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by Constance Reid 







by Constance Reid 

"There is no royal road to geometry," Euchd 
told Ptolemy. But now Constance Reid tells us, 
"Modem geometry is a royal road." And she 
proves it with her fascinating introduction to the 
wonderland of twentieth-century mathematics, 
A Long Way from Euclid. 

This book will delight anyone who has ever 
felt the spell of the Queen of the Sciences. Based 
in part on the author's previous success, Intro- 
duction to Higher Mathematics, it concentrates 
on the role played by the Elements of Euclid in 
the last two thousand years. The reader needs 
no mathematical background beyond his recol- 
lection of elementary algebra and plane geom- 
etry. The author's clear and simple explanations, 
aided by more than 80 drawings integrated with 
the text, will take him step by step from ideas 
familiar since childhood to some of the most 
exciting outposts of contemporary mathematics: 
the arithmetic of the infinite, the paradoxes of 
point sets, the "knotty" problems of topology, 
the "truth tables" of symbolic logic. 

Constance Reid begins with the ancient 
Greeks' disturbing discovery that the real world 
did not fit the system of numbers they had so 
carefully laid out. There are, they found, quanti- 
ties that cannot be specified in terms of whole 
numbers. The Greeks' beautifully neat theory 

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A long way from Euclid 



By the Author 

From Zero to Infinity 
Introduction to Higher Mathematics 
A Long Way from Euclid 






by Constance Reid 

Thomas Y. Crowell Company New York 
A Established 1834 

Copyright 1963 by Constance Reid 

AH rights reserved. No part of this book may 

be reproduced in any form, except by a reviewer, 

without the permission of the publisher. 

Designed by Judith Woracek Barry 

Manufactured in the United States of America 

Library of Congress Catalog Card No. 63-18418 

123456789 10 

This book, which is based in part on the author s Intro- 
duction to Higher Mathematics, has been written for 
those whose first, and last, contact with real mathematics 
was with plane geometry and the Elements of Euclid. 
In a sense mathematics as we know it today began with 
the Elements. In more than two thousand years it has, of 
course, come a long way from Euclid. But it has never 
left him behind. 

It is the hope of the author that the reader of this 
book will be able to glimpse through his own misty 
memories of Euclid's geometry the outline of some of the 
more imposing edifices of modern mathematics. 


1 The Golden Knot in the Golden 
Thread . . . The direction of Western mathematics was 
set by a question it could not answer ... 1 

2 Nothing, Intricately Drawn Nowhere 
. . . Seeking sanctuary from the unsolvable, the Greeks 
laid down a system that was to remain for more 
than twenty centuries the logical ideal of all scientific 
thought . . . 16 

3 The Inexhaustible Storehouse . . . 
Some of the greatest minds of all time have fallen under 
the spell of the simple series which begins with 
0, 1, 2, 3, ... SO 

4 A Number for Every Point on the 
Line ... At the very foundation of mathematics lies 
an assumption which it took two thousand years to 
justify . . . 46 

5 Journey That Begins at O . . . With 
a simple combination of algebraic and geometric ideas, 
problems that eluded the greatest of the ancients fall be- 
fore modern schoolboys ... 60 


6 How Big? How Steep? How Fast? 
... It took genius of the highest order to recognize the 
mathematical relationship which exists among the answers 
to these three questions ... 78 

7 How Many 'Numbers Are Enough? 
... To make the ordinary operations of arithmetic always 
possible, mathematicians were forced to begin an exten- 
sion of the number system which was to lead them inevi- 
tably to the "wonderful square root" of 1 ... 97 

8 Enchanted Realm, Where Thought Is 
Double . . . Artists and mathematicians together create 
a geometry in which most of Euclid's theorems no longer 
hold . . . 115 

9 The Possibility of Impossibility . . . 
In a revolutionary change of attitude mathematicians no 
longer try to do the impossible but instead try to prove 
that is it impossible ... 133 

10 Euclid Not Alone . . . When mathe- 
maticians broke the shackles of Euclid's "self-evident" 
truths, they discovered the meaning of truth . . . 148 

11 Worlds We Make ... An everyday 
method of mapping by coordinates enables mathemati- 
cians to move freely in worlds of more than three dimen- 
sions . . . 163 

12 Where Is In and Where Is Out? . . . 
The earliest-grasped mathematical concepts are basic to 
the most modern geometry, in which we find that simple 
questions have unbelievably complex answers and the 
obvious is almost impossible to prove . . , 174 

13 What a Geometry IS ... A new 
way of thinking enables mathematicians to bring together 
under one definition all the many and varied "geometries" 
which have been developed since Euclid first compiled 
the Elements . , . 188 

14 Counting the Infinite . . . The in- 
finiteproblem . . . paradox . . . and paradise has been 
with mathematics since its beginning . . . 204 

15 A Most Ingenious Paradox . . . 
When the smallest and the greatest are combined in one 
branch of mathematics, paradoxes abound . . . 226 

16 The New Euclid . . . The German 
asked a question the Greek would never have asked, and 
got an answer he did not expect . . . 245 

17 Of Truth Tables and Truths . . . 
Methods of mathematical proof are analyzed by an in- 
genious method which utilizes only a handful of the most 
everyday ideas of language . . . 258 

18 Mathematics, the Inexhaustible . . . 
Mathematics has come a long way from Euclid to a con- 
clusion which he, of all men, would find most satisfy- 
ing ... 271 

Index . . . 283 

There is no royal road to geometry. 


Modern geometry is a royal road. 




Golden Knot 
in the 
Golden Thread 

mathematics began, there was no ques- 
tion among mathematicians but that 
the gods themselves were mathemati- 
cians too. But were the gods arithmeti- 
cians, or were they geometers? 

Number ruled the Universe, ac- 
cording to Pythagoras in 500 B.C. Two 
centuries after Pythagoras, at about 
the same time that Euclid was com- 
piling the Elements, Plato was asked, 
"What does God do?" and had to reply, 
"God eternally geometrizes." The 
choice of God as geometrician rather 
than arithmetician had quite literally 
been forced upon Plato and the other 
Greeks by two of the profoundest 
achievements of pre-Euclidean mathe- 
matics, both of them ironically due 
to Pythagoras and his followers. 

These two achievements deter- 
mined the decisive choice of form 
over number and set Western mathe- 
matics on the path it would follow for 
twenty centuries. The first was the 
discovery and proof that the square 
on the hypotenuse of a right triangle 
is equal to the sum of the squares on 
the other two sides. The second was 
the discovery and proof that when 
the sides of a right triangle are equal 
there is no number which exactly 
measures the length of the hypotenuse. 

Specific instances of what we now call the Pythagorean 
theorem were known long before the Greeks in such far 
and separated parts of the world as India and China, 
Babylon and Egypt. In early Egypt, as the pyramids were 
being erected, basic right triangles were formed on the 
knowledge of the most familiar instance of the theorem: 
32 + 42 = 52 

A rope was divided into twelve units by knots tied at 
equal intervals, and pegs were placed in the third, 
seventh, and final knots. When the rope was stretched and 
pegged into place, it formed of necessity the desired right 
triangle : 

Although the Egyptians knew 3 2 + 4 2 = 5 2 and other 
similar relationships obtained by multiplying or dividing 
this one, we do not know if they were aware that the 
equation gave no mere approximation but a theoretically 
exact right triangle.* 

Whether this general truth was 'actually known 

* The Rhind Papyrus contains only such equivalents as 
(X) + P = (ix) and 12 2 + 16 3 = 20 2 , which are obtained, 
respectively, by dividing and multiplying the original by 4. 


earlier, history has left the discovery of the general 
theorem to the Greeks, and traditionally to Pythagoras. 
Pythagoras was in his youth a pupil of Thales, who had 
measured the height of the great pyramid by comparing 
the length of its shadow with that of a vertical stick. Later, 
as a teacher himself, Pythagoras opened a school of his 
own in his native town, where he attracted only one 
pupil, also named Pythagoras, whom he had to pay to 
keep in class. Justifiably discouraged by this lack of 
appreciation at home, he set out, as Thales had once 
advised him, for Egypt. He came at last, after years of 
travel and study, to southern Italy. Here he opened a 
school which, in contrast to his first, was one of the most 
wildly successful schools in history. Crowds flocked to 
hear Pythagoras. Besides the youths whom he instructed 
during the day, the business and professional leaders of 
the community attended his evening lectures and to hear 
Pythagoras maiden and matron alike broke the law 
which prohibited them from attending public meetings.* 

The teachings of Pythagoras were something of a 
mixture almost equal parts of morality, mysticism and 
mathematics. He saw life as a precarious balance of ten 
somewhat random but nevertheless fundamental pairs of 
opposites: odd and even, limited and unlimited, one and 
many, right and left, male and female, rest and motion, 
straight and curved, light and darkness, good and evil, 
square and oblong. It was a particularly happy circum- 
stance for Pythagoras that the number of these funda- 
mental opposites was 10, for from his point of view 10 was 
the most perfect of numbers, being the sum of 1 (the 
point), 2 (the line), 3 (the plane) and 4 (the solid). 

Pythagoras and his followers were people who saw 
Number in every relationship and very personal attributes 


* One of these pupils, a young and beautiful (and intelligent) 
, married the sixty-year-old teacher. 

in the individual numbers.* Their great discovery of the 
dependence of the musical intervals on certain arithmetic 
ratios of strings at the same tension provided scientific 
support for what they had always intuitively considered 
to be true: 

Number rules the Universe. 

To such a people even their everyday surroundings 
spoke of Number. Quite probably, the first general recog- 
nition of a particular instance of the famous theorem 
about the square on the hypotenuse occurred when some- 
one saw this truth as it was exhibited in the regular 
checkered tiling of a floor. From inspection it would have 
been clear that the square on the diagonal of any tile 
contained as many half-tiles as the squares on both sides 
put together: 

It would also have been clear that this relationship be- 

* The number 1 stood for reason; 2, for opinion. There is no 
record that 3 represented disagreement; but 4, at any rate, was the 
number of justice. 

tween the diagonal ( or the hypotenuse of the right angle ) 
and the sides would remain true regardless of the size of 
the individual squares. 

A square cut by a diagonal represents only one partic- 
ular kind of right triangle that in which the two sides 
containing the right angle are equal. But no one who is at 
all mathematically inclined, today or twenty-five hundred 
years ago, could observe such a truth about isosceles right 
triangles without wondering if it applied as well to all 
right triangles. Thus the general theorem would be sug- 

THEOREM: The square on the side of the hypotenuse of a 
right triangle is equal to the sum of the squares on the 
other two sides. 

To make such a statement about right triangles, either 
we must verify it by actually examining all right triangles 
(which is impossible, since there are an infinite number 
of them) or we must prove that it is a necessary conse- 
quence of right triangle-ness and, therefore, has to be true 
of all right triangles. 

In the centuries since the discovery of this theorem, 
there have been literally hundreds of proofs of the fact 
that the square on the hypotenuse of any right triangle is 
equal to the sum of the squares on the other two sides.* 
At one time, a completely new proof was a requirement 
for a master's degree in mathematics. 

