientific sense, that space is three- 
dimensional, or for that matter, that it is four-, five-, 
six-, or anything but rz-dimensional. Space cannot be 
proved three-dimensional by geometry considered as 
pure mathematics, because pure mathematics is concerned 
only with its own logical consistency and not with space 
or anything else. Nor is this the province of applied math¬ 
ematics, which does not generally inquire into the nature 
of space, but assumes its existence. All that we have 
learned from applied mathematics is that it is convenient, 
but not obligatory, to consider the space of our sense per¬ 
ceptions as three-dimensional. 

To the objection that a fourth dimension is beyond 
imagination we may reply chat what is common sense 
today was abstruse reasoning—even wild speculation— 
yesterday. For primitive man to imagine the wheel, or 
a pane of glass, must have required even higher powers 
than for us to conceive of a fourth dimension. 

Someone may still object: “You tell me that four¬ 
dimensional geometry is a game. I will believe you. 
But it seems to be a game that doesn’t concern itself with 
anything real, with anything I have ever experienced.” 
We may answer in the Socratic way with another ques¬ 
tion. “If a four-dimensional geometry treats of nothing 
real, what does the plane geometry of Euclid consider? 
Anything more real? Certainly not! It doesn’t describe 
the space accessible to our senses which we explain in 
terms of sight and touch. It talks about points that have 
no dimensions, lines that have no breadth, and planes 
that have no thickness—all abstractions and idealiza- 


Assorted Geometries—Plane and Fancy 11 7 

tions resembling nothing we have ever experienced or 
encountered.’* 


The notion of a fourth dimension, although precise, is 
very abstract, and for the greatest majority beyond imagi¬ 
nation and in the purest realm of conception. The de¬ 
velopment of this idea is as much due to our rather child¬ 


ish desire for consistency as to anything more profound. 
In this same striving after consistency and generality, 
mathematicians developed negative numbers, imaginar- 
ies, and the transcendentals. Since no one had ever seen 


minus three cows, or the square root of minus one trees 
It was not without a struggle that these now rather com¬ 
monplace ideas were introduced into mathematics. The 
same struggle was repeated to introduce a fourth dimen¬ 
sion, and there are still skeptics in the camp of the opposi¬ 
tion. 


Every possible allegory and fiction was proposed to 
coax and cajole the doubters, to make the idea of a 
fourth dimension more palatable. There were the ro¬ 
mances which described how impossible a three-dimen¬ 
sional world would seem to creatures in a two-dimen¬ 
sional one, there were stories of ghosts, table-tipping, 
and the land of the dead. It required illustrations from 
the land of the living, which were still less comprehen¬ 
sible than the fourth dimension, to win even a partial 
victory. From this, it should not be inferred that a greater 
absurdity was enlisted in support of a lesser one. 

Beginning as usual with Aristotle, it was proved again 
and again that a fourth dimension was unthinkable and 
impossible. Ptolemy pointed out that three mutually per¬ 
pendicular lines could be drawn in space, but a fourth, 
perpendicular to these, would be without measure or 


118 Mathematics and the Imagination 

depth. Other mathematicians, unwilling to risk a heresy 
greater even than going contrary to the Bible—that is, 
contradicting Euclid—advised that to go beyond three 
dimensions was to go “against nature.” And the English 
mathematician, John Wallis, of whom one might prop¬ 
erly have expected better things, referred to that “fansie,” 
a fourth dimension, as a “Monster in Nature, less possible 
than a Chimera or a Centaure.” 

Unwittingly, a philosopher, Henry More, came to the 
rescue, although mathematicians today would hardly 
acknowledge his support. His suggestion was not an 
unmixed blessing. Ghosdy spirits, said More, surely have 
four dimensions. But Kant delivered an earthly blow by 
laying down his intuitive notions of space which were 
hardly compatible with either a four-dimensional or a 
non-Euclidean geometry. 

In the nineteenth century several leading mathemati¬ 
cians espoused the apparently hopeless cause, and be¬ 
hold—a new mathematical gusher. The great paper of 
Riemann On the Hypotheses Which Underlie the Foundations 
of Geometry^ followed by the works of Cayley, Veronese, 
Mobius, Pliicker, Sylvester, Bolyai, Grassmann, Lobach¬ 
evsky, created a revolution in geometry. The geometry 
of four and even higher dimensions became an indis¬ 
pensable part of mathematics, related to many other 
branches. 

When finally, there came, as for some mysterious rea¬ 
son they always come, direct uses and applications of 
four-dimensional geometry to mathematical physics, to 
the physical world, when the unwanted child was sud¬ 
denly recognized and rechristened “Time, the fourth di¬ 
mension!” the rejoicing made the cup flow over. Curious 


Assorted Geometries—Plane and Fancy 11 g 

and marvelous things were said. The fourth dimension 
would solve all the awful mysteries of the universe, and 
ultimately might prove a cure for arthritis. So far in the 
general jubilation did the mathematicians forget them¬ 
selves that some of them began to refer to it as *‘’‘the fourth 
dimension/* as though, instead of being merely an idea 
shaken loose from the ends of their pencils, only the fourth 
in a class of infinite possibilities, it was a physical reality, 
like a new element. Thus the lamentable confusion spread 
from mathematics to grammar, from the principles of the 
2 -}- 2 to the science of the proper uses of the definite and 
indefinite article. 

* 

Physicists may consider time to be a fourth dimension, 
but not the mathematician. The physicist, like other sci¬ 
entists, may find that his latest machine has just the right 
place for some new mathematical gadget; that does not 
concern the mathematician. The physicist can borrow 
new parts for his changing machine every day for all the 
mathematician cares. If they fit, the physicist says they 
are useful, they are true, because there is a place for them 
in the model of his world in the making. When they no 
longer fit, he may discard them or “destroy the whole ma¬ 
chine and build a new one as we are ready to buy a new 
car when the old one doesn’t run well.** ^ 

The practice of calling time a dimension points to the 
necessity of explaining what is meant by that troublesome 
word. In this way, too, we shall arrive at a clearer image 
of four-dimensional geometry. 

Instead of referring to “a space,” or to “spaces,” we 
shall use the more fashionable and more general term— 
manijold* A manifold bears a rough resemblance to a 


120 


Mathematics and the Imagination 

class. A plane is a class composed of all those points 
uniquely determined by two co-ordinates. It is therefore 
a two-dimensional manifold. 


FIG. 28(a).—A two-dimensional manifold. Each point 
requires a pair of numbers to individualize it. 

A = (3, 2) 

B = (-5>-. 4) 

C = {x,y) 

D = (0, -3) 

E - (0, 0) 


FIG. 28(b).—The same idea can be extended to a three- 
dimensional manifold (space). Each point requires 3 
numbers to individualize it. 

Thus, P = {x,y, z) 

The space studied in three-dimensional analytical 
geometry may be regarded as a three-dimensional mani¬ 
fold, because exactly three co-ordinates are required to 


I2I 


Assorted Geometries—Plane and Fancy 

fix every point in it. Generally, if n numbers are necessary 
to specify, to individualize, each of the members of a 
manifold, whether it be a space, or any other class, it 
is called an n-dimensional manifold. 

Thus, for the word dimension, with its many mysterious 
connotations and linguistic encrustations, there has been 
substituted a simple idea—that of a co-ordinate. And in 
place of the physical word space, the mathematician intro¬ 
duces the more general and more accurate concept of 
class, or manifold. 

♦ 

It is now possible, as a consequence of these refine¬ 
ments, to introduce an idea already familiar from our 
discussion of analytical geometry, which shall serve to 
uniquely characterize space manifolds. We shall use 
some geometrical reasoning. 

The Pythagorean theorem states that,'in a right-angle 


122 Mathematics and the Imagination 

triangle, the length of the hypothenuse equals the square 
root of the sums of the squares of the other two sides. 


FIG. 30.—The Pythagorean theorem in three di¬ 
mensions. 

^2 = ^2 + ^2 -I- f2 

For ^ c^-\- {eY 
and {ey = ^ 

When this is carried over into analytical geometry of 
two dimensions, the result is the well-known distance for¬ 
mula, according to which the distance between any two 
points in the plane, having the co-ordinates and 

(x',y') respectively, is^{x — x')^ -f- {jf yV- 


(1) Distance AB = V(x — x'y -h — yy 

(2) Distance AB = \/(x — x'y -h (j — /y -h U — ■c')* 


Assorted Geometries—Plane and Fancy 123 

Similarly, in three-dimensional analytical geometry the 
distance between any two points having the co-ordinates 
z)y {x\ y, respectively, is _ 

y{x-x'y + {y-yy+ u - z')\ 

Now, in either two or three dimensions the concept 
of distance, as both the mathematician and the layman 
understand it, is the same. The layman is satisfied with 
an intuitive grasp; the mathematician demands an exact 
formulation. However, in the higher dimensions, while the 
layman is halted by a blank wall—the natural limitations 
of his senses—the mathematician scales the wall using his 
extended formula as a ladder. Distance in four dimensions 
means nothing to the layman. Indeed, why should it? For 
even a four-dimensional space is wholly beyond ordinary 
imagination. But the mathematician, who rests the con¬ 
cept upon an entirely different base, is not called upon to 
struggle with the bounds of imagination, but only with 
the limitations of his logical faculties. 

Accordingly, there is no reason for not extending the 
above formula to 4, 5, 6, ... or rz dimensions. Thus, 
in a four-dimensional Euclidean manifold, the distance of 
an element, i.e., point, having the co-ordinates (x^y, Zy u) 
from an element with co-ordinates {x\ y\ z\ n') is 

V{x — x'Y {y —y'y {z — zV + (« — uy. 

This method enables us to define in terms of analytical 
geometry a 2, 3, 4, ... or n-dimensional Euclidean 
manifold. An analogous definition can be given for the 
manifolds of other geometries, in which case some other 
distance formula would apply. We have chosen analytical 
geometry and taken the Pythagorean distance formula 
to distinguish the Euclidean manifolds. 

A condensed definition of a three- and four-dimen¬ 
sional Euclidean manifold in terms of analytical geom¬ 
etry reads: ^ 


124 Mathematics and the Imagination 

1. A three-dimensional Euclidean manifold is the class 

of all number triples: z), (x\y, z'), z"), 

etc., to any two of which there may uniquely be assigned 
a measure (called the distance between them) defined by 

the formula - x'Y + {y - y'Y + ( 4 : - z'Y. Cer- 
tain subclasses of this class are called points, lines, and 
planes, etc. The theorems derived from these definitions 
constitute a mathematical system called “Analytical 
Geometry of Three Dimensions.” 

2. A four-dimensional Euclidean manifold is the class 
of all number quadruples: (x, y, z> u), (x', y', z', u'), 

etc., to any two of which there may 
uniquely be assigned a measure^ (called the distance be¬ 
tween them) defined by the formula 

~ + (y ~y'Y {z — z'Y -h (« — u'y. 

Certain subclasses of this class are called points, lines, 
planes, and hyperplanes. Analytical four-dimensional Eu¬ 
clidean geometry is the system formed by theorems de¬ 
rived from these definitions. 

Note that nothing has been said in either of these defini¬ 
tions about space; neither the space of our sense percep¬ 
tions, nor the space of the physicist, nor that of the philos¬ 
opher. All that we have done is to define two systems of 
mathematics which are logical and self-consistent, which 
may be played like checkers, or charades, according to 
stated rules. Anyone who finds a resemblance between his 
game of checkers or charades and the physical reality of 
his experience is privileged to point morals and to make 
capital of his suggestion. 

* 

But having established that we are in the realm of pure 
conception, beyond the most elastic bounds of imagina¬ 
tion, who is satisfied? Even the mathematician would like 


Assorted Geometries—Plane and Fancy 125 

to nibble the forbidden fruit, to glimpse what it would be 
like if he could slip for a moment into a fourth dimension. 
It’s hard to grub along like moles down here below, to 
hear someone tell of a fourth dimension, to make careful 
note of it, and then to plow along, giving it no further 
thought. To make matters worse, books on popular sci¬ 
ence have made everything so ridiculously simple—rela¬ 
tivity, quanta, and what not—that we are shamed by our 
inability to picture a fourth dimension as something more 
concrete than time. 

Graphic representations of four-dimensional figures 
have been attempted: it cannot be said these efforts have 
been crowned with any great success. Fig. 31(a) illus¬ 
trates the four-dimensional analogue of the three-dimen¬ 
sional cube, a hypercube or tesseract: Our difficulties in 
drawing this figure are in no way diminished by the 
fact that a three-dimensional figure can only be drawn 
in perspective on a two-dimensional surface—such as this 
page—, while the four-dimensional object on a two di¬ 
mensional page is only a perspective of a “perspective.” 

Yet since equals the area of a square, the volume 


FIG. 31 (a).—Two views of the tesseract. 


126 Mathematics and the Imagination 

of a cube, we feel certain that describes something, 
whatever that something may be. Only by analogy can 
we reason that that “something” is the hypervolume (or 
content) of a tesseract. Reasoning further, we infer that 
the tesseract is bounded by 8 cubes (or cells), has 16 
vertices, 24 faces and 32 edges. But visualization of the 
tesseract is another story. 

Fortunately, without having to rely on distorted dia¬ 
grams, we may use other means, using familiar objects 
to help our limping imagination to depict a fourth dimen¬ 
sion. 

The two triangles A and B in Fig. 32 are exacdy alike. 


Geometrically, it is said they are congruent, * meaning 
that by a suitable motion, one may be perfecdy super¬ 
posed on the other. Evidendy, that motion can be carried 
out in a plane, i.e., in two dimensions, simply by sliding 
triangle A on top of triangle 5.1 But what about the two 
triangles C and D in Fig. 33? 

One is the mirror image of the other. There seems to 
be no reason why by sliding or turning in the plane, C 

* See the chapter on paradoxes for an exact definition. 

^ Actually, “sliding on top oP’ would be impossible in a physical 
two-dimensional world. 


Assorted Geometries—Plane and Fancy 127 

cannot be superimposed on D. Strangely enough, this 
cannot be done. C or D must be lifted out of the plane, 
from two dimensions into a third, to effect superposition. 
Lift C up, turn it over, put it back in the plane, and then 
it can be slid over D. 

Now, if a third dimension is essential for the solution 
of certain two-dimensional problems, a fourth dimension 
would make possible the solution of otherwise unsolvable 
problems of three dimensions. To be sure, we are in the 


realm of fancy, and it need hardly be pointed out that 
a fourth dimension is not at hand to make Houdinis of 
us all. Yet, in theoretical inquiries, a fourth dimension 


FIG. 34.—This is no blueprint but an actual 
house in Fladand. 


128 Mathematics and the Imagination 

is of signal importance, and part of the warp and woof 
of modem theoretical physics and mathematics. Ex¬ 
amples chosen from these subjects are quite difficult and 
would be out of place, but some simpler ones in the 
lower dimensions may prove amusing. 

If we lived in a two-dimensional world, so graphically 
described by Abbott in his famous romance, Flatland, 
our house would be a plane figure, as in Fig. 34. Entering 
through the door at we would be safe from our friends 
and enemies once the door was closed, even though there 
were no roof over our head, and the walls and windows 
were merely lines. To climb over these lines would mean 
getting out of the plane into a third dimension, and of 
course, no one in the two-dimensional world would have 
any better idea of how to do that than we know how 
to escape from a locked safe-deposit vault by means of a 
fourth dimension. A three-dimensional cat might peek 
at a two-dimensional king, but he would never be the 
wiser. 

When winter comes to Flatland, its inhabitants wear 
gloves. Three-dimensional hands look like this; 


FIG. 35. 


130 Mathematics and the Imagination 

Modem science has as yet devised no relief for the 
man who finds himself with two left gloves instead of a 
right and a left. In Flatiand, the same problem would 
exist. But there, Gulliver, looking down at its inhabitants 
from the eminence of a third dimension, would see at 
once that, just as in the case of the two triangles on page 
127, all that is necessary to turn a right glove into a left 
one is to lift it up and turn it over. Of course, no one in 
Fladand would or could lift a finger to do that, since it 

involves an extra dimension. 

If then, we could be transported into a fourth dimen¬ 
sion, there is no end to the miracles we could perform— 
starting with the rehabilitation of all ill-assorted pairs 
of gloves. Lift the right glove from three-dimensiond 
space into a fourth dimension, turn it around, bring it 
back and it becomes a left glove. No prison cell could 
hold the four-dimensional Gulliver—far more of a men¬ 
ace than a mere invisible man. Gulliver could take a 
knot and untie it without touching the ends or breaking 
it, merely by transporting it into a fourth dimension and 
slipping the solid cord through the extra loophole. 

Or he might take two links of a chain apart without 
breaking them. All this and much more would seem 
absurdly simple to him, and he would regard our he p- 
lessness with the same amusement and pity as we 00 
upon the miserable creatures of Flatiand. 

♦ 

Our romance must end. If it has aided some 
in making a fourth dimension more real and has satis e 
a common anthropomorphic thirst, it has served its pur 
pose. For our own part, we confess that the fables ave 

never made the facts any clearer. . . 

An idea originally associated with ghosts and spirits 


Assorted Geometries—Plane and Fancy 131 

needs, if it is to serve science, to be as far removed as 
possible from fuzzy thinking. It must be clearly and 
courageously faced if its true essence is to be discovered. 
But it is even more stupid to reject and deride than to 
glorify and enshrine it. No concept that has come out 
of our heads or pens marked a greater forward step in our 
thinking, no idea of religion, philosophy, or science broke 
more sharply with tradition and commonly accepted 
knowledge, than the idea of a fourth dimension. 

Eddington has put it very well: ® 

However successful the theory of a four-dimensional world 
may be, it is difficult to ignore a voice inside us which whispers: 
“At the back of your mind, you know that a fourth dimension 
is all nonsense.” I fancy that voice must often have had a busy 
time in the past history of physics. What nonsense to say that 
this solid table on which I am writing is a collection of electrons 
moving with prodigious speed in empty spaces, which relatively 
to electronic dimensions are as wide as the spaces between the 
planets in the solar system! What nonsense to say that the thin 
air is trying to crush my body with a load of 14 lbs. to the 
square inch! What nonsense that the star cluster which I see 
through the telescope, obviously there nowy is a glimpse into a 
past age 50,000 years ago! Let us not be beguiled by this 
voice. It is discredited. . . . 

We have found a strange footprint on the shores of the un¬ 
known. We have devised profound theories, one after another 
to account for its origin. At last, we have succeeded in recon¬ 
structing the creature that made the footprint. And lo! It is our 
own. 

* 

We have emphasized the fact that pure geometry is 
divorced from the physical space which we perceive 
about us, and we are now prepared to tackle an idea 
which is slightly tougher. There is no harm, however, 


132 Mathematics and the Imagination 

in first distinguishing somewhat differently than before 
between space as it is ordinarily conceived and the space 
manifolds of mathematics. Perhaps this distinction will 
help to make our new concept—the non-Euclidean 

geometries—seem less strange. 

We are quite used to thinking of space as infinite, not 
in the technical mathematical sense of infinite classes, 
but simply meaning that space is boundless without 
end. To be sure, experience teaches us nothing of the 
kind. The boundaries of a private citizen rarely go much 
further than the end of his right arm. The boundaries of 
a nation, as bootleggers once learned, do not go beyond 
the twelve-mile limit. 

Most of what we believe about the infinitude of space 
comes to us by hearsay, and another part comes from 
what we think we see. Thus, the stars look as if they were 
millions of miles away, although on a dark night a 
candle half a mile off would give the same impression. 
Moreover, if we imagined ourselves the size of atoms, a 
pea at a distance of one inch would appear mightier 

and far more distant than the sun. 

The distinction between the space of the individual 
and “public space” soon becomes apparent. Our personal 
knowledge of space does not show it to be either infinite, 
homogeneous, or isotropic. We do not know it to be 
infinite because we crawl, hop, and fly around in only 
tiny portions. We do not know it to be homogeneous 
because a skyscraper in the distance seems much smaller 
than the end of our nose; and the feather on the hat o 
the lady in front of us shuts off our vision of the cinema 
screen. And we know it is not isotropic, that is, it does 
not possess the same properties in every direction, 
because there are blind spots in our vision and our sense 


Assorted Geometries—Plane and Fancy 133 


of sight is never equally good in all directions. 

The notion of physical or “public space” which we 
abstract from our individual experience is intended to 
free us from our personal limitations. We say physical 
space is infinite, homogeneous, isotropic, and Euclidean. 
These compliments are readily paid to an ideal entity 
about which very little is actually known. If we were to 
ask the physicist or astronomer, “What do you think 
about space?” he might reply: “In order to carry out 
experimental measurements and describe them with the 
greatest convenience, the physical scientist decides upon 
certain conventions with respect to his measuring appa¬ 
ratus and operations performed with it. These are, strictly 
speaking, conventions with regard to physical objects and 
physical operations. However, for practical purposes, it is 
convenient to assume for them a generality beyond any 
particular set of objects or operations. They then become, 
as we say, properties of space. That is what is meant by 
physical space, which we may define, in brief, as the 
abstract construct possessing those properties of rigid 
bodies that are independent of their material content. 
Physical space is that on which almost the whole of 

physics is based, and it is, of course, the space of everyday 
affairs.” ® 


On the other hand, the spaces, or more generally the 
manifolds, which the mathematician considers are con¬ 
structed without any reference to physical operations, 
such as measurement. They possess only those properties 
expressed in the postulates and axioms of the particular 

geometry in question, as well as those properties dcducible 
from them. 

It may well be that the postulates are themselves 
suggested, in part or in whole, by the physical space of 


134 


Mathematics and the Imagination 

our experience, but they are to be regarded as full-grown 
and independent. If experiments were to show that some, 
or all, of our ideas about physical space are wrong (as 
the theory of relativity has, in fact, done) we would have 
to rewrite our texts on physics, but not our geometries. 

♦ 

But this approach to the concept of space, as well as to 
geometry, is comparatively recent. There has been no 
more sweeping movement in the entire history of science 
than the development of non-Euclidean geometry, a 
movement which shook to the foundations the age-old 
belief that Euclid had dispensed eternal truths. Compe¬ 
tent and accurate as a measuring tool since Egyptian 
times, intuitively appealing and full of common sense, 
sanctified and cherished as one of the richest of intel¬ 
lectual legacies from Greece, the geometry of Euclid 
stood for more than twenty centuries in lone, resplendent, 
and irreproachable majesty. It was truly hedged by 
divinity, and if God, as Plato said, ever geometrized, he 
surely looked to Euclid for the rules. The mathemadcians 
who occasionally had doubts soon expiated their heresy 
by vodve offerings in the form of further proofs in 
corroboration of Euclid. Even Gauss, the “Prince of 
Mathematicians,” dared not offer his criticisms for fear 
of the vulgar abuse of the “Boethians.” 

Whence came the doubts? Whence the inspiration of 
those who dared profane the temple? Were not the postu¬ 
lates of Euclid self-evident, plain as the light of day? And 
the theorems as unassailable as that two plus two equals 
four? The center of the ever-increasing storm, which finally 
broke in the nineteenth century was the famous fifth pos¬ 
tulate about parallel lines. 

This postulate may be restated as follows: “Through 


Assorted Geometries—Plane and Fancy 135 

any point in the plane, there is one, and only one, line 
parallel to a given line.” 

There is some evidence to show that Euclid, himself, 
did not regard this postulate as “quite so self-evident” 
as his others.® Philosophers and mathematicians, intent 
on vindicating him, attempted to show that it was really 
a theorem and thus deducible from his premises. All of 
these attempts failed for the very good reason which Eu¬ 
clid, much wiser than those who followed him, had al¬ 
ready recognized, namely, that the fifth postulate was 
merely an assumption and hence could not be mathe¬ 
matically proved. 

* 

More than two thousand years after Euclid, a German, 
a Russian, and a Hungarian came to shatter two in¬ 
disputable “facts.” The first, that space obeyed Euclid; 
the second, that Euclid obeyed space. Gauss we credit 
on faith. Not knowing the extent of his investigations, in 
deference to his greatness as well as to his integrity, we 
are hospitable to his statement that he had independently 
arrived at conclusions resembling those of the Hungarian, 
Bolyai, some years before Bolyai’s father informed Gauss 
of his son’s work. 

Lobachevsky, the Russian, and Bolyai, both in the 
1830 ’s, presented to the very apathetic scientific world 
their remarkable theories. They argued that the trouble¬ 
making postulate could not be proved, could not be 
deduced from the other axioms, because it was only a 
postulate. Any other hypothesis about parallels could be 
substituted in its place, and a different geometry—just 
as consistent and just as “true”—would follow. All the 
other postulates of Euclid were to be retained, only, in 
place of the fifth, a substitution was to be made: 


136 Mathematics and the Imagination 

“Through any point in the plane, there are two lines 
parallel to any given line.” 

Overnight, mathematics had thrown off its chains, and 
a new line of richly fruitful theoretic and practical inquiry 
was born. 

* 

In the figure are two parallel lines: 


B A c 


FIG. 39. 


How is it possible, you may ask, that another line 
different from BC, yet parallel to DE may be drawn 
through A? The answer is that the reader is talking about 
the physical plane and lines drawn with a pencil. He is 
haunted by the ghosts of common sense instead of 
reasoning in teims of pure geometry. Tou might go 
further and say that in your system, in Euclidean 
geometry, any line different from BC will meet DE if 
sufficiently extended. We would reply that that rule 
holds in your game, not in ours—Lobachevskian geome¬ 
try. Neither of us, if we are mathematicians, are talking 
about physical space, but even if we were, there is 
better ground to believe that we are speaking the truth 
than you. 

Lobachevsky’s geometry may be introduced in this 
way: In Fig. 40 line AB is perpendicular to CD. If we per¬ 
mit it to rotate about A counterclockwise, it will intersect 
CD at various points to the right of B until it reaches a 
limiting position EF^ when it becomes parallel to CD. 


Assorted Geometries—Plane and Fancy 137 

Continuing the rotation, it will start to intersect CD to 
the left of B. Euclid assumed that there is only one 
position for the line, namely £*F, when it would be 
parallel to CD. Lobachevsky assumed that there were 
two such positions, represented by A*B’ and C'D\ and 
further, that all lines falling within the angle 0, while 
not parallel to CD, would never meet it, no matter how 
far extended. 


A'. _ 

r — 


c 

D 


B 

FIG- 

40 . 


Now this is an assumption, and there is no sense in 
arguing from the diagram that it is evident that if A'D\ 
or C'D' were extended sufficiently far, they would eventu¬ 
ally intersect CD. If, as Professor Cohen has pointed out, 
we rely wholly on our intuition of space, which is finite, 
there will always be an angle 0 which grows smaller as 
our space is extended, but which never vanishes, and all 
lines falling within 0 will fail to intersect the given line.*® 

+ 

What happens to the geometry of Euclid when its 
parallel postulate is replaced by that of Lobachevsky? 
Many of its important theorems, those which in no way 
depend upon the fifth postulate, are carried over. Thus, 
in both geometries: 

1. If two straight lines intersect, the vertical angles 
are equal: 


138 


Mathematics and the Imagination 


FIG. 41.—Angle 1 = Angle 2 (because each 
one = 180° — Angle 3). 

2. In an isosceles triangle, the base angles are equal: 


FIG. 42.—If AB — BC, then Angle 1 = Angle 2. 


3. Through a point, only one perpendicular can be 
drawn to a straight line: 


A 

9 


c n _? 

B 

FIG. 43.—Through the point A one and only one 
perpendicular can be drawn to CD. 

On the other hand, some very important theorems 
of Euclidean geometry are changed when another postu¬ 
late is substituted for the fifth, with startling results. Thus, 
in Euclidean geometry, the sum of the angles of every tri¬ 
angle equals 180 degrees, whereas in Lobachevsky’s geomr 
etry, the sum of the angles of every triangle is less than 180 


Assorted Geometries—Plane and Fancy 139 

degrees. Parallel lines in Euclidean geometry never inter¬ 
sect and remain, no matter how far extended, a constant 
distance apart. Parallel lines in Lobachevsky’s geometry 
never meet, but approach each other asymptotically —that is, the 
distance between them becomes less as they are further 
extended. 

To cite one more interesting theorem, two triangles 
in Euclidean geometry may have the same angles but 
different areas; i.e., one may be a magnification of the 
other. But in Lobachevsky’s geometry, as a triangle in¬ 
creases in areay the sum of its angles decreases; thus, only tri¬ 
angles equal in area can have the same angles. (See Fig. 
47b,) 

♦ 

The brilliant Riemann, in his famous inaugural lecture 
On the Hypotheses Which Underlie the Foundations of Geometry, 
proposed still another substitute for Euclid’s fifth postu¬ 
late differing from that of Lobachevsky and Bolyai, 
This assumption holds: “Through a point in the plane, no 
line can be drawn parallel to a given line.” In other 
words, every pair of lines in the plane must intersect. 
It should be noted that this contradicts Euclid’s tacit 
supposition that a straight line may be infinitely ex¬ 
tended. In this connection, Riemann pointed out the 
important distinction between infinite and unbounded: 
Thus, space may be finite though unbounded. Moving in 
any given direction, like the hands of a clock, we can 
keep going forever, forever retracing our steps. As might 
be expected, Riemann’s hypothesis also affects those 
theorems of Euclid dependent on the fifth postulate. 
Both Euclid’s and Lobachevsky’s geometries state that 
only one perpendicular can be drawn to a straight line 
from a given point. But in Riemann’s any number of 


140 Mathematics and the Imagination 

perpendiculars can be drawn from an appropriate point 

to a given straight line. Again, the sum of the angles of 

any triangle is greater than 180 degrees in Riemann’s 

geometry, and the angles increase as the triangle grows 

larger. (See Fig. 47a, page 144.) 

♦ 

We thus have three postulate systems; Euclid’s, Loba¬ 
chevsky’s, and Riemann’s. From these, three geometries 
have been developed: the first, Euclidean, the other two, 
non-Euclidean. The non-Euclidean geometries are, of 
course, vastly indebted to the postulates and the methods 
of Euclid. So far as the postulates are concerned, they dif¬ 
fer only with respect to the parallel postulate. The theo¬ 
rems differ greatly in many respects. 

A little earlier we laid down the criterion for every 
mathematical system—its postulates must be consistent, 
must lead to no contradictions. But how are we to discover 
whether the non-Euclidean geometries of Lobachevsky 
and Riemann are consistent? For that matter, it may well 
be asked, how are we to be certain that the postulates of 
Euclid will engender no contradictions? Evidentiy, we 
may pile up theorem after theorem without encountering 
any, but that is no proof that at some future time one may 
not arise. Is it that we are no better off than if we were 


PIG. 44.—^The pseudosphere. 


Assorted Geometries—Plane and Fancy 141 

verifying an hypothesis in physics or any other experi¬ 
mental science? 

Fortunately mathematicians have devised a trick which 
satisfies their conscience on this score. It consists in show¬ 
ing, for example, in non-EucIidean geometry, that a set 
of entities which exist in Euclidean geometry would sat- 


0)W 


FIG. 45(a).—One way of generating the tractrix. 
The toy locomotive L is tied to the watch 
the string being perpendicular to the track. When 
the locomotive starts pulling, the path of the watch 
is a tractrix. 


142 Mathematics and the Imagination 

isfy the non-Euclidean theorems. It is assumed that these 
entities, themselves, are “free from contradictions, and that 
they in effect, fully embody the axioms,” and the latter 
are therefore shown to involve no inconsistencies. Let us 
take separate examplesfrom Lobachevsky’s and Riemann’s 
geometries to illustrate what is meant. 

Figure 44 illustrates a surface generated by revolving 
the curve known as the tractrix about a horizontal line. 

The tractrix itself may be obtained as follows: On a 
pair of mutually perpendicular axes, as in Cartesian 
geometry, imagine a chain lying along TY\ To one end 
of this chain there is attached a watch; the other end 
coincides with the point of origin 0. Keep the chain 
taut, and pull the free end slowly along the X axis, to 
the right of 0. Then repeat this procedure to the left. 
The path of the watch in either case generates a tractrix. 
If this curve is now revolved about the line XX , a 
“double trumpet surface,” as E. T. Bell calls it, is formed. 


FIG. 45(b).—The tractrix is also that curve which 
is perpendicular to a family of equal circles with 
their centers on a straight line. 


This surface Beltrami named a pseudosphere. We find 
that the geometry applicable to a pseudosphere is that 
of Lobachevsky. For example, on the pseudosphere, 


Assorted Geometries—Plane and Fancy 143 

through a given point two lines may be drawn parallel 
to a third line, which will approach them asymptotically 
without ever intersecting. Thus, Lobachevsky’s geom¬ 
etry is satisfied by an entity from Euclid’s geometry, and 
this complies with the mathematician’s criterion of con¬ 
sistency. 


FIG. 45(c).—If perpendiculars are drawn to the 
curve (called the catenary) formed by a chain held 
at both ends, the curve which just touches all the 
perpendiculars is again the tractrix. 


The geometry of Riemann is applicable to a very famil¬ 
iar object—the sphere. It may be seen from Fig. 46 that a 
plane which passes through the center of a sphere cuts 
the surface in a great circle. 

Although the earth is somewhat oblate, for the purpose 
of this discussion we may consider it spherical. Every 
circle passing through the North and the South Poles 
on the earth’s surface is a great circle (longitude), but 
with the exception of the equator, the circles of latitude 
are not. Straight lines drawn on the surface of the earth 
are always parts of great circles, and even if two suck 
lines are perpendicular to a third line {which, in Euclidean 
geometry, would imply they are parallel), they will always 


144 Mathematics and the Imagination 

intersect at a pair of poles. Thus, the elements for a geometry 
which will satisfy the surface of the earth are identical 


FIG. 46. 


with those of Riemannian geometry. For example, a 
triangle drawn on the surface of the earth will have 


FIG. 47(a).—Triangle A is small compared ynth the 
sphere; thus it is nearly a plane triangle and its angle 

sum is near 180®. . . • i. r 

But let it grow into triangle B, the sides of which he 

on three perpendicular great circles, and the angle 
sum = 90® + 90® + 90® = 270®. 

In the still larger triangle C, the angles of which arc 
all obtuse, the sum is greater than 270®. 


Assorted Geometries—Plane and Fancy 145 

angles totaling more than 180 degrees, and the larger the 
triangle, the greater the sum of the angles. 


FIG. 47(b).— This is the reverse of what happens 
on a sphere, Fig. 47(a). On a pscudosphere, the 
larger the triangle, the smaller the sum of the angles. 

Furthermore, two straight lines drawn on the earth’s 
surface, if sufficiently extended, will always enclose an 
area. It is convenient at this point to recall the im{>ortant 
distinction noted by Riemann that a surface may be 
finite but unbounded, so that straight lines drawn upon 
the surface of the earth can be infinitely extended, al¬ 
though the surface is evidently not infinite, but merely 
unbounded. The Riemannian properties of the sphere 
are amusingly set out by the following riddle: 

A group of sportsmen, having pitched camp, set forth 
to go bear hunting. They walk 15 miles due south, then 
15 miles due east, where they sight a bear. Bagging their 
game, they return to camp and find that altogether they 
have traveled 45 miles. What was the color of the bear? 

♦ 

Our brief discussion of non-Euclidean geometry is 
bound to raise in the mind of the reader many questions 


146 Mathematics and the Imagination 

outside our province, but the literature, even the popular 
literature, is so extensive that anyone sufficiendy inter¬ 
ested and curious need not go begging for answers. 

Yet it is perhaps proper that we should consider one 
very natural question which might take this form. On 
a sphere, two straight lines, even though parallel at one 
place, are certain (if sufficiently extended) to intersect, 
and may enclose an area. Why, then, call such lines 

‘straight’? Are they not really curved?” 

At the outset it is obvious that whether a line is 
straight or not depends on the definition of “straight. 
In mathematics, it has been found convenient to formu¬ 
late such a definition only with reference to the particu¬ 
lar surface under consideration. One way of defining a 
straight line is to say that it is the shortest distance be¬ 
tween two points. On the other hand, everyone knows, 
from the many references in recent times to aeronautical 
exploits, that the shortest route between two points on 
the surface of the earth can be covered by following the 
arc of the great circle lying between them. Conveniently 
enough, through each two points on the surface of a 

sphere there does pass a great circle. 

The great circle, then, on the sphere, corresponds to 
the straight line in the plane—it is the shortest distance 
between two points. Suitable curves may be found for 
other types of surfaces, for instance, the pseudosphere, or 
a saddle-shaped surface which will fulfill the same role. 
Generalizing this notion, a curve which is the shortest 
distance between two points (analogue of the straight 
line in the plane) on any kind of a surface is called a 
geodesic of that surface. When we sought entities that 
would satisfy the geometry of Lobachevsky, and that of 
Riemann, we were really looking for surfaces, the geo- 


Assorted Geometries—Plane and Fancy 147 

desics of which would obey the parallel postulates of these 
geometries. 

In the plane, if we adopt Euclid’s hypothesis, a pair of 
geodesics meet in one point, unless they are parallel, in 
which case they do not meet at all. On a sphere, a pair of 
geodesics (arcs of great circles), even if parallel, always 
meet in two points, and therefore the sphere obeys the 
geometry of Riemann. On a pseudosphere, obeying 
Lobachevsky’s geometry, parallel geodesics may ap¬ 
proach one another asymptotically, but never intersect. 


POSITIVE NEGATIVE 


ZERO 

FIG. 48.—Curvature. 


The geodesics of a surface are determined by its 
curvature. Curvature is not easy to explain, although we 
all have an intuitive notion of what it means. A plane 
has a curvature of 0. A surface like that of a sphere or 
an ellipsoid is one of positive curvature, whereas the 
saddle-shaped surface or the pseudospherc is said to be of 
negative curvature. We can imagine more complicated 


Mathematics and the Imagination 


surfaces, parts of which may have a positive, parts a 
negative, and parts a 0 curvature. The geodesics of a 
surface, as well as its most appropriate geometry, depend 
upon such curvature—positive, negative, or 0. Whence 
the geometry of a surface of constant negative curvature 
is Lobachevskian, that of a surface of constant positive 
curvature Riemannian, and that of a surface of 0 curva¬ 
ture Euclidean. 


All that has been said about non-Euclidean geometry, 
while evident enough when we talk of geometry^ tends to 
become obscure when applied to everyday surroundings. 
We are inclined to pity the inhabitants of a two-dimen¬ 
sional world, as much for their ignorance as for their 
physical limitations. They cannot even dream of doing 
things which to us are perfectly commonplace. Yet we 
tend to show the same intellectual limitations in picturing 
our world to ourselves. Indeed, we go further, for we 
deliberately reject our own experience. Our experience 
is that space is finite but unbounded, and that the 
straight lines we are able to draw on the surface on which 
we live can never recdly be straight, but must be curved. 
(Of course the earth’s curvature differs from 0.) But we 
continue to confuse infinity and unboundedness, to reject 
the latter which constitutes our actual spatial knowledge 
and to embrace the former for religious and aesthetic rea¬ 
sons. And, although every intelligent person knows the 
earth’s surface is curved, and every navigator practices 
great-circle sailing, most of us behave like Seventh-Day 
Adventists in reasoning that our straight lines are drawn 
in a plane of 0 curvature—or, in effect, in a world that 
is flat. From this it is only a step to the belief that Euclid s 
fifth postulate is sacred and any substitute is “against 


Assorted Geometries—Plane and Fancy 149 

nature.” A little curvature, even more than a little learn¬ 
ing, has its disadvantages. 

Although we know a good deal more about the surface 
we inhabit than about the physical space in which we 
live, there is hardly any choice between the absurdities 
of our beliefs about either one. The geometry of Euclid, 
which considers surfaces of 0 curvature, in the strictest 
sense (disregarding convenience in computation) does 
not suit the surface on which we live as well as that of Rie- 
mann. Unmistakably, our geometries, though suggested 
by our sense perceptions, are not dependent upon them. 

The geometries we have discussed are only three of an 
infinite number of possible ones. Any geometry, whatever 
its postulates (provided they lead to no contradictions), 
will be just as “true” as the geometry of Euclid. For 
every surface, however complex its curvature, there is a 
peculiarly suited geometry. It is true we start our geom¬ 
etries as purely logical structures, but, as in other branches 
of mathematics, we find that Nature has anticipated us, 
and that a surface often waits upon our inventiveness. 
For that reason, the non-Euclidean mathematics has found 
enormously important fields of application in the weird 
topsy-turvy of modern physics. 

While we have considered the applications of two- 
dimensional non-Euclidean geometries to familiar sur¬ 
faces, the mathematical physicist studies the application 
of higher-dimensional non-Euclidean geometries to 
higher-dimensional space manifolds. In attempting to 
discover experimentally what space we actually live in, 
scientists have obtained results which lead them to believe 
that space is curved rather than straight. Having emanci¬ 
pated ourselves from the primitive idea that we live on a 
plane surface, curved space should not be so hard to take. 


150 Mathematics and the Imagination 

There is a final point: If we consider the geometries of 
Euclid, Lobachevsky and Riemann as applied, and not 
as pure, mathematics, if we ask which one is most 
suitable to the space immediately surrounding us and the 
surface on which we live, what shall our answer be? Ex¬ 
periment and measurement alone can answer that ques¬ 
tion. It turns out that Euclid’s geometry is the most con¬ 
venient, and the one, in consequence, which we shall 
continue to use to build our bridges, tunnels, skyscrapers, 
and highways. The geometries of Lobachevsky, or Rie¬ 
mann, properly handled, would do just as well.^® Our 
skyscrapers would stand it, and so would our bridges, 
tunnels, and highways; our engineers might not. The 
geometry of Euclid is easier to teach, fits in more readily 
with misguided common sense, but above everything, 
is easier to use. And we are concerned, after all, in such 
matters with living, and not with logic. 

Yet our vistas have widened and our vision is clearer. 
Mathematics has helped us to transcend those sense 
impressions which we now say “deceive us never, while 
lying ever.” 


FOOTNOTES 

1. St. Augustine, Confessions. —P. 12. 

2. An illustration of pure mathematics: * 

Consider the following propositions, which are the axioms 
for a special kind of geometry. 

Axiom 7. If A and B are distinct points on a plane, there is at 
least one line containing both A and B. 

Axiom 2. If A and B cire distinct points on a plane, there is 
not more than one line containing both A and B. 

* Morris Raphael Cohen and Ernest Nagel, An Introduction to 
Logic and Scientific Method {Nc^Yot]l\ Harcourt Brace, 1936), pp- 133- 
139. 


Assorted Geometries—Plane and Fancy 151 

Axiom 3. Any two lines on a plane have at least one point of 
the plane in common. 

Axiom 4. There is at least one line on a plane. 

Axiom 5. Every line contains at legist three points of the plane. 

Axiom 6. Ail the points of a plane do not belong to the same 
line. 

Axiom 7. No line contains more than three points of the plane. 

These axioms seem clearly to be about points and lines on a 
plane. In fact, if we omit the seventh one, they are the assump¬ 
tions made by Veblen and Young for “projective geometry” 
on a plane in their standard treatise on that subject. It is un¬ 
necessary for the reader to know anything about projective 
geometry in order to understand the discussion that follows. 
But what are points, lines and planes? The reader may think he 
“knows” what they are. He may “draw” points and lines with 
pencil and ruler, and perhaps convince himself that the axioms 
state truly the properties and reladons of these geometric things. 
This is extremely doubtful, for the properdes of marks on paper 
may diverge noticeably from those postulated. But in any case 
the quesdon whether these actual marks do or do not conform 
is one of applied and not of pure mathematics. The axioms them¬ 
selves, it should be noted, do not indicate what points, lines, and 
so on “really” are. For the purpose of discovering the implicadons 
of these axioms, it is unessential to know what we shall understand 
by points, lines, and planes. These axioms imply several theorenns, 
not in virtue of the visual representation which the reader may 
give them, but in virtue of their logical form. Points, lines, and 
planes may be any entities whatsoever, undetermined in every 
way except by the relations stated in the axioms. 

Let us, therefore, suppress every explicit reference to points, 
lines, and planes, and thereby eliminate all appeal to spatial 
intuition in deriving several theorems from the axioms. Suppose, 
then, that instead of the word “plane,” we employ the letter S; 
and instead of the word “point,” we use the phrase '^element of 
•S’.” Obviously, if the plane (5) is viewed as a collection of points 
(elements of 5), a line may be viewed as a class of points (ele¬ 
ments) which is a subclass of the points of the plane (i*). We 
shall therefore substitute for the word “line” the expression 
^'L-class'* Our original set of axioms then reads as follows: 

Axiom V. If A and B are disdnet elements of S, there is at least 
one L-class containing both A and B. 