No one knows exactly how Pythagoras himself proved 
the general theorem. The proof which appeared a few 
hundred years later in the Elements is definitely not 

* A Mason who saw in the Pythagorean society the beginnings 
of Masonry made a classified collection of more than two hundred 
proofs of the famous theorem (E. S. Loomis, The Pythagorean 
Proposition) and gave the publication rights to the Masters and 
Wardens Association of the 22nd Masonic District of the Most 
Worshipful Grand Lodge of Free and Accepted Masons of Ohio. 


Pythagorean, being the only theorem in the book which 
tradition universally ascribes to Euclid himself. 

It would be pleasant to think that Pythagoras first 
established this great truth with one of those ingenious 
arrangements which bring the idea to eye and mind in the 
instant of seeing. Such a proof would be given by the two 
equal squares below with sides (a-\- b). These show with- 
out a word that 

since both sides of the equation, when subtracted from the 
two original and equal squares, leave as remainders four 
right triangles, all of the same size. 

Although we do not know how the theorem was actu- 
ally proved, tradition tells us that Pythagoras himself was 
so delighted (and certainly any true mathematician 
would have been! ) that he sacrificed to the gods a heca- 
tomb (100) of oxen, causing the theorem to be known 
during the Middle Ages as inventum hecatomb digrium.* 

* There is no specific written evidence that Pythagoras himself 
discovered or proved the theorem which bears his name. It was the 
custom for all discoveries of the school to be attributed to the 
master himself, regardless who made them. However, early writers 
are quite definite about "one famous figure" discovered by Pythag- 
oras and "a famous proposition on the strength of which he offered 


Thus, five hundred years before the birth of Christ, 
mathematics had in hand its famous theorem about the 
square on the hypotenuse of the right triangle a theorem 
which was destined, in the words of E. T. Bell, to run 
"like a golden thread" through all of its history. This 
theorem would serve in trigonometry, which is entirely 
based on it as the tool for measurement lying beyond the 
immediate use of tape measure and ruler. In analytic 
geometry, it would serve as the basic distance formula for 
space in any number of dimensions. In its arithmetical 
generalization (a n + b n = c n ), it would provide mathe- 
matics with its most famous unsolved problem, known as 
Fermat's Last Theorem.* In the most revolutionary 
mathematical discovery of the nineteenth century, it 
would be revealed as the equivalent of the distinguishing 
axiom of Euclidean geometry; and in our own century it 
would be further generalized so as to be appropriate to 
and include geometries other than that of Euclid. Twenty- 
five hundred years after its first general statement and 

a splendid sacrifice of oxen." That the famous figure and the 
famous proposition were one and the same, and that both referred 
to the theorem about the square on the hypotenuse, is not certain. 
Tradition, however, has always insisted upon ascribing the theorem 
to the man Pythagoras. 

* Toward the end of the sixteenth century, an "amateur" French 
mathematician named Pierre Fermat noted in the margin of a 
book of problems the theorem that a n -j- b n = c n is solvable if, and 
only if, n = 2 (i.e., as in the Pythagorean theorem) . He did not 
prove his theorem but added regretfully to his note, "I have dis- 
covered a truly marvelous proof of this, which, however, the margin 
is not large enough to contain." Today it is generally thought that 
the theorem is true, but that Fermat was mistaken when he said 
he had discovered a proof. Efforts to prove Fermat's Last Theorem 
have resulted in the development of many extremely valuable 
mathematical methods; and it has been said that, if the margin of 
Fermat's book had been wider, the whole history of mathematics 
might have been different! 

proof, the theorem of Pythagoras would be found, firmly 
embedded, in Einstein's theory of relativity. 

But we are getting ahead of our story. For the moment 
we are concerned only with the fact that the discovery 
and proof of the Pythagorean theorem was directly re- 
sponsible for setting the general direction of Western 

We have seen how the Pythagoreans lived and discov- 
ered their great theorem under the unchallenged assump- 
tion that Number rules the Universe. When they said 
Number, they meant whole number: 1, 2, 3, .... Although 
they were familiar with the sub-units which we call frac- 
tions, they did not consider these numbers as such. They 
managed to transform them into whole numbers by con- 
sidering them, not as parts, but as ratios between two 
whole numbers. (This mental gymnastic has led to the 
name rational numbers for fractions and integers, which 
are fractions with a denominator equal to one. ) Fractions 
disposed of as ratios, all was right with the world and 
Number (whole number) continued to rule the Universe. 
The gods were mathematicians aritlimeticians. But, all 
the time unsuspected, there was numerical anarchy afoot. 
That it should reveal itself to the Pythagoreans through 
their own most famous theorem is one of the great ironies 
of mathematical history. The golden thread began in a 

The Pythagoreans had proved by the laws of logic 
that the square on the hypotenuse of the right triangle is 
equal to the sum of the squares on the other two sides. 
They had also discovered the general method by which 
they could obtain solutions in whole numbers for all three 
sides of such a triangle. Although these whole num- 
ber triples (the smallest being the long-known 3, 4, 5) 
still bear the name of "the Pythagorean numbers," the 

Pythagoreans themselves knew that not all right triangles 
had whole-number sides. They assumed, however, that the 
sides and hypotenuse of any right triangle could always 
be measured in units and sub-units which could then be, 
expressed as the ratio of whole numbers. For, after all, did 
not Number whole number rule the Universe? 

Imagine then the Pythagoreans' dismay when one of 
their society, observing the simplest of right triangles, 
that which is formed by the diagonal of the unit square, 
came to the conclusion and proved it by the inexorable 
processes of reason, that there could be no whole number 
or ratio of whole numbers for the length of the hypotenuse 
of such a triangle: 

When we look at any isosceles right triangle and re- 
member that the size is unimportant, for the length of one 
of the equal sides can always be considered the unit of 
measure it is clear that the hypotenuse cannot be meas- 
ured by a whole number. We know by the theorem of 
Pythagoras that the hypotenuse must be equal to the 
square root of the sum of the squares of the other two 
sides. Since I 2 + I 2 = 2, the hypotenuse must be equal 


to V2. Some number multiplied by itself must produce 2. 
What is this number? 

It cannot be a whole number, since 1X1 = 1 and 
2X2 = 4. It must then be a number between 1 and 2. 
The Pythagoreans had always assumed that it was a 
rational "number." When we consider that the rational 
numbers between 1 and 2 are so numerous that between 
any two of them we can always find an infinite number 
of other rational numbers, we cannot blame them for as- 
suming unquestioningly that among such infinities upon 
infinities there must be some rational number which when 
multiplied by itself would produce 2. Some of them 
actually pursued V2 deep into the rational numbers, con- 
vinced that, somewhere among all those rational numbers, 
there must be one number one ratio, whole number to 
whole number which would satisfy the equation we 
would write today as 

The closest they came to such a number was 1 %2, which 
when multiplied by itself produces 28 %44, or 2^44- 

But one of the Pythagoreans, a man truly ahead of his 
time, stopped computing and considered instead another 
possibility. Perhaps there is no such number. 

Merely considering such a possibility must be rated 
as an achievement. In some respects it was even a greater 
achievement than the discovery and proof of the famous 
theorem that produced the dilemma! 

Perhaps there is no such number. How does a mathe- 
matician go about proving that there isn't a solution to 
the problem he is called upon to solve? The answer is 
classic. He simply assumes that what he believes to be 


false is in actuality true. He then proceeds to show that 
such an assumption leads to a contradiction, usually with 
itself, and of necessity cannot be true. This method has 
been vividly called proof per impossibile or, more com- 
monly, reductio ad absurdum. "It is," wrote a much more 
recent mathematician than the Pythagorean, "a far finer 
gambit than a chess gambit: a chess player may offer the 
sacrifice of a pawn or even a piece, but a mathematician 
offers the game," * 

The most recent proof f to shake the foundations of 
mathematical thought was based on a reductio and so, 
twenty-five hundred years ago, was the first. We shall 
present this proof, which is a fittingly elegant one for so 
important an idea, in the notation of modern algebra, 
although this notation was not available to the man who 
first formulated the proof. 

Let us assume that, although we have never been 
able to find it, there actually is a rational number a/b 
which when multiplied by itself produces 2. In other 
words, let us assume there exists an a/b such that 

5** = ' 

We shall assume (and this is the key point in the proof) 
that a and b have no common divisors. This is a perfectly 
legitimate assumption, since if a and b had a common 
divisor we could always reduce a/b to lowest terms. Now, 
saying that 

* G. H. Hardy, A Mathematician's Apology (Cambridge, 
England: Cambridge University Press, 1941). 

fThis proof, by the twentieth-century mathematician Kurt 
Godel, will be discussed in the last chapter. 


is the same as saying that 

If we multiply both sides of this equation by b 2 (which 
we can, since b does not equal and since we can do any- 
thing to an equation without changing its value as long 
as we do the same thing to both sides), we shall obtain: 

or, by canceling out the common divisor b 2 on the left- 
hand side: 

a 2 = 26 2 

It is obvious, since a 2 is divisible by 2, that a 2 must be an 
even number. Since odd numbers have odd squares, a also 
must be an even number. If a is even, there must be some 
other whole number c which when multiplied by 2 will 
produce a; for this is what we mean by a number being 
"even/' In other words, 

If we substitute 2c for a in the equation a 2 = 2b 2 , which 
we obtained above, we find that 


4c 2 - 26 2 

Dividing both sides of this equation by 2, we obtain 
2c 2 = fo 2 

Therefore, b 2 , like a 2 in our earlier equation, must also be 
an even number"; and it follows that b, like a, must be 


BUT (and here is the impossibility, the absurdity 
which clinches the proof) we began by assuming that afb 
was reduced to lowest terms, If a and b are both even, 
they must by the definition of evenness have the com- 
mon factor 2. Our assumption that there can be a rational 
number a/b which when multiplied by itself produces 2 
must be false, for such an assumption leads us into a 
contradiction: we begin by assuming a rational number 
reduced to lowest terms and end by proving that the 
numerator and the denominator are both divisible by 2! 

We can only imagine with what consternation this 
result was received by the other Pythagoreans. Mysticism 
and mathematics were met on a battleground from which 
there could be no retreat and no compromise.* If the 
Universe was indeed ruled by Number, there must be a 
rational number a/b equal to V2. But by impeccable 
mathematical proof one of their members had shown that 
there could be no such number! 

The Pythagoreans had to recognize that the diagonal 
of so simple a figure as the unit square was incommensura- 
ble with the unit itself. It is no wonder that they called 
V2 irrational! It was not a rational number, and it was 
contrary to all they had believed rational, or reasonable. 
The worst of the matter was that V2 was not by any 
means the only irrational number. They went on to prove 
individually that the square roots of 3, 5, 6, 7, 8, 10, 11, 
12, 13, 14, 15 and 17 were also irrational. f Although they 
worked out a very ingenious method of approximating 

* "He is unworthy of the name of man who is ignorant of the 
fact that the diagonal of the square is incommensurable with the 
side." Plato, quoted by Sophie Germain, Memoite sur les surfaces 

f The general theorem states that the square root of any num- 
ber which is not a perfect square is an irrational number. Accord- 
ing to an even more general theorem, the mth root of any number 
which is not a perfect mth power is irrational. 


such irrational values by means of ratios (detailed on 
pages 14-15), they had to face the fact that there was not 
just one, there were many (in fact, infinitely many) 
lengths for which they could find no accurate numerical 
representation in a Universe that' was supposedly ruled by 

Tradition tells us that they tried to solve their 
dilemma by persuading the discoverer of the unpleasant 
truth about V2 to drown himself. But the truth cannot be 
drowned so easily; nor would any true mathematician, 
unconfused by mysticism, wish to drown it. The Pythag- 
oreans and the mathematicians who followed them, from 
Euclid to Einstein, had to live and work with the irrational. 