Axiom 2'. If A and B are distinct elements of S, there is not 
more than one L-class containing both A and B. 


Mathematics and the Imagination 

Axiom 3'. Any two L’classes have at least one element of S' in 
common. 

Axiom 4'. There exists at least one L-class in .S'. 

Axiom 5'. Every L-<lass contains at least three elements of S. 

Axiom 6'. All the elements of S do not belong to the same 
L-class. 

Axiom 7'. No L<lass contains more than three elements of S. 

In this set of assumptions no explicit reference is made to any 
specific subject matter. The only notions we require to state them 
are of a completely general character. The ideas of a “class,” 
“subclass,” “elements of a class,” the relation of “belonging to 
a class” and the converse relation of a “class containing ele¬ 
ments,” the notion of “number,” are part of the fimdamental 
equipment of logic. If, therefore, we succeed in discovering the 
implications of these axioms, it cannot be because of the proper¬ 
ties of space as such. (As a matter of fact, none of these axioms 
can be regarded as propositions; none of them is in itself either 
true or false. For the symbols 5*, Lrclassy Ay By and so on are 
variables. Each of the V 2 iriables denotes any one of a class of 
possible entities, the only restriction placed upon it being that it 
must “satisfy,” or conform to, the formal relations stated in the 
axioms. But until the symbols are assigned specific values, the 
axioms are propositional functions, and not propositions.) 

Our “assumptions,** therefore, consist in relations considered 
to hold between undefined terms. But the reader will note that 
although no terms are explicitly defined, an implicit definition of 
them is made. They may denote anything whatsoever, provided 
that what they denote conforms to the stated relations between 
themselves. This procedure characterizes modem mathemadcal 
technique. In Euclid, for example, explicit definitions arc given 
of points, lines, angles, and so on. In a modem treatment of ge¬ 
ometry, these elements are defined implicitly through the axioms. 
As we shall see, this latter procedure makes it possible to give a 
variety of interpretations to the undefined elements, and so to ex¬ 
hibit an identity of structure in different concrete settings. . . . 

STRUCTURAL IDENTITY OR ISOMORPHISM 

We want to show now that an abstract set such as the one 
discussed in the previous section may have more than one concrete 
representation, and that these different representations, though 
extremely unlike in material content, will be identical in logical 
structure. 


Assorted Geometries—Plane and Fancy 153 

Let us suppose there is a banking firm with seven partners. 
In order to assure themselves of expert information concerning 
various securities, they decide to form seven committees, each of 
which will study a special field. They agree, moreover, that each 
partner will act as chairman of one committee, and that every 
partner will serve on three, and only three, committees. The fol¬ 
lowing is the schedule of committees and their members, the first 
member being chairman: 


Domestic railroads 
Municipal bonds 
Federal bonds 
South American securides 
Domesdc steel industry 
Continental securities 
Public udlities 


Adams 

Brown 

Smith 

Brown 

Murphy Ellis 

Murphy 

Smith 

Jones 

Smith 

Ellis 

Gordon 

Ellis 

Jones 

Adams 

Jones 

Gordon 

Brown 

Gordon 

Adams 

Murphy 


An examination of this schedule shows that it “satisfies” the 
seven axioms if the class S is interpreted as the banking firm, 
its elements as the partners, and the L-<lasses as the various 
committees. . . . 


Paraphrasing: 

One further interpretation illustrates the same seven formal 
relations. 


In the diagram there are seven points lying by threes on seven 
lines, one of which is bent.” If each point represents an element 
of S and each set of three points lying on a line an L-class, then 
all the seven assumptions are satisfied. Thus, for example, the 
three-termed relation between Adams, Brown, and Smith, by 
virtue of which they are on the same committee, holds for the 
points A, By Dy in virtue of which they lie on the same line. And in 
general, what may be deduced for A from the assumptions holds 


154 Mathematics and the Imagination 

for Mr. Adams, what may be deduced for B holds for Mr. Brown, 
and so on.—P. 114. 

3. Forsyth, Geometry oj Four Dimensions. —P. 119. 

4. It should be emphasized that a manifold, as usually defined, is 
stripped of every attribute, except that it is a class. Accordingly, 
it is easy to think of many familiar kinds of manifolds which 
have nothing to do with either space or geometry. A three- 
dimensional manifold would be a class of elements, each of , 
which would require exaedy three numbers to identify it—to 
distinguish it from every other element in the class. Think of a 
cylinder containing a quantity of three gases which have been 
thoroughly mixed so that the volume of gas or any portion of it, 

is uniquely determined by three numbers, x, y^ and Zy each of 
which represents the percentage of the three respective gases 
in the mixture. Or, one further instance: A group of people 
might be thought of as a manifold. If we find that five numbers 
are necessary and sufficient to individualize each one, say x 
equals age, y equals bank balance, z equals telephone number, 
u equals height, v equals weight, then they constitute a five¬ 
dimensional manifold. Other examples of manifolds can be 
devised: (a) four-dimensional: particles of air, 3 dimensions to 
fix them in space, 1 to fix their density; (b) four-dimensional: 
all conceivable spheres in space, 3 dimensions to fL^ their centers, 

1 to determine their radii.—P. 119. 

5. Nobeling, “Die vierte Dimension und der krumme Raum,’* in 
Krise und Neuaufbauy Leipzig: Deudeke, 1933.—P. 123. 

6. Eddington, .y/xzfe, Timey and Gravitation. —P. 131. 

7. Lindsay and Margenau, Foundation of Physics. —P. 132. 

8. Op. cit.~~P. 133. 

9. Young, Fundamental Concepts of Algebra and Geometryy New York: 
Macmillan, 1911.—P. 135. 

10. Morris Raphael Cohen, Reason and Nature. —P. 137. 

11. Cohen and Nagel, Introduction to Logic and Scientific Methody New 
York: Harcourt Brace, 1934.—P. 142. 

12. The diagram illustrates somewhat in detail what is meant. 

A perpendicular is drawn to the line G on the pseudosphere; 
through the point O two parallels are to be drawn to the line G. 
Mark off the distance S on Gy terminating in Q. At Q, erect a 
perpendicular to G. If, then, about the point O with iS" as radius 
we draw a circle, this circle will cut QT at S\ and »S' 2 . These two 
points, together with the point O, determine the two parallels 
to G, P\ and P-i. All of the lines through O making an angle 
smaller than 0 do not intersect G, although they arc not parallel 


Assorted Geometries—Plane and Fancy 


155 


FIG. 49. 


to it. This diagram is found in Colerus, Vom Punkt zut vierten 
Dimension, Vienna; Zsolnay, 1935.—P. 143. 

13. These geometries are indispensable in the physics of the atom and 
the stars, in regions of space which are not a part of our im¬ 
mediate experience.—P. 150. 


Pastimes of Past and Present Times 


Work consists of whatever a body is obliged to dOy and 
play consists oj whatever a body is not obliged to do. 

-MARK TWAIN 


It has been said, “It is not by amusing oneself that one 
learns,” ^ and in reply: “It is only by amusing oneself 
that one can learn.” Wherever the truth may lie, some¬ 
where between those extremes, it is undeniable that math¬ 
ematical recreations furnish a challenge to imagination 
and a powerful sdmulus to mathematical activity. The 
theory of equations, of probability, the infinitesimal cal¬ 
culus, the theory of point sets, of topology—all are fruits 
grown from seeds sown in the fertile soil of creative im¬ 
agination—all have grown out of problems first expressed 
in puzzle form. 

Puzzles and paradoxes have been popular since antiq¬ 
uity, and in amusing themselves with these playthings 
men sharpened their wits and whetted their ingenuity. 
But it was not for amusement alone that Kepler, Pascal, 
Fermat, Leibniz, Euler, Lagrange, Hamilton, Cayley, and 
many others devoted so much time to puzzles. Researches 
in recreational mathematics sprang from the same desire 
to know, were guided by the same principles, and re¬ 
quired the exercise of the same faculties as the researches 
leading to the most profound discoveries in mathematics 
and mathematical physics. Accordingly, no branch of in- 

156 


Pastimes of Past and Present Times 157 

tellectual activity is a more appropriate subject for dis¬ 
cussion than puzzles and paradoxes. 

+ 

The field is enormous. Puzzles have been in the making 
since Egyptian times and probably before. From the 
cryptical utterances of the oracle of Delphi, through the 
time of Charlemagne, down to the golden age of the 
crossword, paradoxes and puzzles, like the creatures of 
the earth, have assumed every shape and form and have 
multiplied. We can examine only a few of the dominat¬ 
ing species, those which have survived in one shape or 
another and continue to thrive in streamlined form. 

Most of the famous puzzles invented before the 17th 
century may be found in the first great puzzle book, 
Les probCkmes plaisants et dHectables, qui se font par les nombres^ 
by Claude-Gaspard Bachet, Sieur de Meziriac. Although 
it appeared in 1612, two years before Napier’s work on 
logarithms, it is still a delightful book and a quarry of in¬ 
formation. Many collections have appeared since then,’^ 
Bachet’s volume alone having been enlarged to almost 
five times its original size. 

All we can hope to do is to follow the illustrious 
example of Mark Twain in a similar predicament. He 
attempted to reduce all jokes to a dozen primitive or 
elementary forms (mother-in-law, farmer’s daughter, 
etc.). We shall attempt to present a few of the typical 
puzzles that illustrate the basic ideas from which all 
are evolved. We shall restrict our interest to puzzles 
and problems, reserving for another chapter some of the 
more celebrated logical and mathematical paradoxes. 
Although it may not always be easy to distinguish 
between a puzzle and a paradox, for our purposes it is 
sufficient to consider a puzzle as an ingenious game or 


158 Mathematics and the Imagination 

problem, and a paradox as an apparendy fallacious and 
self-contradictory proof or statement. 

* 

Puzzles often seem difficult because they are not easy 
to interpret in precise terms. In attempting the solution 
of a problem, the method of trial and error is not only 
more natural, but generally easier than the mathemadcal 
attack. It is common experience that often the most 
formidable algebraic equations are easier to solve than 
problems formulated in words. Such problems must 
first be translated into symbols, and the symbols placed 
into the proper equadons before the problems can be 
solved. 

When Flaubert was a very young man, he wrote a 
letter to his sister, Carolyn, in which he said: “Since you 
are now studying geometry and trigonometry, I will give 
you a problem. A ship sails the ocean. It left Boston with 
a cargo of wool. It grosses 200 tons. It is bound for Le 
Havre. The mainmast is broken, the cabin boy is on deck, 
there are 12 passengers aboard, the wind is blowing East- 
North-East, the clock points to a quarter past three in the 
afternoon. It is the month of May—How old is the cap¬ 
tain?” Flaubert was not only teasing, he was uttering a 
complaint shared by that large and respectable company 
“not good at puzzles,” that the average puzzle both con¬ 
fuses and overwhelms with superfluous words.® For that 
reason, the following puzzles have been stripped of all in¬ 
essential elements so as to exhibit their underlying mathe¬ 
matical structure. And we understand by the term “math¬ 
ematical structure” not necessarily something expressed 
by numbers, angles, or lines, but the essential internal 
relationship between the component elements of the puz- 


Pastimes of Past and Present Times 159 

zle. For, at bottom, that is all that mathematical analysis 
can reveal, all that mathematics itself signifies. 

+ 

Among the oldest problems are those which involve 
ferrying people and their belongings across a river under 
somewhat trying conditions. Alcuin, the friend of Char¬ 
lemagne, suggested a problem which has since been re¬ 
stated and complicated in many ways. A traveler comes 
to a riverbank with his possessions: a wolf, a goat, and a 
head of cabbage. The only available boat is very small 
and can carry no more than the traveler and one of his 
possessions. Unfortunately, if left together, the goat will 
eat the cabbage and the wolf dine on the goat. How shall 
the traveler transport his belongings to the other side of 
the river, keeping his vegetables and animals intact? ^ 
The solution may be attempted with the aid of a match 
box, representing the boat, and four slips of paper for its 
occupants. 

A more elaborate version of this problem was suggested 
in the sixteenth century by Tartaglia. Three beautiful 
brides with their jealous husbands also come to a river. 
The small boat which is to take them across holds only 
two people. To avoid any compromising situations, the 
crossings 2 ire to be so arranged that no woman shall be 
left with a man unless her husband is present. Eleven 
passages are required. Five passages would be required 
for two couples, but with four or more couples the cross¬ 
ing under the conditions stated would be impossible. 

Similar problems involve shunting. In Fig. 50 there 
is a locomotive, L, and 2 freight cars, IT, and W 2 . The 
common portion of the rails of the two sidings on which 
W\ and W 2 are standing, DA, is long enough to hold 


i6o Mathematics and the Imagination 

Wi or W 2 , but not both, nor the locomotive L. Thus, a 
car on DA can be shunted to either siding. The engineer’s 
job is to switch the positions of W\ and W 2 - How can this 
be done? Although this problem presents no particular 


L 

FIG. 50- 


difficulties, the same theme in more complex form niay 
demand of the engineer mathematical talents of a high 

order. 

* 

Simeon Poisson’s family tried to make him everything 
from a surgeon to a lawyer, the last on the theory that he 
was fit for nothing better. One or two of these professions 
he tackled with singular ineptitude, but at last he found 
his metier. It was on a journey that someone posed him 
a problem similar to the one below. Solving it immedi¬ 
ately, he realized his true calling and thereafter devoted 


Pastimes of Past and Present Times i6i 

himself to mathematics, becoming one of the greatest 
mathematicians of the nineteenth century.^ 

Two friends who have an eight-quart jug of wine wish 
to share it evenly. They also have two empty jars, one 

6 qti 


1 / 


PIG. 51. Solution to the problem of the 3 jars. 


162 Mathematics and the Imagination 

holding five qucirts, the other three. The diagram il¬ 
lustrates how they were able to divide their wine into 
two portions of four quarts each,® 

This brings to mind another ‘‘pouring problem,” per¬ 
haps not exactly germane at this point, but a good exer¬ 
cise in logical rigor and liquid refreshment. 

THE INTERNATIONAL BEER-DRINKING PUZZLE 

In a certain town lying on the border between Mexico and 
the United States a peculiar currency situation exists. In 
Mexico a U. S. dollar is worth only ninety cents of their money, 
while in the United States the value of the Mexican dollar is 
only ninety cents of our money. One day a cowhand strolls 
into a Mexican cantina and orders a ten-cent beer. He pays for 
it with a Mexican dollar, receiving for change an American 
dollar, worth just ninety cents there. After drinking his beer, 
he strolls over the border to an American saloon and orders 
another. This he pays for with an American dollar receiving a 
Mexican bill for change. He takes this back across the border 
and repeats the process, drinking beer merrily all day, and 
ends up as rich as he started, with a dollar. 

The question; Who paid for the beer? 

The moral: Visit sunny Mexico on your vacation. 

* 

The mystifying nature of so many arithmetic triclp 
lies, as we have indicated, in their structure, not their 
content. With a strainer to sift out the essential ideas 
hidden between dozens of useless ones, every man could 
be his own magician. A silly little riddle, oft repeated 
among mathematicians, comes to mind. “How shall one 
catch the lions in the desert?” it is asked. Since there is 
so much sand and so few lions, simply take a strainer, 
strain out the sand, and there are the lions! Such a 


Pastimes of Past and Present Times 163 

strainer, then, or perhaps a scalpel, is needed to get at 
the rudiments. When the verbiage has been swept away 
the puzzle skeletons succumb to simple arithmetic or 
algebra. The parlor tricks of guessing numbers which 
others have selected, or cards which someone has chosen 
seem almost as wonderful as instances of “extra sensory 
perception.” But after we have learned to separate the 
lions from the sand, caging them is comparatively simple. 

Card tricks are usually arithmetic puzzles in disguise. 
Generally, they are amenable to mathematical analysis, 
and are not, as is commonly believed, performed by 
sleight of hand. One important principle, easily over¬ 
looked, is that “cutting a pack of cards never alters the 
relative positions of the cards, provided that, if necessary, 
we regard the top card as following immediately after 
the bottom card in the pack.” ^ Once this is understood, 
many tricks cease to be baffling. 

Seven poker players have invested in a new deck of 
cards. In keeping with tradition the cards are cut, not 
shuffled, on the first deal. The dealer, pretending to 
cheat, takes his second and fourth cards from the bottom 
of the deck. This lapse is noticed by everyone, as in¬ 
tended. However, when the other players pick up their 
cards, they are reluctant to demand a new deal, each one 
finding that he has a full house. But still fearful that the 
dealer has fixed up a better hand for himself, they insist 
that he discard his 5 cards and take the next 5 from the 
top of the deck. Feigning indignation he acquiesces— 
and wins with a straight flush. Try it. Ninety-nine times 
out of one hundred you will succeed in cheating your 
friends—but then, you can't cheat an honest man. 

Frequently, the arithmetic tricks of guessing a number 
selected by another depend on the “scale of notation.” 


164 Mathematics and the Imagination 

When a number is expressed in the decimal system, such 
as 3976, what is actually meant is 

(3 X 10^) + (9 X 10^) + (7 X 101) + (6 X 10°). 

The table ® further illustrates other numbers written to 
the base 10. 


EXAMPLE 10® 10* 


10 * 10 * 10 * 


469 

s 

9 X 

10® 

+ 

6 X 10« 

+ 

4 X 10* 

1 


9 

+ 

1 


+ 


1 


7901 

s 

1 

X 

10“ 

+ 

0 X 10* 

+ 

9 X 10* 

+ 

7 X 10* 


1 

1 


0 

+ 


+ 

7000 


30,000 

s 

0 

X 

10“ 

+ 

0 X 10> 

+ 

0 X 10* 


0 X 10* 


3 X 10“ 

B 

0 

+ 

0 

+ 

0 

+ 


+ 

30.000 

21148 


8 

X 

10“ 

+ 

4 X 10* 

+ 

1 X 10* 

+ 

1 X 10* i 

+ ' 

2X 10“ 

1 

8 

+ 

40 

+ 


+ 

1000 

+ 


Among the wide variety of problems which arise from 
the use of the decimal system, the following are of some 
interest: 

A useful device for checking multiplication goes by the 

copybook title of “casting out nines.” 

Consider 1234 X 5678 = 7006652. Add the digits of 
the multiplier, multiplicand, and product, thus obtaining 
10, 26 and 26 respectively. Since each of these numbers is 
greater than 9, add the digits of the individual sums 
once more,* obtaining 1, 8, and 8. (If, after the first 
repetition a sum greater than 9 remains, the digits may 
be added once again.) Now, take the product of the 
integers corresponding to the multiolier, and the multi- 

*Thus 10 = 1 + 0 = 1 

26 = 2 + 6 = 8, etc. 


Pastimes of Past and Present Times 165 

plicand, i.e., 1X8, and compare this with the integer 
corresponding to the sum of the digits of the product, 
which is also 8. Since they are the same, the result of the 
original multiplication is correct. 

Using the same rule, let us test whether the product of 
31256 and 8427 is 263395312. Again the sum of the digits 
of the multiplicand, multiplier, and the product are re¬ 
spectively 17, 21 and 34; repeating, the sum of these digits 
is 8, 3, and 7. The product of the first two equals 24 which 
has 6 for the sum of its digits. But the sum of the digits of 
the product equals 7. Thus, we have two different re¬ 
mainders, 6 and 7, whence the multiplication must be in¬ 
correct. 

Closely connected with the rule of casting out nines 
is the following trick, which reveals a remarkable prop¬ 
erty common to all numbers. 

Take any number and rearrange its digits in any order 
you please to form another number. The difference be¬ 
tween the first number and the second is always divisible 
by 9.® 

Another type of problem dependent on the decimal 
scale of notation involves finding numbers which may 
be obtained by multiplying their reversals by integers. 
Among such numbers with 4 digits, 8712 equals 4 times 
2178, and 9801 equals 9 times 1089. 

The binary or dyadic notation (using the base 2) is 
hardly a new concept, having been referred to in a 
Chinese book believed to have been wTitten about 3000 
B.c. Forty-six centuries later, Leibniz rediscovered the 
wonders of the binary scale and marveled at it as though 
it were a new invention—somewhat like the twentieth- 
century city dweller, who, upon seeing a sundial for (he 
first time, and having it explained, remarked with awe: 


166 Mathematics and the Imagination 

“What will they think of next?” In its use of only two 
symbols, Leibniz saw in the dyadic system something of 
great religious and mystic significance: God could be 
represented by unity, and nothingness by zero, and since 
God had created all forms out of nothingness, zero and 
one combined could be made to express the entire 
universe. Anxious to impart this gem of wisdom to the 
heathens, Leibniz communicated it to the Jesuit Grim¬ 
aldi, president of the Tribunal of Mathematics in China, 
in the hope that he could thus show the Emperor of 
China the error of his ways in clinging to Buddhism 
instead of adopting a God who could create a universe 
out of nothing. 

Whereas the decimal scale requires ten sumbols: 0, 1, 
2, 3, 4, . . . , 9, the binary scale uses only two: 0 and 1. 
Below are the first 32 integers given in the binary scale. 


DECIMAL 


BINARY 

1 


1 

2 


10 

3 

= 

11 

4 

= 22 = 

100 

5 

= 

101 

6 


no 

7 


111 

8 

= 23 = 

1000 

9 

SS 

1001 

10 

s 

1010 

11 

— 

1011 

12 

— 

1100 

13 

— 

1101 

14 


1110 

15 


nil 

16 

= 2^ = 

10000 


DECIMAL 


BINARY 

17 


10001 

18 


10010 

19 


10011 

20 


10100 

21 


10101 

22 


10110 

23 


10111 

24 


11000 

25 


11001 

26 


11010 

27 


non 

28 

= 

11100 

29 


11101 

30 


lino 

31 


inn 

32 

= 2^ = 

100000 


Since 2® = 1, it may readily be seen that any number can 
be expressed as the sum of powers of 2, just as any number 


Pastimes of Past and Present Times 167 

in the decimal system can be expressed as the stm of 
powers of 10. For example, the number expressed in the 

^ e n^ as 25. is expressed in the binary system, 
using only the two symbols 1 and 0, by 11001. 

DECIMAL DYADIC 

25 = 11001 

I I 

(2 X 10*) + (5 X 10®). (1 X 20 + (1 X 20 + (0 X 20 

+ (0 X 2‘) + (I X 2®). 

Because numbers can be more briefly written in the 
decimal scale than in the binary, it is more convenient, 
although in every other respect the latter is just as ac¬ 
curate and efficient. Even fractions have their place in the 
dyadic notation. The fraction for example, given by 
the nonterminating decimal, .33333 - . . , is represented 
in the binary notation by a nonterminating binary, 
.01010101 . . 

The binary system easily makes understandable the 
solution of problems such as: 

I. In many sections of Russia, the peasants employed 
until recently what appears to be a very strange method 
of multiplication. In substance, this was at one time in 
use in Germany, France, and England, and is similar to 
a method used by the Egyptians 2000 years before the 
Christian era. 

It is best illustrated by an example: To multiply 45 
by 64, form two columns. At the head of one put 45, at 
the head of the other, 64. Successively multiply one 
column by 2 and divide the other by the same number. 
When an odd number is divided by 2. discard the re¬ 
maining fraction. The result will be: 


168 Mathematics and the Imagination 


DIVIDE 

MULTIPLY 

45 

64 

22 

128 

11 

256 

5 

512 

2 

1024 

1 

2048 


Take from the second column those numbers which ap¬ 
pear opposite an odd number in the first. Add them and 
you obtain the desired product: 

64 .... 64 = 2® X 64 
128 =2^X64 

256 .. . .256 = 2^ X 64 
512 .. . .512 » 2® X 64 
1024 = 2* X 64 

2048 . . . 2048 = 2^ X 64 
2880 = 45 X 64 

The relation of this method to the dyadic system may be 
seen upon expressing 45 in the dyadic notation. 

45 =(1 X 2^) 4- (0 X 2") + (1 X 2^) + U X 2^) 4 (0 X 2*) 
4 (1 X 2®) 

= 101101 

= 32 4048444041 
Therefore, 

45 X 64 = (25 + 2® 4 22 4 2®) X 64 

= (25 X 64) 4 (25 X 64) 4 (2“ X 64) 4 (2® X 64). 

Since 2? and 2^ do not appear in the dyadic expression 
for 45, the products (2^ X 64) and (2^ X 64) are not in¬ 
cluded in the numbers to be added in (B). Thus, what 
the peasant does in multiplying 45 by 64 is to multiply 
2°, 2^, 2^, 2®, successively by 64, and then take the sum. 

11. Another well-known problem, already mentioned by 
Cardan, consists in the removal of a number of rings 
from a bar. The puzzle can best be analyzed by the use 


Pastimes of Past and Present Times 169 

of the dyadic system, although the actual manipulation 
of the rings is at all times extremely difficult. 


The rings on the bar are so connected that although tlie 
end one can be removed without difficulty, any other 
ring can be put on or removed only when the one next 
to it, toward the end {A in the figure) is on the bar, and 
all the rest of the rings are off. Thus, to remove the fifth 
ring, the first, second, third must be off the bar, and the 
fourth must be on. If the position of all the rings on or 
off the rack are written in the dyadic notation, 1 des¬ 
ignating a ring which is off, and 0 designating a ring 
which is on, the mathematical determination of the num¬ 
ber of moves required to remove a given numl)er of rings 
is not too hard. The solution without the aid of the dyadic 
notation, as the rings increase in number, would b(‘ 
wholly beyond one’s imaginative powers. 

III. The problem of the Tower of Hanoi is similar in 
principle. The game consists of a board with three ])egs, 
as illustrated in Fig. 53. 

On one of these pegs rest a number of discs of various 
sizes, so arranged that the largest disc is on the bottom, 
the next largest rests on that one, the next largest on that. 


170 Mathematics and the Imagination 

and so on, up to the smallest disc which is on top. The 
problem is to transfer the entire set of discs to one of the 
other two pegs, moving only one disc at a time, and 
making certain that no disc is ever permitted to rest on 
one smaller than itself. If the removal of a disc from one 


FIG. 53. 


peg to another constitutes one transfer, the following 
table shows the number of transfers required for various 
numbers up to n discs: 


TABLE FOR TRANSFERS " 


DISCS 

1 

2 

3 

4 

5 

6 

7 

• 

n 


TRANSFERS 

1 

3 

7 

15 

31 

63 

127 

.L, 


There is a charming story about this toy:^^ 

In the great temple at Benares beneath the dome which 
marks the center of the world, rests a brass plate in which are 


Pastimes of Past and Present Times i 71 

fixed three diamond needles, each a cubit high and as thick 
as the body of a bee. On one of these needles, at the creation, 
God placed sixty-four discs of pure gold, the largest disc 
resting on the brass plate and the others getting smaller and 
smaller up to the top one. This is the Tower of Brahma. 
Day and night unceasingly, the priests transfer the discs from 
one diamond needle to another, according to the fixed and 
immutable laws of Brahma, which require that the priest on 
duty must not move more than one disc at a time and that 
he must place this disc on a needle so that there is no smaller 
disc below it. When the sixty-four discs shall have been thus 
transferred from the needle on which, at the creation, God 
placed them, to one of the other needles, tower, temple, and 
Brahmans alike will crumble into dust, and with a thunderclap, 
the world will vanish. 

The number of transfers required to fulfill the prophecy 
is 2®^ - 1, that is 18,446,744,073,709,551,615. If the 
priests were to effect one transfer every second, and work 
24 hours a day for each of the 365 days in a ycar,^’ it 
would take them 58,454,204,609 centuries plus slightly 
more than 6 years to perform the feat, assuming they 
never made a mistake—for one small slip would undo 
all their work. 

IV. One other game may be mentioned in connection 
with the dyadic system—Nim. In this game, two players 
play alternately with a number of counters placed in 
several heaps. At his turn, a player picks up one of the 
heaps, or as many of the counters from it as he pleases. 
The player taking the last counter loses. If the number of 
counters in each heap is expressed in the binary scale, 
the game readily lends itself to mathematical analysis. 
A player who can bring about a certain arrangement of 
the number of counters in each heap may force a w in.^^ 


172 Mathematics and the Imagination 

It is interesting to note that the number 2^^18,446,- 
744^073,709,551,616—represented in the dyadic system 
by a number with 64 digits, appears in the solution of 


FIG 54.—The diagram illustrates how to force a win at 
the game of Nim. Assume each player at his turn must 
pick up at least one match and may pick up as many as 
five. The rule is that the player picking up the last match 
loses. Then, for example, imagine that the original heap 
consists of 21 matches. In that case, the one playing hrst 
can force a win by mentally dividing the rnatches into 
• groups of 1, 6, 6, 6, and 2 (as in B). Since he plays first, he 
picks up 2 matches. Then, however many his oppo^e^ 
'picks, the first player picks up the complement ot o. 
is shoNvn in /I: If the second player takes 1, the first player 
takes 5; if the second player takes 2, the first piay^ 

4, and so on. Each of the three groups of 6 is thus ex- 
hausted, and the second player is left with the last ma c 
Had there been 47 matches, say, the grouping to Jorce a 
win for the first player would have been: 1, 6, 6, 6, 6, , > 
6, and 4. Rules for any other variation of Nim can also oe 

easily formulated. 


Pastimes of Past and Present Times 173 

a puzzle connected with the origin of the game ol 
chess. 

According to an old tale, the Grand Vizier Sissa Ben 
Dahir was granted a boon for having invented chess for 
the Indian King, Shirham. Since this game is played on a 
board with 64 squares, Sissa addressed the king: “Majesty, 
give me a grain of wheat to place on the first square, 
and two grains of wheat to place on the second square, 
and four grains of wheat to place on the third, and eight 
grains of wheat to place on the fourth, and so. Oh, King, 
let me cover each of the 64 squares of the board.” “And 
is that all you wish, Sissa, you fool?” exclaimed the as¬ 
tonished King. “Oh, Sire,” Sissa replied, “I have asked 
for more wheat than you have in your entire kingdom, 
nay, for more wheat than there is in the whole world, 
verily, for enough to cover the whole surface of the earth 
to the depth of the twentieth part of a cubit.” Now the 
number of grains of wheat which Sissa demanded is 
2®4 — 1, exactly the same as the number of disc transfers 
required to fulfill the prophecy of Benares related on page 
171. 

Another remarkable way in which 2®4 arises is in com¬ 
puting the number of each person’s ancestors from the 
beginning of the Christian era—just about 64 generations 
ago. In that length of time, assuming that each person 
has 2 parents, 4 grandparents, 8 great-grandparents, 
etc., and not allowing for incestuous combinations, every¬ 
one has at least 2*^-* ancestors, or a little less than eighteen 
and a half quintillion lineal relations alone. A most de¬ 
pressing thought 

* 

The Josephus problem is one of the most famous and 
certainly one of the most ancient. It generally relates a 


174 Mathematics and the Imagination 

story about a number of people on board a ship, some of 
whom must be sacrificed to prevent the ship from sinking. 
Depending on the time that the version of the puzzle 
was written, the passengers were Christians and Jews, 
Christians and Turks, sluggards and scholars, Negroes 
and whites, etc. Some ingenious soul with a knowledge of 
mathematics always managed to preserve the favored 
group. He arranged everyone in a circle, and reckoning 
from a certain point onward, every nth person was to be 
thrown overboard —n being a specified integer. The ar¬ 
rangement of the circle by the mathematician was such 
that either the Christians, or the industrious scholars, or 
the whites,—in other words, the assumedly superior group 
—were saved, while the rest were thrown overboard in 

accordance with the Golden Rule. 

Originally, this tale was told of Josephus who found 
himself in a cave with 40 other Jews bent on self-extinc¬ 
tion to escape a worse fate at the hands of the Romans. 
Josephus decided to save his own neck. He placed every¬ 
one in a circle and made them agree that each third 
person, counting around and around, should be killed. 
Placing himself and another provident soul in the 16th 
and the 31st position of the circle of 41, he and his 
companion, being the last ones left, were conveniendy 
able to avoid the road to martyrdom. 

A later version of this problem places 15 Turks and 
15 Christians on board a storm-ridden ship which is cer¬ 
tain to sink unless half the passengers are thrown over¬ 
board. After arranging ever^'one in a circle, the Chris¬ 
tians, ad majorem Dei gloriam^ proposed that every ninth 
person be sacrificed. 

Thus, every infidel was properly disposed of, and all 
true Christians saved. 


Pastimes of Past and Present Times 175 

Among the Japanese, the Josephus problem assumed 
another form: Thirty children, 15 of the first marriage, 
and 15 of the second, agree that their father’s estate is 
too small to be divided among all of them. So the second 


O 

% 


ecce 


Fic. 55. 

C = Christian T = Turk 


wife proposes that all the children be arranged in a 
circle, in order to determine her husband’s heir by a 
process of elimination. Being a prudent mathematician, 
as well as the proverbially wicked stepmother, she ar¬ 
ranges the children in such a way that one of her own is 
certain to be chosen. After 14 of the children of the lirst 
marriage have been eliminated, the remaining child, 


Oii\ 


176 Mathematics and the Imagination 

evidently a keener mathematician than his stepmother, 
proposes that the counting shall start afresh in the oppo¬ 
site direction. Certain of her advantage, and therefore 
disposed to be generous, she consents, but finds to her 


FIG. 56.—The Josephus problem, from Miyake 
Kenryu’s Shojutsu. (From Smith and Mikami, A 
History oj Japanese Matfumatics.) 


dismay that all 15 of her own children are eliminated, 
leaving the one child of the first marriage to become e 

heir.^’ . 

Elaborate mathematical solutions of more difficult an 

generalized versions of the Josephus problem were given 

by Euler, Schubert, and Tait. 


Pastimes of Past and Present Times i 77 

No discussion of puzzles, however brief, can afford to 
omit mention of the best-known of the many puzzles 
invented by Sam Lloyd. “15 Puzzle,” “Boss Puzzle,” 
“le Jeu de Taquin,” are a few of its names. For several 
years after its appearance in 1878, this puzzle enjoyed a 
popularity, particularly throughout Europe, greater than 
swing and contract bridge combined enjoy today. In Ger¬ 
many, it was played in the streets, in factories, in the royal 
palaces, and in the Reichstag. Employers were forced to 
post notices forbidding their employees to play the “15 
Puzzle” during business hours under penalty of dismissal. 
The electorate, having no such privileges, had to watch 
their duly elected representatives play the “Boss Puzzle” 
in the Reichstag while Bismarck played the Boss. In 
France, the “Jeu de Taquin” was played on the boule¬ 
vards of Paris and in every tiny hamlet from the Pyrenees 
to Normandy. A scourge of mankind was the “Jeu de 
Taquin,” according to a contemporary French journalist, 
—worse than tobacco and alcohol—“responsible for un¬ 
told headaches, neuralgias, and neuroses.” 

For a time, Europe was “15 Puzzle” mad. Tourna¬ 
ments were staged and huge prizes offered for the solution 
of apparently simple problems. But the strange thing was 
that no one ever won any of these prizes, and the ap¬ 
parently simple problems remained unsolved. 

The “15 Puzzle” (figure below) consists of a square 
shallow box of wood or metal which holds 15 little 
square blocks numbered from 1 to 15. There is actually 
room for 16 blocks in the box so that the 15 blocks can 
be moved about and their places interchanged. 1 he num¬ 
ber of conceivable positions is 16! = 20,922,789,888.000. 
A problem consists of bringing about a specified ar¬ 
rangement of the blocks from a given initial position, 


178 Mathematics and the Imagination 

which is frequently the normal position illustrated in 
Fig. 57. 

Shortly after the puzzle was invented, two American 
mathematicians proved that from any given initial 
order only halj of all the conceivable positions can actu- 


FiG. 57.—The 15 Puzzle (also Boss Puzzle orjeu 
de Taquin) in normal position. 


ally be obtained. Thus, there are always approximately 
ten trillion positions which the possessor of a “15 Puzzle’* 
can bring about, and ten trillion that he cannot. 

The fact that there are impossible positions makes it 
easy to understand why such generous cash prizes were 
offered by Lloyd and others, since the problems for 
which prizes were offered always entailed impossible 
positions. And it is heart-breaking to think of the head¬ 
aches, neuralgias, and neuroses that might have been 
spared—to say nothing of the benefits to the Reichstag 
if The American Journal oj Mathematics had been as widely 
circulated as the puzzle itself. With ten trillion possible 
solutions there still would have been enough fun left 
for everyone. 

In the normal position (Fig. 57),. the blank space is in 


Pastimes of Past and Present Times 179 

the lower right-hand corner. \Vhen making a mathemati¬ 
cal analysis of the puzzle, it is convenient to consider that 
a rearrangement of the blocks consists of nothing more 
than moving the blank space itself through a specific 
path, always making certain that it ends its journey in 
the lower right-hand corner of the box. To this end, the 
blank space must travel through the same number of 
boxes to the left as to the right and through the same 
number of boxes upwards as downwards. In other words, 
the blank space must move through an even number of boxes. 
If, starting from the normal position, the desired one 
can be attained while complying with this requirement, 
it is a possible position, otherwise it is impossible. 

Based upon this principle, the method of determining 
whether a position is possible or impossible is very simple. 
In the normal position every numbered block appears 
in its proper numerical order, i.e., regarding the boxes, 
row by row, from left to right, no number precedes any 
number smaller than itself. To bring about a position 
different from the normal one, the numerical order of 
the blocks must be changed. Some numbers, perhaps all, 
will precede others smaller than themselves. Every in¬ 
stance of a number preceding another smaller than itself 
is called an inversion. For example, if the number 6 pre¬ 
cedes the numbers 2, 4, and 5, this is an inversion to 
which we assign the value 3, because 6 precedes three 
numbers smaller than itself. If the sum of the values of 
all the inversions in a given position is even, the position 
is a possible one—that is, it can be brought about from 
the normal position. If the sum of the values of the in¬ 
versions is odd, the position is impossible and cannot be 
brought about from the normal configuration. 

The position illustrated in figure 58 can be created 


13 


18o Mathematics and the Imagination 

from the normal position since the sum of the values of 
the inversions is six—an even number. 


1 

4 

3 

1 ^ 

5 

8 

7 

Ijia 

9 

12 

11 

|l3 

14 

15 

■ 


FIG. 58 . 


But the position shown in Fig. 59 is impossible, since, 
as may readily be seen, the sum of the value of the in¬ 
versions brought about is odd: 


1 ^ 

3 

2 

Tt 

1 ^ 

7 

6 

5 1 

|]2 

11 

10 

91 

1 

14 

13 

n[ 


FIG. 59 . 

Figures 60 a, b, c illustrate three other positions. Are 
they possible, or impossible to obtain from the normal 
order? 


1 

2 

^3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

15 

U 

13 


11 

7 

4 


8 

13 

1 

2 

5 

10 

3 

9 

15 

12 

14 

6 


2 

4 

6 

8 

10 

11 

12 

13 

3 

5 

7 

9 

15 

1 

14 


FIGS. 60 (a, b, c). 


Pastimes of Past and Present Times 


i8i 


SPIDER AND FLY PROBLEM 

Most of US learned that a straight line is the shortest 
distance between two points. If this statement is supposed 
to apply to the earth on which we live, it is both useless 
and untrue. As we have seen in the previous chapter, 
the nineteenth-century mathematicians Ricmann and 
Lobachevsky knew that the statement, if true at all, 
applied only to special surfaces. It does not apply to a 
spherical surface on which the shortest distance between 
two points is the arc of a great circle. Since the shape of 
the earth approximates a sphere, the shortest distance be¬ 
tween two points anywhere on the surface of the earth is 
never a straight line, but is a portion of the arc of a great 
circle. (See page 146.) 

Yet, for all practical purposes, even on the surface of 
the earth, the shortest distance between two points is 
given by a straight line. That is to say, in measuring 
ordinary distances with a steel tape or a yardstick, the 
principle is substantially correct. However, for distances 
beyond even a few hundred feet, allowance must be 
made for the curvature of the earth. When a steel rod 
over 600 feet in length was recently constructed in a 
large Detroit automobile factory, it was found that the 
exact measurement of its length was impossible without 
allowing for the earth’s curvature. We indicated that 
the determination of a geodesic is very difficult for com¬ 
plicated surfaces. But we can give one puzzle showing 
how deceptive this problem may be for even the simplest 
case—the flat surface. 

In a room 30 feet long, 12 feet wide, and 12 feet high, 
there is a spider in the center of one of the smaller walls, 

1 foot from the ceiling; and there is a fly in the middle 


182 Mathematics and the Imagination 

of the opposite wall, 1 foot from the floor. The spider 
has designs on the fly. What is the shortest possible 
route along which the spider may crawl to reach his 


FIG. 61.—The spider, his kind invitation to the 
fly having been rebuffed, sets out for dinner along 
the shortest possible route. What path is the geo¬ 
desic for the hungry spider? 

* 

prey? If he crawls straight down the wall, then in a 
straight line along the floor, and then straight up the 
other wall, or follows a similar route along the ceiling, 
the distance is 42 feet. Surely it is impossible to imagine a 
shorter route! However, by cutting a sheet of paper, 
which, when properly folded, will make a model of the 
room (see Fig. 61), and then by joining the points rep- 


Pastimes of Past and Present Times 183 

resenting the spider and the fly by a straight line, a geo¬ 
desic is obtained. The length of this geodesic is only 40 
feet, in other words, 2 feet shorter than the “obvious” 
route of following straight lines. 

There are several ways of cutting the sheet of paper, 
and accordingly, there are several possible routes, but 
that of 40 feet is the shortest; and remarkably enough, 
as may be seen from cut D in Fig. 61, this route requires 
the spider to pass over 5 of the 6 sides of the room. 

This problem graphically reveals the point emphasized 
throughout—our intuitive notions about space almost 
invariably lead us astray. 

* 


RELATIONSHIPS 


Ernest Legouve,^® the well known French dramatist, 
tells in his memoirs that, while taking the baths at Plom- 
bieres, he proposed a question to his fellow bathers: “Is it 
possible for two men, wholly unrelated to each other, to 
have the same sister?” “No, that’s impossible,” replied a 
notary at once. An attorney who was not quite so quick 
in giving his answer, decided after some deliberation, that 
the notary was right. Thereupon, the others q uickly agreed 
that it was impossible. “But still it is possible,” Legouve 
remarked, “and I will name two such men. One of them 
is Eugene Sue, and I am the other.” In the midst of cries 
of astonishment and demands that he explain, he called 
the bath attendant and asked for the slate on which the 
attendant was accustomed to mark down those who had 
come for their baths. On it, he wrote: 


( 


Mrs. Suc^Mr. Sue 


Eugene Sue 


means married to; | means offspring of) 

Mrs. Sauvais^Mr. Sue Mrs. Sauvais^Mr. Legouvrf 

Flore Sue Ernest Legouv^ 


184 Mathematics and the Imagination 

“Thus, you see,” he concluded, “it is quite possible for 
two men to have the same sister, without being related 
to each other.” 

Most of the puzzles treated hitherto required four 
steps for their solution: 

1. Sifting out the essential facts. 

2. Translating these facts into the appropriate sym¬ 
bols. 

3. Setting up the symbols in equations. 

4. Solving the equations. 

To solve the problems of relationship two of these steps 
must be modified. A simple diagram replaces the alge¬ 
braic equation; inferences from the diagram replace the 
algebraic solution. Without the symbols and diagrams, 
however, the problems may become extremely confusing, 

Alexander MacFarlane, a Scotch mathematician, de¬ 
veloped an “algebra of relationships” which was pub¬ 
lished in the proceedings of the Royed Society of Edin¬ 
burgh, but the problems to which he applied his calculus 
were easily solvable without it. McFarlane used the-well- 
known jingle: 

Brothers and sisters have I none, 

But this man’s father is my father’s son, 

as a guinea pig for his calculus, although the diagram¬ 
matic method gives the solution much more quickly. 

An old Indian fairy tale creates an intricate series of 
relationships which would probably prove too much even 
for MacFarlane’s algebra. A king, dethroned by his rela¬ 
tives, was forced to flee with his wife and daughter. 
During their flight they were attacked by robbers; while 
defending himself, the king was killed, although his wife 
and daughter managed to escape. Soon they came to a 


Pastimes of Past and Present Times 185 

forest in which a prince of the neighboring country and 
his son were hunting. The prince (a widower) and his 
son (an eligible bachelor) noticing the footsteps of the 
mother and daughter decided to follow them. The father 
declared that he would marry the woman with the large 
footsteps—undoubtedly the older—and the son said he 
would marry the woman with the small footsteps who 
was surely the younger. But on their return to the castle, 
the father and his son discovered that the small feet be¬ 
longed to the mother and the big feet to the daughter. 
Nevertheless, mastering their disappointment, they mar¬ 
ried as they had planned. After the marriage, the mother, 
daughter-in-law of her daughter, the daughter, mother- 
in-law of her mother, both had children—sons and 
daughters. The task of disentangling the resulting re¬ 
lationships we entrust to the reader, as well as the 
explanation of the following verse found on an old grave¬ 
stone at Alencourt, near Paris: 

Here lies the son; here lies the mother; 

Here lies the daughter; here lies the father; 

Here lies the sister; here lies the brother; 

Here lie the wife and the husband. 

Still, there are only three people here. 

♦ 

In Albrecht Durer’s famous painting, “Melancholia,’* 
there appears a device about which more has been 
written than any other form of mathematical amusement. 
The device is a magic square. 

A magic square consists of an array of integers in a 
square which, when added up by rows, diagonals, or 
columns, yield the same total. Magic squares date back 
at least to the Arabs. Great mathematicians like Euler 


186 Mathematics and the Imagination 

and Cayley found them amusing and worth studying. 
Benjamin Franklin admitted somewhat apologetically 
that he had spent some time in his youth on these 
“trifles”—time “which,” he hastened to add, “I might 
have employed more usefully.” Mathematicians have 
never pretended that magic squares were anything more 
than amusement, however much time they spent on 
them, although the continual study devoted to this 
puzzle form may incidentally have cast some light on 
relations between numbers. Their chief appeal is still 
mystical and recreational. 

There are other puzzles of considerable interest not 
discussed here because we treat them more fully in their 
proper place.^® Among these are problems connected 
with the theory of probability, map-coloring, and the 
one-sided surfaces of Mobius. 

Only one extensive group of problems remains—those 
connected with the theory of numbers. The modern the¬ 
ory of numbers, represented by a vast literature, engages 
the attention of every serious mathematicizin. It is a 
branch of study, many theorems of which, though exceed¬ 
ingly difficult to prove, can be simply stated and readily 
understood by everyone. Such theorems are therefore 
more widely known among educated laymen than the¬ 
orems of far greater importance in other branches of 
mathematics, theorems which require technical knowl¬ 
edge to be understood. Every book on mathematical 
recreations is filled with simple or ingenious, cunning or 
marvelous, easy or difficult puzzles based on the behavior 
and properties of numbers. Space permits us to mention 
only one or two of those significant theorems about num¬ 
bers which, in spite of their profundity, can be easily 
grasped. 


Pastimes of Past and Present Times 187 

Ever since Euclid proved 21 that the number of primes 
is infinite, mathematicians have been seeking for a test 
which would determine whether or not a given number 
IS a prime. But no test applicable to all numbers has 
been found. Curiously enough, there is reason to believe 
that certain mathematicians of the seventeenth century, 
who spent a great deal of time on number theory, had 
means of recognizing primes unknown to us. The Trench 
mathematician Mersenne and his much greater con¬ 
temporary, Fermat, had an uncanny way of determin¬ 
ing the values of />, for which 2^ - 1 is a prime. It has 
not yet been clearly determined how completely they 
had developed their method, or indeed, exactly what 
method they employed. Accordingly, it is still a source of 
wonder that Fermat replied without a moment’s hesi¬ 
tation to a letter which asked whether 100895598169 
was a prime, that it was the product of 898423 and 
112303, and that each of these numbers was prime. 
Without a general formula for all primes, a mathema¬ 
tician, even today, might spend years hunting for the 
correct answer. 

One of the most interesting theorems of number 
theory is Goldbach s, which states that every even num¬ 
ber is the sum of two primes. It is easy to understand; and 
there is every reason to believe that it is true, no even 
number having ever been found which is not the sum 
of two primes; yet, no one has succeeded in finding a 
proof valid for all even numbers. 

But perhaps the most famous of all such propositions, 
believed to be true, but never proved, is “Fermat’s Last 
Theorem.” In the margin of his copy of Diophantus, 
Fermat wrote: “If n is a number greater than two, there 
are no whole numbers, a, b, c such that = c^. 


188 Mathematics and the Imagination 

I have found a truly wonderful proof which this margin 
is too small to contain.” What a pity! Assuming Fermat 
actually had a proof, and his mathematical talents were 
of such a high order that it is certainly possible, he would 
have saved succeeding generations of mathematicians 
unending hours of labor if he had found room for it on 
the margin. Almost every great mathematician since 
Fermat attempted a proof, but none has ever succeeded. 

Many pairs of integers are known, the sum of whose 
squares is also a square, thus: 

32 + 42 = 5^ or, 62 -h 8^ = 102. 

But no three integers have ever been found where the 

sum of the cubes of two of them is equal to the cube of the 

third. It was Fermat’s contention that this would be true 

for all integers when the power to which they were raised 

was greater than 2. By extended calculations, it has been 

shown that Fermat’s theorem is true for values of n up to 

617. But Fermat meant it for every n greater than 2. Of 

all his great contributions to mathematics, Fermat s most 

celebrated legacy is a puzzle which three centuries of 

mathematical investigation have not solved and which 

skeptics believe Fermat, himself, never solved. 

* 

Somewhat reluctantly we must take our leave of puz 
zles. Reluctantly, because we have been able to catch 
only a glimpse of a rich and entertaining subject, an 
because puzzles in one sense, better than any other sing e 
branch of mathematics, reflect its always youthful, un 
spoiled, and inquiring spirit. When a man stops wonder 
ing and asking and playing, he is through. Puzzles are 
made of the things that the mathematician, no less t an 


Pastimes of Past and Present Times 189 

the child, plays with, and dreams and wonders about, for 

they are made of the things and circumstances of the world 
he lives in. 


FOOTNOTES 

1. Anatole France, The Crime of Sylvestre Bonnard 156. 

2. W. W. R. Ball, Mathematical Recreations and Essays, 11th ed. New 
York: Macmillan, 1939. 

W. Lietzmann, Lustiges und Merkwixrdiges von Zahlen und Formen, 
Breslau: Hirt, 1930. 

Helen Abbot Merrill, Mathematical Excursioris, Boston; Bruce 
Humphries, 1934. 

W. Ahrens, Matherhatische Unterhaltungen und Spiele, Leipzig; B. G. 
Teubner, 1921, vols. I and II. 

H. E. Dudeney, Amusements in Mathematics, London; Thomas 
Nelson, 1919. 

E. Lucas, Recreations Mathematiques, Paris; Gautier-VilJars 
1883-1894, vols. I, II, III and IV.—P. 157. 

3. Here is an example of a type of puzzle quite fashionable of late, 
which, though apparently wordy, contains no unessential facts: 


THE ARTISANS 

There are three men, John, Jack and Joe, each of whom is 
engaged in two occupations. Their occupations classify each 
of them as two of the following: chauffeur, bootlegger, musi¬ 
cian, painter, gardener, and barber. 

From the following facts find in what two occupations each 
man is engaged: 

1. The chauffeur offended the musician by laughing at his 
long hair. 

2. Both the musician and the gardener used to go fishing 
with John. 

3. The painter bought a quart of gin from the bootlegger. 

4. The chauffeur courted the painter’s sister. 

5. Jack owed the gardener S5. 

6. Joe beat both Jack and the painter at quoits.—P. 158. 

4. There are two different ways, both of which are symbolized in 

the following table.—P. 159. 


1 go Mathematics and the Imagination 


FIRST SOLUTION SECOND SOLUTION 

W = WOLF C = CABBAGE 
G = GOAT -> = CROSSING 


5. At least so says his biographer, Arago. Not only was the quality 
of Poisson’s work extremely high, but the output was enormou^ 
Besides occupying several important official posidons, he turned 
out over 300 works in a comparadvely short lifetime, (1781 

1840). “La vie, c’est le travail,” said this erstwhile shadow on the 
Poisson household, though oddly enough, a puzzle brought him 
to a life dedicated to unceasing labor.—P. 161. 

6. Fill the 5 quart jar from the 8 quart jar and pour 3 quarts from 
the 5 quart jar into the 3 quart jar. Then pour the 3 quarts 
back into the 8 quart jar. Pour the remaining 2 quarts from the 
5 quart jar into the 3 quart jar. Now fill the 5 quart jar again. 
Since there are 2 quarts in the 3 quart jar, one additional qu^t 

will fill this jar. Pour enough wine from the 5 quart jar to c 

3 quart jar. The 5 quart jar will then have 4 quarts rernaining 
in it. Now pour the 3 quarts from the 3 quart jar into quar 
jar. This, together with the 1 quart remaining in the 8 quart 
jar, will make 4 quarts.—P. 162. 

7. W. W. R. Ball, op. cit.—?. 163. v th t 

8. Other bases have been suggested. There is reason to believe a 
the Babylonians employed the base 60, and in more recerU 

the use of the base 12 has been urged rather strongly. 

9. Hall and Knight, Hightr Algebra. —P. 165. 


Pastimes of Past and Present Times 191 

10. Arnold Dresden, An Invitation to Mathematics, New York- Henrv 

Holt & Co., 1936.—P. 167. ' ^ 

11. W. Ahrens, op. cit. —P. 170. 

12. W. W. R. Ball, op. dt.—P. 170. 

13. (Making allowance for leap years.—Ed.)—P. 171. 

14. See W. Ahrens, op. cit., and Bouton, Annals oj Mathematics, series 

(1901-1902), pp. 35-39, for the mathematical proof 
of Nim.—P. 171. 

15. One-twentieth of a cubit is about one inch.—P. 173. 

16. The general rule for solution of all such problems may be found 
in P. G. Tait, Collected Scientijic Papers, 1900.—P. 174. 

17. Smith & Mikami, A History of Japanese Mathematics, p. 83.—P. 

176. 

18. Johnson & Story, American Journal of Mathematics, vol. 2 (1879).— 

P* 178. 

19. Ahrens, op. cit., volume 2.—P. 183. 

20. There are also puzzles which though very amusing and deceptive, 
present no mathemadcal idea which has not been already con¬ 
sidered—and such puzzles have, therefore, been omitted. We 
may, nevertheless, give three examples, chosen because they 
are so often solved incorrecdy: 

(a) A glass is half-filled with wine, and another glass half- 
filled with water. From the first glass remove a teaspoon¬ 
ful of wine and pour it into the water. From the mixture 
take a teaspoonful and pour it into the wine. Is the 
quandty of wine that was removed from the wine glass 
greater, or less than the quantity of water removed from 
the glass of water? To end all quarrels—they are the same. 

(b) The following puzzle troubled the delegates to a dis¬ 
tinguished gathering of puzzle experts not long ago. A 
monkey hangs on one end of a rope which passes through 
a pulley and is balanced by a weight attached to the other 
end. The monkey decides to climb the rope. What hap¬ 
pens? The astute puzzlers engaged in all sorts of futile 
conjectures and speculations, ranging from doubts as to 
whether the monkey could climb the rope, to rigorous 
“mathematical demonstrations” that he couldn’t. (We 
yield to a shameful and probably superfluous urge to point 
out the solution—the weight rises, like the monkey!) 

(c) Imagine we have a piece of string 25,000 miles long, Just 
ong enough to exactly encircle the globe at the equator. 
We take the string and fit it snugly around, over oceans, 
deserts, and jungles. Unfortunately, when we have com- 


192 


Mathematics and the Imagination 

pleted our task we find that in manufacturing the string 
there has been a slight mistake, for it is just a yard too long. 

To overcome the error, we decide to tie the ends to¬ 
gether and to distribute this 36 inches evenly over the en¬ 
tire 25,000 miles. Naturally (we imagine) this will never 
be noticed. How far do you think that the string will 
stand off from the ground at each point, merely by virtue 
of the fact that it is 36 inches too long? 

The correct answer seems incredible, for the string 
will stand 6 inches from the earth over the entire 25,000 
miles. 

To make this seem more sensible you might ask your¬ 
self: In walking around the surface of the earth, how much 
further does your head travel than your feet?—P. 186. 

21. Euclid’s proof that there is an infinite number of primes is an 
elegant and concise demonstration. If P is any prime, a prime 
greater than P can always be found. Construct /*! + !. This 
number, obviously greater than P, is not divisible by P or any 
number less than P. There are only two alternatives: (1) It is 
not divisible at all; (2) It is divisible by a prime lying between 
P and P\ -j- 1. But both of these alternatives prove the existence 
of a prime greater than P. Q.E.D.—P. 187. 

22. Ball, op. cit.—P. 187. 


Paradox Lost and Paradox Regained 


How quaint the ways of paradox — 

At common sense she gaily mocks. 

-W. S. GILBERT 


Perhaps the greatest paradox of all is that there are par¬ 
adoxes in mathematics. We are not surprised to discover 
inconsistencies in the experimental sciences, which peri¬ 
odically undergo such revolutionarychanges that although 

only a short time ago we believed ourselves descended 
from the gods, we now visit the zoo with the same friendly 
interest with which we call on distant relatives. Similarly, 
the fundamental and age-old distinction between matter 
and energy is vanishing, while relativity physics is shatter¬ 
ing our basic concepts of time and space. Indeed, the testa¬ 
ment of science is so continuously in a flux that the heresy 
of yesterday is the gospel of today and the fundamental¬ 
ism of tomorrow. Paraphrasing Hamlet—what was once 
a paradox is one no longer, but may again become one. 
Yet, because mathematics builds on the old but does not 
discard it, because it is the most conservative of the sci¬ 
ences, because its theorems are deduced from postulates 
by the methods of logic, in spite of its having undergone 
revolutionary changes we do not suspect it of being a dis¬ 
cipline capable of engendering paradoxes. 

Nevertheless, there are three distinct types of paradoxes 
which do arise in mathematics. There arc contradictory 

'93 


194 Mathematics and the Imagination 

and absurd propositions, which arise from fallacious rea¬ 
soning. There are theorems which seem strange and in¬ 
credible, but which, because they are logically unassail¬ 
able, must be accepted even though they transcend intui¬ 
tion and imagination. The third and most important class 
consists of those logical paradoxes which arise in connec¬ 
tion with the theory of aggregates, and which have re¬ 
sulted in a re-examination of the foundations of mathe¬ 
matics. These logical paradoxes have created confusion 
and consternation among logicians and mathematicians 
and have raised problems concerning the nature of 
mathematics and logic which have not yet found a satis¬ 
factory solution. 

PARADOXES-STRANGE BUT TRUE 

This section will be devoted to apparently contradic¬ 
tory and absurd propositions which are nevertheless 
true.^ Earlier, we examined the paradoxes of Zeno. 
Most of these were explained by means of infinite series 
and the transfinite mathematics of Cantor. There are 
yet others involving motion, but unlike Zeno’s puzzles, 
they do not consist of logical demonstrations that motion 


FIG. 62. 


Paradox Lost arid Paradox Regained 195 

is impossible. However, they graphically illustrate how 
false our ideas about motion may be; how easily, for 
example, one may be deceived by the path of a moving 
object. 

In Fig. 62, there are two identical coins. If we roll the 
coin at the left along half the circumference of the other, 
following the path indicated by the arrow, we might sus¬ 
pect that its final position, when it reaches the extreme 
right, should be with the head inverted and not in an up¬ 
right position. That is to say, after we revolved the coin 
through a semicircle (half of its circumference), the head 
on the face of the coin, having started from an upright 
position, should now be upside down. If, however, we 
perform the experiment, we shall see that the final posi¬ 
tion will be as illustrated in Fig. 62, Just as though the 
coin had been revolved once completely about its own 
circumference. 

The following enigma is similar. The circle in Fig. 63 
has made one complete revolution in rolling from A to 
B. The distance AB is therefore equal in length to the 


circumference of the circle. The smaller circle inside the 
larger one has also made one complete revolution in trav¬ 
ersing the distance CD. Since the distance CD is equal to 
the distance AB and each distance is apparently equal to 
the circumference of the circle which has been unrolled 
upon it, we are confronted with the evident absurdity 


*4 


196 Mathematics and the Imagination 

that the circumference of the small circle is equal to the 
circumference of the large circle. 

In order to explain these paradoxes, and several others 
of a similar nature, we must turn our attention for a 
moment to a famous curve—the cycloid. (See Fig. 64.) 


FIG. 64.—The cycloid. 


The cycloid is the path traced by a fixed point on the 
circumference of a wheel as it rolls without slipping upon 
a fixed straight line. 

In Fig. 65, as the wheel rolls on the line MN^ the points 
A and B describe a cycloid. After the wheel has made 
half a revolution, the point A \ is at ^4 3, and B\ is at B%. At 
this juncture, there is nothing to indicate that the point 


Paradox Lost and Paradox Regained 197 

A and the point B have not traveled throughout at the 
same speed, since it is evident that they have covered the 
same distance. But, if we examine the intermediary 
points A 2 and B 2 which show the respective positions of 
A and B after a quarter-turn of the wheel, it is clear that 
in the same time A has traveled a greater distance than 
B. This difference is compensated for in the second 
quarter turn in which 5, traveling from B^ to Bz, covers 
the same distance that A covered moving from Ai to A^', 
it is obvious that the distance along the curve from B 2 
to Bz is equal in length to the distance from Ai to A^. 
Hence in one-half revolution, both A and B have 
traversed exactly the same distance. 


FJG- 65.—When the rolling wheel is in the dotted 
position it has completed one-quarter of a turn, 
and A has traveled from /I, to ^ 2 , but B only from’ 
Bx to B-i. The shaded circle indicates the wheel has 
completed three-quarters of a revolution. 


This strange behavior of the cycloid explains the fact 
that when a wheel is in motion, the part furthest from 
the ground, at any instant, actually moves along hori¬ 
zontally faster than the part in contact with the ground. 
It can be shown that as the point of a wheel in contact 
with the road starts moving up, it travels more and more 
quickly, reaching its maximum horizontal speed when its 
position is furthest from the ground. 


198 Mathematics and the Imagination 

Another interesting property of the cycloid was dis¬ 
covered by Galileo. It was pointed out in the chapter on 
Pie that the area of a circle could only be expressed with 
the aid of tt, the transcendental number. Since the nu¬ 
merical value of TT can only be approximated (although as 
closely as we please, by taking as many terms of the 
infinite series as we wish), the area of a circle can also 
only be expressed as an approximation. Remarkably 
enough, however, with the aid of a cycloid, we may 
construct an area exactly equal to the area of a given circle. 
Based upon the fact that the length of a cycloid, from 
cusp to cusp, is equal to four times the length of the di¬ 
ameter of the generating circle, it may be shown that the 
area bound by the portion of the cycloid between the two 
cusps and the straight line joining the cusps, is equal to 
three times the area of the circle. From which it follows 
that the enclosed space (shaded in Fig. 66) on either side 
of the circle in the center is exactly equal to the area of the 
circle itself. 


FIG. 66. —When the rolling circle is in the indicated 
position, the shaded areas on each side are exactly 
equal to the area of the circle. 


The paradox resulting from the pseudo-proof that the 
circumference of a small circle is equal to that of a larger 
circle can be explained with the aid of another member 
of the cycloid family—the prolate cycloid (Fig. 67). 

An inner point of a wheel which rolls on a straight line 


Paradox Lost and Paradox Regained 199 

describes the prolate cycloid.'Thus, a point on the cir¬ 
cumference of a smaller circle concentric with a larger 
one will generate this curve. The small circle in Fig. 63 
makes only one complete revolution in moving from C 
to D and a point on the circumference of this circle will 
describe a prolate cycloid. However, by comparing the 
prolate cycloid with the cycloid, we observe that the 


FIG. 67.—The prolate cycloid is generated by the 
point P on the smaller circle as the larger circle rolls 
along the line MN. 


small circle would not cover the distance CD merely by 
making one revolution as the large circle docs. Part of 
the distance is covered by the circle while it is unrolling, 
but simultaneously, it is being carried forward by the 
large circle as this moves from A to B. This may be seen 
even more clearly if we regard the center of the large 
circle in Fig. 63. The center of a circle, being a math¬ 
ematical point and having no dimensions, docs not re¬ 
volve at all, but is carried the entire distance from A to B 
by the wheel. 

With regard to the problems arising from a wheel roll¬ 
ing on a straight line, we have discussed the trajectory 
(path) of a point on the circumference of the wheel and 
found this path to be a cycloid, and we have considered 
the curve traced by a point on the inside of the wheel and 
discovered the prolate cycloid. In addition, it is interest- 


200 


Mathematics and the Imagination , 

ing to mention the path "traced by a point outside of the 
circumference of a wheel, such as the outermost point of 
the flanged wheels used on railway trains. Such a point 
is not actually in contact with the rail upon which the 
wheel is revolving. The curve generated is called a curtate 
cycloid (Fig. 68) and explains the curious paradox that, 
at any instant of time, a railroad train never moves 
entirely in the direction in which the engine is pulling. 
There are always parts of the train which are traveling 
in the opposite direction!* 


FIG. 68.—The Curtate Cycloid. 

A point on the flange of a moving railway wheel 
generates this curve. The part of a railway train 
which moves backward when the train moves for¬ 
ward is the shaded portion of the wheel. 

* Other such parts would be the crosshead and connecting rod of 
the locomotive. 


201 


Paradox Lost and Paradox Regained 

Among the innovations in mathematics of the last 
quarter-century, none overshadows in importance the de¬ 
velopment of the theory of point sets and the theory of 
functions of a real variable. Based entirely upon the new 
methods of mathematical analysis, a greater rigor and 
generality in geometry was achieved than could have 
been imagined had science been developed entirely by 
intuitive means. It was found that all conventional geo¬ 
metrical ideas could be redefined with increased accuracy 
by drawing upon the theory of aggregates and the power¬ 
ful new tools of analysis. In rubber-sheet geometry', as 
we shall see, curves are defined in such a way as to elimi¬ 
nate every naive appeal to intuition and experience. A 
simple closed curve is defined as a set of points possessing 
the property that it divides the plane into exactly two 
regions: an inside and an outsidey where inside and outside 
are precisely formulated by analytical methods without 
reference to our customary notions of space. By just such 
means, figures far more complex than had ever before 
been studied were developed and investigated. Indeed, 
although analytical geometry is limited to contours 
which can be described by algebraic equations whose 
variables are the co-ordinates of the points of the con¬ 
figuration, the new analysis made possible the study of 
forms which cannot be described by any algebraic equa¬ 
tion. Some of these we shall encounter in the section on 
Pathological Curves. 

Extended studies were also undertaken of certain classes 
of points—like the points in space—and the notion of 
dimensionality was freshly re-examined. In connection 
with this study, one of the great accomplishments of 
recent years has been to assign to each configuration a 
number: 0, 1, 2, or 3, to denote its dimensionality. The 


202 


Mathematics and the Imagination 

established belief had been that this was a simple and ob¬ 
vious matter which did not require mathematical analysis 
and could be solved intuitively. Thus, a point would be 
said to have zero dimension, a line or a curve—one di¬ 
mension, a plane or a surface—two dimensions, and a 
solid—three dimensions. It must be conceded that the 
problem of determining whether an object has 0, 1,2, or 
3 dimensions does not look very formidable. However, 
one remarkable paradox which was uncovered is suffi¬ 
cient in itself to show that this is not the case at all and 
that our intuitive ideas about dimensionality, as well as 
area, are not only lacking in precision, but are often 
wholly misleading. 

The paradox appeared in trying to ascertain whether 
a number (called a measure) could be uniquely assigned 
to every figure in the plane so that the following three 
conditions would be satisfied: (1) the word “congruent’’ 
being used in the same sense in which it was learned in 
elementary geometry,^ two congruent figures were to 
have the same measure; (2) if a figure were divided into 
two parts, the sum of the measures assigned to each of 
the two parts was to be exacdy equal to the measure 
assigned to the original figure; (3) as a model for de¬ 
termining the method of assigning a measure to each 
figure in the plane, it was agreed that the measure 1 
should be assigned to the square whose side has a length 
of one unit. 

What is this concept of measure? From the foregoing, 
it would seem to follow that the measure to be assigned 
to each figure in the plane is nothing more than the area 
of that figure. In other words, the problem is to ascertain 
whether the area of every figure in the plane, regardless of 
its complexity, can be uniquely determined. It need 


Paradox Lost and Paradox Regained 203 

hardly be pointed out that this was intended as a general 
and theoretical exercise and not as the vast and ob¬ 
viously impossible undertaking of actually measuring 
every conceivable figure. The problem was to be con¬ 
sidered solved if a theoretical proof were given that every 
figure could be assigned a unique measure. But it should 
be noted that the principal aim was to keep this in¬ 
vestigation free from the traditional concepts of class¬ 
ical geometry—the notion of area understood in the old 
way was taboo, and the customary methods of determin¬ 
ing its specifically excluded; the approach was to be 
analytic (by means of point sets), rather than geometric. 
Adhering to just such restrictions, it was proved that no 
matter how complex a figure is, no matter how many 
times the boundary crosses and recrosses itself, a unique 
measure can be assigned to it. 

Then came the debacle. For the amazing fact was 
uncovered that the same problem, when extended to 
surfaces, was not only unsolvable but led to the most 
stunning paradoxes. Indeed, the very same methods 
which had been so fruitful in the investigations in the 
plane, when applied to the surface of a sphere proved 
inadequate to determine a unique measure. 

Docs this really mean that the area of the surface of 
a sphere cannot be uniquely determined? Docs not the 
conventional formula Arrr^ give correctly the area of the 
surface of a sphere? Unfortunately, we cannot undertake 
to answer these questions in detail, for to do so would 
carry us far afield and require much technical knowledge. 
We admit that the area of a spherical surface as deter¬ 
mined by the old classical methods is Airr^. But the old 
methods were lacking in generality; they were found to be 
inadequate to determine the area of complex figures; fur- 


204 Mathematics and the Imagination 

thermore, we already gave warning that the naive con¬ 
cept of area was deliberately to be omitted from the meas¬ 
uring attempt. While the advance in function theory and 
the new methods of analysis did overcome some of these 
difficulties, they also introduced new problems closely con¬ 
nected with the infinite, and as mathematicians have long 
realized, the presence of that concept is by no means an 
unmixed blessing. Though it has enabled mathematics 
to make great strides forward, these have always been in 
the shadow of uncertainty. One may continue to employ 
such formulae as 47rr^, for the very good reason that they 
work; but if one wishes to keep pace with the bold and 
restless mathematical spirit, one is faced with the com¬ 
fortless alternatives of abandoning logic to preserve the 
classical concepts, or of accepting the paradoxical results 
of the new analysis and casting horse sense to the winds. 

The conditions for assigning a measure to a surface 
are similar to the conditions for assigning a measure to 
figures in the plane: (1) The same measure shall be 
assigned to congruent surfaces; (2) The sum of the 
measures assigned to each of two component parts of a 
surface shall be equal to the measure assigned to the 
original surface; (3) If S denotes the entire surface of a 
a sphere of radius r, the measure assigned to S shall be 
47rr^. 

The German mathematician Hausdorff showed that 
this problem is insoluble, that a measure cannot uniquely 
be assigned to the portions of the surface of a sphere so 
that the above conditions will be satisfied. He showed 
that if the surface of a sphere were divided into three 
separate and distinct parts: d, 5, C, so that A is congruent 
to B and B is congruent to C, a strange paradox arises 
which is strongly reminiscent of, and indeed, related to 


Paradox Lost and Paradox Regained 205 

some of the paradoxes of transfinite arithmetic. For 
Hausdorff proved that not only is .1 congruent to C (as 
might be expected), but also that .-I is congruent to B 
4- C. What are the implications of this startling result? 

If a measure is assigned to zl, the same measure must 
must be assigned to B and to C, because A is congruent 
to By B is congruent to C and A is congruent to C. But, 
on the other hand, since A is congruent to 5 + C, the 
measure assigned to A would also have to be equal to 
the sum of the measures assigned to B and C. Obviously, 
such a relationship could only hold if the measures 
assigned to d, By and C were all equal to 0. But that is 
impossible by condition (3), according to which the sum 
of the measures assigned to the parts of the surface of a 
sphere must be equal to Airr-. How then is it possible to 
assign a measure? 

From a slightly different viewpoint, we see that if /I, 
By and C are congruent to each other and together make 
up the surface of the entire sphere, the measure of any 
one of them must be the measure of one-third of the sur¬ 
face of the entire sphere. But if A is not only congruent to 
B and C, but also to 5 + C (as Hausdorff has shown), the 
measure assigned to A and the measure assigned to B 
+ C must each be equal to half the surface of the sphere. 
Thus, whichever way we look at it, assigning measures 
to portions of the surface of a sphere involves us in a 
hopeless contradiction. 

Two distinguished Polish mathematicians, Banach 
Tarski, have extended the implications of Haus¬ 
dorff s paradoxical theorem to three-dimensional space, 
with results so astounding and unbelievable that their 
like may be found nowhere else in the whole of mathe¬ 
matics. And the conclusions, though rigorous and unim- 


2 o 6 Mathematics and the Imagination 

peachable, are almost as incredible for the mathema¬ 
tician as for the layman. 

Imagine two bodies in three-dimensional space: one 
very large, like the sun; the other very small, like a pea. 
Denote the sun by iS* and the pea by S'. Remember now 
that we are referring not to the surfaces of these two spher¬ 
ical objects, but to the entire solid spheres oj both the sun and 
the pea. The theorem of Banach and Tarski holds that the 
following operation can theoretically be carried out: 

Divide the sun iS" into a great many small parts. Each 
part is to be separate and distinct and the totality of the 
parts is to be finite in number. They may then be des¬ 
ignated by Si, S 2 y J 3 , . . . Sn, and together these small 
parts will make up the entire sphere S. Similarly, S '—the 
pea—is to be divided into an equal number of mutually 
exclusive parts— s'l, j' 2 , -^^ 3 , • • • s'm which together 
will make up the pea. Then the proposition goes on to 
say that if the sun and the pea have been cut up in a suit¬ 
able manner, so that the litde portion Si of the sun is con¬ 
gruent to the litde portion of the pea, ^2 congruent to 
s' 2 , ^3 congruent to r' 3 , up to congruent to ^'n, this 
process will exhaust not only all the little portions of the 
pea, but all the tiny portions oj the sun as well. 

In other words, the sun and the pea may both be di¬ 
vided into a finite number of disjoint parts so that every 
single part of one is congruent to a unique part of the otherj and 
so that after each small portion of the pea has been matched with 
a small portion of the run, no portion of the sun will be left over. 

* We recognize this, of course, to be a simple one-to-one cor¬ 
respondence between the elements of one set which make up the sun, 
and the elements of another set which make up the pea. The paradox 
lies in the fact that each element is matched with one which is com¬ 
pletely congruent to it (at the risk of repeating, congruent means 
identical in size and shape) and that there are enough elements in the 


Paradox Lost and Paradox Regained 207 

To express this giant bombshell in terms of a small fire¬ 
cracker. There ts a a)ajy oj dividing a sphere as large as the 
sun into separate partSy so that no two parts will have any points 
in commony and yet without compressing or distorting any part, 
the whole sun may at one time be fitted snugly into one^s vest 

pocket. Furthermore the pea may have its component 
parts so rearranged that without expansion or distortion, 
no two parts having any points in common, they will fill 
the entire universe solidly, no vacant space remaining either in 
the interior oj the pea, or in the universe. 

Surely no fairy tale, no fantasy of the Arabian nights, 
no fevered dream can match this theorem of hard, mathe¬ 
matical logic. Although the theorems of Hausdorff, 
Banach, and Tarski cannot, at the present time, be put 
to any practical use, not even by those who hope to 
learn how to pack their overflowing belongings into a 
week-end grip, they stand as a magnificent challenge to 

imagination and as a tribute to mathematical concep¬ 
tion.^ 

♦ 

As distinguished from the paradoxes just considered, 
there are those which are more properly referred to as 
mathematical fallacies. They arise in both arithmetic 
and geometry and are to be found sometimes, although 
not often, even in the higher branches of mathematics as, 
for instance, in the calculus or in infinite series. Most 
mathematical fallacies are too trivial to deserve attention; 
nevertheless, the subject is entitled to some consideration 
because, apart from its amusing aspect, it shows how a 

chain of mathematical reasoning may be entirely vitiated 
by one fallacious step. 

set making up the pea to match exactly the elements which make up 
the sun. ^ 


2o8 


Mathematics and the Imagination 


ARITHMETIC FALLACIES 

I. A proof that 1 is equal to 2 is familiar to most of 
us. Such a proof may be extended to show that any two 
numbers or expressions are equal. The error common to 
all such frauds lies in dividing by zero, an operation 
strictly forbidden. For the fundamental rules of arith¬ 
metic demand that every arithmetic process (addition, 
subtraction, multiplication, division, evolution, involu¬ 
tion) should yield a unique result. Obviously, this re¬ 
quirement is essential, for the operations of arithmetic 
would have little value, or meaning, if the results were 
ambiguous. If 1 *4" 1 vvere equal to 2 or 3; if 4 X 7 were 
equal to 28 or 82; if 7 2 were equal to 3 or 3^, mathe¬ 

matics would be the Mad Hatter of the sciences. Like 
fortunetelling or phrenology, it would be a suitable sub¬ 
ject to exploit at a boardwalk concession at Coney Island. 

Since the results of the operation of division are to be 
unique, division by 0 must be excluded, for the result of 
this operation is anything that you may desire. In gen¬ 
eral, division is so defined that if a, b, and c are three 
numbers, a b ~ only when c ^ b = a. From this 
definition, what is the result of 5 -i- 0? It cannot be any 
number from zero to infinity, for no number when mul¬ 
tiplied by 0 will be equal to 5. Thus 5 -i- 0 is meaning¬ 
less. And even 5-i-0 = 5-i-0isa meaningless expres¬ 
sion. 

Of course, fallacies resulting from division by 0 are 
rarely presented in so simple a form that they may be 
detected at a glance. The following example illustrates 
how paradoxes arise whenever we divide by an expres¬ 
sion, the value of which is 0: 

Assume A B = and assume ^4 = 3 and 5 = 2. 


Paradox Lost and Paradox Regained 209 

Multiply both sides of the equation A B = C hw 
{A + B). 

We obtain A- + 2s\B + 5- = C{A + B). 

Rearranging the terms, we have 

A^ A- AB - AC = ~AB - B- BC. 

Factoring out (A A- B ~ C), we have 

A{AA~B-C)= -B(-\- a a- B - C). 

Dividing both sides by {A A- B ~ C), that is, dividing 
by zero, we get /I = -B, or A A- B = 0, which is 
evidently absurd. 

II. In extracting square roots, it is necessary to re¬ 
member the algebraic rule that the square root of a pos¬ 
itive number is equal to both a negative and a positive 
number. Thus, the square root of 4 is —2 as well as +2 

(which may be written -\/4 = ±2), and the square root 
of 100 is equal to -f- 10 and — 10 (or, = ±10). 

Failure to observe this rule may generate the following 
contradiction: * 

(a) {n -f 1)2 = + 2« + 1 

(b) {n A- - {2n + 1 ) = ^2 

(c) Subtracting n{2n + 1) from both sides and fac¬ 
toring, we have 

(d) (n + 1)2 - (n A- l)(2tt -h 1) = - n(2n + 1) 

(e) Adding j(2n ± 1)^ to both sides of (d) yields 
(n A- - {ri A- l)(2n + 1) + i(2« + 1)^ = 

- n{2n ± 1) + i(2« + 1 ) 2 . 

This may be written: 

(f) [(« + 1) - h(2n A- \)V = [« - i(2n + 1)]^ 


21 o Mathematics and the Imagination 

Taking square roots of both sides, 

(g) n \ — ^{2n + 1) = n — §(2« + 1) 

and, therefore, 

(h) n = w + 1. 

III. The following arithmetic fallacy the reader may 
disentangle for himself: ^ 

(1) V^XVb = Va xT ..true 

(2) X \/(-l) X (-1) .true 

(3) Therefore, — = \/\; i.e., —1 = I ? 

IV. A paradox which cannot be solved by the use of 
elementary mathematics is the following: Assume that 
log (—1) = X, Then, by the law of logs, 

log i-iy = 2 X log (-1) = 2x, 

But, on the other hand, log ( — 1)^ == log (I), which is 
equal to 0. Therefore, 2* = 0. Therefore, log ( —1) = 

which is obviously not the case. The explanation lies in 
the fact that the function that represents the log of a 
negative, or complex, number is not single~valued, but is 
many'Valued. That is to say, if we were to make the usual 
functional table for the logarithm of negative and com¬ 
plex numbers, there would be an infinitude of values 
corresponding to each number.® 

V. The infinite in mathematics is always unruly unless 
it is properly treated. Instances of this were found in the 
development of the theory of aggregates and further ex¬ 
amples will be seen in the logical paradoxes. One instance 
is appropriate here. 

Just as transfinite arithmetic has its own laws differing 
from those of finite arithmetic, special rules are required 


21 I 


Paradox Lost and Paradox Regained 

for operating with infinite series. Ignorance of these rules, 
or failure to observe them brings about inconsistencies. 
For instance, consider the series equivalent to the natural 
logarithm of 2: 

Log 2 = 1 - 1 + I - i + i _ 1 . , . 

If we rearrange these terms as we would be prompted to 
do in finite arithmetic, we obtain: 

Log2 = (1 + i + i + i...) _ (1 q-i + i + .| . . 
Thus, 

Log2 = {(1 +1 + 1 + 1...)+ (1 + 1+ 1 + 1)! 

— 2(^ + ? + i + I •..) 

~ {^+i + T + i + i + -- -! 

= 0 

Therefore, log 2 = 0. 

On the other hand, 

log 2 = 1 - i + i - 1 -f i - i . . . = 0.69315, 

an answer that can be obtained from any logarithmic 
table. 

Rearranging the terms in a slightly different way: 
log 2 = 1 + 1 - 1 + 1 + -^ _ 1 + J + .J.. . _ I . . . 

— X 0.69315 or, in other words, 
log 2 = f X log 2. 

A famous series which had troubled Liebniz is the 
beguilingly simple: 4-l-l + l- i + i_i-f-i ... 
By pairing the terms differently, a variety of results is 
obtained; for example: (1 — 1 ) -}- (i — i) + (i — i) -j- 
• • * — 0, butl (1 — l)-f'(l — 1) . , . = 1 

GEOMETRIC FALLACIES 

Optical illusions concerning geometric figures arroiint 
for many deceptions. We confine our attention to fal- 


212 


Mathematics and the Imagination 

lacies which do not arise from physiological limitations,’^ 
but from errors in mathematical argument. A well-known 
geometric “proof’ is that every triangle is isosceles. It 
assumes that the line bisecting an angle of the triangle 
and the line which is the perpendicular bisector of the 
side opposite this angle intersect at a point inside the tri¬ 
angle. 

The following is a similarly fallacious proof, namely, 
that a right angle is equal to an angle greater than a right 
angle.® 


FIG. 69. 

In Fig. 69, ABCD is a rectangle. If H is the midpoint 
of CB, through //draw a line at right angles to CBy which 
will bisect DA at J and be perpendicular to it. From A 
draw the line AE outside of the rectangle and equal to 
AB and DC. Connect C and E, and let K be the midpoint 
of this line. Through K construct a perpendicular to CE. 
CB and CE not being parallel, the lines through H and K 
will meet at a point O. Join OAy OE, OB, OD and OC. It 
will be made clear that the triangle ODC and OAE are 
equal in all respects. Since KO is the perpendicular bi¬ 
sector of CE and thus any point on KO is equidistant 
from C and E^ OC is equal to OE. Similarly, since HO is 
the perpendicular bisector of CB and DAy OD equals OA. 


Paradox Lost and Paradox Regained 213 

As AE was constructed to equal DC, the three sides of the 
triangle ODC are equal respectively to the three sides of 
the triangle OAE. Hence, the two triangles are equal, and 
therefore, the angle ODC is equal to the angle OAE. But 
angle ODA is equal to angle OAD, because side AO is 
equal to side OD in the triangle OAD and the base angles 
of the isosceles triangle are equal. Therefore, the angle 
JDC, which is equal to the difference of ODC and ODJ, 
equals JAE, which is the difference between OAE and 
OAJ. But the angle JDC is a right angle, whereas the 
angle JAE is greater than a right angle, and hence the 
result is contradictory. Can you find the flaw? Hint: Try 
drawing the figure exactly. 

LOGICAL PARADOXES 

Like folk tales and legends, the logical paradoxes had 
their forerunners in ancient times. Having occupied 
themselves with philosophy and with the foundations 
of logic, the Greeks formulated some of the logical co¬ 
nundrums which, in recent times, have returned to 
plague mathematicians and philosophers. The Sophists 
made a specialty of posers to bewilder and confuse their 
opponents in debate, but most of them rested on sloppy 
thinking and dialectical tricks. Aristotle demolished them 
when he laid down the foundations of classical logic—a 
science which has outworn and outlasted all the i)hil- 
osophical systems of antiquity, and which, for the most 
part, is perfectly valid today. 

But there were troublesome riddles that stubbornly 
resisted unraveling.® Most of them are caused by wlut is 
known as “the vicious circle fallacy,” whicli is “due to 
neglecting the fundamental principle that what invoi\es 


214 Mathematics and the Imagination 

the whole of a given totality cannot itself be a member 
of the totality.” Simple instances of this are those pon¬ 
tifical phrases, familiar to everyone, which seem to have 
a great deal of meaning, but actually have none, such as 
“never say never,” or “every rule has exceptions,” or, 
“every generality is false.” We shall consider a few of the 
more advanced logical paradoxes involving the same 
basic fallacy, and then discuss their importance from 
the mathematician’s point of view. 

(A) Poaching on the hunting preserves of a powerful 
prince was punishable by death, but the prince further 
decreed that anyone caught poaching was to be given 
the privilege of deciding whether he should be hanged 
or beheaded. The culprit was permitted to make a state¬ 
ment—if it were false, he was to be hanged; if it were 
true, he was to be beheaded. One logical rogue availed 
himself of this dubious prerogative—to be hanged if he 
didn’t and to be beheaded if he did—by stating: “I 
shall be hanged.” Here was a dilemma not anticipated. 
For, as the poacher put it, “If you now hang me, you 
break the laws made by the prince, for my statement is 
true, and I ought to be beheaded; but if you behead me, 
you are also breaking the laws, for then what I said was 
false and I should, therefore, be hanged.” As in Frank 
Stockton’s story of the lady and the tiger, the ending is 
up to you. However, the poacher probably fared no worse 
at the hands of the executioner than he would have at the 
hands of a philosopher, for until this century philosophers 
had little time to waste on such childish riddles—es¬ 
pecially those they could not solve. 

(B) The village barber shaves everyone in the village 
who does not shave himself. But this principle soon in¬ 
volves him in a dialectical plight analogous to that of 


Paradox Lost and Paradox Regained 215 

the executioner. Shall he shave himself? If he does, then 
he is shaving someone who shaves himself and breaks 
his own rule. If he does not, besides remaining unshaven, 
he also breaks his rule by failing to shave a person in the 
village who does not shave himself. 

(C) Consider the fact that every integer may be ex¬ 
pressed in the English language without the use of sym¬ 
bols. Thus, (a) 1400 may be written as one thousand, 
four hundred, or (b) 1769823 as one million, seven hun¬ 
dred and sixty-nine thousand, eight hundred and twenty- 
three. It is evident that certain numbers require more 
syllables than others; in general, the larger the integer, 
the more syllables needed to express it. Thus, (a) requires 
7 syllables, and (b) 21. Now, it may be established that 
certain numbers will require 19 syllables or less, while 
others will require more than 19 syllables. Furthermore, 
it is not difficult to show that among those integers re¬ 
quiring exactly 19 syllables to be expressed in the English 
language, there must be a smallest one. Now, “it is easy 
to see that “77^^ least integer not nameahle in jewer than 
nineteen syllables"^ is a phrase which must denote the spe¬ 
cific number, 111777. But the italicized expression above 
is itself an unambiguous means of denoting the smallest 
integer expressible in nineteen syllables in the English 
language. Yet, the italicized statement has only eighteen 
syllables! Thus, we have a contradiction, for the least 
integer expressible in nineteen syllables can be expressed 
in eighteen syllables. 

(D) The simplest form of the logical paradox which 
arises from the indiscriminate use of the word all may be 
seen in Fig. 70. 

What is to be said about the statement numbered 3? 1 
and 2 are false, but 3 is both a wolf dressed like a sheep 


216 Mathematics and the Imagination 

and a sheep dressed like a wolf. It is neither the one thing 
nor the other: It is neither false nor true. 