Here was the golden thread impossibly knotted at its 
very beginning! 

It was at this point that the Pythagoreans, rather than 
struggling to unravel arithmetically what must have 
seemed to them a veritable Gordian knot, took the way 
out that a great soldier was to take in a similar situation. 
They cut right through the knot. If they could not repre- 
sent V2 exactly by a number, they could represent it 
exactly by a line segment. For the diagonal of the unit 
square is V2. 

With a choice of two mathematical roads before them, 
the Greeks, long before the time of Euclid, chose the 
geometric one; and 

"That has made all the difference." 


Today we customarily approximate the value V2 by 
extracting die square root of 2 to as many decimal places 
as we feel necessary for accuracy. In this way, from one 
side, we approach closer and closer to that single point, 
which is represented by the non-terminating and non- 

repeating decimal 1.41421. . . . Using rational representa- 
tions rather than decimals, the Pythagoreans worked out 
a method of approaching this same point from both sides 
with successively closer approximations. 

They began a ladder with a pair of 1's and by the ad- 
ditions indicated below obtained the number pairs on 
the right: 

1 - + 1 1 1 

2 - + 3 

5 <- + 7 


12 17 


294- + 41 29 41 

^o -"^ t q ** "7& ^ v 
The reader should try to determine the next rung of 

the ladder. If he will then square the fractions obtained 
by taking the numerator from the right and the denomi- 
nator from the left, he will find that although he will 
never reach 2 exactly he will approach it in a continuously 
narrowing zigzag as the fractions he is squaring approach 

OZ, s ! *& 



Thus begins the most durable and 
influential textbook in the history of 
mathematics. Thus, in fact, begins 
modern mathematics. 

It has been more than two thou- 
sand years since the Greek Eukleides, 
whom we know better as Euclid, 
gathered together the mathematical 
work of his predecessors into thirteen 
books which he entitled, simply, the 
Elements. During this time the Ele- 
ments of Euclid, in addition to serving 
as a mathematical textbook for ado- 
lescents, has also served as Western 
man's final, and first, bulwark against 
ignorance. Newton cast his Principia in 
the already hallowed form of the 
Elements. Kant called on the axioms 
of the Elements as "the only immutable 
truths." On the first few pages of this 
seemingly spare and formal work, 
bloodless battles have been waged. It 
was here, at the middle of the nine- 
teenth centuiy, that mathematics made 
its greatest self-discovery; and it was 
here, at the beginning of the twentieth, 
that it made its great and final stand 
to establishto prove, ,in fact its own 
internal consistency. We have come, in 
the last two thousand years, a long 
way from Euclid; but we have also 
taken his Elements with us, all the 


The man Euclid and the facts of his life and career* 
were lost very early on the journey. We are told that he 
"flourished" about 300 B.C., that he founded a school at 
Alexandria in the time of Ptolemy I. There are about him 
only two traditional anecdotes, both of which are also 
recounted of other Greek mathematicians. In the years 
after his death various writers confused him with another 
Euclid, the philosopher of Megara; and the Arabs put 
forth a claim that he had really been an Arab all along. 
It can be said that in the history of mathematics there is 
no Euclid; there is only the Elements. Probably within his 
own time ( in the words that Auden used of Yeats ) he had 
become his admirers. 

The Elements, from the beginning, was immediately 
recognized for what it was a masterpiece. The form of 
the book was not original. The logical ladder of defini- 
tions, axioms, theorems and proofs was first erected by 
some earlier Greek than Euclid, perhaps a priest. The 
subject matter was not original. The masterly treatment 
of proportion which enabled the later Greeks to handle 
incommensurable as well as commensurable magnitudes, 
is that of Eudoxus; and the other books are frankly based 
on the known work of other men. ( "The picture has been 
handed down of a genial man of learning, modest and 
scrupulously fair, always ready to acknowledge the 
original work of others," H. W. Turnbull wrote in The 
Great Mathematicians.) Only one proof that of the 
Pythagorean theorem is traditionally ascribed to Euclid 
himself, although it is apparent that to fit theorems into 
his new arrangement he must have had to create other 
new proofs. Even the title, the Elements, was not original. 
This term did not refer, as we might think, merely to the 
elementary aspects of the subject but rather according to 
an early mathematical historian to certain leading 


theorems in the whole of mathematics which bear to those 
which follow the relation of a principle, furnishing proofs 
of many properties. Such theorems were called by the 
name of elements; and their function was somewhat like 
that of the letters of the alphabet in the language, letters 
being called by the same name in Greek. There had been 
many Elements before Euclid. That there was none after 
him is an unequivocal tribute to the sheer genius of his 

As a mathematician, Euclid falls far behind Eudoxus, 
who preceded him, and Archimedes and Apollonius, who 
followed. The Encyclopaedia Britannica admits regret- 
fully that he was not even a "first-rate" mathematician, 
but adds that there is no question but that he was a first- 
rate teacher. What he brought to the already great mathe- 
matics of his time was a genius for system. And system 
was exactly what was needed! There were many fine 
single works on specialized subjects. Many editors had 
gathered together what seemed to them important. There 
were definitions, axioms, theorems and proofs galore; and 
an almost equal number of organized and disorganized, 
overly complete and incomplete arrangements, all called 
the Elements. Euclid took these. He selected, substituted, 
added, rearranged; and what came out in his Elements 
was a distillation of all those that had come before a 
model of systematic thought. 

We have no copy of this original work. Oddly enough, 
we have no copy made even within a century or two of 
Euclid's time. Until recently the earliest known version of 
the Elements was a revision with textual changes and 
some additions by Theon of Alexandria in the fourth cen- 
tury after Christ, a good six centuries after Euclid com- 
piled it in Alexandria. Early in the nineteenth century, a 
Greek manuscript in the Vatican was discovered by 


internal evidence to be a pre-Theonine text.* The tradi- 
tional textbook version of the Elements, which was used 
almost completely without change until very recently, was 
based, of course, on the text of Theon. In a quite literal 
sense, Euclid has become his admirers; for when we say, 
"Euclid says," we are speaking of a compiler much closer 
to us than the original compiler of the Elements. This is 
unimportant at this time. We are not concerned with what 
Euclid himself actually wrote in the Elements, but with 
what has served mathematics for so many centuries as the 
Elements of Euclid. 

What, then, is this work which has played such an 
influential role in the history of mathematics and of 
thought itself? Most of us are probably not familiar with 
a translation of Theon's traditional version of the master- 
piece. Our high school geometry textbook, however, was 
probably based directly upon it. After a few introductory 
remarks and simple explanations in modern terms most 
authors in the past fell back very quickly upon the orig- 
inal. If we were to examine at this time a translation of 
the Elements, such as Sir Thomas Heath's, available now 
in paperback (Dover Press), we" would find it unexpect- 
edly familiar. 

The Elements we would find is composed of thir- 
teen sections, or "books," arranged according to subject 
matter: the first few to plane geometry, the last to solid, 
and the books between to proportion and number. We 

* It is interesting to note that the Romans never translated the 
Elements into Latin. "Among the Greeks," Cicero wrote con- 
descendingly, "nothing was more glorious than mathematics. But 
we have limited the usefulness of this art to measuring and calculat- 
ing." The earliest extant Latin translation (c. 1120) is one by the 
Englishman Athelhard, who obtained an Arabic copy of the Ele- 
ments by going Jo Spain disguised as a Moslem student, and made 
his translation from that copy. 


would meet again the famous pom asinormn, or Bridge of 
Asses, as the fifth proposition in Book I: 

THEOREM: In isosceles triangles the angles at the base are 
equal to one another, and, if the equal straight lines he 
produced further, the angles under the base will be equal 
to one another. 

This is the theorem which traditionally separates mathe- 
matical boys from mathematical men, since the asses 
supposedly cannot get through the proof, or across the 
bridge. In the Middle Ages the mastering of this theorem 
and its proof marked the culmination of the mathematical 
training required for a degree. 

At the end of Book I we would find our old friend, 
the famous theorem about the square on the hypotenuse 
of the right triangle, which laymen know as the theorem 
of Pythagoras and which loving geometers have called for 
over two thousand years merely "I, 47," because of its 
position as the forty-seventh proposition in the first book 
of Euclid's Elements. The proof of this theorem is the only 
one in the Elements which is specifically credited to 
Euclid himself. Although the philosopher Schopenhauer 
dismissed it contemptuously as a "mouse-trap proof" and 
"a proof walking on stilts, nay, a mean, underhand proof," 
Sir Thomas Heath, the English editor of the Elements, 
calls it "a veritable tour de force which compels admira- 
tion." It is Heath's contention that Euclid found the 
theorem proved by the incomplete theory of proportion 
of the Pythagoreans (incomplete because it was not ap- 
plicable to the yet undiscovered incommensurable mag- 
nitudes), and that this proof by proportion suggested to 
him the method of I, 47. Although his plan for the 
Elements did not call for the treatment of proportion 
until Book V, according to the Heath theory, he managed 
to transform the Pythagorean proof by proportion into 


one based on Book I only. "A proof extraordinarily in- 
genious," insists Heath and a fig to the philosopher who 
expects an intuitive proof of the "look-see" type from the 
compiler of the Elements! 

In Book V, we would find what is without question 
the finest mathematics in the Elements the theory of 
proportion as expounded by Eudoxus. It was this theory, 
applying as it did to incommensurable as well as to com- 
mensurable magnitudes, which allowed Greek mathema- 
ticians, after the shattering discovery of the irrational, to 
move forward again. Because of its importance to our 
story as a whole, we shall treat it separately in Chapter 

After Book VI, which also deals with problems of 
proportion, we would find the three books on the theory 
of numbers. Although the "numbers" seems strangely 
unf amiliar, since they are all represented by straight lines 
"in continued proportion," we would find here many 
familiar truths of our own school arithmetic. Proposition 
1 of Book VII, for instance, gives us the standard method 
still known as "Euclid's algorithm" * for finding the 
greatest common divisor of two numbers, although in the 
Elements, with its generally geometric approach, it is "the 
greatest common measure." As Proposition 20 of Book X, 
the third and final book on numbers, we would find that 
most important and interesting truth: that the number of 
primes is infinite; in Euclid's words, "Prime numbers are 
more than any assigned multitude of numbers." (Proved 
in the next chapter. ) 

At the end of the thirteenth and final book of the 
Elements we would meet again the five regular solids, 
those bodies with which the Platonists identified all crea- 
tion. In their philosophy the cube represented the earth; 
the octahedron, the air; the tetrahedron, fire; the icosahe- 

* Detailed at the end of this chapter. 


dron, water; and the dodecahedron, the Universe itself. 
Good Platonists always maintained that Euclid organized 
the Elements solely for the purpose of presenting the 
construction of the perfect figures, but this is obviously 
not true. The Elements contains a great deal, including 
the three books on arithmetic, which contributes nothing 
to these final constructions. 