An elaboration appears in the famous paradox of Rus¬ 
sell about the class of all classes not members of them- 


Fic. 70. 


selves. The thread of the argument is somewhat elusive 
and will repay careful attention: 

(E) Using the word class in the customary sense, we 
can say that there are classes made up of tables, books, 
peoples, numbers, functions, ideas, etc. The class, for 
instance, of all the Presidents of the United States has for 
its members every person, living or dead, who was ever 
President of the United States. Everything in the world 
other than a person uho was or is a President of the 
United States, including the concept of the class itself is not 
a member of this class. This then, is an example of a class 
which is not a member of itself. Likewise, the class of all 
members of the Gestapo, or German secret police, which 
contains some, but not all, of the scoundrels in Germany; 
or the class of all geometric figures in a plane bounded by 
straight lines; or the class of all integers from one to 
four thousand inclusive, have for members, the things 


Paradox Lost and Paradox Regained 21 7 

described, but the classes are not members of themselves. 

Now, if we consider a class as a concept, then the class 
of all concepts in the world is itself a concept, and thus is 
a class which is a member of itself. Again, the class of all 
ideas brought to the attention of the reader in this book 
is a class which contains itself as a member, since in men¬ 
tioning this class, it is an idea which we bring to the 
attention of the reader. Bearing this distinction in mind, 
we may divide all classes into two types: Those which 
are members of themselves and those which are not mem¬ 
bers of themselves. Indeed, we may form a class which is 
composed of all those classes which are not members of them¬ 
selves (note the dangerous use of the word “all”). The ques¬ 
tion is presented: Is this class (composed of those classes 
which are not members of themselves) a member of itself, 
or not? Either an affirmative or a negative answer in¬ 
volves us in a hopeless contradiction. If the class in ques¬ 
tion is a member of itself, it ought not be by definition, for 
it should contain only those classes which are not mem¬ 
bers of themselves. But if it is not a member of itself, it 
ought to be a member of itself, for the same reason. 

It cannot be too strongly emphasized that the logical 
paradoxes are not idle or foolish tricks. They were not 
included in this volume to make the reader laugh, unless 
it be at the limitations of logic. The paradoxes arc like 
the fables of La Fontaine which were dressed up to look 
like innocent stories about fox and grapes, pebbles and 
frogs. For just as all ethical and moral concepts were 
skillfully woven into their fabric, so all of logic and math¬ 
ematics, of philosophy and speculative thought, is inter¬ 
woven with the fate of these little jokes. 

Modern mathematics, in attempting to avoid the 
paradoxes of the theory of aggregates, was squarel>' faced 


218 Mathematics and the Imagination 

with the alternatives of adopting annihilating skepticism 
in regard to all mathematical reasoning, or of reconsider¬ 
ing and reconstructing the foundations of mathematics 
as well as logic. It should be clear that if paradoxes can 
arise from apparently legitimate reasoning about the 
theory of aggregates, they may arise anywhere in mathe¬ 
matics. Thus, even if mathematics could be reduced to 
logic, as Frege and Russell had hoped, what purpose 
would be served if logic itself were insecure? In proposing 
their “Theory of Types” Whitehead and Russell, in the 
Principia Mathematical succeeded in avoiding the contra¬ 
dictions by a formal device. Propositions which were 
grammatically correct but contradictory, were branded as 
meaningless. Furthermore, a principle was formulated 
which specifically states what form a proposition must 
take to be meaningful; but this solved only half the diffi¬ 
culty, for although the contradictions could be recognized, 
the arguments leading to the contradictions could not be 
invalidated without affecting certain accepted portions of 
mathematics. To overcome this difficulty, Whitehead and 
Russell postulated the axiom of reducibility which, however, 
is too technical to be considered here. But the fact re¬ 
mains that the axiom is not acceptable to the great ma¬ 
jority of mathematicians and that the logical paradoxes, 
having divided mathematicians into factions unalterably 
opposed to each other, have still to be disposed of.^^ 

♦ 

It has been emphasized throughout that the mathe¬ 
matician strives always to put his theorems in the most 
general form. In this respect, the aims of the mathemati¬ 
cian and the logician are identical—to formulate prop- 
sitions and theorems of the form: if A is true, B is true, 
where A and B embrace much more than merely cab- 


Paradox Lost and Paradox Regained 219 

bages and kings. But if this is a high aim, it is also dan¬ 
gerous, in the same way that the concept of the infinite is 
dangerous. When the mathematician says that such and 
such a proposition is true of one thing, it may be inter¬ 
esting, and it is surely safe. But when he tries to extend 
his proposition to everything^ though it is much more inter¬ 
esting, it is also much more dangerous. In the transition 
from one to a//, from the specific to the general, mathe¬ 
matics has made its greatest progress, and suffered its 
most serious setbacks, of which the logical paradoxes con¬ 
stitute the most important part. For, if mathematics is to 

advance securely and confidently it must first set its affairs 
in order at home. 


FOOTNOTES 

1. Strictly speaking, mathematical propositions are neither true nor 

false; they are merely implied by the axioms and postulates which 

vve assume. If we accept these premises and employ legitimate 

logical arguments, we obtain legitimate propositions. The 

postulates are not characterized by being true or false; we simply 

ag^ee to abide by them. But we have used the word true without 

any of its philosophical implications to refer unambiguously to 

propositions logically deduced from commonly accepted axioms. 
—P. 194. ^ t' 

2. Two point sets (configurations) are called congruent if, to every 
pair of points /*, of one set, there uniquely corresponds a pair of 
points P i OJ o{ the other set, such that the distance between 
P' and QJ equals the distance between P and (^.—P, 202. 

3. In the version given of the theorems of Hausdorff, Banach, and 
Tarski, we have made liberal use of the lucid explanation given 
by Karl Menger in his lecture: “Is the Squaring of the Circle 
Solvable? in Alte Probleme—Neue Uisungen, Vienna: Dcuticke, 
1934.—P. 207. 

4. Lietzmann, Lustiges und Merkwurdiges von ^ahlen und Formen, 
Breslau: Ferd. Hirt, 1930.—P. 209. 

5. Ball, op. cil.~P. 210. 

6. Weismann, Einjuhrung in das matfumatisch Denken, Vienna, 1937. 

—P. 210. 


220 


Mathematics and the Imagination 


7, The following optical illusions, while not properly part of a book 
on mathematics, may be of some interest—at least to the imagina- 


FIG. 71. —Are the three hori¬ 
zontal lines parallel? 


FIG. 72.—T 
le black 


se large 


Paradox Lost and Paradox Regained 


22 I 


FIG. 74.—Which of the two pciuils is longer? 
Measure them and find out. 


FIG. 75.—What do you see? Now look atjaiii. 


8. Ball, 0 /). aV.—p. 212 . 

9. for instance, the riddle of the r.piincnitl<-s concet nint' the ( tci.in 
who says that all Oretnns are liars (Cdiapiei II i I’ 21 v 

10. Ramsay, frank Plumpttjn. Articles t>ti ‘■M.idtcni.itu " and 

Logic,” Encyclopedia liriiannica, 1.5th edititjii.- P. 214 

11. This expression may, perhaps, be taken in the sense in whuh 
Laplace employed it. When he wrote his momnnental Mi,'ifti'j'ic 
Celfite, he made abundant use of the expr<'svi(>ii. ‘ It js <\ivv nj 
see often prefixing it to a mathematical formula whitli li<‘ had 
arrived at only after months (jf lafjor. 1 he result \%as tiiat 
scientists who read his work almost invariably recogni/ed the 

expression as a danger signal that theie was very roueh eoing 
ahead.—P. 215. 

12. As was pointed out in the cliapter f)n the googol, there are tlie 
ftillowers of Russ<'ll who are satisfi<-fl with ihf ilieor'.' (>f t\p<-s 
and the axiom of rrducibility; there are th<- Inimtioiiists, led h\ 
Brouwer and Weyl, who reject the axiom and whose skeptic i.sin 


222 Mathematics and the Imagination 

about the infinite in mathematics has carried them to the point 

where they would reject large portions of modem mathematics 

as meaningless, because they are interwoven with the infinite; 

and there are the Formalists, led by Hilbert, who, while opposed 

to the beliefs of the Intuitionists, differ considerably from 

Russell and the Logistic school. It is Hilbert who considers 

mathematics a meaningless game, comparable to chess, and he 

has created a subject of metamathematics which has for its 

program the discussion of this meaningless game and its axioms. 

—P. 218 . 


Chance and Chanceability 


There once was a brainy baboon 
Who always breathed down a bassoon^ 

For he saidy “// appears 
That in billions ojyears 
I shall certainly hit on a tune." 

-SIR ARTHUR EDDINGTON 


Holmes had been seated for some hours in silence with his 
long, thin back curved over a chemical vessel in which he 
was brewing a particularly malodorous product. His head 
was sunk upon his breast, and he looked from my point of view 
like a strange, lank bird, \vith dull gray plumage and a black 
topknot. 

“So Watson,*’ said he, suddenly, “you do not propose to 
invest in South African securities?” 

I gave a start of astonishment. Accustomed as I was to 
Holmes* curious faculties, this sudden intrusion into my most 
intimate thoughts was utterly inexplicable. 

“How on earth do you know that?” I asked. 

He wheeled round upon his stool with a steaming test tube 
in his hand, and a gleam of amusement in his deep-set eyes. 

“Now, Watson, confess yourself utterly taken aback,” said 
he. 

“I am.” 

“I ought to make you a sign a paper to that effect.” 

“Why?” 

“Because in five minutes you will say that it is all so absurdly 
simple.” 

“I am sure that I shall say nothing of the kind.” 

“You see, my dear Watson”—he propped his test tube in 

223 


224 Mathematics and the Imagination 

the rack, and began to lecture with the air of a professor 
addressing his class—“It’s not really difficult to construct a 
series of inferences, each dependent upon its predecessor and 
each simple in itself. If, after doing so, one simply knocks out 
all the central inferences and presents one’s audience with the 
starting point and the conclusion, one may produce a startling, 
though possibly a meretricious, effect. Now, it was not really 
difficult, by an inspection of the groove between your left 
forefinger and thumb, to feel sure that you did not propose to 
invest your small capital in the gold fields.” 

“I see no connection.” 

“Very likely not; but I can quickly show you a close connec¬ 
tion. Here are the missing links of the very simple chain. 
1. You had chalk between your left finger and thumb when 
you returned from the club last night. 2. You put chalk 
there when you play billiards, to steady the cue. 3. You never 
play billiards except with Thurston. 4. You told me, four 
weeks ago, that Thurston had an option on some South African 
property which would expire in a month, and which he de¬ 
sired you to share with him. 5. Your check book is locked in 
my drawer and you have not asked for the key. 6. You do 
not propose to invest your money in this manner.” 

“How absurdly simple!” I cried. 

“Quite so!” said he, a little nettled. “Every problem becomes 
very childish when once it is explained to you. . . . ” ^ 

This excerpt from the adventures of Mr. Sherlock 
Holmes, distinguished consulting detective, is an excel¬ 
lent caricature of reasoning by probable inference. Such a 
method of reasoning, while it resembles the formal pro¬ 
cedure of the syllogism, is more loose-jointed and less con¬ 
fined to an exact framework. Accordingly, it is better 
suited to daily thinking. 

Reasoning of the type: * 

* Cohen and Nagel, op. cit. 


Chance and Chanceability 225 

A. No fossil can be crossed in love. 

An oyster can be crossed in love. 

Therefore, oysters are not fossils. 

B. No ducks waltz. 

No officers ever decline to waltz. 

All my poultry are ducks. 

Therefore, my poultry are not officers. 

carries with it great compulsion. It is clear, exact and pre¬ 
cise, securing for our thoughts the maximum of formal 
validity. Just as in mathematics, certain fundamental as¬ 
sumptions are made and we deduce conclusions from 
them. But most of our thinking is non-mathematical, most 
of our beliefs are not certain, only probable. As Locke 
once wrote, “In the greatest part of our concernment God 
has afforded us only the twilight, as I may so say, of Prob¬ 
ability, suitable, I presume, to that state of Mediocrity 
and Probationership He has been pleased to place us in 
here.” 

It is then the relation of probability, not certainty, that 
obtains between most of our premises and conclusions. 
We are certain that a coin will fall after being tossed. We 
are equally certain that a black ball cannot be drawn 
from an urn containing only white ones. But most of our 
beliefs fall short of certainty, though they may range from 
very weak to very strong. Thus, we arc nearly certain 
that an ordinary penny will not fall heads 100 times in 
succession. Or we may faintly believe that we will win 
the grand prize in the next sweepstakes. 

Perhaps it is possible to explain this attitude. Some 
things in the world happen in conformity with natural 
laws, which (unless we believe in miracles) operate in¬ 
exorably. Thus, because of gravitation, pennies when 
tossed in the air will fall. The sun will rise tomorrow 


226 Mathematics and the Imagination 

because the planets follow regular courses. All men are 
mortal because death is a biological necessity—and so on. 

But about most of the phenomena which surround us 
we know very little. We know neither the laws they obey, 
nor indeed, whether they obey any laws. One given to 
pointing morals about man’s limitations would not have 
to go beyond trivial instances for startling confirmation. 
We are able to predict the motions of planets millions of 
miles off in space, but no one can predict the outcome of 
tossing a penny or throwing a pair of dice. Events in 
this category, and countless others, we ascribe to chance. 
But chance is merely a euphemism for ignorance. To 
say an event is determined by chance is to say we do not 
know how it is determined. 

Nevertheless, even within the realm of chance we 
sense a certain regularity, a certain symmetry—an order 
within disorder—and so even about events which we as¬ 
cribe to chance we form various degrees of rational belief. 
The theory of probability considers what are paradoxi¬ 
cally called “the laws of chance.” Part of its critical anal¬ 
ysis is an attempt to formulate rules about when and how 
mathematics may be employed to measure the relation of 
probability. However, the intrinsic meaning of probabil¬ 
ity must be made clear before it is possible to turn to a 
consideration of its rules. 

* 

Though most of our judgments are based upon prob¬ 
ability rather than certainty, careful thought is rarely 
given to the mechanics of this method of reasoning. In 
the laboratory, in business, as jurors, or at the bridge 
table, judgments are formed by probable inference. Few 
have the powers of a Sherlock Holmes, or can point to 
such successful deductions. Nevertheless, in almost all 


Chance and Chanceahility 227 

out daily thinking, everyone is called upon to play the 
part of amateur detective, logician, and mathematician. 

When it is cloudy and warm, wc say “It will probably 
ram.” The meteorologist may require better evidence be¬ 
fore venturing a prediction. He will want to know about 
barometric pressure, isobars, and tables of precipitation. 
But the average man makes his prediction with much less 
to go on. Money quickly, abundantly, and mysteriously 
earned during prohibition (it was judged, without con¬ 
sulting Bradstreet’s) was probably the fruit of bootlegging. 
And the man who gets a few kicks under the bridge table 
infers that he \s probably playing the wrong suit, whether 
he is a businessman or a scientist. 

And so do we reason about matters ranging from the 
most trivial to the most important, making frequent use 
of words and expressions such as: “probable,” “the prob¬ 
ability is,” or “the chances are” without, however, hav- 
irig a precise idea of what is meant by probability. Yet, 
this is not for want of definitions. It is true that practical 
scientists have generally left the job of defining and in¬ 
terpreting probability to the philosophers, mindful, per¬ 
haps, of the Gallic aphorism that science is continually 
making progress because it is never certain of its results. 
But while scientists have been satisfied to enlarge upon 
the uses of mathematical probability and to perfect its 
methods, philosophers and mathematicians have repeat¬ 
edly attempted to define it. 

Out of many conflicting opinions and theories three 
principal interpretations have crystallized. 

* 

The subjective view of probability^ though now somewhat 
outmoded, at one time (particularly during the last 
century) held a very respectable position. One of its 


228 Mathematics and the Imagination 

chief adherents and expositors was Augustus De Morgan, 
the celebrated logician and mathematician. He thought 
that probability referred to a state of mind^ to the degree of 
certainty or of uncertainty which characterizes' our 
beliefs. This is not entirely an erroneous view; the prin¬ 
cipal difficulties which it entails, as we shall see, arise 
when we attempt to justify a calculus of probability upon 
such foundations. 

A proposition is either true or false, ^ but our knowledge 
is for the most part so limited as to make it impossible to 
be rationally certain of either its truth or falsity. To form 
a rational beliefs we must have some pertinent knowledge. 
Occasionally, such knowledge may be sufficient to jus¬ 
tify our certainty that the proposition is true or false. Thus 
we are certain that Socrates was not an American citizen; 
and we are equally certain that Hitler should have re¬ 
mained a house painter. On the other hand, between the 
extremes of certainty there is a rainbow of shadings of 
belief corresponding to the degree of our knowledge. 

In a sense, it is undoubtedly true that our rational 
beliefs are subjective. Still, if we are convinced of the 
objective truth or falsity of all propositions, we cannot, 
if we wish to be rational, permit ourselves to be guided 
by mere intensity of belief. As a matter of principle, faulty 
conclusions based on limited knowledge and correct rea¬ 
soning, are infinitely preferable to correct results obtained 
by faulty reasoning. It is only thus that we faintly ap¬ 
proach the life of reason. 

Moreover, we feel that if the relation of probability is 
to be treated mathematically, it must furnish us with 
better material for measurement than mere strength of 
belief. In most instances a numerical magnitude cannot 
be assigned to the relation of probability, yet it can only 


229 


Chance and Chanceability 

be considered by the mathematician when it is measur¬ 
able and countable. If probability is to serve in describ¬ 
ing certain aspects of the world in terms of fractions, it 
must be expressible as a number. When a thing cannot 
happen, its probability is 0; if it is certain to happen its 
probability is 1. Every probability between these extremes 
is expressible by a fraction between zero and one. But to 
form these fractions entails measurement and counting, 
and how is the mathematician to measure “intensity of 
belief’? At best this is a problem for the psychologist. 
Even if an instrument could be devised to measure in¬ 
tensity of belief, its value would be little more than that 
of the lie detector, that gem of jurisprudence. People dif¬ 
fer widely in their beliefs based upon the same set of facts. 
What is perfectly evident to one man is thoroughly un¬ 
convincing to another; and our beliefs often vaguely con¬ 
ceived and loosely drawn are too interwoven with our 
emotions and our prejudices to justify considering one 
without the other. 

One of the difficulties arising out of the subjective 
view of probability results from the principle of insufficient 
reason. This principle, the logical basis upon which the 
calculus of probability must rest according to the sub¬ 
jective view, holds that if we are wholly ignorant of the 
different ways an event can occur and therefore have no reason¬ 
able ground for preference, it is as likely to occur one way as an¬ 
other. Since first enunciated by James Bernoulli, this 
principle has been exhaustively analyzed by mathe¬ 
maticians. As the principle rests on ignorance, it would 
seem to follow that the calculus of probability was most 
effective when used by those who had an “equally 
balanced ignorance.” Howev'cr well men approximate to 
this ideal, philosophers and mathematicians hold them- 


230 Mathematics and the Imagination 

selves in higher esteem, and so the principle has fallen on 
lean days. 

Nevertheless, it contains an element of truth, and no 
consistent calculus of probability can be developed with¬ 
out in some measure being dependent on it. Mainly, it has 
value as a negative criterion, in the sense that it cannot be 
said that two events are equally probable if there is ground 
for preferring one to the other. 

When the principle of insufficient reason is used with¬ 
out great caution, it gives rise to contradictions. Two 
examples: Take the case of an ape, who is given a num¬ 
ber of cards, each with an English word written upon it. 
Is it equally probable that any way he arranges the cards 
will, or will not, produce a meaningful English sentence? 
By the principle of insufficient reason this would seem to 
follow, although it is evidently absurd. Or, having no 
evidence relevant to whether Mars is inhabited or not, we 
could conclude that the probability is J that it is ex¬ 
clusively inhabited by Nazis, and we could just as well 
conclude that the probability of each of the propositions, 
“Mars is exclusively inhabited by jackasses” and “Mars 
is exclusively inhabited by termites,” is also §. But this 
confronts us with the impossible case of three exclusive 

alternatives all as likely as not.® 

* 

A much more workable and widely held theory which 
avoids some of these difficulties is the relative frequency^ or 
statistical interpretation. In a large measure, this view is 
responsible for the advance in applying probability, not 
only to physics and astronomy, but also to biology, to 
the social sciences, and to business. The statistical in¬ 
terpretation comes close to the view expressed by Aris¬ 
totle that the probable is that which usually happens. 


Chance and Chanceability 231 

Probability is considered to be the relative frequency with which 
an event occurs in a certain class of events. Thus, the probabil¬ 
ity of an event is expressed as a definite mathematical 
ratio which is hypothetically assigned. The hypothesis 
may be verified either rationally; by showing, for ex¬ 
ample, from our knowledge of mechanical causes, that a 
penny, or a pair of dice must fall in a certain way; or 
experimentally, by showing that the penny, or the pair of 
dice, do^ in fact, fall in that way. 

Suppose a penny is tossed in a random manner. Hav¬ 
ing no special information, there is no reason to predict 
how the coin will fall, either head or tail. If it is tossed a 
great many times and the ratio of heads to tails recorded, 
let us assume the following frequencies arc obtained: 


TOSSES RESULTS 

15. 6 heads; 9 tails 

20. 9 heads; 11 tails 

30.16 heads; 14 tails 

40.21 heads; 19 tails 

80.41 heads; 39 tails 

150.74 heads; 76 tails 


We notice that the ratio of heads to the total number of 
tosses, as these increase, approaches more and more 
closely to the fraction This represents the relative 
frequency of the class of heads in the larger class of tosses. 
We then advance to a general prediction from a large 
number of particular instances, and assume the future 
will be consistent with the past. 

However, consider for the moment: What justification 
is there for such a step? Having performed our experi¬ 
ment and determined the relative frequency, we now 
say that the probability of getting a head is Evidently, 
that statement is a hypothesis. Further experiments may 
serve to strengthen our belief in that hypothesis or cause 


232 Mathematics and the Imagination 

us to either modify or abandon it. The assumption (based 
on our experiment) is that in a great number of cases, 
heads will appear as often as tails. If the results do not 
corroborate the hypothesis, we conclude that the coin is 
perhaps heavier on one side than on the other. But it is 
important to remember that since the proof is not logical, 
but only experimental, it is never complete, it is subject 
always to further experiments. A logical proof is only pos¬ 
sible if every cause that affects an event is known. Ob¬ 
viously, such an occasion cannot arise outside of mathe¬ 
matics itself. Thus the verification of an hypothesis by 
experiment can only show that in actual practise, the rela¬ 
tive frequency approaches the predicted probability—that 
our assumptions are borne out by experience. 

It is appropriate to point out how the logical or de¬ 
ductive method of proof differs from the experimental 
one. “The process of induction, which is basic in all 
experimental sciences, is forever banned from rigorous 
mathematics . . In order to prove a proposition in 
mathematics, even a vast number of instances of its 
validity would not be sufficient, whereas one exception 
will suffice to disprove it. The propositions of mathe¬ 
matics are true only if they lead to no contradictions. But 

• • • 

outside of mathematics, in all other human activities, 
such a restriction would have a paralyzing effect. Scien¬ 
tific procedure rests on the same convenient rule of thumb 
as that which guides us in practical affairs: A hypothesis 
is valuable if it leads to correct results more often than 
not; experimental verifications are quite final—until the 

next day’s experiments upset them. 

+ 

“The Adventure of the Dancing Men,” from which was 
selected the incident at the beginning of this chapter, 


Chance and Chanceability 233 

may serve again to illustrate how the statistical method 
serves probable inference. 

Holmes is confronted with a cryptogram composed of 
several messages; (see Fig. 76): 


The solution of most cryptograms depends to a large 
extent upon certain statistical knowledge as well as upon 
shrewd inferences. Holmes derived his method of solu¬ 
tion from a method already referred to by Edgar Allan 
Poe in The Gold Bug. 

Having once recognized, however, that the symbols stood 
for letters, and having applied the rules which guide us in all 
forms of secret writings, the solution was easy enough. The 
first message submitted to me was so short that it was impossible 
for me to say with confidence, that the symbol X stood for 
E. As you are aware, E is the most common letter in the Englisli 
alphabet, and it predominates to so marked an extent that 


234 Mathematics and the Imagination 

even in a short sentence one would expect to find it most 
often. Out of fifteen symbols in the first message, four were 
the same, so it was reasonable to set this down as E. It is true 
that in some cases the figure was bearing a flag and in some 
cases not, but it was probable, from the ways in which the 
flags were distributed, that they were used to break the 
sentence up into words. I accepted this as a hypothesis, and 
noted that E was represented by ^ . 

But now came the real difficulty of the inquiry. The order 
of the English letters after E is by no means well marked, and 
any preponderance which may be shown in an average of a 
printed sheet may be reversed in a single short sentence. 
Speaking roughly, T,A,0,I,N,S,H,R,D, and L are the nu¬ 
merical order in which letters occur, but T,A,0, and I are 
nearly abreast of each other, and it would be an endless task 
to try each combination until a meaning was arrived at. I 
therefore waited for fresh material. In my second interview 
with Mr. Hilton Cubitt he was able to give me two other 
short sentences and one message, which appeared—since there 
was no flag—to be a single word. Here are the symbols. 
Now, in the single word I have already got the two E’s second 
and fourth in a word of five letters. It might be “sever” or 
“lever,” or “never.” There can be no question that the latter 
as a reply to an appeal is far the most probable, and the 
circumstances pointed to its being a reply written by the lady. 
Accepting it as correct, we are now able to say that the symbols 
^ "p stand respectively for N,V, and R. 

Even now I was in considerable difficulty, but a happy 
thought put me in possession of several other letters. It occurred 
to me that if these appeals came, as I expected, from someone 
who had been intimate with the lady in her early life, a 
combination which contained two E’s with three letters be¬ 
tween might very well stand for the name “ELSIE.” On 
examination I found that such a combination formed the 
termination of the message which was three times repeated. 

It was certainly some appeal to “Elsie.” In this way I had 


Chance and Chanceability 235 

got my L,S, and I. But what appeal could it be? There were 
only four letters in the word which preceded ‘ Elsie,” and it 
ended in E. Surely the word must be “COME.” I tried all 
other four letters ending in E, but could find none to fit the 
case. So now I was in possession of C,0, and M, and I was 
in a position to attack the first message once more, dividing 
it into words and putting dots for each symbol which was 
still unknown. So treated, it worked out in this fashion: 

.M .ERE ..E SL.NE. 

Now the first letter can only be A which is a most useful 
discovery, since it occurs no fewer than three times in this 
short sentence, and the H is also apparent in the second word. 
Now it becomes: 

AM HERE A.E SLANE. 

Or filling in the obvious vacancies in the name: 

AM HERE ABE SLANEY 

♦ 

In spite of the brilliant successes achieved by the 
statistical method, it is open to serious objections. While 
some of the difficulties can be remedied without greatly 
imparing its usefulness, others are not so easily disposed 
of. 

The concept of the limit, which plays such an impor¬ 
tant role in many branches of mathematics, is also used 
in statistics, although its use here can hardly be defended, 
for this concept arises properly only in connection with 
infinite processes. The statistician uses it in saying that 
frequencies approach a limiting ratio, but the statistician, 
and also the physicist, do not deal with infinity—rather 
with phenomena which, however vast and complex, 
are finite and limited. Because an experiment yields the 
same result a thousand times is no proof that the results 
to follow will be consistent. Even Scheherezade may 
tell an unpleasing tale on the thousand and second night. 


236 Mathematics and the Imagination 

Relative frequencies can hardly be said to approach a 
mathematical limit. The limiting concept as it is used in 
the theory of relative frequency bears roughly the same 
relation to the mathematical concept of limit as reason¬ 
ing by probable inference bears to the syllogism. 

Reference is often made to the probability of past 
events, although such probability in terms of the relative 
frequency view has apparently no meaning. “It is im¬ 
probable that John Wilkes Booth escaped the federal 
soldiers after the assassination of Lincoln”; or “Henry 
VIII was probably not so much interested in reform 
when he broke with the Pope as in getting rid of Cather¬ 
ine of Aragon.” How shall such statements be evaluated 
if probability is the relative frequency of an event within 
a class of events? Indeed, whether the event be past or 
future, what is meant by the probability of any single event? 

Whatever interpretation of probability is advanced, 
this problem is particularly troublesome. Perhaps sad 
necessity accounts for the best accredited opinion that 
probability has no meaning whatsoever when applied to 
a single event, either past or future. 

According to the statistical interpretation, probability 
can refer to a single ev'ent only in relation to a class of 
similar events. But this often makes for confusion. Every¬ 
one would agree that the following reasoning is absurd; 
In a certain community records of births for the past 10 
years indicate a ratio of 51 females to 50 males. The first 
35 children born in a particular month are all girls. Mr. 
Jones, expectant father, is therefore quite certain that the 
odds are heavily in his favor that his wife will present him 
with a boy, because of the “law of averages.” * 

* Not to leave the reader in suspense we can tell him that Jones is 
just as well off as though he were starting from scratch. 


Chance and Chanceability 237 

On the other hand, it is a very common misapprehen¬ 
sion of the very same kind to which we still cling in¬ 
tuitively that if X throws five sevens in a row at dice, the 
chance of his tossing another seven the next throw is 
much less than his chance of throwing some other 
particular number. We find it hard to believe that the 
mathematical probability, the chance of a future ev'ent, 
where the events are independent, is unaffected by what 
has already happened. 

In our daily lives we instinctively and deliberately 
reject this principle. When logic says ‘‘You must,” we 
often reply “Not this time.” Charles S. Peirce, the famous 
pragmatist, illustrates the point exceedingly well: “If a 
man had to choose between drawing a card from a pack 
containing 25 red cards and a black one, or from a pack 
containing 25 black cards and a red one; and if the 
drawing of a red card were destined to transport him to 
eternal felicity and that of a black one to consign him to 
everlasting woe, it would be foolish to deny that he 
ought to prefer the pack containing the larger portion of 
red cards, although from the nature of the risk, it could 
not be repeated. It is not easy to reconcile this with our 
analysis of the conception of chance. But suppose he 
he should choose the red pack and should draw the 
black card. What consolation would he have? He might 
say that he had acted in accordance with reason, but 
that would only show that his reason was absolutely 
worthless. And if he should choose the red card, how 
could he regard it as anything but a happy accident? He 
could not say that if he had drawn from the other pack 
he might have drawn the wrong one, because an hypo¬ 
thetical proposition such as: ‘If A, then B’ means nothing 
with reference to a single case.”“ 


238 Mathematics and the Imagination 

Finally, a brief allusion to an interpretation of prob¬ 
ability, accredited chiefly to Peirce, which seems to 
avoid some of the difficulties inherent in the interpreta¬ 
tions already examined.® 

Peirce holds that probability refers not to events but to 
propositions. With some modifications, his view is adhered 
to by John Maynard Keynes in his remarkable Treatise 
on Probability. According to Peirce, probability has nothing 
to do with either intensity of belief or with statistical fre¬ 
quencies. “Instead of talking about such an event as ‘heads,’ 
the truth frequency theory discusses propositions such as: This 
coin will fall head uppermost on one toss.” The proba¬ 
bility of the truth of this proposition must be the same as 
the relative frequency with which the event “head” oc¬ 
curs in a series of tosses. 

This interpretation of probability is better able to take 
care of single events. The statement, “It will probably 
rain tomorrow” means that the propositions about the 
state of the weather, temperature, barometric pressure and 
so on, more often than not imply propositions of the type: 
“It will probably rain tomorrow.” In other words, if from 
our knowledge of the weather we conclude this latter 
proposition, we will be right more often than wrong. 

Before passing to a consideration of a few of the 
theorems of the calculus of probability, there is one 
further caution. Everything said thus far points to one 
fact unmistakably: No proposition has any probable truth 

_ _ • 4 

except in relation to other knowledge. To say that a proposition 
is probable, when the knowledge on which it is based is 
either obscure or nonexistent, is absurd. To be sure, 
we often make elliptical statements about probability, 
where it is clearly understood to what body of knowledge 
we refer. This is just as permissible as to say that San 


Chance and Chanceability 239 

Francisco is 3000 miles away, it being evident that what 
is meant is “San Francisco is 3000 miles away from New 
York.” As already emphasized, it is more laudable to 
adhere to a statement which turns out to be wrong, so 
long as the evidence from which we reach our conclusion 
is the best available, than to advance a true proposition 
on the basis of faulty reasoning or incorrect facts. Herod¬ 
otus says: “There is nothing more profitable for a man 
than to take good counsel with himself; for even if the 
event turns out contrary to one’s hopes, still one’s decision 
was right even though fortune has made it of no effect; 
whereas if a man acts contrary to good counsel, although, 
being lucky, he gets what he had no right to expect, his 
decision was not any the less fallacious.” 


THE CALCULUS OF CHANCE 

In moderation, gambling possesses undeniable virtues. 
Yet it presents a curious spectacle replete with contradic¬ 
tions. While indulgence in its pleasures has always lain 
beyond the pale for fear of Hell's fires, the great labo¬ 
ratories and respectable insurance palaces stand as monu¬ 
ments to a science originally born of the dice cups. 

The Chevalier de Mere, euphemistically called a “gam- 
ing philosopher” of the seventeenth century, desired some 
information about the division of stakes at games of dice. 
He directed his inquiries to one of the ablest mathema¬ 
ticians of all times—the gentle and devoutly religious 
Blaise Pascal. Pascal, in turn, wrote to an even more cele¬ 
brated mathematician, the Parliamentary Town Coun¬ 
cilor of Toulouse, Pierre de Fermat, and in the corre- 
spondenee that ensued, the theory of probability first saw 
the light of day. 


240 


Mathematics and the Imagination 


Pascal could not forbear from a mild rebuke of De 
Mere, not because he was a gambler, but for the more 
serious reason that De Mere was not a mathematician: 
“Car, il a tres bon esprit,'' (he wrote to Fermat) '"rnais 2 / 
n'est pas geometre; c'est comme vous savez un grand deJauL 
Indeed, the Chevalier deserved worse, for the answer 
to his question evidently interfered with his business so 
that he took the occasion to write a diatribe on the 
worthlessness of all science, in particular arithmetic. 
And that was the fate of the first brain trust. 

Interest in probability grew, encouraged by the re¬ 
searches of such eminent mathematicians as Leibniz, 
James Bernoulli, De Moivre, Euler, the Marquis de Con- 
dorcet, and above all, Laplace. The latter’s epochal work 
on the analytic theory of probability brought the calculus 
to the point where Clerk Maxwell could say that it is 
“mathematics for practical men,” and Jevons could wax 
quite lyrical (quoting without acknowledgment from 
Bishop Butler) that the mathematics of probability is the 
very guide of life and hardly can we take a step or make a 
decision without correctly or incorrectly making an esti¬ 
mation of probability.” And these opinions were ottered 
even before the calculus had achieved its most brilhan 
successes in physics and genetics as well as in more practi¬ 
cal spheres." It was indeed remarkable, as Laplace wrote, 
that “a science which began with the considerations o 
play has risen to the most important objects of human 


knowledge.” 


In developing a calculus of probability it is necessary 
to make certain ideal assumptions. Particularly since a 
great many things to which we w'ould like to app y t 
are not measurable, we must be doubly careful that the axi 


241 


Chance and Chanceability 

oms and postulates which we formulate are precise, so 
that their range of application may be readily judged. VVe 
have already referred to the fact that the mathematical 
probability of an event lies between 0 and 1. The proba¬ 
bility of an impossible event is 0, that of a certain event, 1. 

We must now define what is meant by “equiprobable” 
(equally probable). This is a rather difficult task; for our 
purposes we can shorten the road by employing a rough 
definition. 

Two contingent events will be considered equiprobable if, 
either in the absence oj any evidence or after considering all the 
relevant evidence^ one event cannot be expected in preference to 
the other. 

Perhaps the reader detects an incongruity. Had he not 
been cautioned that no probability can bo estimated where 
there is no appropriate or relevant knowledge? Yet here 
it is said that two propositions, or events, can be equally 
probable, even if we have no knowledge about them 
whatsoever. But therein lies the clue! A little knowledge 
is dangerous. None at all is much more satisfactory. For 
our purposes we invoke the principle of insufficient rea¬ 
son, according to which, in the absence of any knowl¬ 
edge about two events, they are considered equally likeh'. 
The reader must bear in mind that our definition is rough 
—very rough. And also, that it is possible to know that 
two quantities are equal without knowing what they are. 
Thus, one may know from a general knowledge of gann's 
that in chess both sides start with equal forces without 
knowing what these are, or anything else about the game. 

If we assume, then, that a penny is symmcirical, it is 
equiprobable that it will fail heads or tails, since there is 
no more reason to anticipate one result than the other. 

If there are a number of equiprobable ways in which 


242 Maikematics and the Imagination 

an event can happen and a number of equiprobable ways 
in which it cannot happen, the probability of the occur¬ 
rence of the event is the ratio of the number of ways in 
which the event can happen to the total number of ways 
in which it can and cannot happen. The coin may fall 
heads or tails. The probability of its falling heads is thus 

—In general, if we call the ways in 

H + T 2 ^ 


the ratio 


which an event can happen, favorable, and the ways m 
which it cannot happen, unfavorable, the probability of an 

event is the fraction ^ ■ 

F -f U 

That branch of mathematics which considers permuta¬ 
tions and combinations is concerned with the number 
of different ways in which an event can happen. It is the 
study of mathematical possibility, and furnishes an ideal 
framework for the mathematics of probability. 

The typical problems of permutations and combina¬ 
tions have a dry and dreary look. At first it is hard to 
believe that information gained in solving problems of 
this type can be of much service in other studies; ‘ Four 
travelers arrive at a town where there are 5 inns. In how 
many ways can they take up their quarters, each at a dif¬ 
ferent hotel?” Nor does it seem that a theory which is used 
to determine in how many different ways the letters of 
the word Mississippi ® may be arranged, would be useful 
in determining either the physics of the atom or in fixing 
insurance rates. Nevertheless, the theorems of combina¬ 
torial analysis are the basis for the calculus of probabil 
ity. We have to know how to calculate the total number 
of different ways an event can happen before aspiring to 

predict how it is likely to happen. 

Our overworked penny again furnishes an example. 


Chance and Chanceability 243 

A penny is tossed three times in succession. The possible 
results are: 

O- HEADS 
• = TAILS 

o#o#o«o# 

oo%%oo%% 

oooo#### 

1 2 3 4 5 6 7 8 

IV V 

FIG. 77.—The possible results of tossing a penny 
three times. The arrows indicate the cases of two 
heads and one tail. 


These eight possible results answer all the questions which 
might be asked in permutations and combinations. But, 
further, any others that arise in the calculus of probabil¬ 
ity can also be answered by referring to the diagram.Thus, 

F 

the probability of getting 3 heads, the ratio ^ ^ is 

The probability of getting 2 heads and 1 tail is the ratio 
of cases 2, 3, 5 to all the possible cases, i.e., -g-. 

Now it is plain that the enumeration of all possible cases 
becomes both tedious and unwieldy as these increase in 
number. For that reason the calculus contains many theo¬ 
rems taken from combinatorial analysis which make di¬ 
rect enumeration unnecessary. 

MUTUALLY EXCLUSIVE EVENTS 

I. Since there are four aces in a deck, the probability 
of drawing an ace from 52 cards is But what 

is the probability of drawing either an ace or a king from 


17 


244 Aiathematics and the Ifnagination 

a deck of cards in one draw? This is the probability of 
mutually exclusive^ or alternative^ events; if one of the two 
events occurs, the other cannot. A theorem in the calculus 
states that the probability of the occurrence of one of sev¬ 
eral mutually exclusive events is the sum of the probabilities 
of each of the single events. The probability of getting an ace 
or a king is therefore, Jg- + ^ 

What is the probability of obtaining either a 6 or a 7 in 

throwing a pair of dice? We may enumerate the number 

of cases favorable to either 6 or 7 and then check our re¬ 
sults with the theorem. 

FIRST DIE SECOND DIE 

1 6 

2 5 

3 VII 4 

4 3 

5 2 

6 1 

There are 36 possible combinations of the dice and 11 
are favorable to the event; therefore, the probability of 
obtaining either a 6 or a 7 is 

Had we used the theorem, we would have taken the 
sum of the separate probabilities, i.e., ^ and of 

course, obtained the same result. 

INDEPENDENT EVENTS 

II. Two events are said to be independent of each other 
if the happening of one is in no wav connected with the 
happening of the other. A penny is tossed twice in suc¬ 
cession. What is the probability of getting 2 heads in a 
row? The appropriate theorem states that the probability 
of the joint occurrence of two independent events is the prodiu^t of 
the separate probabilities of each of the events. The probability 


FIRST DIE 

1 

2 

3 

4 

5 


VI 


SECOND DIE 

5 

4 

3 

2 

1 


Chance and Chanceabiliiy 245 

of getting 2 heads in succession is, therefore, 2 X § “ 4* 
And, as we saw above, by direct enumeration, the proba¬ 
bility of getting three heads in a row is Checking this 
against the theorem gives i X J X | 


• 

• 

• 

# 

• # 

# • 

• • 

• 

1 # # 

1 • #1 

1 • 


VI 

Vll 

1 # 

1 • 


1 

VI 

Vll 

! 


H 


VI 

Vll 


VI 

Vll 


• • 

• 

VI 

Vll 


A 


• # 

• • 

# # 

Vll 


FIG. 78.—Each square represents an equiprob- 
able result. For instance, the square marked A 
represents getting a 4 with one die and 5 with the 
other. Of the thirty-six possibilities, five result in a 
6, and six result in a 7. 

Consider now a problem slightly different in form: 

In tossing a coin twice in succession, what is the 
probability of getting at least one head? This problem 
may be solved easily without enumeration, by ascertain¬ 
ing the probability of the desired event not happening and 
subtracting this fraction from 1. Since the probability of 
getting two tails in succession, which is the sole alternative 
to getting at least one head, is the probability of at 
least one head is 1 — j = f. 


246 Mathematics and the Imagination 

D’Alembert, in his article on probability in the famous 
Encyclopedie ^ revealed that he did not understand the 
theorem of multiplying independent probabilities. He 
doubted that the last-named probability was f, reason¬ 
ing that if a head appeared at the first throw, the gcime 
was finished and there was no need for a second. Enu¬ 
merating only three possible cases: H, TH, and TT, he 
arrived at the probability of f. What he failed to consider 
was that the first alternative was in itself no more likely 
than the alternative of getting a tail. 

Although D’Alembert consistently misunderstood the 
fundamentals of probability, some of his ideas fore¬ 
shadowed the statistical interpretation. He suggested that 
by making experiments approximations of desired proba¬ 
bilities could be estimated. 

Long before the wave of enthusiasm for statistics swept 
over Europe in the middle of the nineteenth century, 
experiments of the kind suggested by D’Alembert were 
carried out. The eighteenth-century naturalist, Count Buf- 
fon, carried on many experiments, the most famous of 
which is his “Needle Problem.” A plain surface is ruled 
by parallel lines (as in Fig. 79), the distance between 
the lines being equal to H. Taking a needle whose length, 
L, is less than //, Buffon dropped it, permitting it to fall 
each time on the ruled surface. He considered the toss 
favorable when the needle fell across a line, unfavorable 
when it rested between two lines. His amazing discovery 
was that the ratio of successes to failures was an expression 
in which tt appears. Indeed, if L is equal to H, the proba- 

2 

bility of a success is —. The larger the number of trials, 

TT 

the more closely did the result approximate the value of 
TT, even to three decimal places. 


Chance and Chanceability 247 

Elaborate experiments were performed in 1901 by an 
Italian mathematician, Lazzerini, who made 3,408 tosses, 
giving a value for tt equal to 3.1415929, an error of only 
0.0000003. One could scarcely expect to find a better ex- 


pjc 79.—Count Buffon’s needle problem. 


aunple of the interrelatedness of all mathematics. Thus far 
we have seen tt in three guises: as the ratio of the circum¬ 
ference of a circle to its diameter; as the limit of infinite 
series; and as a measure of probability. 


248 


Mathematics and the Imagination 


COMPOUND PROBABILITY 

III. The theorem dealing with the probability of 
independent events may sometimes be usefully extended 
to deal with cases where the probabilities are not actually 
independent. 