As we continue our re-examination of the Elements, 
we would note a certain pattern in the arrangement. Each 
of the thirteen books begins with a list of definitions of 
the terms which will be needed in it; the first book is pre- 
ceded as well by a group of more or less obvious state- 
ments, or axioms; and each of the thirteen books consists 
of a related series of theorems which are proved by ap- 
pealing to the authority of previously stated theorems, 
axioms, and definitions, all of these derived logically by 
the accepted rules of reason. 

This is the ladder by which the Greeks believed that 
man could ascend to truth and they believed it to be the 
only ladder: 

L Proofs L 




As Euclid is reputed to have told the first Ptolemy when 
asked if there were no other, easier way than that of the 
Elements: "There is no royal road to geometry." Today 
we call Euclid's ladder the axiomatic method, and we still 
find it the ladder by which man can ascend most surely 
to truth. If our concept of the truth we reach is somewhat 
different from that of the Greeks, that is a story for a later 
chapter; for the moment we must concentrate on examin- 


ing the rungs ot the ladder with the eyes and minds of the 
men who built it. 

To the Greeks, the definitions given by Euclid at the 
beginning of each book of the Elements were not state- 
ments of existence but merely descriptions. Existence of 
that which was defined had to be established by construc- 
tions which met the specifications laid out in the defini- 
tions. In the words of Aristotle: "Thus, what is meant by 
triangle the geometer assumes, but that it exists he has to 
prove." Accordingly, in Book I Euclid begins by produc- 
ing the equilateral triangle which he has described in 
Definition 20. In Proposition 11 he constructs a right angle 
(Definition 10) and in Proposition 46, a square (Defini- 
tion 22). Until these figures are actually constructed on 
the authority of the axioms and previously proved 
theorems, they are never used in the Elements. 

There are, however, certain terms defined at the be- 
ginning of Book I which Euclid never produces "from 
scratch." These are terms the existence of which is spe- 
cifically implied by the postulates: the point, the straight 
line and the circle in short, his "subject matter." These 
are the objects in terms of which all the others have been 
defined. Among the other definitions, Euclid does describe 
these objects, but just for the record: 

A point is that which has no part. 

A straight line is a line which lies evenly with the 
points of itself. 

A circle is a plane figure contained by one line such 
that all the straight lines falling upon it from one point 
among those lying within the figure are equal to one 
another; and this point is called the center of the 

He clearly recognizes that he will never be able to pro- 


duce a point, a straight line or a circle unless lie assumes 
before he begins that he can produce them. 

"Let the following be postulated," he announces at the 
beginning of Book I; 

To draw a straight line from any point to any point. 

To produce a finite straight line continuously in a 
straight line. 

To describe a circle with any center and distance. 

On the arbitrarily assumed ability to do these three 
things, the ladder rests. We can join any two points, ex- 
tend any straight line, describe about any center a circle 
of any size because we have agreed that we can. To those 
who may object that any point which we put on paper 
will have by the nature of the instrument with which we 
must make it some "part"; that for the same reason any 
line which we draw cannot lie evenly on all its partless 
points; that the points on the boundary of any circle can- 
not be all the same distance from the center to all those 
who object, we have in the postulates our unanswerable 
answer: we can because we have begun by agreeing that 
we can. 

"It is ignorance alone that could lead anyone to try to 
prove the axioms." * 

But we must never forget that the Choice of the as- 
sumptions on which we are to rest our ladder to truth is 
a purely arbitrary one. Just as in a game we could, by 
agreement of all the players, make different rules under 
which to play (making, of course, a different game of it), 
so Euclid could have chosen other axioms, as we shall see 
in a later chapter. It was his choice, more than anything 
else, which was indicative of his genius. 

What constitutes a well-chosen set of axioms? Since 


long before Euclid chose his, men have discussed this 
question, and they have always been pretty well agreed. 
There is one absolute requirement: consistency. The 
axioms that we have chosen must never lead us into a 
contradiction. Beyond this essential requirement there are 
others that are more of a practical or an esthetic nature. 
A well-chosen set of axioms should exhibit such virtues as 
simplicity, economy, sufficiency, and a certain indefinable 

We could discuss more precisely the characteristics of 
these characteristics; but the reader can probably get a 
much quicker and much more vivid picture of the require- 
ments if he imagines himself in the following game situa- 
tions and considers, not what constitutes a well-chosen set 
of axioms, but rather what is wrong with the rules of the 
game which he is playing: 

He finds the rules hard to play by because they list 
many exceptions. (Not simple.) 

He finds that one of the rules is unnecessary since it 
is already stated, although in quite different words, by 
another rule. (Not economical.) 

He finds that under the rules he cannot make a move 
which seems necessary if the game is to be really in- 
teresting. (Not sufficient.) 

He finds that there is a rule which forbids a certain 
move which is permitted by another rule. (Not con- 
sistent. ) 

He finds that the game played according to the rules 
is so uninteresting that, even when he wins, he feels 
very little satisfaction. (Not important.) 

If we substitute for "rules of the game," "set of axioms" 
and for "moves," "theorems," we see that the requirements 


are very much the same; and the axioms Euclid chose so 
well in Alexandria long before the birth of Christ have 
provided Western man for more than twenty centuries 
with a very good game indeed. 

Before we leave the subject of the axioms, we should 
point out that Euclid distinguished between two types of 
assumptions, "common notions" and "postulates." The 
common notions include such statements as "the whole is 
greater than the part"; while one of the postulates states 
that "all right angles are equal." ( All the common notions 
and postulates are listed on page 27, since we shall be 
referring to them again from time to time.) Probably no 
one has been able to say exactly what distinction Euclid 
himself made between the two; but if anyone is well quali- 
fied to make an educated guess, it is Sir Thomas Heath, a 
career civil servant in the British government, who will go 
down in history as the ultimate and complete editor of 
the Elements. 

Heath writes on the two different types of axioms: "As 
regards the postulates we may imagine him [Euclid] say- 
ing, 'Besides the common notions there are a few other 
things which I must assume without proof, but which 
differ from the common notions in that they are not self- 
evident. The learner may or may not be disposed to agree 
with them; but he must accept them at the outset on the 
superior authority of his teacher, and must be left to con- 
vince himself of their truth in the course of the investiga- 
tion which follows/ " 

Having defined our terms and agreed upon them and 
to our axioms (common notions and postulates alike), we 
are now ready to climb, rung by rung, the ladder of math- 
ematical truth., guided always by the accepted laws of 
logic. Each rung of this ladder is a proposition (which 
may be either a problem or a theorem) and its proof; and 
by the rules of the game each rung may utilize in its con- 


struction only the rungs below. This means that the first 
proposition must depend for its proof only upon the 
axioms and definitions already given, but the second may 
utilize as well the now proved first proposition, and so on. 
By the time we arrive at the famous fifth, the pons 
asinorum, we find that to prove it we need Propositions 3 
and 4, which we have already proved, as well as Postu- 
lates 1 and 2. This process continues. The proof of I, 47, 


1. Things which are equal to the same thing are also 
equal to one another. 

2. If equals be added to equals, the wholes are equal. 

3. If equals be subtracted from equals, the remainders 
are equal. 

4. Things which coincide with one another are equal 
to one another. 

5. The whole is greater than the part. 


Let the following be postulated: 

1. To draw a straight line from any point to any point. 

2. To produce a finite straight line continuously in a 
straight line. 

3. To describe a circle with any center and distance. 

4. That all right angles are equal to one another. 

5. That if a straight line falling on two straight lines 
makes the interior angles on the same side less than two 
right angles, the straight lines, if produced indefinitely, 
will meet on that side on which the angles are less than 
two right angles. 

* This version is given by Sir Thomas Heath in The Elements 
of Euclid. 


relies upon five previously proved propositions as well as 
on two of the common notions which we agreed to before 
we started. From the moment, on the first page of the 
Elements, when we placed our hand on the first rung ( "A 
point is that which has no part" ) , we have been climbing. 

His own time considered the Elements of Euclid as 
near to perfect as work of man could be. The succeeding 
centuries of the Christian era were, as we shall see in a 
later chapter, troubled by one small flaw which they 
struggled valiantly to eliminate, only to find in the end 
that it supported the entire edifice (something Euclid 
himself had apparently known when he laid the founda- 
tions). At the beginning of the twentieth century, the 
men who looked hard and long at the logical bases of 
mathematics were to find the Elements riddled with fal- 
lacies and unstated assumptions. Yet the Elements remain, 
less perfect than they originally appeared to their com- 
piler's contemporaries, frankly imperfect by the rigorous 
standards of modern mathematics, but still on the throne. 
For every domain of mathematics today is ruled by the 
axiomatic method, the system of Euclid's Elements. 

Twenty-three hundred years after the Greek Eukleides 
lived and taught on the shores of the Mediterranean, 
mathematicians and scientists from all over the world 
gathered in Berkeley, California, under the shadow of the 
cyclotron, for a week-long international symposium on 
the axiomatic method and its relation to modern science. 
The ladder to truth was set on a far different shore, but 
the rungs were still the same: Definitions. Axioms. 
Theorems. Proofs. 


Eucli.d's algorithm is one of the oldest techniques in 
arithmetic, probably even older than Euclid. 

To find the greatest common divisor of two numbers 


a and b by this method, we divide the smaller a into the 
larger b. If we obtain a remainder c, we divide c into a 
and so on, the remainder d being divided into c, e into d. 
Eventually we shall come to one of two possible situa- 

1. Our division comes out even, in which case our last 
positive remainder is the greatest common divisor 
of the two numbers; or 

2. Our remainder is 1, in which case the two numbers 
are relatively prime and their greatest common 
divisor is 1. 

Both of these situations are illustrated in the simple ex- 
amples below, to find the g.c.d. of 26 and 94 and of 26 
and 101. 

_J5 3 

26 794 26 )IOl 

78 1 78 1 

16 )26 23 J26 

16 1 23 7 

10)16 "3)23 

10 1 21 1 

"6)10 ~~2 )3 

61 2 

4 ye g.c.d. = 1 I 

4 2 
2 ]4 
g.c.d. = 2 

The reader may now enjoy using this same method to 
find the greatest common divisor for some larger pairs: 
116 and 280; 507 and 1862; 280 and 882; 2475 and 19404 


66 >T 'T > 


Mathematics began with these two 
basic concerns, and for centuries the 
subject was defined simply as "the 
science of form and number." Yet form 
has never been completely distinct 

3 from number. When, after the discov- 

ery of the irrational, mathematics 
found itself forced into the guise of 
form, it did not leave number behind. 
The beginnings of what we know to- 
day as the theory of numbers lie in 
The Books VII, VIII and IX of Euclid's 

Inexhaustible Elements. 

Storehouse The theory of numbers, or the 

higher arithmetic as it is often called, 
limits itself entirely to the whole num- 
bers 0, 1, 2, 3, ... and the relationships 
that exist among them. These numbers 
are a very simple sequence, formed by 
making each member one unit larger 
than the one that precedes it and con- 
tinuing without end. They have chal- 
lenged the minds of men for centuries 
because under their simple surface 
characteristics lie layer after layer of 
increasingly complex and utterly un- 
expected relationships. 

This challenge was felt by Euclid. 
It has been felt, regardless of their in- 
dividual specialties, by almost all the 
mathematicians who have followed 

"The higher arithmetic," wrote 


Karl Friedrich Gauss (1777-1855), known today and in 
liis own lifetime as the Prince of Mathematicians, "pre- 
sents us with an inexhaustible storehouse of interesting 
truths of truths, too, which are not isolated, but stand in 
the closest relation to one another and between which, 
with each successive advance of the science, we continu- 
ally discover new and wholly unexpected points of con- 

In this chapter we shall try to glimpse some of the 
treasures of tin's inexhaustible storehouse by examining 
a few of the mathematically interesting relationships 
which exist between two kinds of numbers the primes 
and the squares. Both the primes and the squares were 
studied extensively in the Elements of Euclid; yet mathe- 
maticians are still discovering in the words of Gauss 
"new and wholly unexpected points of contact" between 

Although the classification into even and odd is the 
most ancient, the most mathematically suggestive classi- 
fication of the whole numbers greater than 1 is into those 
which can be divided by some number besides themselves 
and 1 (called composite numbers) and those which can 
be divided only by themselves and 1 ( called prime num- 
bers). The first few prime numbers are easily recogniz- 
able, for they are those the units of which cannot be 
arranged except in straight lines: 

2 00 7 0000000 

3 000 11 00000000000 

5 00000 13 0000000000000 

The units of all other, composite numbers can always be 
arranged into rectangles as well as straight lines : 


4 00 9 000 

00 000 


6 000 10 00000 
000 00000 

8 0000 12 000000 0000 

0000 000000 or 0000 


It is difficult to believe that no matter how high we go 
among the numbers, we shall continue to find numbers 
that can be arranged only in straight lines. Yet in Book IX 
(Prop. 20) of the Elements, Euclid proved that these 
essentially indivisible numbers the primes are infinite. 

Euclid's proof is, of course, distinctly geometric in 
flavor. His numbers are straight lines, "beginning from a 
unit and in continued proportion," and his primes are 
lines "measured by the unit alone." The truth that he 
establishes, however, is the one above all others which 
makes numbers so interesting. 

Euclid's proof rests upon the fact that if we multiply 
together any group of prime numbers, the number which 
is 1 more than the number we get as our answer will be 
either (1) another prime not in our original group or 
(2) a composite number which has, as one of its factors, 
a prime not in the group of primes we multiplied. This is 
because all of the primes we have multiplied must leave a 
remainder of 1 when divided into this next number; 

2 X 3 X 5 = 30 (30 + 1) divided by 2, 3 or 

5 leaves a remainder of 1 

Euclid showed, therefore, that it would be impossible to 

have a finite set which contained all the primes because 
by multiplying them and adding one to our answer we 
could always produce a prime not in our set of "all." 

The relationship which exists between the divisible 
composite numbers and the indivisible primes is such a 
key to unlocking the secrets of numbers that the theorem 
which expresses it is universally acclaimed the Funda- 
mental Theorem of Arithmetic. 

Before stating this theorem, let us recall that by defi- 
nition every composite number is divisible by some 
number other than itself and 1. This number which di- 
vides it must be prime or composite and, of course, smaller 
than the original. If it is composite, it must be divisible in 
turn by some number other than itself and 1, and so on. 
This process ends only when we come to a number which 
is not divisible by any other: a prime factor of the original 
composite number. It follows, then, that every composite 
number can be produced by the multiplication of primes 
or, conversely, can be factored into primes. 

The Fundamental Theorem of Arithmetic states 
simply that this prime factorization for any composite 
number is unique. 

This means tbat when we reduce a number like 36 
to its prime factors (2X2X3X3), we know that al- 
though it has other factors (4X9, for instance, and 
6X6), it can be reduced to no other combination of 
prime factors. By the Fundamental Theorem we know 
that the same thing will be true of a number like 18,674,392 
or any other number, no matter how large. We can thus 
work with any number n as a unique individual among 
the numbers. Not only do we know that it has a unique 
place hi the sequence of numbers (between n 1 and 
n + 1 ) , but also we know that it is a unique combination 


of certain prime factors pi k *p2 k - - p r kr , where the p's 
represent different primes, and the k's how many times 
each prime appears as a factor. 

The numbers which, next to the primes, have received 
the most attention from mathematicians are the squares. 
Their name comes to us from, the eye-minded Greeks who 
noted that the units of a number when multiplied by itself 
always form a perfect square. They also noted something 
else of great interest about these squares when they were 
built up by successive borders of units : 










1 + 3 



= 4 

1 + 3 + 5 


= 9 


= 18 


= 25 

Between the primes and the squares there are many 
interesting "points of contact," deep, intricate and com- 
pletely unexpected. Yet the primes and the squares are 
basically very different numbers. 

On page 35 we have printed a table of the first fifty 
numbers in each classification. Let us first examine only 
the last digits of these numbers. Among the squares we 
see immediately that not one of them ends in 2, 3, 7 or 8; 
in fact, the last digits follow a pattern 0, 1, 4, 9, 6, 5, 6, 9, 
4, 1 which repeats indefinitely. Since, when we multiply a 
number by itself, the last digit of the product depends 
only upon the last digit of the number being multiplied, 
any number ending in 3 will have a square ending in 9, 
and so on. Obviously, there are infinitely many squares 
ending in each of the digits 0, 1, 4, 5, 6, 9 and none what- 







































































































soever ending in 2, 3, 7 or 8. But when we examine the 
last digits of the primes, we find that aside from 2 and 5 
all primes end in 1, 3, 7 or 9. Since all even numbers are 
by definition divisible by 2 and all numbers ending in 5 
divisible by 5, it is apparent that primes can end only in 


1, 3, 7 or 9. But the primes, unlike the squares, are very 
unpredictable in their appearance among the numbers. 
We know by Euclid's proof that the number of primes is 
infinite, but are there as with the squares infinitely many 
primes ending in each of the possible digits? 

The answer is given affirmatively by a very deep 
theorem proved over a hundred and fifty years ago by 
P. G. Lejeune Dirichlet (1805-1859). He showed that 
every arithmetic progression of numbers 

a, a + d, a + 2,d, a + 3d, a -f- 4d, a + 5d, . . . 

contains infinitely many primes when a and d have no com- 
mon factor. If we take a = 1, 3, 7 or 9 (the only possible 
endings for primes ) and d = 10, we know that in each of 
the four resulting progressions there are infinitely many 
primes: infinitely many primes ending in 1; infinitely many 
ending in 3; infinitely many ending in 7, and infinitely 
many ending in 9. 

1, 11, 21, 31, 41, 51, 61, 71, 81, 91, 101, . . . 

3, 13, 23, 33, 43, 53, 63, 73, 83, 93, 103, . . . 

7, 17, 27, 37, 47, 57, 67, 77, 87, 97, 107, . . . 

9, 19, 29, 39, 49, 59, 69, 79, 89, 99, 109, . . . 

If we look again at our table of primes and squares, 
we can see that it is no problem to write down the next 
entry in the column of squares : we simply multiply 50 by 
50 and put down 2500.* But, to make the next entry in the 
column of primes, the best we can do is to examine the 
next odd number, 231. By inspection we see that it is di- 
visible by 3, so we move on to the next odd number, 233. 
We try to divide it, in turn, by 3, 5, 7, 11 and 13 (all the 
primes which are less than its square root) and since none 
divides it we can conclude that it is prime, and write it 

* We can also add together the first fifty odd numbers. 

down as our next entry. This is the only general method 
for finding out whether a given number is prime.* 

The classifications of the numbers which we have men- 
tioned so far even, odd; prime, composite; square and 
non-square are so obvious that even if we do not usually 
think of all of them by name we cannot remember when 
we were not aware of them. Yet, among these groups of 
numbers there exist, in the words of the great Gauss, 
"wholly unexpected" points of contact. On the surface we 
have a not-unexpected relationship between the prime 
numbers and the odd. All the primes with one exception 
are odd, since every even number is by definition divisible 
by the only even prime, 2. When we separate the odd 
prunes on the basis of their remainders when divided by 
4, all are either of the form 4n + 1 or 4n -f 3. Certainly 
we have no particular reason for expecting that these 
prunes, falling into two mutually exclusive groups because 
of their relation to the first non-trivial square number, 
should present us with any significant and unvarying dif- 
ference in their relation to the squares. Yet they do. This 
difference becomes apparent when we attempt to repre- 
sent each of the first few primes as the sum of two squares. 
With 3, 7, 11, 19, 23, 31, and 43, we have no success at all; 
but we find that 

17 = p + 4 2 
29 = 2 2 + 5 2 
37 = I 2 + 6 2 
and so on. 

It is immediately suggested that every prime of the 

* The largest known prime at the date of writing is 2 9941 1, 
found prime by D. B. Gillies, on Illiac II at the University of 
Illinois, April 21, 1963. 


form 4n + 1 can be represented as the sum of two squares, 
while not one prime of the form 4n + 3 can be so repre- 
sented. The theorem which expresses this relationship is 
even more specific, for it further states that the 4n -f- 1 
primes can be represented as the sum of two squares in 
only one way. This is the classic Two Square Theorem of 
Pierre Fermat. Although it involves no mathematical 
concepts which are not familiar to a bright child, it ex- 
presses a profound point of contact among the numbers, 
and one of the most '"beautiful" relationships in all num- 
ber theory. 

Fermat wrote to a fellow mathematician that he had 
proved the Two Square Theorem by what he called "the 
method of infinite descent." He began with the assumption 
that there existed a prime of the form 4n + 1 which could 
not be represented as the sum of two squares; proved that 
if there were such a prime, there would have to be a 
smaller prime of the same form which could not be so rep- 
resented; and continued in this way until he got to 5, the 
smallest prime of the form 4n -f- 1. Since 5 can be repre- 
sented as the sum of two squares, the original assumption 
was false; the theorem, as stated, was true. The extreme 
difficulty of this proof can be grasped from the fact that 
although Fermat detailed it roughly to the extent we have 
here, it was not until almost a hundred years after his 
death that a mathematician was actually able to prove the 
Two Square Theorem. 

In addition to the Two Square Theorem, we have a 
Three Square Theorem and a Four Square Theorem, both 
of which reveal interesting relationships between the 
square numbers and all the numbers. Both theorems deal 
with the same relationship, the representation of numbers 
as the sums of squares; but the Three Square Theorem 
penetrates much more deeply into the relationship than 
the Four Square Theorem. 


FOUR SQUARE THEOREM: Everij number can be represented 
as the sum of four squares. 