A bag contains one white ball (W) and two black 
balls (5); the probability of drawing a black ball is f; 
a white ball Assume two successive drawings from 
the same bag, with the ball replaced after each drawing. 
Now the probability of drawing two in succession 
is J X ^ = -g-, of drawing two B's in succession f X | = 

However, if after each drawing the balls are not re¬ 
placed, the drawings are no longer independent but depend¬ 
ent on each other. After each drawing the new proba¬ 
bility must be calculated to form the correct compound 
probability. After one ball has been drawn, the proba¬ 
bility of drawing two B'% in succession, with no re¬ 
placements is I X J — J. That the probability of the 
second drawing depends on the outcome of the first is 
also shown by the fact that the probability of drawing 
two Ws in a row is 0, if no replacement is made, whereas 
it is ^ if the W is replaced on being drawn the first time. 

IV. Thus far we have considered the probability of 
events that are mutually exclusive, dependent, and inde¬ 
pendent. If these factors are varied and combined, new, 
interesting methods result. 

A bag contains 6 and 6 B's. If one ball is drawn, 
two events are equiprobable—either W or B. This may be 
denoted by 

(a) ( 1 ) W, (2) B = 2^ 

The possible results in two drawings are: 


Chance and Chanceahilxty 249 

(1) IV^, (2) H’B, (3) BIV, (4) BB = 2* 

In three drawings there are eight possible results; 


(1) WWW 

( 2 ) WBW 

(3) WBB 

(4) WWB 

(5) BWB 

( 6 ) BWW 

( 7 ) BBW\ _ 

(8) BBB j 

2 * 

In four drawings there are sixteen: 


( 1 ) wwww 

(2) WWWB 
' (3) WBWW 

(4) WWBW 

(5) BWWW 

( 6 ) WWBB 

( 7 ) WBBW 

(8) BBWW 

( 9 ) BWBW 

( 10 ) WBWB 

( 11 ) BWWB 

( 12 ) WBBB 

( 13 ) BBBW 

( 14 ) BBWB 

( 15 ) BWBB 

( 16 ) BBBB 

. = 2* 


In general, then, in n drawings there are 2" possible 
results. 

But this information is the clue to a most valuable 
method! Let us avail ourselves of an important theorem 
from another branch of mathematics the Binomial The¬ 
orem. 

Let IV denote the drawing of a white ball, and B the 
drawing of a black. Expanding the expression (JV + 
in accordance with the binomial theorem, we obtain 

PT2 4- 2WB + B\ 

Now this algebraic expression pictures compactly 
what was already explicitly set forth in (b) above, 
namely; every possible result of two drawings from a bag 
containing the same number of black and white balls. 
Thus,3 

( 1 ) WW = 

( 2 ) WB^ ^ 2WB 

(3) BW] 

(4) BB = B'^ 

Three drawings from such a bag is represented by the 

expression 


4_ 3H/25 _p 2>WB^ + B^ 


250 Mathematics and the Imagination 

for again 

(1) WWW = W^ 

(2) WWB] 

(3) WBW} = 3W^B 

( 4 ) BWWj 

(5) BBW] 

( 6 ) BWB\- = 3WB^ 

( 7 ) WBBj 

( 8 ) BBB = B^ 

There are, therefore, eight possible results, one way of get¬ 
ting three Whites, three ways of getting two Whites and 
a Black, three ways of getting two Blacks and a White, 
and one way of getting three Blacks. 

The respective probabilities are f, f, and 
For n successive drawings the general binomial theorem 
gives:^® 

{W + By = m + nW'-'B + 

I 

^ n(n-l)in-2) ^ _ 

3! 

One further illustration may be considered of the 
application of the binomial theorem: A bag contains 
3 Whites and 2 Blacks. After each drawing the ball is 
replaced. What is the probability of 3 W^s and 2 B^s in 
5 drawings? 

Now, for each drawing the probability of a W = §, 
of a B — g. Expanding: 

(W + B)^ = W^ + 5W^B -h lOW^B^ + lOW^B^ 

H- 5WB^ -i- B\ 

The result, the probability of which we are seeking, is 
W^B^ since this represents 3 IT’s and 2 B’s. There are ten 


Chance and Chanceability 251 

such possible results since the coefficient of the term 
is 10. The desired probability, which is compound, must, 
therefore, be 

10 X ar X (1)^ = m- 

* 

It should be even more evident now how limited are 
the instances when the calculus of probability is appli¬ 
cable. In none of the several examples which appeared on 
page 227, however properly they may have illustrated 
the concept of probability, is our mathematical appara¬ 
tus of any use. Indeed, the calculus of probability, like 
all other mathematical disciplines, cannot be regarded 
as a source of information about the physical world. 
Furthermore, speaking mathematically, it may be pos¬ 
sible to define what is meant by equiprobable, but it is 
without doubt impossible to find two events in the phys¬ 
ical world which actually are equiprobable. 

Equiprobability in the physical world is purely a hy¬ 
pothesis. We may exercise the greatest care and use the 
most accurate of scientific instruments to determine 
whether or not a penny is symmetrical. Even if we are 
satisfied that it is, and that our evidence on that point is 
conclusive, our knowledge, or rather our ignorance, about 
the vast number of other causes which affect the fall of 
the penny is so abysmal that the fact of the penny s sym¬ 
metry is a mere detail. Thus, the statement head and 
tail are equiprobable” is at best an assumption. 

Yet the calculus of probability is only helpful after we 
have made such an assumption—an assumption which, 
like all hypotheses in science, must justify its existence 
by its usefulness and which we must be prepared to 
modify or reject when experience fails to corroborate it. 


252 Mathematics and the Imagination 

By following such a bold procedure, the mathematics 
of probability has been remarkably successful in science 
and in commerce. In the eighteenth and nineteenth cen¬ 
turies, when science and philosophy were almost entirely 
under the spell of mechanistic ideas, it was enthusiasti¬ 
cally supposed that the calculus of probability would sup¬ 
plement every “ignorance and weakness of the human 
mind.” The calculus would help to illuminate those re¬ 
gions of knowledge in which the beacon of science did not 
yet burn too brightly. 

It is readily understandable that a convenient and dog¬ 
matic philosophy of materialism was popular in a world 
which had witnessed the parade of scientific achievements 
from Kepler and Galileo to Newton and Laplace. The 
materialistic concept is based on a naive faith in the all- 
pervading regularity and the recurrent order of natural 
phenomena, from the behavior of atoms to our own be¬ 
havior on arising in the morning. Men hoped, and the 
history of science until recently encouraged them to be¬ 
lieve, that science would explain all miracles and disclose 
all secrets, that the future was contained in and would 
therefore resemble the past, and that consequendy the ex¬ 
periences of the past would help in predicting the future. 

As a leading exponent of this view, Laplace had far 
greater hopes for the limits of knowledge than the modest 
twilight of mediocrity in which Locke felt the human 
mind would forever have to grope. 

“We ought then,” Laplace wrote, “to regard the pres¬ 
ent state of the universe as the effect of its anterior state 
and as the cause of the one which is to follow. Given for 
one instant an intelligence which could comprehend all 
the forces by which nature is animated and the respective 
situation of the beings who comjjose it—an intelligence 


Chance and Chanceabiliiy 253 

sufficiently vast to submit these data to analysis—it would 
embrace in the same formula the movements of the great¬ 
est bodies of the universe and those of the lightest atom; 
for it, nothing would be uncertain and the future, as the 
past, would be present to its eyes.” “ 

When Napoleon asked Laplace where in his monu¬ 
mental Mecanique celeste there was any reference to the 
Deity, he is said to have replied, “Sire, I have no need of 
that hypothesis.” On hearing Napoleon recount tliis 
story, Lagrange remarked “That, Sire, is a wonderful 
hypothesis.” Modern physics, indeed all of modern sci¬ 
ence, is as humble as Lagrange, and as agnostic as La¬ 
place. Professing no God, it attributes to itself neither di¬ 
vine omniscience, nor the possibility of achieving it. 

♦ 

It was expected then, in the eighteenth and nineteenth 
centuries, that a Utopia in morals and politics as well as 
in the physical sciences was not far off. If exact natural 
laws in these spheres had not yet been uncovered, it was 
not doubted that they existed. In the meantime the cal¬ 
culus of probability would meet the deficiency. Though 
social phenomena had not yet been mastered in detail, 
as the motions of many of the planets had been, it was 
certain they would exhibit the same regularities when 
studied on the grand scale. Probability was to be a tem¬ 
porary expedient, an ordnance map which scientists 
would fill in in due time. 

Hopes were high, and among those who expected the 
most was the Marquis dc Condorcet. The theory of prob¬ 
ability, he thought, might be applied effectively to the 
judgments of tribunals in order to minimize the danger ot 
erroneous decisions. To that end he proposed that a large 
increase in the number of judges on any tribunal would 


254 Mathematics and the Imagination 

assure a great many independent opinions, which, when 
combined, would safeguard the truth by neutralizing op- 
posingly extreme and prejudicial views. Unfortunately, 
Condorcet failed to take numerous other factors into con¬ 
sideration. Not the least of these was the logic of the guil¬ 
lotine. For it was to this, ironically and tragically enough, 
that the judgment of a revolutionary tribunal, composed 
of many judges, all of whom held the same extreme views, 
eventucdly consigned him. 

In the less heated atmosphere of the nineteenth century 
some of Condorcet’s views were vindicated—if not in 
morals and politics, then in science and industry. The 
statistical view of nature changed the map of science in 
both the nineteenth and twentieth centuries as much, 
perhaps, as the inventions and the discoveries of the lab¬ 
oratory. Indeed (and this point cannot be emphasized too 
strongly), the statistical view has so permeated and pene¬ 
trated modern scientific thinking and method that it has 
gone far beyond anything that even Condorcet could have 
imagined. But the fundamental materialism of his time 
which accompanied this faith in probability has largely 
vanished. 

Instead of serving as an expedient, as a substitute for 
natural laws as yet unrevealed, statistical inference has 
come in time to supplant them almost completely. This 
signifies a change in the interpretation of physical reality 
comparable in intellectual importance to the Renais¬ 
sance. With this in mind contemporary physicists often 
refer to the Renaissance of Modern Physics. 

* 

In his great work on the Analytical Theory of Heat, 
Fourier stated the principle which best exemplifies what 
we have already referred to as the classical view of phys- 


Chance and Chanceability 255 

ics—in fact of all natural laws. “Primary causes arc un¬ 
known to us, but are subject to simple and constant laws, 
which may be discovered by observation, the study of them 
being the object of natural philosophy.” And he went on 
to add : “Profound study of nature is the most fertile source 
of mathematical discoveries . . . There cannot be a lan¬ 
guage more simple, more free from errors and obscurities, 
that is to say, more worthy to express the relations of nat¬ 
ural things ... It brings together phenomena the most 
diverse, and discovers the hidden analogies which unite 
them.” 

The scientist of the present day, particularly the phys¬ 
icist would be in complete agreement with the latter part 
of this quotation. He would agree that mathematics is 
the ideal language in which to express the results of his 
observations and even the uncertainties of his predictions. 
He would, however, differ with Fourier sharply when he 
says that the laws governing natural phenomena arc simple 
and constant.'*'* 

Instead of holding to the opinion that nature obeys 
perfect and certain laws, which it is the job of the scientist 
to discover and explain, the physicist is now content to 
make hypotheses and to perform experiments, to carry 
on a kind of scientific bookkeeping, with the aid of which 
he strikes a balance from time to time. That balance 
bears no relation to eternal verities. It refers only to cur¬ 
rent assets and liabilities. Instead of pinning his faith 
on uncovering in all natural phenomena a general all- 
pervading, regular, and recurrent order, he is content 
to hope that there is occasional method in the madness 
of the physical world, that in the large, if not in the small, 
there is some semblance of a scheme. 

The old materialistic dogmatism seemed to foreclose fur- 


256 Mathematics and the Imagination 

ther metaphysical speculations about the nature of reality 
and was “comfortable and complete.” It had the “com¬ 
pelling power of the old logic.” The outlines of the world 
were hard and fast, and the mysteries of the universe, its 
apparent uncertainties, were confessions of our own in¬ 
competence, our own limitations. When we said that the 
fall of a penny was determined by chance, “we regarded 
this confession of uncertainty as due to our own igno¬ 
rance, and not the uncertainties of nature.” 

But the new physics and the new logic have changed 
our outlook as profoundly as they have changed our basic 
distinction between matter and energy. “We start prej¬ 
udiced against probability, grudging it as a makeshift, 
and in favor of causality,” and we end convinced that 
the outlines of the world are “not hard, but fuzzy,” and 
that our most exact scientific laws are merely approxi¬ 
mations good enough for our crude senses. Thus, in place 
of the syllogism and the rules of formal logic our ideas 
about the physical universe must be gauged entirely by 
the rules of probable inference. We translate “Socrates is 
a man; all men are mortal, therefore Socrates is mortal, 
as a statement about the world of fact, into, “Socrates 
will probably die, because so far as we know all men be¬ 
fore him have died.” “The uncertainties of the world we 
now ascribe not to the uncertainties of our thoughts, but 
rather to the character of the world around us. It is a 
more sensible, more mature and more comprehensible 
view.” 

Here we recall the moving words of Charles Peirce. 
“All human affairs rest upon probabilities, and the same 
thing is true everywhere. If man were immortal, he could 
be perfecdy sure of seeing the day when everything m 
which he had trusted should betray his trust, and, in 


Chance and Chanceability 257 

short, of coming eventually to hopeless misery. He would 
break down, at last, as every good fortune, as every dy¬ 
nasty, as every civilization does. In place of this we have 
death. 

“But what, without death, would happen to every man, 
with death must happen to some man ... It seems to 
me that we are driven to this, that logicality inexorably 
requires that our interests shall not be limited. They must 
not stop at our own fate, but must embrace the whole 
community.” 


APPENDIX 

A discussion of the theory of probability can ill afft)rd 
to omit some applications. They are, however, generally 
quite technical, but the more persevering reader will surely 
find these few, chosen at random, of interest. 

KINETIC THEORY OF GASES AND PROBABILITY CURVE OF 

ERROR 

The law of gases was arrived at experimentally by the 
English physicist and chemist Robert Boyle (1627-1691;, 
whose most important work bears the title The 
Chymist: or Chymico—Physical Doubts and Paradoxes^ touch¬ 
ing the experiments whereby vulgar Spagirists are wont to en¬ 
deavour to evirue their Salty Sulphur and Atercury to be the true 
Principles of Things. His law of gases states that the [pres¬ 
sure of a gas is inversely proportional to its volume. I bus: 
Pressure X Volume = Constant. But any volume oi gas 
is made up of vast numbers of moving molecules, eath 
of which has a velocity proportional to its energy. Nat¬ 
urally, molecular collisions occur in great numbers at 


258 Mathematics and the Imagination 

every instant. It has been estimated that in “ordinary 
air each molecule collides with some other molecule 
about 3000 million times every second and travels an 
average distance of about 1 e 6^6 6 0 between succes¬ 
sive collisions.” * 

Assuming these collisions occur with perfect elasticity, 


i.e. no energy is lost, it can be inferred, based on the ideas 
of change, that at any instant there will be some mole¬ 
cules moving in all directions and with all velocities. 
Mathematicily, it was shown first by Clausius, and later 
by Maxwell and Boltzmann, that P = \ nmV^ where P 
is pressure, w, the number of molecules in unit volume, 
m, the mass of each, zind the average value of the 
square of the velocity. 

To the problem of the distribution of velocities among 
the molecules, Maxwell applied Gauss’ law of error (of 
importance in many branches of inquiry) derived from 
the theory of probability. 

• Sir James Jeans, The Universe Around Us (Cambridge University 
Press, 1930). 


Chance and Chanceahility 259 

The normal curve of error (see Fig. 80) may be ob¬ 
tained by the binomial expansion as n —* co 

This curve shows that in ordinary observation, small 
errors occur with larger frequency than great ones. 

“The (kinetic) theory shows that molecules subject to 
chance collisions may be divided into groups, each group 
moving within a certain range of velocity in a manner 
illustrated in the diagram.^’ * (See Fig. 81.) The resem¬ 
blance of this curve to the normal curve of error is ap¬ 
parent. 


FIG. 81.—Velocity of molecules of a gas. 


“The horizontal ordinate measures the velocity and the 
vertical ordinate, the number of molecules which move 
with it. The most probable velocity is taken as unity. It 
will be seen that the number of molecules moving with a 
velocity only three times the most, probable velocity is 
almost negligible. Similar curves may be drawn to illus¬ 
trate the distribution of shots on a target, or errors in a 
physical measurement, of men arranged in groups ac- 

• Sir William Dampier, A History of Scierue and its Relations with 
Philosophy and Religion (London: Macmillan). 


18 


200 Mathematics and the Imagination 

cording to height or weight, length of life, or ability as 
measured by examination 


FIG. 82.—This distribution curve tells the chest meas¬ 
urement of Scottish soldiers. Incidentally, it also serves 
to describe phenomena as diverse as the following: 

1. Age distribution of pensioners of a large concern. 

2. Runs at roulette. 

3. Scattering of bullets about a target. 


STATISTICS IN ANTHROPOLOGY 

The Belgian astronomer, L. A. J. Quetelet (1796—1874) 
showed that the theory of probability could be applied 

* Sir William Dampier, op. cit. 


Chance and Chanceability 261 

to human problems. Thus, the same distribution is found 
for “runs” at roulette, or in the distribution of bullets 
around the center of a target, as in the chest measure¬ 
ments of Scottish soldiers, or in the velocities of molecules 
in a gas. * 

STATISTICS AND PAST EVENTS t 

One of the most ancient problems in probability is 
concerned with the gradual diminution of the probabil¬ 
ity of a past event, as the length of the tradition increases 
by which it is established. Perhaps the most famous solu¬ 
tion of it is that propounded by Craig in his Theologiae 
Christianae Principia Mathematical published in 1699. He 
proves that suspicions of any history vary in the duplicate 
ratio of the times taken from the beginning of the history 
in a manner which has been described as a kind of paiody 
of Newton’s Principia. “Craig,” says Xodhuntcr, con¬ 
cluded that faith in the Gospel, so far as it depended on 
oral tradition, expired about the year 880, and that, so 
far as it depended on written tradition, it would expire 
in the year 3150. Peterson, by adopting a different law of 
diminution, concluded that faith would expire in 1789! 

In the Budget of Paradoxes De Morgan quotes Lee, the 
Cambridge orientalist, to the effect that Mohammedan 
writers, in reply to the argument that the Koran has not 
the evidence derived from Christian miracles, contend 
that, as evidence of Christian miracles daily glows 
weaker, a time must at last arrive when it will fail of al- 
fording assurance that they were miracles at all: whence 
the necessity of another prophet and other miracles. 

* Ibid. 

t John Maynard Keynes, A Treatise on Probability (New \ ork and 
London: Macmillan, 1921), chapter XVI, p. 184. 


202 


Mathematics and the Imagination 


STATISTICS OF AIR-RAID CASUALTIES 

Professor J. B. S. Haldane in a communication to 
Nature (October 29, 1938) discussed the mathematics of 
air-raid protection. A more bitter commentary on con¬ 
temporary society would be hard to find, though it is 
coldly dispassionate and purely scientific in tone and pur¬ 
pose. It reads in part: 

In view of the discussion which is occurring on this subject, 
it seems desirable to have some quantitative measure of the 
degree of protection afforded by a given shelter. In order to 
limit the problem we may consider only risks of death, and 
further confine ourselves to high-explosive bombs. Incendiaries 
have proved a negligible danger to life in Spain, and gas is 
also negligible except for babies and those whose respirators 
do not fit. 

Consider a given type of bomb, say a 250 kilo, bomb, which 
is commonly used on central areas of Spanish cities, and a 
man in a given situation, whether in the street or in a shelter. 
Let n be the expected number of bombs falling in his neighbor¬ 
hood (say 1 square kilometer) during a war, the distribution 
of bombs over this area being supposed even, since aim is 
poor when cities are bombed. Let p be the probability that a 
single bomb falling at the point (x^y) in this area will kill 
him. Then the probability that he will be killed in the course 
of the war is P = ^n/A pdxdy, integration being taken over 

the whole neighborhood of area A. 

The values of n and p will, of course, be different for each 
type of bomb, and the different expressions so obtained must 
be summed. Further, the man will be in different places 
during the war, and thus another summation is necessary. 
Finally, P must be summed for the whole nation. 

The policy of evacuation is intended to reduce the value 
of n, even though it may increase that of />, as when a child is 
evacuated from a fairly solid house into a flimsy hut. The 


Chance and Chanceability 263 

policy of dispersal within a dangerous area does not, of course, 
reduce either n or p. It merely ensures that no single bomb 
will kill a large number of people, while increasing the proba¬ 
bility that any given bomb will kill at least one. It is likely to 
save a few lives by equalizing the numbers of wounded to be 
treated in different hospitals; and the psychological effect of 
having 20 killed in each of 10 areas may perhaps be less than 
that of 200 killed in one area. But, as it may actually increase 
the mean value of p by encouraging people to stay in a number 
of flimsy buildings rather than one strong one, it is at least 
as likely to increase the total casualties as to diminish them. 
The argument that a number of people must not be con¬ 
centrated in one place in order that a single bomb should not 
kill hundreds is clearly fallacious when applied to a war in 
which the total casualties will be large. It is, however, true 
that a small group of key men each of whom can replace 
another should not be grouped together. 

FOOTNOTES 

1. A. Conan Doyle, The Return ojSherlock Holmes, “The Adventure of 
the Dancing Men.”—P. 224. 

2. It may also be that certain sentence structures which look like 
propositions are neither true nor false, but meaningless. There are, 
for example, propositional functions like “x is a or wholly 
meaningless statements like “A snark is a boojum. But neither 
of these need concern us at this point.- P. 228. 

3. The following paradox which arises from the principle of in¬ 
sufficient reason is quoted by Keynes from the German mathema¬ 
tician von Kries (Keynes, A Treatise on Probability, London: 
Macmillan, 1921). Suppose that we know the specific volume of 
a substance to lie between 1 and 3; but we have no information 
as to the exact value. The principle of indifference would justify 
us in placing the specific volume between 1 and 2; or between 
2 and 3 with equal likelihood. The specific density of a substance 
is the reciprocal of the specific volume; if the specific volume is V, 
the specific density is \/V, so that wc know that the specific 
density must lie between 1 and Again, by the principle of in¬ 
sufficient reason, it is as likely to lie between 1 and \ as between 


>64 Mathematics and the Imagination 

§ and but the specific volume being a function of the specific 
density, if the latter lies between 1 and the former lies between 
1 and and if the latter lies between | and the former lies 
between Ij and 3. Whence it follows that the specific volume 
is as likely to lie between 1 and between \\ and 3, which is 
contrary to our first assumption that it is as likely to lie between 
1 and 2 as between 2 and 3.—P. 230. 

4. Dantzig, Number^ the Language of Science^ p. 67. P. 232. 

5. Charles S. Peirce, Chance, Love, and Logic. P. 237. 

6. For a lucid and admirably refreshing discussion of this and other 
problems of probability see Cohen and Nagel, An Introduction to 
Logic and Scientific Method, New York: Harcourt Brace, 1936.— 
P. 238. 

7. See Appendix to this chapter.—P. 240. 

8. As a matter of interest, there are 36,568 ways of arrangring the 

letters of the word “Mississippi.”—P. 242. 

9. The reader should not be disturbed by the fact that WB and 
BW are represented simply by 2WB. 2WB simply means two 
drawings in each of which there is one Black ball and one White, 
regardless of the order in which they appear. P. 249. 

10. Without troubling to remember the general formula, by the 

of the famous triangle of Pascal, one can at once ascertain the 

coefficients of any binomial expansion: 

1 

1 2 1 
13 3 1 
1 4 6 4 1 
1 5 10 10 5 1 
1 6 15 20 15 6 1 
1 7 21 35 35 21 7 1 
1 8 28 56 70 56 28 8 1 

By examining this array, the reader may determine for himself 
how each new line is formed.—P. 250. 

11. Laplace, Essai philosophique sur la probabilite. P. 253. n ■ ■ l 

12. Quoted from C. G. Darwin. Presidential Address to the British 

Association, 1938.—P. 256. 


Rubber-Sheet Geometry 


It s-t-T-e-t-c-h-e-s. 

-POPULAR ADVERTISEMENT 


Once upon a time seven bridges crossed the riv'ei Pregcl 
as it twisted through the little German university town 
of Kdnigsberg. Four of them led from opposite banks to 
the small island, Kneiphof. One bridge connected Rnci- 
phof with another island, the other two joined this with 
the mainland. These seven bridges of the eighteenth 
century furnished the material for one of the celebiaied 
problems of mathematics. 

Seemingly trivial problems have given rise to the devel¬ 
opment of several mathematical theories. Probability rat¬ 
tled out of the dice cups of the young noblemen of France; 
Rubber-Sheet Geometry was brewed in the gemutliche air 
of the taverns of Konigsberg. The simple German folk 
were not gamblers, but they did enjoy their walks. Over 
their beer steins they inquired: “How can a Sunday after¬ 
noon stroller plan his walk so as to cross each of our seven 
bridges without recrossing any of them?” 

Repeated trials led to the belief that this was impos¬ 
sible, but a mathematical proof is based neither on beliefs 
nor trials. 

Far away in St. Petersburg, the great Euler shivered 
in the midst of honors and emoluments, as mathemati¬ 
cian at the court of Catherine the Great. To Euler, home- 

265 


266 Mathematics and the Imagination 

sick and weary of pomp and circumstance, there came in 
some strange fashion news of this problem from his father- 
land. He solved it with his customary acumen. Topology, 
or Analysis Situs was founded when he presented his solu¬ 
tion to the problem of the Konigsberg bridges before the 


Fic. 83. 


Russian Academy at St. Petersburg in 1735. This cele¬ 
brated memoir proved that the journey across the seven 
bridges, as demanded in the problem, was impossible. 

Euler simplified the problem by replacing the land 
(in Fig. 83) by points, and the bridges by lines connecting 
these points. Once this simplification has been effected, 
can Fig. 84 be drawn with one continuous sweep of the 
pencil, without lifting it from the paper? For this is the 
equivalent of physically traversing the seven bridges in 
one journey. Mathematically, the problem reduces to 
one of traversing a graph. A “graph,” as the term is used 
here, is simply a configuration consisting of a finite num¬ 
ber of points called vertices and a number of arcs. The 
vertices are the end points of the arcs, and no two arcs 
have a common point, except, perhaps, a common vertex. 


Rubber-Sheet Geometry 267 

A vertex is odd or even, according as the number of arcs 

fornaing the vertex is odd or even. 

A graph is traversed by passing through all the arcs 
exactly once. Euler discovered that this can be done, 
starting and finishing at the same point, if the graph con¬ 
tains only even vertices. Further, he discovered that if 
the graph contains at most two odd vertices, it may also 
be traversed, but it is not possible to return to the start¬ 
ing point. In general, if the graph contains 2n odd ver¬ 
tices, where n is any integer, it will require exactly n 
distinct journeys to traverse it.^ 


34 _—graph with four vertices and seven 
arcs, illustrating the Konigsberg bridges. 

Figure 84 is the graph of the seven bridges of Konigs¬ 
berg. Since all four vertices arc oddj that is, each one is the 
end point of an odd number of arcs, 2n = 2 X 2, and, 
therefore, two journeys are required to traverse the graph 
—a single journey will not suffice. 

If, as in Fig. 85, an additional arc is drawn from A to 
C, representing another bridge, and the arc BD is re¬ 
moved, all the vertices become even; and C of order 

4, and D of order 2, and the graph can be traversed in a 
single journey. If the arc BD is not removed, the stroller 
may take his walk, cross all the bridges only once, but will 


268 


Mathematics and the Imagination 

find that he cannot finish at the point where he started. 
Thus, if he starts at Z), he will finish at B, and vice versa. 
(Note: he must start his walk at an odd vertex.) 

A 


D 


FIG. 85.—A graph with four vertices and eight 
arcs. 

The problem of the seven bridges is representative of 
a group of problems, some dating back to antiquity. They 
exemplify the difficulty of mentally grasping the true geo¬ 
metric properties of cdl but the simplest figures. 


FIG. 86. 


In the history of magic and superstition, the figure 
shown above (Fig. 86) has played an important part as 
a talisman against all forms of misfortune. Known to the 
Mohammedans and the Hindus, to the Pythagoreans and 


Rubber-Sheet Geometry 269 

the Cabalists, it was sometimes carved on babies’ cribs 
to fend off evil, while in more practical countries it was 
painted on animals’ stalls. It is possible to traverse this 
star, returning to the starting point, with a single pencil 
stroke. 


Euler’s rule explains why the figure in Fig. 87 cannot 
be traversed with a single stroke, for there are 5 vertices, 
4 of which are the terminal points of three arcs, in other 
words, of an odd order, and thus two journeys are re¬ 
quired. 


A 


The pentagon in Fig. 88, far more complicated in 
appearance, can be traversed in a single journey. Start¬ 
ing at the point the journey would successively pass 
through points ABCDEFGBHJDKFAGHCJKEA. 


270 Mathematics and the Imagination 

Even Fig. 89 yields to a single journey, for example: 
ABCcc 'CDEee 'EFAaa 'AbBDdEJFBb 'Cd^DFfA, 

* 

In grappling with the problem of the seven bridges, Eu¬ 
ler did much more than merely solve a puzzle. He rec¬ 
ognized the existence of certain fundamental properdes 


FIG. 89. 

of geometric figures in no way dependent upon, or re¬ 
lated to, size or shape. These properties are functions 
solely of the general position of the lines and points of a 
figure. For example, on a line ABCy the fact that the point 
B lies between the points A and C is just as important as the 
fact that the line ABC is straight or curved, or has a cer¬ 
tain length. Again (Fig. 90), when an interior point of a 


Rubber-Sheet Geometry 271 

triangle is connected with a point outside, the line joining 
them must cut one side of the triangle—a fact which is 
just as important as that the angles of a triangle equal 
180®. It is the study of such properties, properties which 
remain unaffected when the figpjre is distorted^ which con¬ 
stitute the science of topology. Topology is a geometry of 
place, of position (which accounts for the name Analysis 
Situs)^ as distinguished from the metric geometries of Eu¬ 
clid, Lobachevsky, Riemann, etc., which treat of lengths 
and angles. 


Fic. 90.—The line joining the interior point A 
to the exterior point B cuts the triangle at C. No 
matter how the line is drawn, it must cut the tri¬ 
angle at some point. 

In topology we never ask “How long?” “How far?” 
“How big?”; but we do ask, “Where?” “Between what?” 
“Inside or outside?” A traveler on a strange road 
wouldn’t ask “How far is the Jones farm?” if he didn’t 
know the direction, for the answer, “Seven miles from 
here,” would not help him. He is more likely to inquire, 
“How do I get to the Jones farm?” Then, an answer like, 
“Follow this road till you come to a fork, then turn to 
your right,” will tell him just what he wants to know. 
Because this answer says nothing about distances and docs 


272 Mathematics and the Imagination 

not describe whether the path is straight or curved, it may 
seem nonmathematical, yet it bears the same relation to 
the first answer that topology bears to metric geometry. 

Topology is a non-quantitative geometry. Its propositions 
would be as true of figures made of rubber as of the ordi¬ 
nary rigid figures encountered in metric geometry. For 
that reason it has been picturesquely named Rubber-Sheet 

Geometry. 

+ 

Geometry was a very fashionable subject in the nine¬ 
teenth century. The eighteenth century had been devoted 
to the calculus and to analysis. The nineteenth belonged 
in large part to the geometers. It was inevitable that to¬ 
pology, then in its infancy, should receive its share of at¬ 
tention. 

The first systematic treatise appeared in 1847, the 
work of the German mathematician Listing, entitled 
Vorstudien znr Topologte. Topology today is still concerned 
with the same thing as when Euler invented it, although 
its language, as befits a grown-up science, has become 
more abstruse. It is now defined as the study of properties 
of spaces, or their configurations, invariant under con¬ 
tinuous one-to-one transformations; it remains the study 
of the position and relation of the parts of a figure to each 
other without regard to shape or size. Indeed, although 
topology was weaned on bridges, it now feeds on pretzels 
and doughnuts as well as upon other more curious and 
less digestible objects. 

* 

Even a glance at one or two of the theorems of this 
bizarre branch of mathematics requires the introduction 
of a new terminology. 

Poincare pointed out that the propositions of topology 


Rubber-Sheet Geometry 273 

have a unique feature: “They would remain true if the 
figures were copied by an inexpert draftsman who should 
grossly change all the proportions and replace the straight 
lines by lines more or less sinuous.” ^ In mathematical 
language, the theorems are not altered by any continuous 
point-to-point transformation. Figure 91 is an example of 
a plane triangle drawn by an expert draftsman, Fig. 92 its 
distorted surrealist twin. Nevertheless, topologically, 92 is 
a perfect copy of 91. The straight lines are curved, the an¬ 
gles changed and distorted and the lengths of the sides al¬ 
tered; but there remain geometric properties common to 
both figures. These properties which hav'e been unaffected 
by the distortion are invariants.^ 


FIG. 91.—A plane triangle. fig. 92.— Its surrealist twin. 


In Fig. 91, the point D lies between points A and C, 
and E, between A and B. In Fig. 92 that order has been 
preserved. The order of the points is, therefore, invariant 
under the transjormation which brought about this dis¬ 
tortion. Figure 91 could have been transformed in some 
other way. If it had been cut from a sheet of rubber and 
the rubber triangle twisted, stretched and distorted in 
every possible way without tearing, the order of the points 
would still remain invariant. 


2 74 Mathematics and the Imagination 

The invariants of rigid bodies under ordinary motion 
are even more familiar, but they are so much a part of our 
lives that we never give them much thought. Yet our ex¬ 
istence would be quite unthinkable without them. A rigid 
body suffers no change in size or shape when moved 
about. Its metric properties are invariant. In simple 
terms, ordinary motion has no distorting effect. The 
derby, bought in London, still fits when transported to 
New York. A measuring rod is the same in length after 
being moved from the top of a mountain to the bottom of 
the sea. A latch key fits a lock whether the door is swung 
open or shut. A steamer appears smaller on the horizon; 
but no one would maintain seriously that it shrinks as it 
steams away. And the philosopher’s armchair fits him in 
every corner of the room, regardless of how he changes its 
position or his philosophy. 

Such invariants we take for granted. To the math¬ 
ematician, however, obvious things serve as valuable 
clues, and he rarely dismisses the obvious as unimportant. 
He carefully notes that the size and shape of rigid bodies 
are unaffected by motion, and reports in technical lan¬ 
guage that the metric properties of rigid bodies are invariant 
under the transformation of motion. He then considers those 
bodies which are not rigid, and which do change in size 
and shape when moved about, and seeks their geometric 
invariants. Topology embraces these invariants and inte¬ 
grates them into a mathematical system. 

* 

According to an ancient tale, a Persian Caliph, with a 
beautiful daughter, was so troubled by the number of 
her suitors that he was forced to set up qualifying rounds 
to determine the finals. The aspirants for his daughter s 
hand were presented with a problem (Fig. 93 ): To con- 


Rubber-Sheet Geometry 275 

nect the corresponding numbers of the symmetric figures 
by lines which do not intersect. 

That was simple. But the Caliph’s daughter was not 
so easily won, for her father also insisted that the surviv- 


Fic. 93.—Connect 1 with 1, 2 with 2, and 3 with 
3 by nonintersecting lines. 


ing suitors join the corresponding numbers shown in Fig. 
94. 


FIG. 94.—Try connecting the corresponding 
numbers by noninterscciing lines. 

Unless the Caliph relented, we may assume that his 
daughter died an old maid, for this problem cannot be 
solved. Two lines may be drawn connecting any two 
corresponding numbers, but the third cannot be traced 
without crossing one of the other two. Again, we see why 
the mathematician never rejects the obvious. The prob¬ 
lem of Fig. 93 is easy. That of Fig. 94 seems just as easy, 


276 Mathematics and the Imagination 

but it is actually impossible. In what essential respects do 
the two differ? 

As early as the nineteenth century, the physicist Kirch- 
hoff recognized the importance of investigations in to¬ 
pology in order to aid in the solution of problems con¬ 
nected with the branching out and intertwining of wires 
or other conductors carrying an electric current. And, 
curiously enough, many important effects in physics have 
since been found exactly analogous to the spatial relation¬ 
ships displayed in the Caliph’s problem. 

The first real step in the systematic attack on all such 
problems was taken in the nineteenth century by the 
French mathematician Jordan. His theorem is as fund¬ 
amental and important for topology as the Pythagorean 
theorem for metric geometry. It bears no resemblance to 
any previously stated mathematical theorem. It says simply 
that Every closed curve in the plane which does not cross itself 
divides the plane into one inside and one outside^ 

Doubtless this strikes you as being either idiotic or 
wonderful. Had mathematicians labored for centuries to 
bring forth such a mouse? But Jordan’s theorem only 
seems idiotic, for when expressed in formal terms it looks 
so obvious as to be hardly worth repeating. In truth, it is 
a wonderful theorem, because it is so simple, so un¬ 
assuming, and so important. 

A curve which divides the plane into one inside and 
one outside is called simple. This is a simple curve: 


Rubber-Sheet Geometry 


277 


But these aren’t: 


FIG. 96(a). 


Nor is this: 


FIG. 97. 


The three curves in Figs. 96a, 96b, and 97 do not fall 
within Jordan’s definition of simple connectivity. The 
first has two insides and one outside, the second, several 
insides and one outside, and the area enclosed by the 
smaller curve in the third is also considered “outside,” 
and not inside. It must be conceded that Jordan’s theo¬ 
rem seems trivial when applied to easy figures. But it is 
not so easy to believe that the curve in Fig. 2, in spite of 
its tortuous appearance and labyrinthlike character, has 
only one inside. Strange as it may seem, such a curve may 
be regarded as a deformed circle. This might be demon¬ 
strated quite easily if it were made of a piece of string or a 
rubber band, for it could then be retransformed into a cir¬ 
cle merely by smoothing out the twists and kinks. In 
metric geometry, a circle is defined as the locus of all 
points equidistant from a given point, which means that 


278 Mathematics and the Imagination 

all the radii of a circle are of equal length. But in topology 
“equal length” has no significance. Thus a circle is 
thought of as a curve which has the fundamental property 
of dividing the entire plane into one inside and one out¬ 
side. Any curve, however deformed, which has this prop¬ 
erty, may be regarded as the topological equivalent of a cir¬ 
cle, It follows that every simple curve in the plane is topologically 
equivalent to a circle, 

♦ 

Jordan’s theorem, when extended to three dimensions, 
states that any closed surface^ any two-dimensional mani¬ 
fold"* which does not cross itself, divides space into an 
inside and an outside. 

Think of the room you are sitting in. The air in the 
room, all the furnishings, and you, are inside. The rest of 
the entire universe, from Vesuvius to the core of the 
earth, from Times Square to the Rings of Saturn and 
beyond, are outside. The gas in a balloon is inside, while 
everything else, in all possible directions, including the 
hopes and fears in the head of the balloonist, are outside. 
The circulatory system of the body is a two-dimensional 
manifold difficult to visualize. Nevertheless it is simply 
connected. It divides space into one inside and one out¬ 
side. Inside flows the bloodstream, outside there are the 
countless cells of the body that twine and intertwine with 
the blood vessels, and beyond, the entire universe. 

The restriction that the two-dimensional manifold 
shall not cross itself does not recall to mind any that do. 
Yet such manifolds are the center of attraction at the 
Institute for Advanced Study at Princeton where learned 
and famous mathematicians discourse strangely, almost 
like Alice’s Walrus, on pretzels, knots, and doughnuts. 

The pretzel is an object of interest, not only for its 


Rubber-Sheet Geometry 279 

gastronomical properties, but also for its topological ones. 
It is an example of a two-dimensional manifold which 
does not obey Jordan^s theorem, for it crosses itself. But 
the pretzel is too difficult for our modest mathematical 


FIG. 97(a, b, c, d)— Not the creations of Walt Disney nor Picasso’s 
impressions of the human form divine, but the objects of serious mathe¬ 
matical lucubrations at Princeton. 

equipment. We must be content with manifolds which do 
obey Jordan’s theorem. They cause enough trouble. 

Figure 98 shows a ring —the portion of the plane 
bounded by two concentric circles, A ring is a figure 


FIG. 98. 


28 o 


Mathematics and the Imagination 

which is not simply connected since its boundary consists 
of two curves rather than one. How can we differentiate 
the inside from the outside? 

Many of the difficulties we experience in explaining 
and analyzing spatial problems spring from the limita¬ 
tions of language revealed by such a question. One is apt 
to sympathize with the bibulous gendeman who was stag¬ 
gering around a cylindrical column on a Paris boulevard, 
weeping bitterly. “For heaven’s sake,” asked a curious 
passerby, “what’s wrong?” “I’m walled in,” wailed the 
toper, “walled in.” 


FIG. 99.—The man is walking counter-clockwise 
on the boundary of the curve. To the left of him is 
inside, to the right, outside. 

Purely relative terms, such as “inside” and “outside,” 
may confuse the mathematician as well as the melancholy 
boulevardier. The sole recourse is to agree upon a formal 
definition. A familiar analogy readily comes to mind: 
All parts of New York City lying on one side of Fifth 
Avenue are labelled “East,” while all parts lying on the 
opposite side are labelled “West.” 

Intuitively, everyone knows the difference between the 
inside and the outside of a circle. But can this intuitive 
notion be translated into precise terms? Since no one has 


28 i 


Rubber-Sheet Geometry 

the slightest difficulty in distinguishing between left and 
right, and the notions of clockwise and counterclockwise 
also occasion litde confusion, “inside,” and “outside” 
may be redefined in terms of left, right, clockwise, and 
counterclockwise. Thus, for instance, starting on the cir¬ 
cumference of a circle and proceeding in a counterclock¬ 
wise direction, “inside” is defined as the region to the 
left, “outside,” the region to the right. 

The application of this definition to a nonsimply-con- 
nected manifold, such as the ring, requires a slight arti¬ 
fice. By cutting any nonsimply-connected manifold, it may 
be transformed into one which is simply connected. 


Thus, while inside and outside appear to have little 
significance in relation to the ring (Fig. 98), the simple 
operation of cutting transforms the ring into a new mani¬ 
fold (Fig. 100) to which the definition is plainly appli¬ 
cable. The mathematician agrees that those regions 
which are “inside” after the ring is cut were “inside” be¬ 
fore it was cut; and those regions which are “outside” 
after the cut were “outside” before. The doughnut pre¬ 
sents the same problem in three dimensions as the ring in 
two. “Is the hole part of the inside or the outside of the 
doughnut?” If we relied entirely on the experience gained 


282 


Mathematics and the Imagination 

at the breakfast table, we might assert that the hole is in¬ 
side. But the few facts thus far gathered would give rise to 
some doubts. It turns out that the hole inside the doughnut 
is outside. Of course, the first impression was not an optical 
illusion. The conclusion that the hole is outside is purely 
conceptual and must be regarded as the logical con¬ 
sequence of certain definitions. 


FIG. 100(a).—A triply-connected curve. It takes 
three cuts to make it simply connected. 

As in two dimensions any simply-connected manifold 
is the equivalent of a circle, so in three dimensions any 
simply-connected surface is the equivalent of a sphere. By 
a gradual deformation, without tearing, any simply-con¬ 
nected three-dimensional object can be transformed into 
a sphere. A doughnut cannot be so transformed whence it 
follows that a doughnut is not simply connected. But an 
operation similar to the one performed on the ring—a 
simple cut—turns the doughnut into a sausage, which is 


FIG. 101.—^The doughnut becomes a sausage. 


Rubber-Sheet Geometry 283 

simply connected and the topological equivalent of a 
sphere. 

The pretzel, together with the other objects shown (Fig. 
102 ) are some of the more difficult manifolds studied in 
topology. None is simply connected, none can be trans¬ 
formed into a sphere. But by a number of cuts, similar to 
the ones performed on the ring and doughnut, these com¬ 
plex manifolds may be transformed into simply-connected 
configurations. Thus, with a sufficient number of cuts it 


FIG. 102— Weird topological manifolds—ex¬ 
alted relatives of the pretzel. 


284 Mathematics and the Imagination 

is possible to change even the most tortuous pretzel into 
the equivalent of a sphere. 

The number of cuts necessary to effect such a trans¬ 
formation is not a matter of chance, but perfectly determi¬ 
nate, and depends upon the connectivity of the manifold. 
A general rule may be formulated which will apply to 
both fantastic objects and easy ones. As in all math¬ 
ematical inquiries, only such a rule will reveal the imder- 
lying principle; accordingly, topologists do not stop with 
the consideration of three-dimensional manifolds, how¬ 
ever complicated and forbidding. They go far beyond 
the reaches of the imagination and devise theorems valid 
even for r?-dimensional pretzels. 

* 

One of the curios of topology is the Mobius strip. A 
Mobius strip is easily constructed. Take a long rectangle 
{ABCD) made of paper (Fig. 103), give it a half-twist and 
join the ends so that C falls on B, and D on A (Fig. 104). 
This is a one-sided surface, and if a painter agreed to 


FIGS. 103, 104.—The Mobius strip—a one-sided, 
two-sided surface. 


Rubber-Sheet Geometry 285 

paint only one side of it, his union would interfere be¬ 
cause in painting one side he would be painting both 
sides. ^ If the strip had not been twisted before gluing the 
ends together, a cylinder would have resulted—which 
is evidently a /ttjo-sided surface. However, the half-twist 
eliminated one of the sides. Incredible? You may con¬ 
vince yourself. Draw a straight line down the center of 
the strip, extending it until you return to the point at 
which you started. Now separate the ends of the strip 
and you will find that both sides are covered by the straight 
line even though in drawing it you did not cross any 
edges. Had you followed this same procedure with a 
cylinder, you would have had to cross over the edge, to get 


FIG. 105. 


from one side to the other. Although every dictate of 
common sense indicates that the strip with the half twist 
has two bounding edges, we have proved it has only one. 
For any two points on the Mobius Strip may be con¬ 
nected by merely starting at one point and tracing a path 
to the other without lifting the pencil or carrying it over 
any boundary. 

There is a good bit of amusement and interest in mak¬ 
ing such a strip for yourself. When you have studied the 
properties described, cut it in half with a pair ol stissois 
along a line drawn down the center. The result will be 


286 


Mathematics and the Imagination 

astounding I'And you can continue twisting and cutting a 
few more times for still further surprises. 

Two interlocking iron rings are shown in Fig. 105. It 
is perfectly evident that they cannot be separated unless 
one of the rings is broken. But being perfecdy evident, 
how shall we prove it? Before topology was invented, 
none of the existent tools of mathematics was suited for 
such a job. Only the creation of special tools made it 
possible to give an analytic proof of so evident a fact. 

Here is a similar problem. Tie a piece of string to each 
of your wrists. Tie a second piece of string to each of the 
wrists of a partner in such a way that the second piece 
loops the first (Fig. 106). 


Do you think you can separate yourself from your partner 
without tearing the string? Although this looks like the 
problem of separating the two rings, which we agreed is 
impossible, this feat can be accomplished. Try working it. 

With a topologist and a pair of scissors handy (for 
accidents), you might try removing your vest without re- 
moving your coat. No fourth dimension is required. Merely 


Rubber-Sheet Geometry 287 

remember the conditions of the problem. The coat may 
be unbuttoned, but at no point during the removal of 
the vest may your arms slip out of the coat sleeves. 


FIG. 107. —The above portrays the trade-mark used by 
a well-known brewer. The three rings have this strange 
relation to one another. If any one ring is removed, the 
other two are found not to be joined. Thus no two rings are 
joined, but all three are. To put it more simply, no two rings 
are joined, but each holds the other two. 

Topology is one of the youngest members of the math¬ 
ematics family, but still it claims its problem child. While 
some mathematicians have been content to concentrate 
on the pretzels, knots, and doughnuts of analysis situs, 
a determined band of mathematical pediatricians have 
focused their attention exclusively on the Four-Color 
Problem. For a short while in the nineteenth century, it 
was thought that the child had been cured and its prob¬ 
lem unraveled, but these were vain hopes, and the four- 
color puzzle continues to baffle the leading topologists. 

At one time or another everyone has had experience in 
map coloring. Maps depicting the Holy Roman Empire, 
the cotton states of the South before the Civil War, or 
the rescrambling of Europe by the Treaty of Versailles, 
are painfully outlined every school day. Recently the 


288 Mathematics and the Imagination 

business has become more hectic than ever. Stout crayons 
and a good eraser must always be at hand. Students 
discover early in their cartographical career that if a 
map is to be colored, countries having a common bound¬ 
ary, such as France and Belgium, must be colored dif- 
ferendy so that they can be distinguished at a glance. 
The generalization of that idea led to the question “How 
many colors are necessary to color a map, with any num¬ 
ber of countries, so that no two countries which adjoin 
on a frontier shall have the same color?” This problem 
had troubled cartographers for many years. 


FIGS. 108, 109, 110. 


Figure 108 illustrates an island in the sea. Each of two 
countries owns part of the island. Three colors are re¬ 
quired for this map—one for the sea and one for each of 
the two countries. 

For depicting the island in Fig. 109 four colors are re¬ 
quired. The map with more regions, as in Fig. 110, also 
requires only four colors. The reason is not hard to find, 


Rubber-Sheet Geometry 289 

for the country in the center, marked 1, may be the same 
color as the sea without causing any confusion. 

Figures 111 and 112 respectively require three and 
four colors, even though they contain many more regions 
than any of the maps above. 


FIG. 111.—An island 
owned by five countries, 
requiring only three 
colors to map. 


Fic. 112.—An island wilh nine¬ 
teen counties. Only four colors are 
needed to map it. 


It is quite natural to suppose that as maps grow more 
complicated, depict more countries, additional colors will 
be required to differentiate any two adjoining territories. 
Strangely enough, mathematicians have thus far found it 
impossible to construct a plane map for which four colors 
would not suffice. At the same time no one has been able 
to prove that four colors would be sufficient for any possible 

map. 

The classical problem is concerned with the number of 
colors required to map any number of regions on a 
sphere. Though it has been shown that four colors are 
necessary, and five colors sufficient, the standard math¬ 
ematical requirement, which is to find the one condition 
both necessary and sufficient, has not yet been satislied. 


290 Mathematics and the Imagination 

Paradoxically, the problem has not been solved for a 
sphere or for a flat surface, although it has been solved 
for much more complicated surfaces, like the torus (dough¬ 
nut), or the sphere with handles. 

A. B. Kempe, English mathematician and barrister, au¬ 
thor of the celebrated litde book with the provocative title. 
How to Draw a Straight Line, offered a proof in 1879 that 
four colors are both necessary and sufficient for the con¬ 
struction of any map on a sphere. Unfortunately, Kempe’s 
proof is now known to contain a fatal logical error. 

That five colors are sufficient for any map drawn on a 
sphere, or on a plane, is in itself remarkable. The proof 
rests on Euler’s even more remarkable theorem about sim¬ 
ply-connected solids that states that the sum of the vertices 
and faces of any such solid is equal to the sum of the 
edges plus two: 

V + F = E -\-2 

Euler’s theorem is the simplest universal statement 
about solids. The underlying idea was familiar to Des¬ 
cartes, but very likely his proof was unknown to Euler. 

We know that any three-dimensional solid which is 
simply connected is the topological equivalent of a 
sphere. From this fact and from Euler’s theorem, there is 
one interesting consequence: Consider a hollow cube 
made of rubber. It is bounded by six faces, twelve edges, 
and eight vertices. Inflate this cube until it resembles a 
sphere. The faces of the cube are then regions of the 
sphere; the edges cf the cube, the boundaries of these 
regions; and the vertices, points where three regions meet. 
The exercise of coloring the sphere is thus seen to be gov¬ 
erned by Euler’s theorem. For, if each region represents a 
country; each curved line, the boundary between two 


Rubber-Sheet Geometry 291 

countries; and each vertex the juncture of three countries, 
the number of countries plus the number of points at which 
three countries meet is equal to the number of boundaries 
+ two. In this way we see how Euler’s theorem is ex¬ 
tended to curved figures. 

For a solid with a hole, such as a doughnut, the theo¬ 
rem fails. Indeed, it fails for any nonsimply-connectcd 
solid. In short, Euler’s theorem is applicable in topology 
only when each of the faces of the figure is simply con¬ 
nected, and the entire surface is simply connected. 

* 

Of those who have made essential contributions to to¬ 
pology, L. J. Brouwer, the Dutchman, is one of the great¬ 
est. Particularly in the theory of point sets, Brouwer’s to- 


FIC. 113.—Ai the points 1 and 2, all three coun¬ 
tries, A, B, C, meet. 


pological theorems have proven of signal importance. But 
it is not his technical contributions which concern us here. 
In 1910 he published a problem, based on an idea of 
the Japanese mathematician Yoncyama, which illustrates 
beautifully the difficulties and subtleties of topology. The 
solution of this problem will perhaps leave you dissatis¬ 
fied, but it cannot fail to challenge your imagination. 


20 


202 Mathematics and the Imagination 

Figure 113 is a map of three countries. The points 
marked 1 and 2 are rather singular, for at both of these 
points all three countries meet. Manifesdy such points 
are scarce on any map, no matter how complicated, for 
there are not many geographical instances of three coun- 


A B C 


FIG. 114.—Countries A, B, C are separated by 
unoccupied corridors and D is unclaimed land. 


tries meeting at a single point. But even if there were many 
such points, if it were a very queer map, their number 
would always be small compared to the totality of points 
along all the boundary lines. It is reasonably certain that 
a boundary point, chosen at random, on any map, will be 
the meeting place of at most two countries. 

Now Brouwer concocted an example, at first sight 


Rubber-Sheet Geometry 293 

wholly unbelievable, of a map of three countries, on 
which every single point along the boundary oj each country is a 
meeting place of all three countries.^ 

Consider the map in Fig. 114. 


FIG. 1 15. 


None of the nations borders on any of its neighbors, 
and the white unmarked portion of the map is intended 
to represent unclaimed territory. In keeping with the 
spirit of Lebensraumy Country A decides to extend its 
sphere of influence over the unclaimed land by grabbing 
a substantial portion. Accordingly, it sends out a corridor 
which does not touch the land of either of its neighbors, 
but leaves no point of the remaining, unclaimed land 
more than one mile from some point of the enlarged 


294 Mathematics and the Imagination 

Country A. It has now spread itself over the map as in 
Fig. 115. 

Country B, instead of applying sanctions, decides to 
grab a share before it is too late. With becoming re¬ 
straint, as well as with an eye to its neighbors’ greater 
strength, B extends a corridor to within a half-mile of 
every point of the remaining unclaimed land. This cor¬ 
ridor alters the map like this: 


A B C 


FIG. 116. 

Of course Country C will not be left behind. It builds 
a corridor which approaches within a third of a mile of 
ever>' point of the remaining unclaimed land but, just as 
the other two corridors, touches on no country but its 
own. The new map is shown in Fig. 117. 


Rubhfr-Sheel Geometry 


■^95 


By now cvcrsoric should bo quite' tenucni. On tlu' eon- 
trary; this is only the beginning. C’.ountry .1 has the slioi t- 
est corridor. Intolerable state ol atVairs which must be 
remedied sojoit. It decides upon a new ce>nidt)r to extend 
into the remaining territory which shall approach e\ery 
point of that territory within a ejuarter a mile (Mg. 


FIG. 117. 

Country B follows with a corridor which api)roa( hes 
each unocciqjied pinni within a filth ol a mile. C-ountiy 
Os corridor comes within a sixth of a mil<‘ ol each un¬ 
occupied point, and the merry-go-rouiul ito<-s nnind. 
More and more corridors! Never any (oniact lj(‘iw(‘en 
them, although they continue to come < 1 os{T and elost r, 


\ 1 1 
75 b’ 9’ ■ 

of a mile. 


1 -- 

ru'd’ • * • 1 0oO’ • • * 


1 ' , 
lOooooo’ • • • 


296 Mathematics and the Imagination 

We may assume, in order that this feverish program 
shall be completed in a finite length of time (“Two-Year 
Plan”), that the first corridor of Country A took a year 
to build, the first corridor of 5 a half-year, the first 
corridor of C a quarter-year, the second corridor of 


A B C 


FIG. 118. 


Country A an eighth of a year, and so on; each corridor 
took exactly half as long to build as its immediate pred¬ 
ecessor. 1 he total elapsed time then gives rise to the 
familiar series 


+ i + § + i + ^ + 


Thus, at the end of two years, the once unclaimed ter¬ 
ritory has been entirely occupied, and not a speck of it 


Rubber-Sheet Geometry 297 

remains unclaimed. Over each square inch there flies 
the flag of one of the three countries, either A, B, or C. 

What of the new map which is to depict these bound¬ 
aries? Actually it is impossible to draw but suppose we 
try to conceive what it would look like if it could be 
drawn. This conceptual map is put together out of pieces 
of sober mathematics and sheer fancy. For every single 
boundary point on the map will be a meeting place, a boundary 

point, oj not two, but oj all three countries! 

* 

In an apparently dynamic, incessantly changing world, 
one of perpetual novelty, the search for things which do 
not change constitutes one of the principal objectives of 
science. Philosophers since pre-Socratic times have been 
rummaging about for the unchanging essence of reality. 
Today, that is the job of the scientist. 

In topology, as in other branches of mathematics, it 
takes the form of a search for invariants. Repeatedly, in 
the course of that search, the neccessity arises for aban¬ 
doning intuition, for transcending imagination. The in¬ 
variants of 4, 5, 6, and n dimensions are purely concep¬ 
tual. To fit them into our lives, to find use for them in the 
laboratory, to shape them for duty in the applied sciences 
seem impossible. There is nothing in experience to com¬ 
pare them with, not even a dream in which they could 
play a part. 

Nevertheless, what is gathered by the mathematicians, 
slowly, painfully, bit by bit, in the weird world of beyond- 
the-make-believe, is in reality a part of the world of every¬ 
day, of tides, of cities, and of men, of atoms, of electrons, 
and of stars. All at once, what came from the land of n 
dimensions is found useful in the land of three. Or, per¬ 
haps, we discover that after all we live in a land of n di- 


298 Mathematics and the Imagination 

mensions. It is the reward for the courage and industry, 
for the fine, untrammeled, poetic, and imaginative sense 
common to the mathematician, the poet and the philos¬ 
opher. It is the fulfillment of the vision of science. 


FOOTNOTES 

1. For two distinct journeys, the pencil must be lifted from the 
paper once; for three distinct journeys, twice; for n distinct 
journeys, n — \ times.—P. 267. 

2. Poincare, Science and Hypothesis. —P. 273. 

3. Invariant is a name invented by the English mathematician, 
Sylvester, who was called the mathematical Adam because of the 
many names that he introduced into mathematics. The terms “in¬ 
variant,” “discriminant,” “Hessian,” “Jacobian” are all his. In 
fact, he employed Hebrew characters in some of his mathematical 
papers, which, according to Cajori, caused the German mathe¬ 
matician Weierstrass to abandon him in horror. 

Invariants arise in other branches of mathematics. The theory 
of algebraic invariants, developed by Clebsch, Sylvester, and 
Cayley, lurks in the memory of everyone who studied quadratic 
equations. For example: The discriminant of the quadratic equa¬ 
tion — c = 0 is the classical instance of an algebraic 

invariant. A quadratic equation under a linear transformation 
maintains unchanged a certain relation between its coefficients, 
expressed by the discriminant, A* — Aac. The discriminant of the 
transformed equation remains equal to the discriminant of the 
original equation multiplied by a factor which depends only upon 
the coefficients in the transformation.—P. 273. 

4. See Chap. 4, p. 119 and footnote 4, p. 154.—P. 278. 

5. Osgood, Advanced Calculus. —P. 285. 

6. We avail ourselves here of the version of the problem given by 
the distinguished Viennese mathematician, the late Hans Hahn, 
because it is more satisfying and clearer than Brouwer s own 
statement.—P, 293. 


Change and Changeability—The Calculus 


The evfr-whirling whefle 

Of Change, the which all morlall things doth sway. 

People used to think that when a thing changes, it 
must be in a state of change, and that when a thing moves, 
it is in a state of motion. This is now known to be a mis¬ 
take. 

-BERTRAND RUSSELL 


‘‘Everyone who understands the subject will agree that 
even the basis on which the scientific explanation of 
nature rests is intelligible only to those who have learned 
at least the elements of the differential and integral cal¬ 
culus ” These words of Felix Klein, the distin¬ 

guished German mathematician, echo the conviction of 
everyone who has studied the physical sciences. It is 
impossible to appraise and interpret the interdependence 
of physical quantities in terms of algebra and geometry 
alone; it is impossible to proceed beyond the simplest 
observed phenomena merely with the aid of these mathe¬ 
matical tools. In the construction of physical theories, 
the calculus is more than the cement which binds the 
diverse elements of the structure together, it is the imple¬ 
ment used by the builder in every phase of the construc- 

tion. . 

Why is this branch of mathematics peculiarly suited 

for the precise formulation of natural phenomena? What 

299 


300 Mathematics and the Imagination 

powers can be attributed to the calculus that are not also 
shared by geometry and algebra? 

Our most common impression of the world, whether 
erroneous or not, is its ever-changing aspect. Nature, as 
well as the artifacts we have invented to master it, seems 
to be in constant flux. Even the “absolutes”—space and 
time—contract and expand incessandy. Night and day 
repeatedly flow into one another, setting forth the vicis¬ 
situdes of the seasons. Everywhere there is motion, flow, 
cycles of birth, death and regeneration. Everywhere the 
pattern moves. 

For some strange reason, the subjects already con¬ 
sidered, the many domains of mathematics already sur¬ 
veyed, have neglected this dynamism. With the exception 
of the exponential function, we have not spoken of the 
rate of change of a known or unknown quantity. Indeed, 
our equipment thus far could not have handled this con¬ 
cept. Fortunately, every problem was essentially static. 
Four-dimensional and non-Euclidean geometry treated 
of unchanging configurations; puzzles and paradoxes 
were solvable with the aid of ingenuity, logic and static 
arithmetic; topology sought out the invariant aspects of 
geometric forms independent of size and shape; and the 
concepts developed in the chapters on Pie, the Googol, 
and Probability were, with one or two exceptions, free of 
the ingredient of change. The conclusion is inevitable 
that the one indispensable means of attacking the vast 
majority of phenomena has been neglected—that our 
investigation has been confined to a peripheral aspect of 
the world scene. 

* 

The word “calculus” originally meant a small stone 
or pebble; it has acquired a new connotation. The cal- 


Change and Changeability—The Calculus 301 

cuius may be regarded as that branch of mathematical 
inquiry which treats of change and rate oj change. The 
comfort with which one rides in an automobile is made 
possible, in part at least, by the calculus. While the plan¬ 
ets would continue in their paths without the calculus, 
Newton needed it to prove that their orbits about the 
sun are ellipses. Shrinking from the celestial to the atomic, 
the solution of the very same equation used by Newton 
to describe the motion of the planets determines the tra¬ 
jectory of an alpha particle which bombards an atomic 
nucleus. By means of the formula which relates the dis¬ 
tance traversed by a moving body to the time elapsed, the 
velocity of the body, as well as its acceleration, at every 

instant of time is determined by the calculus. 

Each of the above illustrations, whether simple or com¬ 
plex, involves change and rate of change. Without their 
exact mathematical enunciation none of the problems 
described would have meaning, much less be solvable^ 
Thus, a mathematical theory has been created which 
takes cognizance of immanent and ubiquitous changes ol 
pattern and which undertakes to examine and explain 

them. That theory is the calculus. 

+ 

But had we not previously declared quite fervently that 
we live in a motionless world? And had we not show n at 
great length, by employing the paradoxes of Zeno, that mo¬ 
tion is impossible, that the flying arrow is actually at rest. 
To what shall we ascribe this apparent reversal of position. 

Moreover, if each new mathematical invention rests 
upon the old-established foundations, how is it possible 
to extract from the theories of static algebra and static 
geometry a new mathematics capable of solving prob¬ 
lems involving dynamic entities? 


302 Mathematics and the Imagination 

As to the first, there has been no reversal of viewpoint. 
We are still firmly entrenched in the belief that this is a 
world in which motion as well as change are special cases 
of the state of rest. There is no state of change, if change 
implies a state qualitatively different from rest; that which 
we distinguish as change is merely, as we once indicated, 
a succession of many different static images perceived in 
comparatively short intervals of time. An example may 
help to clarify the idea. Although in the cinema, a series 
of static pictures are projected upon the screen, one after 
the other, in rapid fashion, each picture differing only 
slightly from the one preceding it, there is not the slightest 
doubt in the mind of even the most intelligent moviegoer 
that motion is being portrayed on the screen. A completely 
convincing display of change is presented by a series of 
wholly static images. Let us pursue this with a more tech¬ 
nical illustration. A steel rod, clamped in a horizontal 
position at one end, has a weight attached at the other. 
This system being at rest, it is said that the set of elements 
composing it are in equilibrium. If, when we next ex¬ 
amine it, after some interval of time, we observe the same 
arrangement, the rod bent by the same amount, it is 
apparent that there has been no change. If, however, 
there is a new position of the rod, obviously a change has 
taken place. It is certain that the equilibrium could only 
have been disturbed and the posit on of the rod altered 
by a change in the attached weight. It is not hard to 
convince ourselves that additional weight would bend 
the rod further and that such additions, if made grad¬ 
ually, and just as quickly as motion pictures are projected 
on the screen, would give the impression that the rod is 
in motion. On the other hand, if we are aware of these 
additions of weight, we conclude that what we really have 


Change and Changeability—The Calculus 3^3 

observed is not motion but merely a correlation of amount 
of bend with degree of weight and that for different 
weights there are different positions of the rod. 


119 —Each addition to the weight bends 
the rod a little further. 

Intuitively convinced that there is continuity in the be¬ 
havior of a moving body, since we do not actually see the 
flying arrow pass through every point on its flight, there 


304 Mathematics and the Imagination 

is an overwhelming instinct to abstract the idea of motion 
as something essentially different from rest. But this ab¬ 
straction results from physiological and psychological 
limitations; it is in no way justified by logical analysis. 
Motion is a correlation of position with time. Change is 
merely another name iov function, another aspect of that 
Scime correlation. 

For the rest, the calculus, as an offspring of geometry 
and algebra, belongs to a static family and has acquired 
no characteristics not already possessed by its parents. 
Mutations are not possible in mathematics. Thus, inevi¬ 
tably, the calculus has the same static properties as the 
multiplication table and the geometry of Euclid. The 
calculus is but another interpretation, although it 
must be admitted an ingenious one, of this unmoving 
world. 

♦ 

The historiccil development of the calculus did not fol¬ 
low such clear lines. The philosophic discussions as to the 
meaning of the subject came only after its usefulness had 
been indisputably established. Before that philosophers 
would not have deigned it worthy of attack. Unfortu¬ 
nately we cannot recount (though it would be amusing) 
the pitfalls which every philosopher and mathematical 
analyst from Newton to Weierstrass dug for his adver¬ 
saries—and promptly fell into himself. We may, however, 
sketch the steps that preceded the theory as it is accepted 
today. 

The calculus does not differ from other mathematical 
theories; it did not spring full-grown from the genius of 
any one man. Rather was it developed from a consider¬ 
ation of numerous questions essayed and successfully an¬ 
swered by the predecessors of Newton and Leibniz. “Every 


Change and Changeability—The Calculus 305 

great epoch in the progress of science is preceded by a 
period of preparation and prevision . . . The concep¬ 
tions brought into action at that great time had been long 
in preparation.’^ ' 

The advent of analytic geometry furnished a powerful 
stimlus to the invention of the calculus, for the pictorial 
representation of a function revealed many interesting 
features. Kepler had noticed that as a variable quantity 
approaches its maximum value, its rate of change be- 


FiG. 120.—The rate of change of a variable 
quantity is smaller at a maximum point than else¬ 
where. 

comes less than at any other value. It continues to choke 
off until at the maximum value of the variable, the rate of 
change is zero. 

In the above diagram, the values assumed by a vari¬ 
able quantity are measured by the distance from the 
straight line (the ;c axis) to the curve. The maximum 
value of the variable quantity (the greatest distance from 
the X axis to the curve) is attained at the point labeled A\ 
when moving slightly either to the right or to the left of A, 
for instance to the point the change in the value of 


3 o 6 Mathematics and the Imagination 

the variable quantity is very small, and is measured by 
P.U we move to the right or left of some other point E the 
same distance that we moved from A to B, so that the 
distance EF is equal to the distance ABj the change in 
the value of the variable quantity in the neighborhood of 
E is measured by Q,. But obviously, this second width, Q, 
is greater than the first width, P. In this, which is Kepler s 
contribution, we have a geometric illustration of the 


FIG. 121.—Using the scale, the perimeter of the 
rectangle is clearly 4 units. 

principle of maxima and minima: the rate of change of 
a variable quantity is smaller in the neighborhood of its 
maximum (and minimum) value than elsewhere. In fact, 
at the maximum and minimum values, the rate is zero. 

Pierre de Fermat, who shares with Descartes the dis¬ 
tinction of discovering analytic geometry, was one of the 
first mathematicians to devise a general method appli¬ 
cable to the solution of problems involving maxima and 
minima. His method, used as early as 1629, is substan- 


Change and Changeability—The Calculus 3*^7 

tially that applied today to problems of this type. Let it 
be required to draw a rectangle such that the sum of the 
sides is four inches and such that the area* shall be a 
maximum. If we denote one side of the maximum rec- 


Fic. 122.—The perimeter of each of the seven 
rectangles viz. AAAA, BBBB, CCCC, etc., is the 
same. But obviously the rectangle of maximum 
area is the square DDDD. 


tangle by ;r, the adjacent side, as may be seen from 
Fig. 121, will be 2 - x; and the area of the rectangle will 
be x{2 — x). If the side x is increased by a small amount 
E, the side 2 — x will have to be diminished by E in order 
to maintain a constant perimeter. The new area will 
then be (x + E)(2 - x - E). Since the original area 

•The area of a rectangle is the product of two nclj.tccm sidc^. 


3o8 Mathematics and the Imagination 

was a maximum, this slight alteration in the relation of 
the sides can have produced only a slight change in the 
area. Thus, considering the two areas approximately equal, 
we have 

x{2 -x) ^{x + E){2 -x-E) 
whence 2x — — 2x — x^ — Ex + 2E — Ex — E^. 

Subtracting 2x — x^ from both sides of this equation and 
factoring: 

^ = 2E-2Ex- E? 

0 = E(2 -2x - E). 

But E is not equal to zero, therefore the other factor 
{2 — 2x — E) must be zero: 

0 = 2 - 2x - E. 

As smaller and smaller values are taken for E, (i.e., as 
the altered rectangle approaches closer and closer to the 


FIG. 123.—^The curve is a parabola representing the 
areas of all rectangles whose perimeter is 4 units long. 

Erect a perpendicular at any point n along the x 
to the curve. The length of this perpendicular will 
area of the rectangle, one side of which equals n. The 
ma.\imum area corresponds to the point A on the graph, 
i.e., the perpendicular erected at x = 1. Thus the tcc- 
tangle of maximum area, with a perimeter = 4, has a side 
= 1 and is therefore a square. 


Change and Changeability—The Calculus 309 

original maximum rectangle) the expression on the right- 
hand side of the equation approaches closer and closer 
to the expression obtained by setting E equal to zero, 
namely, 2 — 2x. Solving this resulting equation: 

0 = 2 - 2.V 

we find that: ;c = 1 ; or, in terms of the original problem, 
the rectangle with the maximum area is a square. 

It is well to note that the area of the rectangle is a func¬ 
tion of the lengths of the sides, and this function can be 

portrayed by a curve. (Fig. 123.) 

The highest point of this curve is at .v = 1. This is the 
maximum of the function. To use a crude analogy, since 
this point is neither “uphill” nor “downhill, a small steel 
ball would be in equilibrium, or a ruler could be balanced 
at this point. If we think of a straight line being “bal¬ 
anced” at this point, such a line would be called the ian- 
gent to the curve."^ The interesting fact is that the tangent to 
a curve at its maxima and minima points will ahvays be 
horizontal (Fig. 124). To this idea, so important in the 

calculus, we shall return later. 

Sir Isaac Newton and Baron Gottfried Wilhelm von 


Fio. 124.—The horizontal lines arc tant'ciu to 
the relative maxima and minima of the cur\e. 


31 o Mathematics and the Imagination 

Leibniz share the credit in the history of mathematics as 
independent discoverers of the differential and integral 
calculus. Their conflicting claims gave rise to a contro¬ 
versy which raged in Europe for more than a century. 
This monumental invention made simultaneously by 
these men now commends itself to our attention. 

* 

A tiny flame, lit by Archimedes and his predecessors, 
burst forth into new brilliance in the intellectually hospi¬ 
table climate of the seventeenth century to cast its light 
over the entire future of science. The fertile concept of 
limit revealed its full powers for the first time in the de¬ 
velopment of the differential calculus. 

We are already acquainted with the limit of a variable 
quantity. The sequence of numbers 0.9, 0.99, 0.999, 

0.9999, . . . converges to the limiting value 1. The series 

l+i + i + J-f-jig--!-. . . converges to the limiting 
value 2. Nor are geometric examples unfamiliar. If a reg¬ 
ular polygon is inscribed in a circle, the difference be¬ 
tween the perimeter of the polygon and the circumference 
of the circle can be made as small as one wishes merely by 
taking a polygon with a sufficient number of sides. The 
limiting figure is the circle, the limiting area, the area of 
the circle. 

In these instances, there is no difficulty in determining 
the limit; this is the exception, however, not the rule. 
Usually, a formidable mathematical procedure is re¬ 
quired to determine the limit of a variable quantity. Con¬ 
sider this; Draw a circle with a radius equal to one. In 
it inscribe an equilateral triangle. In the triangle inscribe 
another circle; in the second circle, a square. Continue 
with a circle in this square, and follow with a regular 
five-sided figure in the circle. Repeat this procedure, each 


Change and Changeability—The Calculus 311 

time increasing the number of sides of the regular polygon 
by one. 

At first glance, one might suppose that the radii of the 
shrinking circles approach zero as their limiting value. 


FIG. 125.—The diminishini? radii approach a 
limit approximately ^ that of the radius of the 

first circle. 


But this is not so; the radii converge to a definite limiting 
value different from zero. As an explanatory clue, it need 
only be remembered that the shrinking process itself ap¬ 
proaches a limit as the circles and inscribed polygons be¬ 
come approximately equal. The limiting value of the radii 

is given by the infinite product: 

Radius = cos^ X cos^ X cos^ X ... X cos^— 


312 Mathematics and the Imagination 

Closely related to this problem is the one of circum¬ 
scribing the regular polygons and the circles instead of 
inscribing them. 


FIG. 126.—The increasing radii approach a limit 
approximately 12 times that of the original circle. 


Here it would seem that the radii should grow beyond 
bound, become infinite. This, too, is deceptive, for the radii 
of the resulting circles approach a limiting value given by 
the infinite product: 


Radius = 


Interestingly enough, the two limiting radii are so related 
that one is the reciprocal of the other. 

So much for the limit of a variable quantity. Let us now 
turn to the limit of a function, recalling briefly the meaning 


Change and Changeability 'The Calculus 3*3 

of function.* It is often found that two variable quantities 
are so related that to each value of one there corresponds 


* Though we have done this before, the notion of function is so im¬ 
portant, so all-pervading in mathematics, that it is worth going over 

again. 


314 Mathematics and the Imagination 

a value of the other. Under this condition, the two var¬ 
iable quantities are said to be functions of one another, or 
functionally related. Thus, the force of attraction (or re¬ 
pulsion) between two magnets is a function of the distance 
between them. The greater the distance between the mag¬ 
nets, the less the force; the less the distance, the greater 
the force. If the distance is permitted to assume arbitrary 
values, it may be considered as an independent variable. The 
force then becomes the dependent variable^ dependent upon 
the distance (and the functional relation) and is uniquely 
determined by assigning values to the independent vari¬ 
able. In functional relations, the letter usually denotes 
the independent variable, the letter^ the dependent var¬ 
iable. The dependency is a function of x” is written 
symbolically: 

y = / W • 

The graphic representation of a point has been dis¬ 
cussed in the section on analytic geometry. The equation 
y = J{x) determines a value of^ for every value of jr. Each 
pair of values which satisfies this equation is considered 
as the Cartesian co-ordinates of a point in a plane; the 
curve depicting the function is composed of all such points. 

In discussing the concept “limit of a function,” let us 

study specifically the function^ = represented graph¬ 
ically in Fig. 128. 

The value of the function at the point x = \ is y = 
Jih) — 2. This value is graphically represented by the 
distance from the point on the x axis, § unit to the right 
of the origin, to the curve. Likewise, the value of the func¬ 
tion at each point along the curve is represented by its 
distance from the x axis. 


Change and Changeability—The Calculus 315 

For the function v = take two neighboring points, 

X 

Y 


a: = 5 , and x = ^. As the independent variable moves 
along the x axis from the point x = \ to x = 2 ^ 
pendent variable is “forced’* along the curve from the 
pointy = /(^) = 4 to^ = /(a) = 2. In other words, as 
the independent variable x approaches as its limit the 
value the dependent variable, the function, approaches 
as its limit the value 2. Generally, as an independent 
variable x approaches a value d, its dependent variable r 
(the function of x) approaches a value B. Thus, the limit of 


316 Mathematics and the Imagination 

f{x), as X approaches is B. This is what is meant by 
“limit of a function.’’ 

Recalling the example of the steel rod flexed under a 
weight, we may construct a parallel dictionary of terms. 


MATHEMATICS 

Independent variable, x. 

Dependent variable, _>». 

Function is the relation be¬ 
tween X and^. 

Increase or decrease of x 
(i.e., change). 

Increase or decrease ofy 
(i.e., change). 

Limiting value of y (the 
function of x) equals a 
number. 


PHYSICS 

Amount of weight. 

Amount of bend of steel rod. 

Function is the relation be¬ 
tween the'weight and the 
degree of bend. 

Addition or diminution of 
weight (i.e., change). 

Increase or decrease in the 
degree of bend of the steel 
rod (i.e., change). 

Limiting value of degree of 
bend (function of the 
weight) equals a position. 


With the concepts limit, function, and limit of a func¬ 
tion in mind, there remains to define an idea embracing 

all three—“rate of change.” 

Consider the determination of the speed ^ of a moving 
body at a given instant of time. A bomb is dropped from 
a stationary airship at an altitude of 400 feet. Five seconds 
will elapse before it hits the ground. Its average speed is 


thus = 80 feet per second. Hence, the average 

5 seconds 

rate of change of distance with respect to time is 80. We 
are aware, however, from the most elementary knowl¬ 
edge of physics that a body gathers speed as it falls. 


Change and Changeability—The Calculus 31 7 

Throughout the fall the bomb was not moving at a con¬ 
stant rate of 80 feet per second; the speed with which it 
fell varied from point to point, increasing at each suc¬ 
cessive instant (disregarding air resistance). Suppose, for 
the sake of simplicity, we limit our interest to the speed of 


FIG. 129.—The diagram shows the di'>taiice 
covered by a falling projectile at the end of I, 2, 3, 
4, and 5 seconds. 


218 Mathematics and the Imagination 

the bomb at the exact moment of striking the ground. 
Evidently, its speed during the last second before striking 
will give a fair approximation to its speed at the instant of 
striking. The distance covered during this last second be¬ 
ing 144 feet, the rate of change of distance with respect to 
time is 144. If we now take smaller and smaller intervals 
of time, we may expect to obtain closer and closer ap¬ 
proximations to the speed of the projectile at the moment 
of impact. In the last half second, the distance covered 
was 76 feet, so that the speed was 152 feet per second. The 
table lists the intervals of time, the distance covered in 
those intervals, and the average speed over each interval. 
It is readily seen that as the interval of time approaches 
zero, we obtain the approximation to the speed of the 
body at the instant it hits the ground. 


Interval of time 

1 

§ 

J 

1 


A 

A 

ihxs 

1 6 ^ 

1 a i o'tf 

in seconds. 
Distance covered 

144 

76 

39 

19i 


2m* 

Ttoo 


in feet 

Average speed in 
feet per second 

144 

1 

152 

156 

158 

159 

159J 

1593 

t59A 

159^ 

i59,vy^ 


These approximations approach a limiting value, 160 
feet per second, which is defined as the instantaneous speed 
of the bomb upon striking the ground, or what is the 
same thing, its rate of change of distance with respect 
to time at that instant. 

We may discuss the same example from an algebraic 
standpoint. The distance covered by a falling body is 
given by the function y = 16a:^ where j is the distance, 
and x, the time elapsed. From this formula, merely by 
substituting 5 (seconds) for x, we find that is equal to 
400 (feet). How shall we make use of this formula to 
find the speed at the end of five seconds? Let us fix our 


Change and Changeability—The Calculus 319 

attention upon a short interval of time just before the 
falling object strikes the ground and the correspondingly 
short interval of distance traversed in that period of time. 
We shall call this small interval of time Ax*, and the 


distance traversed in that period Av. Knowing the value 
of Ax, having chosen it arbitrarily, ihe problem is to lind 
the value of A^. At the beginning of the space interval, 
A^, the exact elapsed time since the falling body left the 

* Read “delta x,” not “delta times x " for A is merely n svmbol, 
a direction for performing a certain operation, to wit, taking a small 
portion of jc. 


320 Mathematics and the Imagination 

airship was (5 — seconds. The distance covered in 
the time (5 — Ax) seconds is (400 — Ay) feet. Our func¬ 
tional relation indicates that 

Distance = 16 (Elapsed Time)^ 

Thus, for the entire fall 

400 = 16(5)2, 

and for the incompleted journey 

(400 - Ay) = 16(5 - Ax)\ 

This may be simplified to 

400 - 16(5 - Ax)2 = Ay 
400 - 16(25 - lOAx -{- Ax^) = Ay 
400 - 400 -{- 160Ax - 16Ax^ = Ay 
160Ax — 16Ax2 = Ay. 

The last equation gives the distance Ay in terms of Ax 
units. To find the average speed during the entire time 
interval Ax, we must form the fraction - 

_ , Distance Interval 

Average Speed = 


or 


Average Speed = 


Ay _ 160Ax — 16Ax^ 
Ax Ax 


Thus, 

^ = 160 - 16Ax. 

Ax 

Now as the interval of time Ax is made smaller, that is, 
as we take closer and closer approximations to the speed 
at the instant the body strikes the ground (5 seconds hav¬ 
ing elapsed) the limit of the ratio A^/Ax( = 160 — 16 Ax) 
is 160. In other words, as Ax approaches zero in value, 


Change and Changeability—The Calculus 321 

the function of (the expression 160 — 16 A v) approaches 
160. Thus, the instantaneous speed at the end of five seconds 
is 160 feet per second. We indicate that the ratio Aj /Av 
approaches a limit by writing its limiting value as dy/dx. 
In technical terms 

Lim ^ 

AX-.0 A;t dx 

which may be read, “The limit of Av/Aa- as Av approaches 
zero is dy/dx^ 

* 

Let us pause for a moment to get our bearings. What 
have we accomplished? It may seem trivial that \Nith 
all the elaborate machinery at our disposal we liave 
succeeded only in ascertaining the instantaneous speed 
of a falling body as it strikes the earth. Vet if our accom¬ 
plishment is trivial, then motion is trivial as well, for 
we have whether we realized it or not, trapped Zeno’s ar¬ 
row in its flight and established the changelessness of our 
universe. With the aid of the concepts of limit and func¬ 
tion, we have made meaningful the notion of change and 
rate of change. Change is a Junctional table. As an item (in¬ 
dependent variable) on one side of the table varies, its 
corresponding item (dependent variable) on the other 
side shows a correlative variation. The quotient ol ehiUige. 
i.e., the limiting ratio of the two variations, is denoted by 
rale of change. All the vagaries, the mysteries, and uncer¬ 
tainties indissolubly linked with the idea oi motion, aic 
thus swept away or, more appropriately, traiisioi nied into 
a few precise and definable aspects of tlu‘ idea ot lutK iK)n. 
The limit of a function is exemplified quite simpK by tfu- 
ratio A^/Av as Ar approaches zero. It is ea^y to se<“ th.ii 
A^/Av is a function of A.v, in other words, that rliis laiio is 


322 ■ Mathematics and the Imagimtion 

a function of the independent variable A^c. As we assign 
arbitrary values to tSx, its dependent variable, Ay, assumes 
a corresponding set of values, and as we have seen, that 
ratio approaches a limit. It follows that we have not only 
revealed the meaning of the limit of function but have al¬ 
ready made practical use of this concept. 

It is now possible to define the fundamental process of 
the differential calculus, computing the limit of a func¬ 
tion, or what is the same thing, determining its derivative. 
For, in effect, the rate of change of a function is itself a 
function of that function, and in getting at the limit of the 
rate of change, the derivative, we are getting at the heart 
of the machinery of our primitive function. 

Assume we wish to determine the rate of change of a 
function = f(x) at an arbitrary point xo- The average 
change in the function J(x) over an interval extending 
from xq to xq -\- Ax is the difference in the value of the 
function = f{x) at the two end points, xo and xo + Aar, 
divided by the length between these two end points, 
(Xq + Ax) — Xq. Thus, 

yo = f{Xo) 

and 

yo Ay = f{xo + Ax). 

Whence a change in a function, from the purely algebraic 
standpoint, is given by 

A>’ = f(xo + Ax) —/(xq), 

and the average rate oj change of a function, obtained by 
dividing the change, Ay, by the length of the interval 
over which that change is taken, Ax, is 

Ay _ /(xq + Ax) — f{xo) 

Ax Ax 


Change and Changeability—The Calculus 323 


In order to obtain better approximations to the instan¬ 
taneous rate of change at the point aq, it is only necessary 
to take smaller intervals, that is, to let Aa- approach zero. 

A A L » • A ' 0 + Aat) — /(Ao) 

As Aa approaches zero, the expression '- - -- 

Aa 

approximates as closely as may be desired to the in¬ 
stantaneous rate of change at xq. Thus, in the limit 

as Aa approaches zero, the quotient 

Aa 


approaches a limiting value, denoted by dy/dx. It is this 

which is called the derivative of the function /(a) at the point 

Aq. But since aq is an arbitrary point, the derivative may 

be said to represent the instantaneous rate of change of a function 

as the independent variables ranges through an entire set of 
values. 

For the sake of clarity, a geometric interpretation of 
the derivative may be helpful. Chronologically, the geo¬ 
metric interpretation preceded the analytic. One of the 
outstanding problems of the seventeenth century was that 
of drawing the tangent to a curve at an arbitrary point. 
It was solved by Newton’s predecessor and teacher at 
Cambridge, Isaac Barrow. On the basis of the geometrical 
researches of Barrow, Newton developed the concept of 
the rate of change along analytic lines. The close connec¬ 
tion between algebra and geometry, epitomized by the 
fact that every equation has a graph and every graph an 
equation, thus bore fruit once more. In the Cartesian 
plane, let the graph of the function^ = /(a) be the curve 
in Fig. 131. 

Consider the points Pi and P 2 on this curve; tlieir x 
co-ordinates are denoted by ao and ao + Av. where A.v 
IS the distance between the projection of the twe; points 
on the A axis. The^ co-ordinates of the points P\ and P-i 


22 


324 Mathematics and the Imagination 

are then determined from the equation of the curve 
and are /(atq) and J{xq + A^) respectively. The slope^ 
of the line joining Pi and P 2 (the tangent of the angle 0) 
is precisely the quotient 

/(xq + Ax) -J{xq) 

CSX 


As we let Ax approach zero, the point P 2 is carried along 
the curve so that it approaches the point P 1 , and the 
slope of the line (the above quotient) approaches as its 


FIG. 131. 


limiting value the slope of the tangent to the curve at 
the point Pi. But the slope of the tangent at that point is 

numerically equal to ^^since ^ ~ other 

words, the slope of the tangent at every point along a 
curve is identical with the derivative at that point. Or, 
to put it differently, the slope of the tangent to a curve 
gives the direction the curve is taking (i.e., whether it is 


Change and Changeability—The Calculus 325 

rising or falling), and thus its rate of change. Thus, the 
geometric equivalent of the derivative is the slope of the 
tangent. 

We may now recall our statement that the values for 
which a function attains its maximum or minimum cor¬ 
respond to the points on the curve at which the tangent is 
horizontal. The slope of a horizontal line is, of course, 
zero. Since the derivative is identical with the tangent, we 
may conclude that the maximum and minimum values 
of a function are those for which the derivative of the 
function is equal to zero. Many interesting problems can 
be solved in this way. 