There is no better example in number theory of the 
fact that it is easier to state a truth than to prove it. A little 
computation is enough to suggest that four squares are 
probably sufficient to represent any number. The fact was 
probably known in the early years of the Christian era. It 
was then restated as part of a more general theorem, and 
proved by Fermat. Although Fermat remarked in a letter 
to a friend that no proof had ever given him more pleas- 
ure, he neglected to reveal the details to anyone, and the 
proof died with him. Leonhard Euler (1707-1783), one of 
the greatest, and certainly the most prolific mathematician 
who ever lived, then tackled the part of Fermat's theorem 
pertaining to the squares. In fact, off and on, he devoted 
forty years of his long life to it without success. Eventu- 
ally, though, with the help of much of the work which 
Euler had done, the Four Square Theorem was proved by 
Joseph Louis Lagrange (1736-1813). A few years later 
Euler brought forth a more simple and elegant proof than 
Lagrange's of the theorem which had caused him so much 
difficulty, and it is now the proof generally followed. 

For such representation of all numbers as the sum of 
four squares, we rely extensively upon the use of the square 
of 0, particularly in the case of those numbers which are 
squares to begin with or those numbers, like the primes of 
the form 4n -f- 1, which are the sum of two squares. It is 
obvious from these that four squares are not by any means 
necessary to represent every number as the sum of squares. 
The question which then occurs is whether or not we can 
determine, by any general rule, the particular group of 
numbers for which four squares are necessary. This is ex- 
actly the answer which the Three Square Theorem gave. 
There is, according to the theorem, a particular group of 


numbers, the first of which is 7, that cannot be represented 
by any fewer than four squares; for all other numbers, 
three squares are sufficient. 

THREE SQUABE THEOREM: Every number can be represented 
as the sum of three squares except those numbers of the 

a b + 7).* 

Now the Four Square Theorem is by no means trivial. 
Although the representation of the smaller numbers as the 
sum of four squares is easy to perceive, there is no assur- 
ance that as the numbers get larger more squares will not 
be required. Yet, when compared to the Three Square 
Theorem, which pinpoints the specific type of number 
(and not an obvious, straightforward type, either) requir- 
ing four squares for representation, the Four Square 
Theorem is distinctly inferior "much less deep," in the 
opinion of mathematicians, 

To discover such deep relationships among the num- 
bers, we must not look at them with jaded eyes. Youth, 
freshness, and perhaps mentally standing on one's head 
help. We also need a gift for seeing such relationships. 

There is a relationship between the squares and the 
odd primes which is even more mathematically exciting 
than the one Fermat expressed in the Two Square Theo- 
rem, fully as deep as if not deeper than the relationship 
expressed in the Three Square Theorem. But it would not 
even be observed by anyone who did not have the gift. 
Although this particular relationship had been observed 
earlier, the young Gauss (he was eighteen at the time) 
discovered it wholly on his own and was delighted with 
it. To him it was always the Gem of Arithmetic. More 
formally, it is known as the Law of Quadratic Reciprocity 
(quadratic meaning simply "of or pertaining to the 

* It was proved by Gauss. 

The Law of Quadratic Reciprocity deals exclusively 
with the same kinds of numbers as does the Two Square 
Theorem of Fermat the squares and the primes classified 
according to the remainders they leave when divided by 
4. Let p and q be any pair of odd primes; there exists a 
beautiful and delicately balanced relationship between 
these two apparently unrelated problems: 

1. To find an % such that x 2 p is divisible by q. 

2. To find a y such that y 2 q is divisible by p. 

According to the Law of Quadratic Reciprocity, both 
problems are solvable or both unsolvable unless both p and 
q leave a remainder of 3 when divided by 4, in which case 
one of the problems is solvable and the other is unsolvable. 

"The mere discovery of such a law was a notable 
achievement," writes E. T. Bell in Men of Mathematics. 
"That it was first proved by a boy of nineteen will suggest 
to anyone who tries to prove it that Gauss was more than 
merely competent in mathematics." 

It took Gauss a year to prove the Law of Quadratic 
Reciprocity. "It tormented me and absorbed my greatest 
efforts," he wrote later. His was the first proof of this beau- 
tiful law and he published it proudly in the Disquisitiones 
Arithmeticae under the title of Fundamental Theorem. 
But he was not at all satisfied with his proof: ". . . it pro- 
ceeds with laborious arguments and is overloaded with 
extended operations." In the next seven years he proved 
the Law of Quadratic Reciprocity in four more ways, 
using completely different principles. The first three of 
these four proofs, all of which he conceded were logically 
satisfactory, he dismissed as "derived from sources much 
too remote." The last he published with the frank state- 
ment, "I do not hesitate to say that till now a natural proof 
has not been produced. I leave it to the authorities to 
judge whether [this] proof which I have recently been 


fortunate enough to discover deserves this description." 

The "authorities" apparently decided that it did, for 
this fifth proof (known as "the third" because it was the 
third one he published) is the proof which is universally 
used today. But Gauss himself could not have been satis- 
fied: three more times in his life he proved the Law of 
Quadratic Reciprocity, his Gem of Arithmetic. 

Lest we feel at this point that Gauss himself may have 
singlehandedly exhausted the inexhaustible storehouse of 
interesting truths which he found the natural numbers to 
be, we might mention that he went on to tackle the prob- 
lem of biquadratic reciprocity where x and y are taken to 
the fourth power. A by-product of his solution was the 
creation of the theory of algebraic numbers, which we 
shall touch on in Chapter 7. Perhaps it is too much to 
mention that the general case of x and y taken to the nth 
power still remains in the storehouse! 

It is curious that we usually think of arithmetic as the 
exact science, the science of right answers, the cut-and- 
dried science. But that is because we are thinking of the 
arithmetic of the elementary school, not the "Queen of 
Mathematics." In elementary arithmetic we perform oper- 
ations on the numbers, first with accuracy, and then with 
speed. The ideal is most nearly achieved by the great elec- 
tronic computers which, in spite of the awe they generate, 
can do no more difficult arithmetic than a high school boy 
or girl who is well trained; they can, bowever, do it faster 
and more accurately. An electronic computer is a mere 
drudge of the Queen of Mathematics. Although even 
Gauss loved to compute, he never failed to perceive the 
queen's real challenge. 

"The questions of the higher arithmetic," be wrote, 
"often present a remarkable characteristic which seldom 
appears in more general analysis and increases the beauty 
of the former subject. While analytic investigations lead to 


the discovery of new truths only after the fundamental 
principles of the subject (which to a certain degree open 
the way to these truths ) have been completely mastered, 
on the contrary in arithmetic the most elegant theorems 
frequently arise experimentally as the result of a more or 
less unexpected stroke of good fortune,* while their proofs 
lie so deeply imbedded in the darkness that they elude all 
attempts and defeat the sharpest inquiries. Further, the 
connection between arithmetical truths, which at first 
glance seem of widely different nature, is so close that one 
not infrequently has the good fortune to find a proof (in 
an entirely unexpected way and by means of quite another 
inquiry ) of a truth which one greatly desired and sought, 
in vain, in spite of much effort. These truths are frequently 
of such a nature that they may be arrived at by many dis- 
tinct paths and that the first paths to be discovered are 
not always the shortest. It is therefore a great pleasure, 
after one has fruitlessly pondered over a truth and has later 
been able to prove it in a roundabout way, to find at last 
the simplest and most natural way to its proof." f 

Today, twenty-five hundred years after the Pythag- 
oreans first perceived that the squares and the primes are 
very interesting numbers, there are still many questions 
to be answered about their relationship to one another. 
Is there, for instance, a prime between every pair of con- 
secutive squares? Are there infinitely many primes that 
are just one unit greater than a square (a: 2 + 1)? 

The inexhaustible storehouse awaits. 

* It is interesting to note that Gauss first observed the Law of 
Quadratic Reciprocity when he was computing the decimal rep- 
resentation of all reciprocals through Mooo in an attempt to find a 
general rule for determining the period of a repeating decimal. 

f The quotations from Gauss are translated from the Latin by 
D. H. Lehmer and appear in David Eugene Smith's A Source Book 
in Mathematics. 



The squares are numbers it is fun to play with by eye, 
as the Greeks played with them. If the reader will provide 
himself with a set of counters or just a sheet of graph 
paper, he will find it fun to try to determine why the 
following system of multiplication which is achieved by 
addition, subtraction and division of squares works; 

To multiply 7 by 6- 

we take the sum of 7 and 6, 7 + 6 = 13 

square it, 13 2 = 169 

subtract the square of 7, 169 49 = 120 

subtract the square of 6, 120 36 = 84 

divide by 2, 84 -f- 2 =42 

Why does it work? 



A Number 
for Every Point 
on the Line 


numbers to irrational numbers . . . 

This step-by-step extension of the 
idea of Number was forced upon math- 
ematics by the contemplation of so 
seemingly simple a thing as the straight 
line. Faced with the fact that the 
straight line which is the diagonal of 
the unit square can be measured with 
truly mathematical accuracy only by 
the square root of 2, the Greeks con- 
cluded that there was no such number. 
The men who followed them, how- 
ever, have insisted upon the fact that 
for every point on the line, there must 
he a number. 

It would seem that even in two 
thousand years such diametrically op- 
posed points of view could never be 
brought together. Yet if we begin at 
both ends of the time in the third 
century B.C. and the nineteenth cen- 
tury A.D. we find that the Greek solu- 
tion of this problem, which was a 
revised theory of proportion, and the 
modern solution, which is the concept 
of the real number line or arithmetic 
continuum, are in essence the same. 

The revised theory of proportion 
which allowed Greek mathematics to 
move forward again, although in the 
guise of Form now rather than of Num- 
ber, is contained in the fifth and sixth 
books of the Elements and is consid- 


ered without question the finest mathematics in Euclid. 
It is, almost entirely, the work of Eudoxus. 

Eudoxus was a poor young student who walked every 
day to Athens to sit at the feet of Plato. His genius was 
recognized and he became eventually a great and honored 
teacher himself, with many personal achievements in 
astronomy and geometry. His masterpiece was his theory 
of proportion and, specifically, his redefinition of "in the 
same ratio" so that it could be applied to the newly dis- 
covered incommensurable magnitudes as well as to the 
traditional commensurable magnitudes. 

Under the universal rule of Number, before the dis- 
covery of the irrationality of V2, ratio had been conceived 
by the Pythagoreans as the expression of the relative mag- 
nitude of two whole numbers, or lengths. We might think 
that the need for such rational expressions arose prac- 
tically in measurements where the distance to be meas- 
ured fell between two units, or whole numbers. Actually, 
it arose as a result of Pythagorean interest in the purely 
theoretical relationship between magnitudes. 

Given two magnitudes like A and B below, how can 
we express the relationship between them in whole num- 

A simple way is to multiply, or repeat, these lengths until 
we reach a point where both totals coincide. In the exam- 
ple above, if we take five of the length labeled A and nine 
of the length labeled B, we will find that we have two 


equal lengths. Since 9B = 5 A, the relative magnitude of 
A to B is 9 to 5, or the "rational number" 9/5. (We must 
use quotation marks here, for Greek mathematicians 
from Pythagoras to Diophantus (A.D. 300) did not con- 
sider these rational expressions to be numbers. As far as 
they were concerned, the only real numbers were still the 
whole numbers. ) 

Another way of determining relative magnitude ( also 
known to the Pythagoreans) is the method we still use 
today to find the greatest common divisor of two num- 
bers Euclid's algorithm. If we measure off A by B and 
then measure off JB by the remainder C, we obtain eventu- 
ally a remainder (in our example: D) which exactly 
measures the previous remainder. 