The previously discussed problem of determining the 
rectangle with greatest area and given perimeter falls 
into this category. One side of the rectangle was denoted 
by the adjacent side hy 2 — x, and the area, j', by 
x[2 - x). Since the area is a function of x, its derivative 
will be equal to zero when the function attains its maxi¬ 
mum value. Finding the rectangle with maximum area 
by means of the calculus entails these steps: (1) Differen¬ 
tiate the function, i.c., find its derivative; (2) Set the de¬ 
rivative equal to 0; (3) Solve the resulting equation for 

Step J: 

y = x{2 — x) 

_y + Ay = (jc + Ax) (2 — x — Ax) 

-|- A^) —y = (x -h Ax)(2 — x Ax) — x{2 x) 

A>> = 2x - x^ - xAx H- 2Ax - xAx - Ax^ - 2x -|- x^ 
Ay = 2Ax — 2xAx — Ax- 

^ = 2 - 2 . - A. 


Mathematics and the Imagination 


Sup II: 

^ = 2 - 2x = 0 
dx 


Step III: 

2 - 2;c = 0 
2 = 2x 
1 = 


This checks with the result obtained before without the 
aid of the calculus: the rectangle of maximum area, with a 
perimeter of 4, is a square each of whose sides equals 1. 

More elaborate examples, drawn from the fields of chem¬ 
istry, economics, physics, etc., require a greater sophisti¬ 
cation with respect to mathematical technique, but not 

with respect to the ideas involved. 

♦ 

By considering the derivative at every point of the in¬ 
terval over which it is defined, we have seen that the de¬ 
rivative is in turn a function of the independent variable. 
Differentiation need not stop here, for the derived func¬ 
tion may also have a derivative, the second derivative of 
the original function. The notation for the second deriv¬ 
ative of y = f{x) is —. The nth derivative of a function is 

dx^ 

obtained by differentiating the function n times. Its sym- 

bol is —• What do these higher derivatives mean? 
dx^ 

Usually it is possible to give to the second derivative 
a physical and geometrical interpretation. If the function 
y — J(^x) represents the distance covered by a falling 
body in the time ;t, the first derivative represents the rate 
of change of distance, with respect to time. The second 
derivative is the rate of change of the rate of change of 
distance with respect to time, and is commonly known 
as the acceleration of the body. For a falling body, the 
distance 7 = 16^:^ must be differentiated once to obtain 


Change and Changeability—The Calculus 327 

the speed and once again to obtain the acceleration. 
The mathematical details of both differentiations are. 


(I) ^ = \6x^- 

y Ay = 16 (Ar + Ax)^ 

(y + Ay) —y = 16(x + Ax)^ — 16x- 

= 16(x’ + 2xAx + Ax=) - 16x2 
= 16x2 + 32:vAx + 16Ax2 _ 16^.2 
Ay = 32xAx + 16Ax2 

^ = 32x + 16Ax 


Limit _ dy 

Ax —0 Ax dx 


32x. 


= 32x 

- 32 (x+Ax) 

= 32 (x+Ax) - 32x 
= 32Ax 

* 32 


Limit 
Ax—0 


d\y 

^2 


<Py 

dx^ 


The second derivative is a constant, the number 32. 
This constant is called the gravitational constant of a 


Mathematics and the Imagination 

falling body due to the earth’s gravitational pull. It 
expresses the remarkable fact that any body, regardless of 
its mass, dropped from a height 16 feet above the earth 
(and neglecting air resistance), will strike it in one second, 
moving at a speed of 32 feet per second at the instant of 

impact. 

So far as the geometric interpretation of the second 
derivative goes: For curves drawn in the plane, at every 
point the curvature is directly proportional to the second 
derivative. To determine the curvature of a given arc, 
draw the circle which best fits that arc. 


FIG. 132. 

The radius of that circle is the radius of curvature^ and its 
reciprocal the curvature. 

Let us see how this is applied, for example, to the 
straight line. The curvature of a straight line is zero. 
Any function, the graph of which is a straight line, has 
an equation of the form y = mx b, where m and b 
are constants. 

Differentiating gives dyjdx = m. When m is differenti¬ 
ated, its rate of change or derivative equals zero, since m 
is a constant. Thus, the first derivative tells us that the 
slope of a straight line is a constant; the second derivative, 
that its curvature is zero. 


Change and Changeability—The Calculus 329 

Simple physical or geometric interpretations of third, 
fourth, and higher derivatives do not exist. Higher 
derivatives do occur, however, in many problems arising 
in physics. Automobile engineers are interested in third 


y 


derivatives because they yield information about the 
riding quality of a car. Structural engineers, concerned 
with the elasticity of beams, the strength of columns, 
and any phase of construction where there is shear and 
stress, find first, second, third, and fourth deiivatives 
indispensable; and there exist innumerable othci ex¬ 
amples in the fields of the physical sciences and statistical 

applications to the social sciences. 

♦ 

The questions answered by the integral calculus stein 
from a much earlier period than those of the dilieicniial 


330 Mathematics and the Imagination 

calculus. By this it is not meant that the mathematical 
devices used in the one preceded those used in the other, 
for the concepts of limit, function, and limit of a function, 
as they appear in the calculus, were developed at about the 
same time for both its branches. But the type of problem 
which the integral calculus seeks to solve is easier to pro¬ 
pose and, therefore, it is not surprising to find among the 
writings of the Greek mathematicians problems which we 


FIG. 134.—Squaring the parabola. The shaded 
area = f the area of the rectangle. 

now identifiy as belonging under the head of integration. 

Much more astounding is the close relationship which 
exists between the two divisions of the calculus, the dif¬ 
ferential and the integral. It is one thing to determine the 
rate of change of a function or the slope of the tangent to a 
curve; to compute the area under a given curve appears 
to be an inquiry of a wholly different order. As wonderful 
as the link between these apparently unconnected in¬ 
quiries may seem, it is secondary to the satisfaction that 
the mathematician experiences from the complementary 
character of two such powerful tools. 


Change and Changeability—The Calculus 331 

“Squaring the circle” had challenged Greek mathe¬ 
maticians. Another aspect of this problem, perhaps not 
as well known, but of equal importance, is the rectifica¬ 
tion of the circle. It is concerned with the determination 
of the length of the circumference of a circle in terms of 
the length of the radius. While they never mastered the 
circle, they partially tamed the parabola. In this, as in 
other things, they drew upon their fertile ingenuity. They 
succeeded, with profoundly beautiful methods, in squar¬ 
ing * the parabola, but not in rectifying it.^ 

A discussion of their methods would disclose more of 
the genius of Archimedes than of the general theory of 
the integral calculus. Undoubtedly, Archimedes’ plan 
foreshadowed the technique of the calculus, but in the 
comparatively barren centuries that followed the seed 
he had planted found little nourishment. Not until 
Kepler appeared was there an attempt to deal systemati¬ 
cally with the determination of the areas and volumes of 
curvilinear figures. Melancholy to relate, his incentive 
was less a thirst for learning than the commercial require¬ 
ments of the thirst-slaking industry. “Kepler was origi¬ 
nally led to make the calculations ... by a desire to 
improve upon the crude methods then in use for estimat¬ 
ing the contents of wine casks and other vessels. While 
buying wine he noticed that the vintners determined the 
contents of the cask by passing a measuring rod through 
the bunghole as far as the opposite staves without taking 
account of the curvature of the latter. By rotating the 
longitudinal section of the cask about its axis, a body 
equal in volume to the cask would be formed. Kepler’s 
plan was to divide up such solids of rotation into an in- 

* Squaring the parabola, as wc saw earlier, means computing the 
area bounded by a parabolic segment and a straight line. 


332 Mathematics and the Imagination 

finite number of elementary parts, and to sum these; and 
in his Stereometria he applies this method to some ninety 
special cases. Kepler regarded infinitely small arcs as 
straight lines, infinitely narrow planes as lines, and in¬ 
finitely thin bodies as planes. His conception of infinitely 
small magnitudes was one which the ancients had in 
general avoided, but which a little later formed the basis 

of Cavalieri's method.” ® 

Perhaps it should be stressed at this point that in our 
discussion of the calculus every reference to the infinite, 
whether small or large, has been sedulously avoided. 
Because Weierstrass disposed of the infinitesimal djini^ 
the calculus rests securely on the understandable and 
nonmetaphysical foundations of limit, function, and 
limit of a function. Nothing prevents the extension ot 
these concepts to the integral calculus. Indeed, the ban¬ 
ishment of the infinitely small means more to the integral 
calculus than to the differential. It was precisely this re¬ 
finement of thought which raised the calculus to a very 

exact science. , 

The work of Cavalieri marked progress in showing 

greater generality and a more abstract method of treat¬ 
ment than that of Kepler. One of the leading theorems 
still bears his name. If two solids have the property that 
when they are cut by planes, their areas are exposed in 
constant proportion throughout, their volumes will be 

in the same proportion. 

The initial question, then, of determining the areas 
under curves was well on the way to solution in so far as 
crude machinery made that possible. Yet the design of 
the machinery made it unsuitable for the computation 
of the length of a curved line. A different device was 

required. 


Change and Changeability—The Calculus 333 

All simple problems in mathematics share a common 
feature—one cannot anticipate what difficulties they may 
conceal. Certainly nothing appears easier than measur¬ 
ing the length of a line. Take a piece of paper and mark 
two points on it. If you connect these two points with a 
straight line, all that is required to ascertain its length is 
a ruler. Nor need we be led astray by the yawning re¬ 
gress of philosophical discourse: What means shall be 
employed to measure the length of the ruler; what means 
shall be employed to measure the measuring rod which 
shall measure the ruler, etc., etc. It is agreed that we can 
measure the length of a straight line. Suppose, however, 
that we connect the two points by a curve; finding its 
length presents an altogether different story. One way of 
proceeding might be to take a piece of string, fit it along 
the curve, then remove the string and measure its length 
with a riiler. But this puts us back where we started, ior 
it appears that the only lines that can be measured are 
straight. To measure the length of a curved line, it be¬ 
comes necessary, in effect, to uncurve it. 

By now, another means might suggest itself for measur¬ 
ing curves. Frequent recourse has been had, particularly 
in this chapter, to methods of approximation. T hus, wc 
might divide the arc into a number of smaller parts and 
connect the end points of the small arcs by straight lines. 
The small straight lines will differ in sum from the sum 
of the small arcs less than a single straight line would 
differ from the length of the entire curve. 

In other words, the sum of the lengths of the small 
straight lines will approximate the length of the cui vc. 
By choosing a large enough number of lines (and making 
them individually small), wc should succeed in making 
the sum of their lengths differ from the length ot the curve 


334 Mathematics and the Imagination 

by as little as we please. The more numerous the little 
lines, the more accurate the approximation. 


FIG. 135.—Approximating to the length of a 
curve by straight lines. 


If we conceive of the number of lines as increasing be¬ 
yond bound, it may be said that their sum approaches 
a limit—the length of the curve. Let us try to formulate 

this in terms of limits and limits of functions. 

Suppose y = J{x) is the equation of the cunre which 

connects the two points A and B in a Cartesian p ane. 
Let the a: axis under the curve be subdivided into n equal 
parts. The x co-ordinate of the initial point A is flo; the .v 
co-ordinate of the next point is au of the third point, 
and so on, so that the x co-ordinate of the last point is 
or B. The difference between two adjacent values of a: 
may be denoted by Ax; the difference between wo ad¬ 
jacent values of y (obtained by erecting perpendiculars 
from adjacent values on the x axis) is Ay. In ig* > 


Change and Changeability—The Calculus 335 

every pair of selected points on the curve bounds a hy- 
pothenuseofashaded right-angle triangle, the base of which 
is Ax and the altitude A_y. Thus, the hypothenuse of each 
of these triangles will be an approximation to the length 
of that portion of the curve which bounds it. It follows 
that the sum of the hypothenuses of all the little triangles 
approximates to the length of the curve. By the use of the 
Pythagorean theorem, the value of each hypothenuse is 
easily obtained. Increasing the number of subdivisions 


FIG. 136.—Approximating to the length of a 
curve by the hypothenuses of right-angle triangles. 
The base of each triangle is iis altitude A^* 


will make the approximations more accurate. Thus, as 
Ax approaches 0, as the intervals along the x axis are 
made smaller, the sum of the hypothenuses of the right- 
angle triangles approaches a limit, which is the length of 
the curve. It should be noted that the length of each small 
hypothenuse is a function of its corresponding Ax. 

♦ 

We may now turn to the determination of the area 
under a curve, for it is in this problem that the ideas of the 
integral calculus are first vividly set forth. 


336 Mathematics and the Imagination 

Estimating the area of a figure bounded by straight 
lines, no matter how irregular, is comparatively easy. 
One need only introduce auxiliary lines so that the 
original figure is broken up into a number of triangles. 
By summing the areas of these triangles, the area of the 
original figure is measured. 


Fic. 137.—The area of this irregular polygon is 
determined by forming the triangles indicated and 
computing the area of each. 

When the boundary of a figure is not straight but 
curved, this procedure is inadequate, and one must again 
resort to approximation. If we divide the curved sides of 
the figure into a great many parts, connecting their end 
points by straight lines, exactly as we did above, the 
resulting figure, a polygon bounded by straight edges, has 
an area which may be determined by elementary means. 
By increasing the number of sides of the polygon, its area 
may be made to differ from that of the original figure 
by as little as we wish and thus yield an approximation 

as close as we desire. 


Change and Changeability—The Calculus 337 

But a more effective means of dividing a curvilinear 
figure is by the use of rectangles. Precisely this device was 
invented by Archimedes. Figure 138 illustrates a circle 
divided into rectangular strips. According to the method 
of constructing these strips, it should be noted tliat not 
one, but two approximations can be obtained. The first 
gives the area of the rectangles inscribed in the circle, the 


FIG. 138.—Approximating to the area of a circle 
by the use of rectangles. 


second the area of the rectangles circumscribing the circle. 
The discrepancy between the two rectangulaied areas be¬ 
comes smaller and smaller as the number of rectangles is 
increased, in other words, as they are diminished in 
width. Their common limit, as the inner area increases 
and the outer area decreases, is the area ol the circle. 

Instead of confining ourselves to this sj)ecial example, 
if we discuss the general problem of finding ihc area 
under the segment of an arbitrary curve, the method 
just described can perhaps be made even clearer. We 
wish to find the area of the shaded section in l ig. 130 
below. It is bounded on top by the curve^' = belcnv 


Mathematics and the Imagination 


by the portion of the axis from x A to x = B, and 
on the right and left by two straight lines parallel to the 
y axis. Divide the x axis into n equal subintervals, as in 
Fig. 136. Erect at each of the dividing points a perpen¬ 
dicular from the a: axis to the curve. At each point where a 


FIG. 139. 


perpendicular intersects the curve, draw a horizontal line 
to the adjacent vertical lines. For every little subinterval 
on the a: axis there will be two rectangles, one under the 
curve, the other protruding above it and containing part 
of the area outside. Consider a typical subinterval (see 

Fig. 140). 

The area of the smaller rectangle ABCD is the base AB 
times the altitude AD, where the altitude is the value of 
the function at the initial point of the subinterval, A; the 
area of the larger rectangle ABEF is the product of the 
same base AB, by the altitude BE, the value of the func¬ 
tion at the terminal point of the subinterval, B. The area 
under the curve lies between the areas of these two rec¬ 
tangles. An excellent approximation to the desired area is 
obtained by taking the average value of the two rectangles. 


Change and Changeability—The Calculus 


339 


Repeating this process for each subinterval and forming 
the sum of the average rectangles gives the approximation 
to the entire area under the curve. Again enlisting the aid 
of the concept of limit of a function, it may be seen that as 
the number of subintervals on the x axis is increased, the 


sum of the corresponding areas necessarily approaches 
the area of the shaded figure (Fig. 139). In the limit, this 
sum of the many tiny elements of area is called the definite 
integral of tfie function y = f{x) between the values of x = A 
and of X — B, and in the shorthand of Leibniz is: 


/ 


f{x)dx. 


Briefly recapitulating: Each of the subintervals along 
the X axis is Aj:, which is the base of every one of the tiny 
rectangulated areas. The altitude of the average rectangle 
is represented by a perpendicular line drawn from a typi¬ 
cal interior point of the interval to the curve. Its value 
is, of course, f{x). The area of each such average rec- 


23 


340 Mathematics and the Imagination 

tangle is/(;c)- Aa:, and the sum of these areas is the sum of 
all such products. In technical symbolism the limiting 
area is written l!j{>:)dx, where dx replaces Ax, since 

Ax —* 0. 

* 

Our interpretation of the definite integral is that it is 
an area. To assign such a meaning is always possible, but 
there exist integrals of certain functions which have 
additional physical significance. Mainly this is because 
the definite integral is a number, a sum, as well as an 
area. Whenever, in science, a function is sununed to the 
limit, the definite integral plays a role. One of the achieve¬ 
ments of the integral calculus has been the determination 
of the moment of inertia of all solids. Again, it is to the defi¬ 
nite integral that structural engineers must render thanks 
for the Golden Gate Bridge, for it rests on this even more 
than on concrete and steel. Restraining the force on our 
gigantic dams, with their curved and uneven faces, repre¬ 
sents another problem in the integration of a function. By 
determining the water pressure at an arbitrary point and 
summing it over the whole face of the dam, the total force 
is uniquely determined. The centroid, that is, the center 
of gravity of any plane or solid figure, is easily reckoned by 
means of the integral calculus when applied to the par¬ 
ticular function defining that figure. Such examples 

might be multiplied indefinitely. 

* 

Beyond the concept of the definite integral, with its 
many uses and richly studded field of application, there 
is the notion of the indefinite integral, of even greater in¬ 
trinsic value to the mathematician. Its chief theoretical 
interest is that it enables us to exhibit the astounding 
relationship between the derivative and the Integra . 


Change and Changeability—The Calculus 341 

Consider the function = J{x). Instead of limiting the 
interval as before, from x — A to x = By imagine it to 
extend from x = ^ to at = atq, where xo may assume any 
value. For different values of xoy the definite integral 
will also take on different values. True, we will no longer 
have under consideration a limited area, but we will 
have all the requisites for preparing a functional table. 
On one side there will be listed successive values of .vo; 
on the other, corresponding values of the definite in¬ 
tegral. This correspondence between values of xo and 
values of the definite integral is itself a function called 
“the indefinite integral” of the function y =J{x). Here 
is the crux of the matter; The definite integral of the 
function^ = /(x) is a number determined by an interval 
of definite length and a portion of the curve y = /(-v) 
defined over that interval. When the interval is extended 
from a fixed point through a succession of others, to each 
of these there corresponds a value of the definite integral. 
This correspondence, this function, is the indt'fmitc in¬ 
tegral of the original Junction y = J{x) and is symbolized by 

i J{x)dx. 

From this you may perhaps guess what the two seem¬ 
ingly diverse branches of the calculus have in coininon. 
For the relation between differentiation and integration is 
reminiscent of elementary arithmetic. It is the same 
relation that exists between addition and subtraction, 
multiplication and division, involution and evolution. 
The one operation is the inverse of the other. Starting witli 
the function y = /(x), upon differentiating we oijtain 
dy r • 

What do we get upon integrating ihc function l nc 

C4 ^ 

motif of the calculus is hereby revealed, lor we obtain the 


342 Mathematics and the Imagination 

original function, y = f{x). The indefinite inte^al of 
the function 7 = f{x) is another function of x which we 
shall denote by _>» = F{x). Of course, the derivative of 
y = F{x) isj(x). Every function may thus be regarded as 
the derivative of its integral and as the integral of its 
derivative. 

♦ 

Earlier we alluded to the exponential function,^ = 
and its usefulness in describing the phenomenon of growth. 
It is the only function, the rate of change of which is 
equal to the function itself. Differentiating^ = ^ yields 

^ = e. Integrating yields the same result. It follows that 
dx 

the life history of any organism—amoeba, man, or red¬ 
wood—of any phenomenon which exhibits properties of 
organic growth—is aptly described by the integral of 
This stirring conception is not difficult to visualize. Pro¬ 
portionality of rate of growth to state of growth may be 
embodied in the exponential function. If this is integrated, 
the total growth over any given period is given by the 
definite integral, and the general character of growth suc¬ 
cinctly set forth by the indefinite integral. 

In conclusion, let us re-examine the problem of a 
falling body. We started with the distance that the body 
fell in a period of time and derived its speed at every 
instant by differentiation. Acceleration at every insunt 
was obtained, in turn, by differentiating the first deriva¬ 
tive, finding the rate of change of the speed with respect 
to the time. Galileo and Newton did the same thing 
backwards. They shrewdly guessed that the acceleration 
of a falling body was a constant, the gravitational con¬ 
stant. Upon integrating the function expressing this 


Change and Changeability—The Calculus 343 

hypothesis, they made the classical discovery of the laws 
of motion: 

(1) the speed of a falling body is gt, where g is the 
gravitational constant, 32, and t the time elapsed since 
the body was dropped. 

(2) the distance covered by a falling body is 

This and the other laws of motion governing every 
particle in the universe are derivable simply and elegantly 
by means of the calculus. But this is not all, for the cal¬ 
culus not only helped release some of nature’s most inti¬ 
mate secrets; it gave the mathematician more new worlds 
to conquer than Alexander ever sighed for. 


APPENDIX 

PATHOLOGICAL CURVES 

The curves treated by the calculus are normal and 
healthy; they possess no idiosyncrasies. But mathemati¬ 
cians would not be happy merely with simple, lusty con¬ 
figurations. Beyond these their curiosity extends to psy¬ 
chopathic patients, each of whom has an individual case 
history resembling no other; these are the pathological 
curves of mathematics. We shall try to examine a few in 
our clinic. 

Before we can do so, it will be necessary to introduce 
the idea of a curve as the limit of a sequence of polygons. 
Let an equilateral triangle be inscribed in a circle. This 
triangle may be considered as a curve—Ci. Let C-i be the 
regular hexagon obtained by bisecting the three arcs in 
Fig. 141, and by joining, in order, the six vertices. (Fig. 
142). 


344 


Mathematics and the Imagination 


Cz is the regular duodecagon formed by bisecting the 
six arcs of Fig. 143, and joining the twelve vertices in 
order. Repeat this process each time by bisecting the arcs, 
doubling the number of sides. The curve approached as 


The equilateral tri¬ 
angle is the curve Ci. 


C 3 . 


The regular hex¬ 
agon, curve C 2 . 


C4. 


FIGS. 141, 142, 143, 144.—The circle as the limit 
curve of a sequence of curves. 


limit is the circle. Thus, the circle is described as the 
limit curve of a sequence of curves or polygons. 

(1) The Snowflake Curve. Start with an equilateral tn- 
angle, with a side one unit in length. This triangle is curve 

Cl. (Fig. 145.) 

Trisect each side of the triangle and on each ot tne 
middle thirds erect an equilateral triangle pointing out¬ 
ward. Erase the parts common to the new and the old 
triangles. This simple polygonal curve is called C 2 . 

Trisect each side of C 2 , and again upon each middle 
third erect an equilateral triangle pointing outward. 


Change and Changeability—The Calculus 345 

Erase the part of the curves common to the new and old 
figures. This simple curve is C 3 . 


Repeat this process, as shown in Figs. 148-150. 

What is the limit curve of this sequence of curves? Why is 
it called the Snowflake Curve, and why is it described as 
pathological? 

It derives its name from the shape it assumes in the 
successive stages of its development. Its pathological 
character is borne out by this incredible feature: Al¬ 
though one may conceive that the limit curve can be 
drawn on a piece of paper, it is hard to imagine that this 
is possible, because, though the area is finite, the length 
of its perimeter is infinite! But it is clear that at each stage 
of the construction the perimeter increases, and since the 


Mathematics and the Imagination 


sequence of numbers representing the length of the perim¬ 
eter at each stage does not converge, i.e., does not choke 
off, the perimeter must grow beyond all bounds. We are 


FIG. 146.—The second stage of the Snowflake 
Curve— Cl. 


thus confronted by the amazing fact that a curve of infinite 
length may be drawn on a small sheet of paper—for ex¬ 
ample, on a postage stamp. _ 

The proof is simple: The perimeter of the original 

triangle was 3. The perimeter of curve C 2 is 3 + 1; o 
C3, 3 + 1 + t; of 3 + 1 + f The perimeter 

orC„ is 3 + 1+^ + ^:+ • • • + r- Thus, as « 


Change and Changeability—The Calculus 347 

grows, so grows the sequence, for we are dealing with an 
infinite series which does not converge. 

The fact that the curve remains on the paper proves 
that the area of the snowflake is finite. Explicitly, the 
area of the final curve is if times that of the original 
triangle. And if this is not weird enough, consider that 


FIG. 147.—The third stage—Cj. 


it is not possible to tell at any point on the limit curve 
the direction in which it is going, that is, the tangent line 
does not exist.® 


348 


Mathematics and the Imagination 


(2) The Anti-Snowflake Curve is obtained by drawing 
the triangles inward, not outward, and has many oS the 


FIG. 148.—The fourth stage—C^. 


while its area is finite, and no tangent can be drawn to it 

at any point. (Figs. 151—154.) 

(3) Another pathological curve is the In-And-Out Curve. 

Draw a circle (with radius = 1) and choose six points on 

it so as to divide the circumference into six equal p^s. 

Take three alternate arcs and turn them inw^d. 1 he 

original circle is Ci, the new figure Cj. (Figs. 155 156.; 


Change and Changeability—The Calculus 349 

The perimeter of C 2 is the same as the perimeter of Ci, 
because its length is not altered by turning three arcs 
inward. 

Next, trisect each arc, and turn the middle one out¬ 
ward if it is now turned inward, inward if it is now 


FIG. 149. — The fifth stage— Ci. 


turned outward. This new curve is C.v Its pcriniett'r is also 
equal to that of the original circle. Moreover, the area ul 
C 3 is the same as that of C> because we ahernatclv added 
and subtracted the same size segments. (I'ig. 


350 


Mathematics 


Repeat this process. The limit curve has a perimeter 
equal to the perimeter of the circle. Its area is equal to 
that of C2, which, in turn, is equal to the area of a regular 
hexagon. Like the Snowflake and Anti-Snowflake, this 
curve, too, has its pathological features. 


PIG. 150.—The sixth stage—Ca. 

While the curvature of a circle is computed without 
difficulty, the In-And-Out Curve presents a different 
aspect. Consider an arbitrary point upon it. In which 
direction, toward the center of the circle or away from 


Change and Changeability—The Calculus 35 


FIGS. 151, 152, 153, 154.—The first four stages 
of the Anti-Snowflake Curve. 


FIG. 155.—The In-And-Out Curve 


FIG. 156. — Stage C 2 . 


352 Mathematics and the Imagination 

the center, shall we measure its curvature? We find there 
is no definite curvature. The second derivative does not 
exist. 


FIG. 157.^—Stage Ca. 


FIG. 158.—Stage C 4 . 


( 4 ) Space-Filling Curves: One of the cardinal principles 
of geometry is that a point has no dimensions, and that 
a curve is one-dimensional and can, therefore, never fill 


FIG. 159.—The Space-Filling 
Curve—Stage 1. 


FIG. 160.—Stage 2. 


a given space. This iron conviction must also be shat¬ 
tered. For behold the pathological specimen supreme, 
the Space-Filling Curve, which will not only occupy the 


Change and Changeability—The Calculus 353 

interior of a square, but gobble up the space in an entire 
cubical box. 

The successive stages of such a curve are illustrated 
in Figs. 159-164. Select any point in the square or cube. 


n 

Rl 


n 

IB 

n 

ni 

u 

III 

R 

li 

ill 

Sli 

III 

R 

ai 

II 

1; 

IB 


III 

u 

Hi 

H 

II 

IB 

Si 

ni 

n 

Rl 

H 

II 

IB 

n 

III 

R 

HI 

II 

li 

IB 

J 

B! 

n 

R! 

H 

li 

!B 


B! 

j 

R! 


u 

IB 

ii 


FIG. 161.—Stage 3. 


It can be shown that 
completed, it will 
reasoning extends 
curve must 


FIG. 162.—An advanced stage. 


eventually, when the curve has been 
pass through that point. Since this 
Xo every pointy it follows logically that tlic 
fill the entire square or cube. 


FIGS. 163, 164.—The first two stages of a curv'e 
which fills an entire cubical box. 


(5) The Crisscross Curve: 

This curve has the property that it crosses itself at 


354 


Mathematics and the Imagination 

every one of its points. We are certain that you don't believe 
us—and never will—but here are the directions for mak¬ 
ing it: 

1 st Step: Inscribe a triangle within a triangle as in 
Fig. 165 . Shade the interior triangle. 


c 


FIG. 165, 


2nd Step: Continue the process for each of the three 

remaining triangles as in Fig. 166. 

c 


FIG. 166. 

3rd to nth Step: Repeat the process indefinitely (Fig. 
167 is the 5th stage). Then join the points of the original 
triangle remaining unshaded and distort the original tri- 


Change and Changeability—The Calculus 


355 


angle so that the three points A, B, and C are brought to¬ 
gether. 


FIG. 167. 

There you have the Crisscross Curve. 


FOOTNOTES 

1. Cajori, History of Fluxions. —P. 305. 

2. Protocol on trigonometry for those who have forgotten: 

In the right-angle triangle below the following are the trigonometric 
ratios (functions of an angle): 


Side AB 

^ = Cosine <t> 

Side AC 

^ . - Side BC 

Cosine e = - = Sine <6 


Side AC 


Tangent 6 — 


AB 

Sine 6 AC 


AL 


Cos 9 BC BC 
AC 


= Cotangent <p. 


24 


Mathematics and the Imagination 


In other words, the trigonometric functions are the ratios of the 
sides of a right-angle triangle to one another, and depend in turn upon 
the angles. 

The concept of tangent has immediate application in analytic geom¬ 
etry and the calculus. In the diagram below, the slope of the line AB is 
the ratio P/Q, which is none other than the tangent of the angle 0. 


PIG. 169—The slope of a straight line is the ratio 

P^. 

Q 

But the word tangent has another meaning quite different from the 
one above. This new meaning is essential to the calculus. 

In the Cartesian plane draw the curve ABC. Consider two points P\ 
and P 2 on this curve joined by the straight line passing through them. 
(See Fig. 131, page 324.) As Pz moves along the curve to Pi, the line 
joining these two points approaches a limiting value called the tangent 
to the curve ABC at the point Pi. The slope of this tangent line at the point 
Pi is the derivative of the function, the graph of which is the curve 
^PC.—Pp. 309, 311, 317. 

3. With apologies to the physicist for the use of the term speed, 
instead of the technically correct term “vel city.’*—P. 316. 

4. The length of a parabolic segment is expressible only in terms of 
logarithms and consequently could not be computed by means 
of the elementary methods known to the ancients.—P. 331. 

5. Wolf, History of Philosophy, Science and Technology in the Sixteenth 
and Seventeenth Centuries. —P. 332. 

6 . Thus, we have, in essence, a continuous function without a 
derivative.—P. 347. 


Mathematics and the Imagination 


There is no conclusion. W’hat has concluded that we 
might conclude in regard to it? There are no fortunes to be 
told, and no advice to be given. Farewell. 

— WILLIAM JAMES 

'^Cheshire-Pufs," she began, rather timidly. . . 
“ Would you tell me (dease, which way / ought to go from 
here?'' 

“ That depends a good deal on where you want to get to,'* 
said the Cat. 

“/ don't much care where —” said Alice. 

''Then it doesn't matter which way you go," said the 
Cat. 

— LEWIS CARROLL 


What is mathematics? A large and varied body of 
thought which has grown from the earliest times purports 
to answer this question. But upon examination, the 
opinions which range from those of Pythagoras to the 
theories of the most recent schools of mathematical philo¬ 
sophy reveal the sad fact that it is easier to be clever than 
clear. Particularly of late has there been a tendency to 
present aphorisms in place of straightforward replies, 
aphorisms which, unfortunately, shed little liglu. In the 
method of approaching the problem lies the main ob¬ 
stacle to a satisfactory answer. If one were to ask, “W’hat 
is biology?” it would be comparatively simple to start 
with an etymological definition and then to group to¬ 
gether the great body of imformation compris(‘d in the 
biological sciences. Next would be a conclusion on how 


337 


358 Mathematics and the Imagination 

all the items are synthesized into an integrated science. 
Even a crude explanation, such as “Biology discusses 
horses, bats, daffodils, and whales,” would give a fair 
idea of what it is about. On the other hand, the study of 
mathematics—arithmetic, algebra, geometry, the cal¬ 
culus—does not imply anything more about its nature 
than that it is concerned with numbers and that it is a 
useful technique. So far as the concept of numbers is con¬ 
cerned, no definition has yet been given which in itself 
would simplify the task of defining mathematics. 

Here, then, in mathematics we have a universal lan¬ 
guage, valid, useful, intelligible everywhere in place and 
in time—in banks and insurance companies, on the parch¬ 
ments of the architects who raised the Temple of Solo¬ 
mon, and on the blueprints of the engineers who, with 
their calculus of chaos, master the winds. Here is a dis¬ 
cipline of a hundred branches, fabulously rich, literally 
without limit in its sphere of application, laden with hon¬ 
ors for an unbroken record of magnificent accomplish¬ 
ment. Here is a creation of the mind, both mystic and 
pragmatic in appeal. Austere and imperious as logic, it is 
still sufficiently sensitive and flexible to meet each new 
need. Yet this vast edifice rests on the simplest and most 
primitive foundations, is wrought by imagination and 
logic out of a handful of childish rules. Even though no 
definition thus far has encompassed either its scope or its 
nature, can it be that the question “What is mathe¬ 
matics?” must go unanswered? 

That mathematics enjoys a prestige unequaled by any 
other flight of purposive thinking is not surprising. It 
has made possible so many advances in the sciences, it 
is at once so indispensable in practical affairs and so 


359 


Mathematics and the Imagination 

easily the masterpiece of pure abstraction that the recog¬ 
nition of its pre-eminence among man’s intellectual 
achievements is no more than its due. 

In spite of this pre-eminence, the first significant ap¬ 
praisal of mathematics was occasioned only recently by 
the advent of non-Euclidean and four-dimensional geo¬ 
metry. That is not to say that the advances made by the 
calculus, the theory of probability, the arithmetic of the 
infinite, topology, and the other subjects we have dis¬ 
cussed, are to be minimized. Each one has widened math¬ 
ematics and deepened its meaning as well as our compre¬ 
hension of the physical universe. Yet none has contributed 
to mathematical introspection, to the knowledge of the 
relation of the parts of mathematics to one another and to 
the whole as much as the non-Euclidean heresies. 

As a result of the valiantly critical spirit which en¬ 
gendered the heresies, we have overcome the notion 
that mathematical truths have an existence independent 
and apart from our own minds. It is even strange to us 
that such a notion could ever have existed. Yet this is 
what Pythagoras would have thought—and Descartes, 
along with hundreds of other great mathematicians be¬ 
fore the nineteenth century. Today mathematics is un¬ 
bound; it has cast off its chains. Whatever its essence, we 
recognize it to be as free as the mind, as prehensile as the 
imagination. Non-Euclidean geometry is proof iliat 
mathematics, unlike the music of the spheres, is man’s 
own handiwork, subject only to the limitations imposed 
by the laws of thought. 

+ 

The philosophy which goes under the name of logical 
positivism has prepared a formidable program: first, to 
eliminate metaphysics from philosophy; and second, to 


360 Mathematics and the Imagination 

exhibit the mutual relations between the laws of thought 
(i.e., logic) and mathematics. There are some who believe 
that logical positivism affords an advance, beyond that 
made by non-Euclidean geometry, in evaluating the 
nature of mathematics. Quite modestly the hope has 
been expressed that here at least is a doctrine that faces 
squarely those essential and inherent difficulties blocking 
the road to the summit. 

In purging mathematical philosophy of metaphysics, 
there has been (in our judgment) a real gain. No longer 
is mathematics to be looked upon as a key to the truth 
with a capital T. It may now be regarded as a woefully 
incomplete, though enormously useful, Baedeker in a 
mostly uncharted land. Some of the landmarks are fixed; 
some of the vast network of roads is made understand¬ 
able; there are guideposts for the bewildered traveler. 

On the other hand, one cannot suppress the feeling that 
this new appraisal of mathematics is so incomplete, so 
devoid of color, as to be almost trivial and inconsequen¬ 
tial. In regarding it as merely a handful of primitive, 
undefined propositions, coupled with a methodology for 
manufacturing new ones, something of the spirit, of the 
flavor and color of mathematics seems to have been lost. 
While those who oppose logical positivism grant that 
it serves some purpose, they assail the stultification of 
discourse, the narrowing of horizons, which it inevitably 
entails. We share the feeling that mathematics is more 
than a factory of tautologies, rather that it is a vehicle 
to carry on the highest aspirations of the creative intellect. 

Briefly, here is what the positivists say; Logic is con¬ 
cerned with the formal rules for manipulating the symbols 
of language. Mathematics is concerned only with equa¬ 
tions, i.e., literally statements of equivalence of number. 


Mathematics and the Imagination 361 

All the essential, internal relations of meaning are the 
province of mathematical science. An omniscient being 
would thus require neither logic nor mathematics since 
the relations between all entities would, to him, be self- 
evident. Though he might still find other sciences useful, 
for example, biology to provide him with a catalogue of 
living things, or a telephone directory to help him find 
his friends, his need for logic and mathematics would 
have vanished. For once all meaning and all relationships 
were fully disclosed, these disciplines would be super¬ 
fluous. 

Is there not reason to think that in such an interpreta¬ 
tion, though we have scourged ourselves pitilessly and 
driven out the confusing spirit of metaphysics, we may 
also have drained the vitality of mathematics? May we 
not well have lost ‘‘the spirit in the word”? 

* 

As we have already indicated, the creation of non- 
Euclidean geometry signalized the realization that math¬ 
ematics in no sense depends upon our environment. Al¬ 
though many similarities exist between the behavior of 
the fecund little symbols which we place on paper or jug¬ 
gle in our heads and those phenomena which take place 
in the physical world, mathematics is to be recognized as 
an autonomous discipline, restricted only by the formal 
rules of thinking. The development of mathematics is a 
counterpart of the everlasting struggle for greater com¬ 
prehensiveness and greater freedom: from the particular 
to the general; from configurations bounded by straight 
lines to pathological curves; from the properties ol this or 
that specific figure to the properties of all figures; from 
one dimension to n dimensions; from the finite to the in¬ 
finite. In this march the imagination has played a notable 


362 Mathematics and the Imagination 

role. For imagination has the pragmatic value that it 
leaps ahead of the slow-moving caravan of well-ordered 
thought and often scouts out reality long before its pon¬ 
derous master. Therein lies its essential contribution to 
one of the strangest collaborations of thought, staid math¬ 
ematics and volatile imagination. 

Mathematics is an activity governed by the same rules 
imposed upon the symphonies of Beethoven, the paintings 
of Da Vinci, and the poetry of Homer. Just as scales, as 
the laws of perspective, as the rules of metre seem to lack 
fire, the formal rules of mathematics may appear to be 
without lustre. Yet ultimately, mathematics reaches pin¬ 
nacles as high as those attained by the imagination in its 
most daring reconnoiters. And this conceals, perhaps, the 
ultimate paradox of science. For in their prosaic plodding 
both logic and mathematics often outstrip their advance 
guard and show that the world of pure reason is stranger 
than the world of pure fancy. 


Bibliography 


This descriptive bibliography contains a number of works allied in 
interest to the contents of this book. It is not (nor is it intended to be) 
full, authoritative, or exhaustive. It is merely an expression of per¬ 
sonal tastes which may be helpful to the reader whose curiosity has 
been stimulated. 

Abbott, Edwin A. Flatland—A Rofnance of Alany Dimensiotis. Oxford: 
Blackwell, 1932). 

The celebrated little book about a world of two dimensions. 

Ahrens, W. Mathemalische UnUrhaliun^en und Spiele. Leipzig: B. G. 
Tcubner, Vol. I, 1910; Vol. II, 1918. 

For those who can read German, this work will be a rare treat. 

Although parts of it are quite technical, it is an exhaustive, definitive 

• A * f t % with a wealth ot related stories, anecdotes, and 
historical sketches. 

Ball, VV. W. Rouse. Mathematical Recreations and Essays. London: 
Macmillan. 

This is a standard work, packed with information, but arranged 
more as a handbook than for continuous reading. It is indispensable 
for anyone who likes puzzles. 

Bell, E. T. Men of Mathematics. London: Gollancz, 1937. 

The various portions of this book devoted to non-Eu( lidean geom¬ 
etry make stimulating and engaging reading. On this as well as othei- 
subjects of mathematics, Bell is clear, his style racy and entertaining. 
There are biographical sketches of the leading mathematic ians from 
Zeno to Poincare, sometimes distinguished, olten deliglitlul, always 
readable. The exposition of difficult phases of matliemalics is particu¬ 
larly good. 

j-. The Queen of the Sciences. New \'ork: G. E. .StecluMt. 19.^8. 
This is a reprint of a charming little volume, published originally 
for the World’s tair at Chicago, 1933. Simple, worth reading. 
Black, Max. The Nature of .Mathematics. London: Kegan Paul, 

1933. 

An account of the mathematical philosophies of the three principal 
contemporary schools, concise and clear, but by no m<-ans easy 
reading. 

Bliss, G. A, “Mathematical Interpretations of Geometrical and 
Physical Phenomena,” American Mathematical Monthly. \ ol. 40 
(October, 1933). 

A readable account of the application of mathematics to physical 
phenomena. 

3^3 


364 Mathematics and the Imagination 

Bridgman, Percy W. The Logic of Modern Physics. London: Macmil¬ 
lan. 

Not a mathematics book, but a crystal-clear exposition of modem 
physical theories, the laws of cause and effect, and probability, by a 
distinguished physicist. 

Ball, VV. W. Rouse. History of Mathematics. London: Macmillan. 

Cajori, Florian. History of Mathematics. London: Macmillan. 

Both of these are standard histories of mathematics. The work by 
Cajori is quite detailed, though nothing in comparison to Moritz 
Cantor’s monumental German history of mathematics in four large 
volumes. Ball’s work, while in some respects out of date and discur¬ 
sive, is sufficiently authoritative, and being less detailed, can be read 
more pleasantly with far less of a mathematical background. 

Cajori, Florian. “History of Zeno’s Arguments on Motion,” Ameri¬ 
can Mathematical Monthly, Vol. XXII (1915), Pp. 1-6, 292-297. 

-. A History of Elementary Mathematics. London: Macmillan. 

A standard work, interesting, not too technical, informative. 

Cantor, Georg. Contributions to the Founding of the Theory of Transfinite 
.Numbers. Edited by P. B. Jordain. Chicago: Open Court, 1915. 

A translation of Cantor’s work on the infinite, particularly valuable 
for the historical introduction. 

Carslaw, HoR-rVTio S. The Elements of Non-Euclidean Plane Geometry 
and Trigonometry. London: Longmans Green & Co., 1916. 

.\ good elementary text on non-Euclidean geometry. Technical. 

Cohen and Nagel. Introduction to Logic and Scientific Method. London: 
Routlcdgc, 1934. 

.■\n admirable book in every way. A textbook in a class by itself. 
Refreshing and lucid chapters on the entire problem of scientific 
method and a special chapter on probable inference, well reasoned, 
well written. 

Cooley, Cans, Kline, and Wahlert. Introduction to Mathematics. 
London: Harrap, 1937. 

A clear, unpretentious introduction to many advanced branches of 
mathematics, including infinite classes. It is also suitable for several 
other topics discussed in this book. A text. 

CouRANT, Richard. Differential and Integral Calculus. Vol. I. London: 
Blackie & Son, 1934. 

An excellent text with unusually good examples of the applications 
of the calculus. 

Dantzig, Tobias. Number, the Language of Science. London: Allen & 
Unwin, 1933. 

An interesting non-technical account of the development of the 
number concept up to and including the transfinites. 


Bibliography 365 

De Morgan, Augustus. Budget of Paradoxes. Chicago; Open Court, 
1940. 

A collection of letters, stories, anecdotes, parado.xes, and puzzles, 
all relating to mathematics. 

Dresden, Arnold. An Invitation to Mathematics. New York- Henrv 
Holt & Co., 1936. 

Unusually praiseworthy text on higher mathematics, covering ana¬ 
lytic geometry, number theory, the calculus, projective geometry, the 
mathematics of the infinite. A text with a well-executed idea of 
making mathematics attractive, it presupposes a knowledge of only 
elementary algebra and geometry. 

Eddington, Sir Arthur S. Space^ Time, and Gravitation. Cambridge 
University Press, 1920. 

Brilliant discussion of the problems of relativity including chapters 
on “kinds of space” and “What is geometry."” Non-technical, but by 
no means simple. 

■ . Pathways of Science. Cambridge University Press, 1935. 

Contains interesting discussions on probability by a famous as¬ 
tronomer and equally famous writer on popular science. Ik-yond that, 
the rest is also worth reading, as is anything by Kdtlington. 

Enriques, Federico. Historic Development of Loeic. New York: Henry 
Holt, 1929. « y 

The best work of its kind. The “principles and structure of scienc e 
in the conception of mathematical thinkers.” Large ly non-icchnical 
but occasionally difficult. 

Galileo. Two New Sciences. London: Macmillan. 

Not only one of the greatest books of all times, but a peric cily 
fascinating one. It contains, among much <“lse, (ialilec/s \ i< \vs on 
infinity. I-or anyone interested in the history ol science oi‘ matlie- 
matics, or, for that matter, for anyone who c-tije^ys r«*acling. Besides, 
this edition is a fine piece of bookmaking. 

Granville, William A., Smith, P. F., and LuNcmt^’. \\. R., Tltnutils 
of the Differential and Integral Calculus. London: (iinn Co.. 19-11. 
A good elementary textbook. Well piinted, clear. 

Hardy, G. H. A Course of Pure .Mathematics. Gth Edition. Cambridge: 
Cambridge University Press, 1933. 

A standard work, this elementary introduction to higher m.itlie- 
matics is a tough morsel but will repay whati-ver you invest in it in 
terms of effort. 

Hogben, Lancelot T. Alathematics for the .Million. London; .Allen & 
Unwin, 1936. 

The “Gone With the Wind” of mathematics. Well-written, in¬ 
teresting, this is by no means an easy book. Fhe Marxian intei 7 )reta- 
tion of the history of science and mathematics in Hogben’s {>uiigcni 
style pervades the whole. 


366 Mathematics and the Imagination 

Jeans, Sir James H. The Mysterious Universe. Cambridge University 
Press, 1930. 

Lucid exposition of modern science. Sir James has an inimitable 
gift for making hard things easy, for making gigantic numbers easier 
to remember than telephone numbers, for making tenuous scientific 
theories more transparent than invisible glass. 

Keynes, John Maynard. A Treatise on Probability. London; Mac¬ 
millan, 1921. 

It would be hard to find a more intelligent and comprehensive 
work on the philosophy and mathematics of probability, although 
many portions are only for the trained mathematician. Parts of it, 
nevertheless, are easily understandable, and the style, like Bertrand 
Russell’s, is sparkling. 

Keyser, Cassius, J. Mathematical Philosophy. New York; Dutton, 1922. 

Well-written, not too technical, and clear. In addition. Professor 
Keyser has written many other brilliant essays on mathematics and 
the humanizing of mathematics. He would deserve the title of the 
Grand Old Man of Mathematics—were it not for the fact that it 
would never occur to anyone that Keyser at 78 is anything but 
youthful. 

Klein, Felix. Elementary Mathematics from a Higher Standpoint. 
London: Macmillan, 1932. 

A mathematical classic, although not easy tc follow for the non- 
mathcmatician. Contains a wealth of material, including a discussion 
of transfinite mathematics. 

Levy, Hyman. Modern Science. Hamish Hamilton, 1939. 

This volume devotes considerable portions to the applications of 
mathematics to science. Professor Levy possesses unusual gifts of 
simplification. Highly recommended. 

Lieber, Lillian R. and Hugh G. Non-Euclidean Geometry. New York: 
Academy Press, 1931. 

A delightful little book with charming illustrations, making the 
elements of non-Euclidcan geometry readily understandable. 

Manning, H. P. Non-Euclidean Geometry. Boston: Ginn & Co., 1901. 

A brief, rather simple text. 

. The Fourth Dimension Simply Explained. London: Methuen, 
1921. 

A collection of essays on the fourth dimension, submitted in a prize 
contest run by The Scientific American. Many of the essays are 
amusing, much ingenuity being displayed in finding analogues of 
four-dimensional figures in a three-dimensional world. 

Mark, Thirri.n'g, Nobeling, Hahn, and \1enger. Krise und Neuauf- 
bau in den exakten iVissenschaften. Vienna: Franz Deuticke, 1937. 

A collection of German essays by well-known physicists and 
mathematicians on the revolutionary aspects of modem science. All 


Bibliography 


the contributors to this splendid work were members of the celebrated 
Vienna Circle. The lecture by Hans Hahn is an interesting presen¬ 
tation of some of the more startling paradoxes of modern mathematics. 


Merz, J. T. History of European Thought in the jgth Century. Edinburgh* 
Blackwood & Sons, 4th Edition, 1923. ® 

In this monumental and always eminently readable work, there is 
a^ very full treatment of the development of mathematics in the 
biEty^^"*^ particularly statistics and the theory of proba- 


Peirce, Charles S. Chance, Love, and Logic. New York: Harcourt, 
Brace, Inc., 1923. 

A collection of philosophical essays, particularly on the subject of 

probability, by one of America’s most distinguished philosophers— 
the founder of pragmatism. 


Foundations of Science. Garrison, New York: 
The Science Press, 1913. 

Anything Poincare wrote is worth reading. The pellucid quality of 
s^yle makes all of his non-technical writings readily comprehensible 


Russell, Bertrand. Introduction to Mathematical Philosophy. London* 
Allen and Unwin, 1919. 

A standard work, but sometimes tough sledding. No discussion of 
this subject is easy, but Russell is always readable. 


Mysticism and Logic. London: Allen and Unwin, 1932. 


1 he e^ay. Mathematics and Metaphysicians” in this collection is 
* best style: brilliant, impudent, and jaunty. The problems 
o! inhnite cl^es and Zeno’s paradoxes are particularly well treated. 
The essay A Free Man’s Worship” is one of the finest and noblest 
expressions of faith m science and reason in the English language. 

Sm^h, David E. History of Mathematics, Vols. I and 11. London: 
Ginn & Co., Vol. I, 1923; \'ol. II, 1925. 

A g(wd history, profusely illustrated and well-suited for ihe non- 
professional. 


Snapshots. New York: G. E. Sterhert, 

Intended to reveal some of the more unusual aspects of mathe- 

ma ICS, including mathematical paradoxes. Requires no mathematical 
training whatsoever. 

Sullivan, J W. N. Aspects of Science, 2nd Scries. London: W. Collins 
bons, 1926. 

Stimulating essays for the layman on a variety of .subjects, iiu lucliiig 
mathematics. Recommended, by the same author, Limitations of 
octence, Chatto and Windus, London, 1933. 


368 Mathematics and the Imagination 

Swann, W. F. G. The Architecture of the Universe. London: 
Macmillan, 1934. 

A physicist tells how probability is used in studying the laws of 
gases. Well written, non-technical. 

Whitehead, Alfred N. An Introduction to Mathematics. Oxford 
University Press, 1911. 

This might well serve as a model for all popular books on mathe¬ 
matics. Nothing quite as good appeared before or since its publica¬ 
tion. Designed for the layman, it is simple but not condescending, 
witty but free from “epigramitis,” perfectly clear, informative, full 
of verve, good humour and understanding. A first-rate job in every 
respect. 

Young, J. W. Fundamental Concepts of Algebra and Geometry. London: 
Macmillan, 1911. 

A most enjoyable collection of lectures. Suitable for the neophyte. 


Index 


Abbott, Edwin Abbott. 128, 363 
Abel, Niels Henrik, 17 
Advanced Calculus, 298 
Absolute rest, 24 
Absolute zero, 24 
Acceleration, 301, 326, 342-343 
Achilles and the tortoise paradox, 
37, 38, 57-58, 62 

Ahrens, Wilhelm Ernst Martin 
Georg, 189, 191. 363 
Air-raid casualties, statistics of, 
262-263 
Alcuin, 159 

d'Alembert, Jean le Rond. 246 
Aleph-Null, 45, 54 
Alephs, 45, 46-47, 62, 68 
Alexander of Macedon. 313 
Algebraic equations with integer 
coefficients. 6. 49. 64. 71. 79. 84. 
109. Ill 

Algebraic invariant, 298 
Algebraic numbers, 49. 50. 110 
Alte Probleme—Neue Losuneen, 
219 

American Journal of Mathemat¬ 
ics. The. 178. 191 
American Mathematical Monthly, 
363. 364 

Amusements in Mathematics. 189 
Analysis situs, 266. 271-274. 287 
Analytical geometry. 95-99. 103, 
120-123. 305. 306, 314 
Analytical Theory of Heat. The. 
254 


Analytic 

240 


theory of probability. 


Annals of Mathematics, 191 
Anthropology, statistics in, 260- 
261 

Anti-snowflake curve, 318-351 


Apollo, 71 
Apollonius. 12. 13 
Applied mathematics, 114, 116, 
150, 151 

Approximation, methods of. 318. 

333-334. 336-337 
Arabs, 17. 185 

Arago, Dominique Francois Jean. 
190 

Archimedean number (see tt) . 75 
Archimedes, 35-34. 69. 74. 310. 331 
Architecture of the Universe, The, 
367 

Area under a curve, 335-339 
Argand. Jean Robert. 100 
Aristotle. 62. 117. 213. 230-231 
Arithmetic. 28. 43, 46, 68, 81. 92. 
300. 311 

Arithmetic fallacies, 208-211 
Arithmetic tricks, 163-161 
Arithmetical progression. K2 
Arrow in flight paradox, 37. 38- 
39. 58-^0. 62, 301. 321 
Aspects of Science, 367 
Augustine. Saint. 112. 150 
Axiont, rctlucibility. 63. 218, 22! 

Babylonians. 190 

Bachet. Claudc Gasparcl, Sieur dc 
Meziri.jc. 157 

Ball. Walter Willi.im Rouse, 189. 

190. 191. 219. 221. 363 
Banaclt Steplien. 205-207. 219 
Banach and Tarski'.s tlieoreni. 205- 
207. 219 

Barrow. Isaac, 323 
Beethoven. Ludwig \an. 362 
Bell. Lric 1 einpic. i 12, 363 
Beltrami, faigenio, I 1‘2 
Bergson, Ilcnri. 65, 108 


Index 


370 

Berkeley, George, Bishop, 40 
Bernouilli, James, 240 
Binary notation, 165-173 
Binomial theorem, 249-251, 264 
Bismarck, Otto Edward Leopold, 
Prince von, 177 
Black. Max, 363 
Bliss. Gilbert Ames, 363-364 
Bolsheviks, 8 

Boltzmann, Ludwig, 95. 258 
Bolyai, Johann, 118, 135 
Bolzano, Bernhard. 39. 40-^2 
Boml)elli. Raphael, 91 
Booth, John Wilkes, 236 
"Boss puzzle” {see "15 Puzzle”) 
Bouton, 191 
Boyle. Robert, 257-258 
Bridgman. Percy Williams, 84. 364 
Briggs. Henry. 81. 84 
Brouncker. William. Viscount. 78 
Brouwer, Luitzen Egbertus Jan. 

37. 221. 291, 292, 298 
Brouwer’s problem, 287-291 
Brownian movement, 24-25 
Budget of Paradoxes, 79. 261. 365 
Buffon, George Louis Leclerc, 
Count de. 110, 246-247 
Biirgi, Jobst, 110 

Cabalists, 269 

Cajori, Florian, 81. 298. 355, 364 
Calculus (299-343): 
difTercntial. 14. 75. 299, 310, 
322-329. 330, 341 
integral, 75. 299. 310, 329-342 
of probability, 228, 229. 230, 
238-251. 252. 253 
Cantor. Georg. 39, 41, 42-57, 62. 
63. 194. 364 

Cantor’s diagonal array, 52-53 
Cantor’s paradox, 43, 44 
Cantor's theorem. 45-44, 55 
Card tricks, 163 
Cardan, Girolamo, 91. 168 
Cardinality, 44. 45. 46, 48. 49. 54, 
55 

Cardinal numbers. 64 
Cardinal of the continuum. 53, 55 


Carlyle, Thomas, 39 
Camera, Primo, 23 
Carroll, Lewis, 61 
Carslaw, Horatio Scott, 364 
Casting out nines, 164-165 
Catenary, 107 
Catherine of Aragon, 236 
Catherine II of Russia, 265 
Cauchy, Augustin Louis, Baron, 
14 

Cavalieri, Bonaventura, 332 
Cavalieri’s theorem. 332 
Cayley, Arthur, 118, 156, 186, 298 
Centroid, 340 
Ceulen. van, Ludolph, 75 
Chaldeans, 28 
Chance. 223-264 
calculus of. 239-251 
Chance, Love, and Logic, 264, 367 
Change. 300, 301, 302, 304, 321 
Charlemagne, 157, 159 
Chaucer, Geoffrey, 3 
Chess. 32-33. 68, 115, 173, 222 
Chinese, 90 

Chinese ring puzzle. 169 
Circle. 10-14, 17, 41^2, 66-67. 
73. 78. 80. 277, 278. 279, 280, 
282, 310-312, 337, 343-344 
squaring of the, 12, 65, 66-69, 
71-79, 109, 331 

Classes. 28-31, 42-58, 61. 62. 63. 

120. 132, 151-152, 154, 216-217 
Class of all classes not members 
of themselves, 216-217 
Clausius. Rudolf Julius Emman¬ 
uel. 258 

Clebsch, Rudolf Friedrich Alfred. 
298 

Clock, 14 

Coal and vest problem, 286-287 
Coefficients of an equation, 109- 
110 

Cohen. Morris Raphael, 137, 150- 
154, 224. 264. 364 
Colerus. Egmont, 155 
Collected Scientific Papers (of 
P. G. Tail) . 191 
Combinations, 242-244 


371 


Index 


Combinatorial analysis, 242. 243 
Complex numbers, 95, 101-102, 
103, 210 

Complex plane, 101-102 
Compound interest, 86-87, 111 
Compound probability, 248-251 
Corate, Auguste, 67 
Condorcet, Marie Jean Antoine 
Nicolas, Marquis de, 253-254 
Coniessions (of St. Augustine), 
150 

Congruence, 126, 202, 204, 205, 
206, 219 

Conic sections, 17, 107 
Connectivity: 
nonsimple, 279-283. 284 
simple. 276-277, 281, 282. 283, 
290 

Continued fractions, 75, 78, 85-86 
Continuum, 53, 55, 56, 58 
Continuum, cardinal of the, 53, 
55 

Contributions to the Founding of 
the Theory of Transfinite Num¬ 
bers, 364 

Convergence of a series, 64, 69, 
70. 109 

Cooley, Hollis Raymond, 364 
Co-ordinates. 96-99. 102, 120, 121. 

122, 123. 314 
fcosine, 355-356 
Countable classes, 45, 47, 49 
Counting, 27, 28-30, 34 
Courant, Richard, 364 
Course of Pure Mathematics, A, 
365 

Craig, John, 261 
Creative £i'o/u/ion, 108 
Cretans. 62-63, 221 
Crime of Sylvester Bonnard, The, 
189 

Crisscross curve. 353-355 
Cryptograms. 233-235 
Cubic equation. 17 
Cubit. 173. 191 
Curtate cycloid. 200 
Curvature. 28, 147-148, 328 


Curves: 

anti-snowflake, 348-351 
catenary, 107 

circle. 10-14, 17. 41^2. 66-67. 
73. 78. 80. 277. 278. 279. 280, 
282. 310-312, 337, 343-344 
dock, 14 

crisscross. 353-355 
curtate cycloid, 200 
cyde. 11-13 
cycloid. 196-200 
ellipse, 17 
of error. 257-260 
hyperbola. 17 

nonsimply-connected, 7, 281 
parabola. 17. 106-107, 308. 330. 
331 

pathocircle. 13 
pathological, 343-355 
prolate cycloid. 198-199 
simply-connected. 7, 276-282 
snowflake. 344-350 
space-filling, 352-353 
tractrix. 141, 142, 143 
turbine. 9-10 
Cycle. 11-13 
Cydoid, 196-200 

Dampier. William. Sir, 259, 260 
Dantzig. Tol)ias, HI. 264. 364-365 
Darwin, Charles Gallon, 264 
Dasc. Johann Martin Zathaiias,77 
Da Vinci. Leonardo, 362 
Davis. Watson, 25 
Decimal, nomenninating, 51-53, 
64. 167 

Decimal notation, 164, 165, 166, 
167 

Dcdekind, Richard. 41 
Dcducticc method. 232 
Definite integral, 33‘1-341 
Delphic Oracle. 71. 157 
De .Moivre, .Abraham. 103 
De Morgan. .Augustus, 79. 22H. 261, 
365 

Denumerable cl.jsses. 45. 49. 50 
Denumcrably infinite classes. 15, 
49 


26 


372 Index 


Dependent variable, 31-1, 315, 316. 
321 

Derivative, 14. Ill, 322-329. 341- 
342. 3j6 

Descartes. Rene, 65 , 95, 96 . 290, 
300. 3r)9 

Diagonal proof. Cantor’s, 52-53 
Dice. 237, 239, 244. 245 
Dichotomy paradox, 37-38 
Die Paradoxien des Vneridlicncn, 
40 

Differential calculus, 14, 75, 299, 
310. 322-329. 330. 341 
Differential and Integral Calculus, 
304 

DifTcrentiation. 5, 322-323. 325- 
320. 311. 342 
Dimension. 119-121 
Diopliaiuus, 187 
Discriminant. 298 
Divergence of a series, 109 
Doughnut, 281, 282. 287 
Drawing cards from a pack, 237, 
213-244 

Dresden, Arnold. 191, 365 
Doyle, Arthur Conan, Sir, 263 
Dudeney. Henry Ernest, 189 
Durer, Albrecht. 185 
Duodecimal notation. 190 
Duplication of the cube. 12. 66, 
68. 71-72. 109 
Dyadic notation, 105-173 

c, 49. 50. 66. 80. 84-86, 87-88. 89. 
no. Ill 

Eddington. Arthur Stanley. Sir, 
23. 32. 131. 151. 365 
Egyptians, 8. 17, 66, 74, 167 
Einfiihrung in das mathematische 
Dcnken, 219 

Einstein. Albert. 22. 23, 29 
Elementary Mathematics from a 
Higher Standpoint, 366 
Elements, 9-10. 151-152. 154 
Elements (of Euclid), 4. 108. 113 
Elements of the Differential and 
Integral Calculus. 365 


Elements of Non-Euclidean Plane 
Geometry and Trigonometry, 
364 

Ellipse. 17 
Ellipsoid. 147 
Encyclopcdie, 246 
Enriques. Federigo, 365 
Epiinenides paradox, 62-63, 221 
Equations: 
algebraic, 6. 298 
with integer coefficients. 6, 49, 
64. 71. 79. 84. 109, 111 
cubic. 17 
quadratic. 17 
quartic. 17 
quintic. 17-18 

Equiprobability. 241-242, 248, 251 
Error, probability curve of, 257- 
200 

Essai philosophique sur la proba- 
Inlitc. 264 

Euclid. 4. 62. 69. 108. 113-114. 115, 
134, 135. 137-140, 143, 117-150, 
187. 192. 271. 301 

Euclidean geometry. 116. 134. 137, 
138. 139. 140. 143. 149. 150 
Euclidean manifold, 123-121 
Euler. Eeonhard, 85-86. 92. 93. 
103. 156. 185. 265-268. 269. 
290-291 

Euler s theorem. 266-268. 290-291 
Evolution, 6. 93 
Existctice. 61-62. 63 
Exponential function. 88-89, HI. 
300. 342 

Exponents, 82, 110 

Factorial. Ill 
Fallacies: 

arithmetic, 208-211 
geometric, 211-213 
mathematical, 207-213 
Falling bodies. 316-321. 326-328, 
342-343 

Fermat. Pierre de. 68. 156. 187. 
188, 239-240. 306 

Fermat’s last theorem, 68, 187-188 
“15 Puzzle,’’ 177-180 


Index 


373 


Finite classes, 43 

Finite numbers, 19, 22, 23, 32-33, 
47, 63 

Flatland—A Romance of Many 
Dimensions, 128, 363 
Flaubert, Carolyn, 158 
Flaubert, Gustave, 158 
Formalists, 62, 222 
Forsyth, Andrew Russell, 154 
Foundation of Physics, 154 
Foundations of Science. The. 367 
Four-color problem. 287-291 
Four-dimensional geometry, 93, 
113, 115, 116, 118, 119, 124, 300, 
359 

Four-dimensional manifold, 123- 
124, 154 

Fourier, Jean Baptiste Joseph. 
254-255 

Fourth dimension, 116, 117, 118, 
119, 124, 125, 127-128, 130, 131 
Fourth Dimension Simply Ex¬ 
plained, The, 366 
Fractions: 

continued, 75, 78, 85-86 
rational. 48 
France. Anaiolc, 189 
Franklin, Benjamin, 186 
Frege, Gottlob, 218 
Function. 5. 14. 304, 309, 312-316. 
318-327, 332 

exponential, 88-89, 111, 300, 
342 

monogenic, 14 
one-valued, 64 
polygenic, 14 
trigonometric, 355-356 
Functions of a real variable, the¬ 
ory of. 201 

Fundamental Concepts of Algebra 
and Geometry, 154. 368 

Galileo. 41. 198.252, 365 
Galois, Fvariste, 71 
Cans, David. 364 
Gases, kinetic theory of, 257-260 
Gauss, Karl Friedrich. 68. 77. 92. 
101. 108. 134. 135, 258 


Geodesic. 146, 147, 181 
Geometric fallacies, 211-213 
Geometric mean. 99-100 
Geometric progression, 82 
Geometry: 

analytical. 95-99. 103, 120-123, 
305. 306, 314 

Euclidean, 116, 134, 137, 138, 
139. 140, 143, 149. 150 
four-dimensional, 93, 113, 115, 
116, 118. 119, 124, 300. 359 
Lobachevskian, 136-139, 140, 

142-143, 146, 150 
non-Euclidcan, 113, 132, 134- 
150, 155. 300. 359, 360 
nonquantitaiive, 272-297 
Riemannian, 139-140, 142, 143. 

144, 145. 146. 149, 150 
rubber-sheet, 265-298 
Geometry of Four Dimensions, 
154 

Gestapo, 216 

“Goijig to Jerusalem,” 29-80 
Goldbach. C., 187 
Goldl)ach's theorem, 187 
Cold Bug, The. 233 
Googol, 20-25. 27. 32. 33. 47. 81. 
221 

Googolplex, 23-25. 31, 32. 44, 81 
Granville, \N'il!iain Anthony, 365 
Graph (to[K)logital) , 206 
Gra])hic leprescntation of points, 

96-99, 31 1 

Grassinann. Hermann. 118 
Gravitational constant, 327-328, 
312-343 

Great circle, 1 13. 146, 147, 181 
Greeks, 8, 16. 31. 40, 66, 71, 90, 95. 
108. 213 

Griin.ildi. Fiancis Maria, 100 
Groups. 4, 5. 8 

Growth, phenomenon of. 88, 342 
Guessing mnnbers, 103, 164 

Hahn, Mans, 298. 306 
Haldane, John Burdon Sanderson, 
262-203 


Index 


374 


Hall, Henry Sinclair, 190 
Hamilton, William Rowan, Sir, 
156 

Hardy, Godfrey Harold, 32, 365 
Hausdorff, Felix, 204-205, 207 
Hausdorff's theorem, 204-205, 207, 
219 

Henry IV of France, 75 

Henry VIII of England, 236 

Heraditus, 27 

Hermite, Charles, 111 

Hessian, 298 

Higher Algebra, 190 

Higher derivatives, 326-329 

Hilbert, David, 222 

Hindus, 17, 90, 268 

Historic Development of Logic, 

365 

History of Elementary Mathemat¬ 
ics, A, 364 

History of European Thought in 
the 19th Century, 367 
History of Japanese Mathematics, 
A, 191 

History of Mathematics, 364, 367 
History of Philosophy, Science 
and Technology in the Six¬ 
teenth and Seventeenth Centu¬ 
ries, 356 

History of Science and its Rela¬ 
tions with Philosophy and Re¬ 
ligion. A. 259 
Hitler, Adolf. 228 
Hogben, Lancelot Thomas, 365- 

366 

Homer, 362 

Hottentots, 19 

Houdini, Harry, 127 

How to Draw a Straight Line, 290 

Hyperbola, 17 

Hypercube {see Tesseract) 

Hyperplane, 124 

Hyperradical, 16 

Hypervolume, 126 

I, 65, 66, 89, 91, 93-95, 99, 100, 
101-103, 110, 210 


Illusions, optical, 211, 220-221 
Imaginary numbers, 65, 66, 90-95, 
99, 100, 103, 117, 210 
Impossibility, mathematical, 67- 
68 

Indefinite integral. 340-342 
Independent events, 244-247 
Independent variable, 314, 315, 
316, 321, 326 

Induction, mathematical, 3S-36, 
232 

Infinite dasses, 31, 43-44, 45-46, 
48-50, 53, 55-58, 61, 62, 63, 132 
Infinite numbers, 19, 20, 22, 23, 
33-36 

Infinite series, 38, 64, 65, 66, 69- 
70, 75, 76, 77, 78, 79, 80. 87, 109, 
346-347 

Infinitesimal, 40, 43, 332 
Infinity. 19, 35, 36, 40, 41, 43. 56, 
61. 63. 132, 222, 332 
Integers. 34-35. 43-45, 46-49, 50, 
51. 52. 54. 63, 92, 108, 110 
Integral calculus, 75, 299, 310, 
329-342 
Integrals: 
definite, 339-341 
indefinite, 340-342 
Integration, 5. 339-340, 341, 342 
Interlocking rings, 285, 286. 287 
Introduction to Logic and 5cjVn- 
tific Method, An, 150-154, 264, 
364 

Introduction to Mathematical Phi¬ 
losophy, 367 

Introduction to Mathematics, 364 
Introduction to Mathematics, An, 
368 

Iniuitionists, 62. 221-222 
Invariants, 273-274, 297. 298, 300 
Invitation to Mathematics, An, 
190, 365 
Involution, 54 
Irrational numbers, 49 
Italians, 17 

Jacobian, 298 
Japanese, 175-176 


Index 


Jeans, James, Sir, 258, 366 
Jeu du Taquin {see ‘*15 Puzzle”) 
Johnson, Samuel, 28 
Johnson, William Woolsey, 191 
Jordan, Camille, 7, 276-279 
Jordan’s theorem, 7. 276, 278, 279 
Josephus, 173, 174 
Josephus problem. 173-176 

Kant. Immanuel. 6, 118 
Kasner, Edward, 23 
Kempe, Alfred Bray, Sir, 290 
Kepler, Johann, 156, 252. 331-332 
Keynes, John Maynard. 238. 263, 
366 

Keyser, Cassius Jackson, 13. 366 
Kinetic theory of gases. 257-260 
Kirchhoff, Gustav Robert, 276 
Klein, Felix. 299, 366 
Kline, Morris, 364 
Knight, Samuel Ratcliffe, 190 
Kries, Johann von. 263 
Krise und Neuaufbau in den ex- 
aAlen JVissenschaflen, 154, 367 
Kronecker, Leopold. 35 

La Fontaine, Jean de. 7, 217 
Lagrange, Joseph Louis. Count, 
156. 253 

Laguerre, Jean Henri Georges, 11, 
12. 13 

Laplace, Pierre Simon, Marquis 
de, 221, 252-253, 264 
Lazzerini, 247 
Lebensrautn, 293 
Lee, Samuel. 261 
Legouve, Ernest. 183-184 
Leibniz, Gottfried Wilhelm. Baron 
von, 39, 40. 75-77, 91. 156. 165, 
166. 240. 301, 309 
Length of a curved line, 331, 333- 
335, 356 

Les problemes plaisants et delec- 
tables, 157 
Levy, Hyman, 366 
Licber, Hugh Gray, 366 
Liebcr, Lillian R.. 366 
Lietzmann, Waliher, 189, 219 


375 

Limit, 64, 69. 70. 235. 310, 332 
of a function, 312-316. 321-327, 
332. 341 

of a variable quantity. 3IO-3I2 
Lincoln, Abraham. 236 
Littdemann, Ferdinand, 71, 72. 79. 
Ill 

Lindsay. Robert Bruce, 154 
Listing. Johann Benedict, 272 
Lloyd. Sam. 177. 178 
Lobachevsky. Nikolai Ivanovich, 
118, 135-143, 146-147. 150, 181, 
271 

Lobachevskian geometry, 136-139, 
140. 142-143, 146, 150 
Locke, John. 225, 252 
Logarithms. 50. 80. 81-84, 110, 
210-211 

Logic. 300, 360-361 
Logic of Modern Physics, The, 
364 

Logical paratioxes, 194, 213-219 
Logical posiii\ism, 359-361 
Logistic school, (52. 63, 222 
Loiigley. William Ravniorid, 365 
Lucas, Edouanl, 189 
Ludolphian number (see -rr) , 75 
Lustiges und Merkwiirdiges von 
Za/ilen und Formen, 189, 219 

Magic sc|uarcs. 185-186 
MacFarlane, y\lexaiuler. 184 
Mathin. Jolin. 75-76, 78, 118 
Manifold. 119-124, 133. 149. 154 
Euclidean, 123-121 
four-ilimensional, 123-124, 154 
nonsiinplv-connecietl. 281 
simpl> conncfted, 282 
three-dimensional, 120-124, 154. 
282-281 

tuo-dimeiisiona!, 278, 279, 282 
Manning. Heniy Parker, 366 
Map-coloring problem, 186, 287- 
291 

Margenau, Henry, 154 
Mark. Hermann, 366 
Maichitig. 28 
Matci iaiisnt, 254-256 


Index 


376 

Mathematical Excursions, 189 
Mathematical fallacies, 207-213 
Mathematical impossibility, 67-68 
Mathematical induction, 35-36, 
232 

Mathematical Philosophy, 366 
Mathematical possibility. 212 
Mathematical recreations, 156-192 
Mathescope, 2.‘)-26 
Mathematical Recreations and Es¬ 
says, 189, 363 

Mathematical Snapshots, 367 
Mathematics for the Million, 365 
Mathematische Unterhaltungen 
und Spiele, 189, 363 
Maxima and minima. 306-309, 
325-326 

Maxwell, James Clerk, 258 
Measuring. 28, 66. 133, 13-1. 202- 
203 

Mecatiiqiie Celeste, 221, 253 
Men of Mathematics, 363 
Mendel. Gregor Johann, 10 
Mcnger, Karl. 219. 366 
Mdr^, Antoine Gombault. Cheva¬ 
lier de, 239-2-10 
Merrill, Helen Abbot, 189 
Mersenne, Marin. 187 
Merz, John Theodore, 367 
Meiamathematics. 222 
Metaphysics, 359-361 
Methods of approximation, 318, 
333-33-1. 336-338 
Mikaini, Yoshio, 191 
Miracles, 21. 261-262 
Mirifici Logarithmorum Canonis 
Descriplio, 80. 110 
Mobius, August Ferdinand, 118, 
186 

Mobius strip, 118, 186, 28-1-286 
Modern Science, 366 
Mohammedans. 261. 268 
Moment of inertia, 3-10 
Monkey and rope puzzle. 192 
Monogenic functions, 14 
More. Henry, 118 
Motion. 37. 58-60, 194-195. 274, 
300. 301-304, 321, 342-343 


Mutually exclusive events, 243-244 
i\Iysterious Universe, The, 366 
Mysticism and Logic, 37, 367 

Nagel. Ernest, 150-154, 224. 264 
Napier, John, 80-84, 110 
Napoleon, 253 
Nature, 262 

Nature of Mathematics, The, 363 
Needle problem, 110, 246-247 
Negative curvature, 147-148 
Negative numbers, 65, 90-91, 92, 
111, 117, 210 
Newcomb, Simon, 78 
New Pathways of Science, 365 
Newton, Isaac, Sir, 75, 80, 252, 261, 
304. 309, 323 
Nim, 172, 191 
Nines, casting out, 164-165 
Nobeling, Georg, 154. 366 
Nondenumerable classes, 49, 50, 
53 

Non-Euclidean geometry, 113, 132, 
134-150, 155, 300, 359. 360 
Non-Euclidean Geometry, 366 
Nonquantitative geometry, 272- 
297 

Nonsimple curves, 7, 281 
Nonterminating decimals, 51-53, 
64. 167 
Notation: 
binary. 165-173 
decimal. 164, 165, 166, 167 
duodecimal. 190 
dyadic. 165-173 
positional. 80. 110 
sexagesimal. 190 

Number, the Language of Science, 
111, 264, 364-365 
Numbers: 
cardinal, 64 

complex. 95. 101-102, 103, 210 
imaginary, 65, 66. 90-95, 99, 100, 
103, 117. 210 

infinite. 19, 20. 22. 23. 33-36 
irrational. 49 

negative, 65, 90-91, 92, 111, 117, 
210 


Index 


Numbers {Continued) : 
ordinal. 64 

prime. 64, 108. 187-188. 192 
rational. 49, 50 
real. 49, 50, 51. 53. 54. 55 
transcendental, 6. 49. 50, 53. 64, 
78. 79. 84. 111. 117 
iransfmite, 45, 47, 49. 53. 54, 55, 
63. 194. 359 

Omar Khayyam, 90, 111 

0/1 the Hypotheses tVhich Under¬ 
lie the Foundations of Geome¬ 
try. 139 

One-sided surface, 118, 186. 284- 
286 

One-to-one correspondence. 29, 31, 
34. 41. 45. 48. 49. 50. 56. 57. 58, 
59. 60. 206 

Onc-valucd functions, 61 

Optical illusions. 211. 220-221 

Ordinal numbers. 61 

Osgood, William Fogg, 298 

Parabola. 17, 106-107, 308. 330. 
331. 356 

Parabolic segment. 356 

Paradoxes. 193-222 

Achilles and the tortoise, 37, 38, 
57-58. 62 

arrow in flight, 37. 38-39, 58— 
60. 62. 301, 321 
Cantor's, 13, -11 
Dichotomy, 37-38 
Epimenides. 62-63. 221 
of infinite classes, 43, 44 
logical. 191, 213-219 
railway train. 200 
rolling circles, 195-196 
rolling coin, 191-195 
Russell’s. 216-217 
Parallel lines. 65. 131-110. 143, 
151-155 

Parhexagon, 11-16 
Pascal. Blaise, 112. 156, 239-240 
Patho-circlc, 13 
Pathological curves. 313-355 
Peirce, Benjamin, 103 


377 

Pcircc. Charles Sanders, 237-238. 

256-257. 264. 367 
Perinutalions, 242-243 
Peterson, 261 

TT, 41-12. 19. 50. 61. 66-67, 71. 72. 
74. 75-79. 89. 109. 1 10. Ill, 246, 
217 

Plane, 120, 124. 147 
Plato, 134 

Pliicker. Julius, 118 
Poe. Edgar .Mian. 233 
Poincare, Henri. 35. 63, 272, 298, 
367 

Point sets, theory of. 201-201. 219, 
291 

Poisson. Simeon Denis, 160, 190 
Polsgenic functions. 11 
Possibility, inathcinalical. 212 
Positional notation, 80, 110 
Positive cursaturc, 147-148 
Pouring problems. 161-162, 190, 

191 

Pretzel. 278. 283. 281. 287 
Prime numbers, 64, 108, 187-188, 

192 

Principia, 80. 261 
Principia Malhematica, 218 
i’rituiple of insufliciciu reason. 

229-230, 263-264 
Probability (223-261): 
coin[>ound. 218-251 
etjuiprobabiliiy. 241-212. 218. 

251 

independent evenis, 214-217 
mutually exclusi\c eseiiis, 213- 
21t 

principle of insufficient reason, 
229-230. 263-261 
relatixe frct|uetir\ view, 230-237 
statistical ijiterpretation. 230- 
237 

subjective view, 227-230 
truth frecjucncy ilieory. 238 

Prol>lcms: 

of \pollonius. 12-13 
Brouwer's. 291-297 
of coat and vest. 2HG-2H7 


Index 


378 

Problems {Continued ); 
drawing cards from a pack, 237. 
2-13-214 

of existence, 61-62 
of falling bodies. 316-321, 326- 
328, 342-343 
four-color, 287-291 
of interlocking rings, 285, 286, 
287 

map-coloring, 186. 287-291 
Mbbius surfaces. 118, 186, 284- 
286 

needle. 110, 246-247 
number theory. 186-188 
pouring. 161-162, 190. 191 
relationship. 183-185 
ring. 168-169 

river-crossing. 159, 189-190 
Russian multiplication. 167-168 
of seven bridges. 265-268. 270 
shunting. 159-160 
spider and fly. 181-182 
string. 191-192 
tower of Hanoi. 169-171 
Projective geometry. 151 
Prolate cycloid. 198-199 
Propositional functions. 263 
Pseudosphere, 140, 142-143. 146. 
151-155 

Ptolemy. 74. 117 

Pure mathematics, 114, 116, 150- 
152 

Puzzles, 156. 157. 158. 162. 168- 
169, 177-180, 189-190, 191-192. 
300 

Pythagoras, 28, 357, 359 
Pythagoreans. 268 
Pythagorean theorem, 121-122, 124 

Quadratic equation, 17. 298 
Quartic equations. 17 
Queen of the Sciences, The, 363 
Qu^ielet, Lambert Adolphe 
Jacques, 260 

Quintic equations, 17-18 

Radical, 16-18 
Radius of curvature, 328 


Railway train paradox. 200 
Ramsay, Frank Plumpton, 221 
Rate of change. 111, 305. 316, 321- 
323. 326 

Rational fractions, 48 
Rational numbers, 49, 50 
Real numbers, 49, 50, 51, 53, 54, 
55 

Reason and Nature, 154 
Reasoning by probable inference, 
224. 226 

Reasoning by recurrence, 36 
Recreations, mathematical, 156- 
192 

Recreations Mathematiques, 189 
Rectifying the parabola, 331, 356 
Reducibility {see Axioms) 
Relationship problems, 183-185 
Relative frequency view of prob* 
ability. 230-237 
Rest, absolute, 24 
Richter. 77 

Riemann. Ceorg Friedrich Bern- 
hard. 139. 140, 142-150. 181.271 
Riemannian geometry, 139-140, 
142. 143, 144, 145, 146, 149, 150 
Ring. 5, 279. 281 

River-crossing problem, 159, 189- 
190 

Rolling circles paradox. 195-196 
Rolling coin paradox, 194-195 
Rubber-sheet geometry, 265-298 
Ruffini. Paolo. 17 
Russell, Bertrand Arthur William, 
37. 39. 56. 60. 218, 221. 222. 299. 
367 

Russell’s paradox. 216-217 
Russian multiplication, 167-168 

Sanctions, 294 
Sand Reckoner, The, 33-34 
Sceptical Chymist, The, 257 
Scheherezade. 235 
Schubert, Hermann Caesar Han¬ 
nibal. 176 

Science and Hypothesis, 298 
Sevenih-Dav Adventists. 148 
Second derivative. 326-328 


Index 


379 


Self-consistency, principle of. 62- 
63, 115-116, 124 
Semantics. 90 

Series, infinite, 38, 64, 65, 66, 69- 
70. 75, 76, 77. 78, 79, 80. 87. 109, 
346-347 

Seven bridges’ problem, 265-268, 
270 

Sexagesimal notation, 190 
Shanks. W., 77-78 
Sharp, Abraham, 77 
Shunting problem, 159-160 
Simple curves, 7. 276-282 
Sine. 355-356 
Skewes, 32 

Skewes’ number, 32, 33 
Slope of a tangent, 324-325, 330, 
356 

Smith, David Eugene. 191. 367 
Smith. Percey Franklyn, 365 
Socrates, 228 

Sophie Charlotte. Queen of Prus¬ 
sia. 39 

Sophists, 213 
Soviet Russia. 93 
Space continuum, 58 
Space, physical, 28. 56, 65. 112- 
113, 117, 119, 124. 131. 132, 133, 
137, 149-150 

Space-filling curve. 352-353 
Space, Time, and Gravitation, 154, 
365 

Spencer, Herbert, 6 
Sphere. 143-145, 147. 181.203-204, 
282, 283 . 284 , 290 
Spider and fly problem, 181-182 
Square root, 108, 209-210 
Squaring of the circle, 12. 65. 66- 
69, 71-79. 109, 331 
Squaring of the parabola, 330, 331 
Statistical interpretation of prob¬ 
ability. 230-237 
Statistical view of nature, 254 
Statistics: 

of air-raid casualties. 262-263 
in anthropology. 260-261 
Sieinhaus, Hugo, 367 
Stereometria, 332 


Story, William Edward, 191 
Straight line. 146, 181-183 
String problem. 191-192 
Subjective view of probability, 
227-230 

Sue, Eug<;ne Joseph Marie, 183 
Sullivan. John William Navin, 367 
Swann, William Francis Gray, 367- 
368 

Sylvester, James Joseph, 118, 298 

Tait, Peter Guthrie. 176, 191 
Tangent of an angle, 355-356 
Tangent to a curve. 324-325, 356 
Tarski. Alfred. 205-207. 219 
Tartaglia, Niccold, 159 
Terminating decimal, 64 
Tesseract. 109. 125. 126 
Thcologiae Christianae Principia 
Malhematica, 261 
Theorems: 

Banach and Tarski's, 205-207, 
219 

binomial, 249-251. 264 
Cantor's, 43--14, 55 
Cavalieri's. 332 
Euler's. 266-268, 290-291 
Fermat's last, 68, 1H7-188 
geometric meai», 99-100 
Goldbach's. 187 
Hausdorlfs. 204-205. 207. 219 
Jordan's. 7. 276, 278, 279 
parhexagon. 15-16 
probability of joint occuncncc 
of two independent events, 
244-215 

Pythagorean. 121-122, 124 
Theory: 

of cycles, 12 
of groups, 5 

of point sets, 201-204, 219, 291 
of probability. 223-264 
of types, 63. 218. 221 
Thirring. Mans, 366 
Three-dimensional geometry, 120- 

123 

Three-dinionsiona! manifohl. 120- 
124, 154, 282-284 


Index 


380 

Three-dimensional space. 56, 115- 
116, 130, 205. 206 
Time continuum, 58 
Todhunier. Isaac, 261 
Topology’, 266. 271-27-1, 276-297. 
300, 359 

Tossing coins, 225, 231, 232, 2-14, 
245 

Tower of Hanoi problem, 169-171 
Iracirix, 141, 142, 143 
Transcendental numbers, 6. 49,50, 
53. 64. 78, 79. 84, 111, 117 
Transfinite numbers, 45, 47, 49, 
53. 54. 55, 63. 194. 359 
Treatise on Probability, A, 238, 
263. 366 

Trigonometric functions. 355-356 
1‘risection of an angle. 12, 68. 72, 
109 

Truth frequency theory of prob¬ 
ability, 238 
Turbine, 9-10 
Twain, Mark, 156, 157 
Two-dimensional space. 128 
Two-dimensional manifold, 278, 
279, 282 

Tu'o New Sciences, 365 
Types, theory of. 63, 218, 221 

I’liraradical. 16-18 
Universe Around Us, The, 258 


Variable quantities, 5, 305-306, 
310-312 
Variables: 

dependent. 314, 315, 316, 321 
independent, 314, 315, 316, 321, 
326 

Veblen. Oswald, 151 
Velocity. 301. 342-343, 356 
Veronese, 118 

Vieta, Francisco. 74-75. 80-81 

Vizetelly, Frank. 3 

Votn Punkt lur vierten Dimension, 

155 

1'orstudien zur Topologie, 272 

Wahlert, Howard E., 364 
Waismann, Friedrich, 219 
Wallis, John. 75-76. 78, 118 
Weierstrass, Karl Theodor Wil¬ 
helm. 39. 43. 298, 332 
Wessel, Caspar, 100 
Wcyl, Hermann. 221 
Whitehead. Alfred North, 218. 368 
^Volf, Abraham, 356 

Yoneyama. 291 

Young, John Wesley, 151, 154, 368 

Zeno. 37-39.57. 194, 301.321 
Zero, absolme. 24 
Zeuxippus, 34 


About the Authors 


Edward Kasner was born in New York City in 1878. He 
was graduated from the College of the City of New York and 
received his Ph.D. in 189Q from Columbia University. After 
study at the University of Gottingen he joined the faculty of 
Columbia, where he has been professor of mathematics since 
1910 and Adrain Professor of Mathematics since 1937. He is 
a member of the National Academy of Sciences and of the National 
Research Council and has published six books on mathematics, 
including '‘^Present Problems of Geometry.^'' 

James Newman was born in New York City in 1907 . He 
is a graduate of Columbia and a member of the New York bar. 
Mr. Newman studied with Dr. Kasner while doing graduate 
work at Columbia. He is a contributor to ^'Scripta .Mathe¬ 
matical'' and the author of numerous papers on literary and 
mathematical subjects.