It is easy to see that D measures both A and B exactly, A 
9 times and B 5 times. Taking D as the common unit, the 
relative magnitude of A and B is, as we also found by our 
first method, 9 to 5 the "rational number" 9/5; 


or, in the familiar language of proportion, A is to B as 
9 is to 5 (A:B :: 9:5). 

This definition of ratio is perfectly adequate if we wish 
to express the relative magnitude of the base and the 


hypotenuse of the ancient 3-4-5 right triangle pictured 
on the left below. But if we try to use the same methods 
to find the ratio between the unit base of an isosceles right 
triangle on the right and its hypotenuse, we are in trouble. 
No matter how many times we take the hypotenuse and 
how many times the side, we will never reach a point 
where our totals coincide. If we try to use the method of 
finding the greatest common measure, which worked so 
well for A and B above, we will never obtain a remainder 

which is exactly contained in the preceding remainder.* 

^ _ ry? _ 

In geometry we say that these two lengths are incom- 
mensurable. In arithmetic, if the only numbers we have 
are the whole numbers, we cannot express the relation- 
ship between them. 

Yet our eye tells us that the base and the hypotenuse 
of the triangle on the right have like any two lines a 
relative magnitude, even if we cannot express it in the 
only numbers we have. It is the "ratio" of 1 to V2; but 
we cannot call this a ratio as long as ratio is defined in the 
traditional sense of relative magnitude expressed by whole 

Eudoxus solved this difficulty like a true mathemati- 
cian. He simply redefined ratio so that it could be ap- 
plied to incommensurables as well as to commensurables. 

* If we appear to be successful, it is only because of an error 
introduced by the thickness of our lines. 


It was this new definition which Euclid used in the 

Eudoxus limited his definition of ratio to finite magni- 
tudes of the same land. He then proceeded to the crux of 
the matter. What do we mean when we say that magni- 
tudes are "in the same ratio"? 

The simplest way to determine that a/b and c/d are 
in the same ratio is to reduce them to lowest terms. We 
say that % and %2 are the same since both when reduced 
to lowest terms are the fraction %. A more complicated 
way, but the one more appropriate to Eudoxus' defini- 
tion of "in the same ratio," is the following: 

We say that a/b and c/d are in the same ratio when 
we can multiply the numerators a and c by some whole 
number m and the denominators b and d by some whole 
number n so that 

ma = rib 

To make this process clearer, let us determine by this 
method whether % and % are in the same ratio. We mul- 
tiply both numerators by the same number (3) and both 
denominators by the same number (2) : 

3X4- 12 3X6 18 

2X6 = 12 2X9 18 

If by such multiplication with whole numbers there is 
some way we can make our new numerators equal, re- 
spectively, to our new denominators, we say our two 
original ratios are in the same ratio. This, we say, is what 
we mean by "in the same ratio." 


The difficulty is that our method is applicable only to 
ratios of whole numbers; in other words, commensurables. 
When we are dealing with the ratios of commensurables, 
we can always find an m/n such that 

ma rib 

we nd 

but when we are dealing with incommensurables, ma and 
me will always be greater than nb and nd, or less. They 
will never, no matter what m/n we select, be exactly 
equal. However and this is the fact that Eudoxus seized 
upon for his masterly redefinition if 

and if we multiply the numerators by the same whole 
number m and the denominators by the same whole num- 
ber n, regardless of whether we are dealing with com- 
mensurables or incommensurables, we can never get the 
result that ma is greater than nb while me is less than nd. 
If our original ratios are actually in the same ratio, the 
numerators of our new ratios will always both be greater 
or both be less than our new denominators, or the new 
numerators and the new denominators will be, respec- 
tively, equal. 

This then, said Eudoxus, is in essence what we mean 
by "in the same ratio." 

In modern notation we can state this definition as 

Consider a/b and c/d. If we multiply a and c by the 
same number m, and b and d by the same number n, and 
if we obtain one of the following situations, and no other: 


ma > nb and me > nd 

ma nb and me nd 

ma<nb and me < nd 
then a/b c/d. 

Unfortunately, when in the fifth definition of Book V 
of the Elements Euclid had to present this definition of 
"in the same ratio," he, like Eudoxus, did not have the 
benefit of algebraic notation; and he had to write: "Mag- 
nitudes are said to be in the same ratio, the first to the 
second and the third to the fourth, when, if any equi- 
multiples whatever be taken of the first and third, and 
any equimultiples whatever of the second and fourth, the 
former equimultiples alike exceed, are alike equal to, or 
alike fall short of the latter equimultiples, respectively, 
taken in corresponding order." 

It was this definition which Isaac Barrow (1630- 
1677), who voluntarily gave up his professorship at Cam- 
bridge to the young Newton, called "that Scare-crow at 
which the over modest or slothful Dispositions of Men 
are generally affrighted." * 

Today, more than two thousand years since Eudoxus 
formulated this definition, it is echoed almost word for 
word in the modern definition of equal numbers, which, 

* He went on to add: "They are modest, who distrust their 
own Ability, as soon as a Difficulty appeais, but they are slothful 
that will not give some Attention for the learning of Sciences; as if 
while we are involved in Obscurity we could clear ourselves with- 
out Labour. Both of which Soits of Persons are to be admonished, 
that the former be not discouraged, nor the lattei refuse a little 
Care and Dilligence when a Thing requires some Study." (We rec- 
ommend to the reader the words of Isaac Barrow. ) 


in very much the same way that Eudoxus' definition of 
"in the same ratio" enabled the Greek mathematicians to 
deal with incommensurable lengths, enables modern 
mathematicians to deal with irrational numbers. There is, 
however, high irony in this. When the Greeks found that 
there were points on the line for which their mathematics 
had no exact numerical expression, they fled from Num- 
ber into Form and took sanctuary in a geometric theory 
of proportion which could handle incommensurables. Yet 
in this same sanctuary, although they never found it, was 
the saving concept of number which they sought a 
unique number for every point on the line. 

In the two thousand years that elapsed between the 
Greek theory of proportion and the modern concept of 
the arithmetic continuum, the irrational numbers led a 
curious "here and not here" existence. They were mostly 
"not here" until the late sixteenth century. At that time 
the decimal notation began to come into common use, 
and mathematicians to their delight saw rational and 
irrational numbers fall into place like well-ordered regi- 
ments! . 

All decimal&jkjm be thought of as never-ending repre- 
sentations of numbers. 

Some, after a certain point, repeat O's indefinitely: 

like l / 2 , or .5000000000000000000000 . . . 
Some repeat another single digit: 

like y 3 , or .3333333333333333333333 . . . 
Some repeat after a certain point a series of digits: 

like tf , or .1428571428571428571428 . . . 
Some never end and never repeat: 

like n, or 3.1415926535897932384626 . . . 

It is very easy to show that all rational numbers in 
their decimal representation will repeat; and, conversely, 


that all repeating decimals are representations of rational 

Consider the rational number %7. To obtain a deci- 
mal representation, we simply divide 17 into 1. Sometime 
within the first 16 steps of this division we must obtain 
a remainder which we have obtained before, since there 
are only 16 possible positive remainders. When we do, 
our quotient must of necessity begin to repeat. In the case 
of %?, the decimal representation actually does have a 
16-place period: 

.0588235294117647058823 . . . 

We can say, then, in general terms that the decimal repre- 
sentation of any rational number a/b will repeat within 
( b 1 ) decimal places. 

Now let us consider the reverse situation where we are 
given a repeating decimal and wish to obtain a rational 
representation of it. We take, for example, the repeating 
decimal .1212121212121212121. ... We multiply this deci- 
mal by 100 so that we have a whole number 12 followed 
by the repeating decimal .121212121212121212121. . . . 
We then subtract our original repeating decimal, which 
is the same as this same decimal tail: 

12.121212121212121212121 . . . 
.121212121212121212121 . . . 

12.000000000000000000000 . . . 

What we have done is to subtract our original decimal 
from a number 100 times as great. The answer of 12 which 
we obtain is thus equal to 99 (i.e., 100 1) times our 
original decimal. The original decimal, therefore, must be 
equal to 12 divided by 99, or %a, which is its representa- 
tion as a rational number reduced to lowest terms. Again, 
it is clear that we can always do exactly this: we can al- 


ways obtain for any repeating decimal an expression as a 
rational number. 

Since, as we have seen above, all rational numbers 
can be represented as periodic decimals and since all 
periodic decimals represent rational numbers, it follows 
that all irrational numbers can be represented by non- 
repeating decimals and that all non-repeating decimals 
represent irrational numbers. Granted that this is not a 
very precise definition of irrational number., it was so 
much more numerical than anything mathematicians had 
seen since the Pythagorean discovery of the irrationality 
of V2 that they welcomed it without question. They pro- 
ceeded to apply the operations of arithmetic to these new 
numbers in the same manner they applied them to the 
whole numbers and the fractions, and nobody worried 
much about the niceties. 

In the late nineteenth century, all of this was changed. 
Certain mathematicians, including Richard Dedekind 
(1831-1916) and Georg Cantor (1845-1918), saw the 
necessity for a truly precise formulation of what mathe- 
maticians call the real numbers, the numbers for the 
points on a line. (The reason for this name will become 
clear in Chapter 7.) Curiously enough, at that time, 
twenty-three hundred years after Euclid compiled the 
Elements., they expressed their ideas not in the terms of 
some recent mathematical development but very much in 
the terms of the Eudoxian theory of proportion as pre- 
sented in the fifth book of the Elements, 

Although there are today several ways in which irra- 
tional numbers can be precisely defined, the most popu- 
lar definition remains that of Dedekind and bears the 
dramatic title of "a Dedekind cut." Dedekind formulated 
this definition with full modem rigor, but we can grasp it 
more easily if we discuss it in a rather rough fashion, 


relying heavily on our intuitive understanding of "num- 
ber" and "line." We begin by thinking of all the rational 
numbers as being paired off on a line with those points 
which they represent all lengths being measured from 
an arbitrary origin point labeled 0. For simplicity's sake, 
we can concern ourselves now with only that part of the 
line which is to the right (or positive side) of 0: 

From an everyday point of view, although this line 
looks as familiar as an ordinary ruler, there are several 
rather unusual things about it. For example, it has no be- 
ginning and no ending. If we select on it any two points 
which have been paired with rational numbers, we can 
always find between these as many more points or num- 
bersas we please. Say that we select two points as 
"close" as %ooo and %ooo. Between these two points lies 
the point %ooo. Between %ooo and % () oo lies %ooo, and 
so on. In general, if we take any two rational numbers a/b 
and cjd, add their two numerators and add their two 
denominators, we shall obtain a rational number: 

which lies between them. There is no "nextness" among 
these rational numbers. 

Now let us make our first cut in this line. Let us say 
that we will cut it at the point which is paired with the 
rational number %. The complete line will have been cut 
into two pieces which together include every point, or 
rational number, on the entire line. If our cut line is to be 
in just two pieces, the cut point & must be in one piece or 
or the other; it cannot, of course, be in both: 


If % is included, as in the upper example, in the left-hand 
segment of the line, it must be the largest number on that 
side. Since there is no "nextness" among the rational num- 
bers, the right-hand segment of the line can have no small- 
est number. But if K is included, as in the lower example, 
on the right-hand side of the cut, then it must be the 
smallest number on that side and the left-hand side now 
can have no largest number. 

Thinking in this manner, we are defining rational 
numbers as cuts which divide the line into two parts in 
such a way that one and only one of the parts has either 
a largest or a smallest number. This is a curious enough 
definition; but like Alice, we find that things become 
curiouser and curiouser. 

What if we break the line at a place where there is no 
point and, hence, no rational number? This is completely 
possible, because although the rational numbers are dense 
upon the line, they are not continuous. If they were, and 
if there were a rational number for every point on die 
line, the Pythagoreans would have found a rational num- 
ber which exactly measured the square root of 2. .What 
happens when we cut the line at the place where there 
"should" be a number-point equal to V2? The line divides, 
as before, into two parts but with an important difference: 

There is now no largest rational number in the left- 
hand part and no smallest rational number in the right- 
hand part. 


Such a cut according to DedekincTs definition is an 
irrational number! 

In a much more formal and precise statement, the 
definition can be put in the following way: 

An irrational number a is defined whenever the ra- 
tional numbers are divided into two classes A and B 
such that every rational number belongs to one, and 
only one, class and (1) every number in A precedes 
every number in B, and (2) there is no last number in 
A and no first number in B; the definition of a being 
that it is the only number which lies between all num- 
bers in A and all numbers in B. 

We shall not here expand in detail upon the similari- 
ties between this definition and the Eudoxian definition 
of "in the same ratio." The reader who is particularly in- 
terested will find a complete statement in Sir Thomas 
Heath's edition of the Elements. Suffice it to say that the 
two definitions, separated by more than two thousand 
years of mathematical thought, are in essence the same. 

From Dedekind's definition of an irrational number 
as a cut in the rationals, we can now proceed to the state- 
ment of the axiom upon which all of arithmetic, and hence 
all of mathematics, rests. If we replace every Dedekind 
cut in the rational numbers with a point and a number 
(a non-rational, or irrational, number) so that regardless 
of where on the line we make a cut, we shall always cut 
at a number-point pair, we can state what is known as the 
Cantor-Dedekind axiom: 

It is possible to assign to any point on a line a unique 
real number, and, conversely, any real number can be rep- 
resented in a unique manner by a point on the line. 

We have come in some twenty-five hundred years 

from the despairing conclusion of the Pythagoreans that 
for some lengths there were no numbers to the completely 
satisfying conclusion of Dedekind and Cantor that for 
every point on the line there is a numher! 

From whole numbers to rational numbers to irra- 
tional numbers! These, taken together, are the real num- 
bers; and once again Number (now real number) rules 
the Universe. 


It is fun to test for oneself the fact that every rational 
number can be represented as a repeating decimal, par- 
ticularly when such rational numbers as those listed below 
are taken for the experiment: 


19' 23' 41 

It is also fun to test for oneself the fact that, conversely, 
every repeating decimal represents a rational number. 
The reader is urged to apply the method given on page 54 
to such decimals as: 

.10101010101010101 . . . 
.23523523523523523 . . . 

.28545454545454545 . . . 
.14285714285714285 . . . 

Then just for fun he should "make up" some repeating 
decimals for himself and discover what rational numbers 
they represent! 



These two lines exercised such a 
fascination over the ancient mind that 
they limited the instruments of mathe- 
matical construction, determined the 
subject matter of most of the mathe- 

5matics, and provided mathematical 
"problems" that were not to be finally 
disposed of for more than two thou- 
sand years. 

Fascination with the straight and 
the round apparently blinded Greek 
JourneijThat eyes to the lines which they actually 
Begins at O saw around them. Their geometry was 

based on an axiom which stated in 
essence that parallel lines never meet, 
their intuitive and undefined idea of a 
straight line being inextricably bound 
up with this axiom; yet the architects 
of the Parthenon built their pillars so 
that they bulged in the middle, for 
they knew that if they made the sides 
straight and parallel they would ap- 
pear to curve in toward each other. 
They must have observed that the 
parallel sides of a straight roadway ap- 
pear to converge as they approach the 
horizon. They must also have observed 
that circles always appear elongated 
except when the eye is on the axis of 
the curve. Nevertheless, the Greek 
mathematicians knew they knew that 
the parallel lines which appeared to 
meet could never by their very nature 

meet and that circles, regardless of appearances, were in 
their actuality round. They limited themselves in their 
mathematics to the perfect essence of these figures, and 
did not concern themselves with the imperfect figures 
which their eyes saw all around them. 

The only geometric constructions which the Greek 
mathematicians considered permissible, or "pure," were 
those made with a straightedge, an unmarked rule which 
was the mechanical equivalent of the straight line, and a 
compass, which was the mechanical equivalent of the 
circle. They then conceived that the "solution" of any 
geometric problem must be effected by these two instru- 
ments, alone. 

This made tilings a lot harder. Problems which would 
yield gracefully to other instruments remained "unsolved" 
for two thousand years! 

Undoubtedly the most dramatic of these ancient 
problems, which were to be with mathematics for so 
many centuries, was the problem of duplicating the cube. 
According to tradition, the people of Athens, suffering the 
ravages of a great plague, consulted the oracle at Delos. 
How could they placate the angry gods who had sent this 
plague upon them? The oracle replied that they should 
double the size of the cubical altar to Apollo. The obedient 
Athenians promptly built a cube with each side twice as 
long as that of the original and thereby produced an altar 
which was by volume eight times the size of the original 
altar. The gods, naturally enough, did not appear to be 
placated and the plague continued. 

A century later, although the plague had long since 
run its normal course, the Greek mathematicians were 
still straggling with the problem of doubling the volume 
of a given cube. Since the unit cube has a volume of 1 
(orlXlXl),a cube with volume twice as great must 


be represented in modern notation by the formula 

x 3 =2 

Solving for x then is the equivalent of extracting the cube 
root of 2. 

The Greek mathematicians assumed that the problem, 
since it was proposed by the gods, required an exact 
answer and one which could be effected in its construction 
by the gods' chosen instruments, straightedge and com- 
pass alone. A rather familial picture of their difficulty is 
presented by Eratosthenes, who lived a while after 
Euclid and is famous for a surprisingly accurate measure- 
ment of the earth and for a "sieve" which is still the basic 
principle of all tables of prime numbers. 

"While for a long time everyone was at a loss," 
Eratosthenes wrote, "Hippocrates of Chios was first to 
observe that if between two straight lines of which the 
greater is double the less it were discovered how to find 
two mean proportionals in continued proportion, the cube 
would be doubled; and thus he turned the difficulty of 
the original problem into another difficulty, no less than 
the former." * 

Eratosthenes went on to report that Menaechmus, 
who was a pupil of Eudoxus, found two solutions to this 
problem, both effected by the intersection of conic sec- 
tions. This is the first mention in mathematical literature 
of those beautiful and ubiquitous curves the hyperbola, 

* In modern notation we would state this problem of Hip- 
pocrates as: 

To find - such that _ = - = JL w ith A and 2A being two 

y x y 2A & 

given straight lines. 

From the equation = -, we have x 2 = Ail. Squaring both 
^ x y J ^. e, 

sides of this equation, we obtain t/ 2 = -j^, 

the parabola and the ellipse. To Menaechmus is given the 
credit for their discovery, although their names were given 
them much later by another mathematician. 

Oddly enough, the names themselves go back as far 
as Pythagoras who, like the Greeks that followed him, 
never paid any attention at all to the generally elliptical 
appearance of the circle, the parabolic paths of projec- 
tiles, or the hyperbolic arches cast by shaded lanterns. 
One of the problems that did, however, interest Pythagoras 
was that of drawing upon a given segment a figure- 
triangle, square or pentagon that was required to be the 
size of some other given figure of a different shape. In the 
course of the solution of this problem, one of three things 
might happen. The given line segment would be too short 
(ellipsis), exactly the right length (parabole) or too long 
( hyperbole ) . These same words have come very generally 
into English in the ellipsis, three dots which mark the 
omission of words; the parable, which tells one story but 
parallels another that although untold is the real story; 
and the extravagant exaggeration of statement which we 
call hyperbole. These names, suggesting as they do the 
arithmetical relations of less than, equal to and more than, 
were given to the conies by Apollonius of Perga, who fol- 
lowed Archimedes. 

The conic sections of Menaechmus seemed to possess 
the Greek mind even though the interdiction against con- 

From the equation - = ^-, we obtain y 2 = 2Ax. 

y "-" 

Substituting this value for t/ 2 in the preceding equation, we 
Hod ft- =8^ or . = (!) =2. 

It follows, therefore, that the desired V 2 is ^. 

(The reader who finds his algebra rusty would do well to recall that 
Hippocrates had none at all to help him formulate his problem!) 


struction by instrument other than straightedge and com- 
pass cast them out of the pure geometry of the day. The 
simplest way to produce the various sections is by cuts 
and cross-cuts of a solid circular double cone. (The 
reader may enjoy producing the various sections by cut- 
ting a cone of light with a piece of cardboard placed at 
varying angles. ) A cut exactly parallel to the angle of the 
cone will give us a parabola. If the angle of our cut is 
within the angle of the cone, we obtain the two branches 
of the hyperbola; while if it is outside, we obtain an 
ellipse. A straight cut parallel to the base will give us the 
circle, or limiting form of the ellipse. 

We can also think of each conic curve-ellipse, parab- 
ola, hyperbola as the path of a point which must move 
according to certain rules which determine the curve it 
makes. This is most intuitively clear when we think of a 
circle as the path of a point which must always be a given 
distance (the radius) from another point (the center of 
the circle): 

An ellipse is the path of a point which must always move 
so that the sum of the distances from two given points 


(called the foci) is always the same: 

A hyperbola is the path of a point which must move so 
that the difference of the distances from two given points 
is always the same: 

A parabola is the path of a point which is always the 
same distance from a given point that it is from a given 

Euclid himself wrote a treatise on the conies, since 
lost. It was not Euclid, however, but Apollonius who de- 
veloped the fundamental properties of the conies in 
remarkable generality. Apollonius first showed that all 
conies are sections of any circular cone, right-angled or 
oblique, and in spite of the truly cumbersome expression 
available to him gave for the first time the fundamental 
property of all conies, which we shall present later in 
modem notation. 

This is the kind of achievement which any mathema- 
tician respects. It is the sort of work the English mathe- 
matician Littlewood was thinking of when he remarked 
to his friend Hardy that the Greek mathematicians were 
not just clever boys, scholarship candidates, but in the 
language of his Cambridge "fellows at another college/' 

Pappus, who has been called the worthiest com- 
mentator on the work of Apollonius, then showed that the 
ratio of distance of any point on any conic from a fixed 
point (the focus) and a fixed line (the directrix) is con- 
stant. This ratio, which we express as e, is called the 
"eccentricity" of the curve. A conic is an ellipse, a parab- 
ola or a hyperbola according as e is less than 1, equal to 
1, or more than 1. In the circle e is 0. 

Of such work by the ancient Greeks, a president of 
the British Association for the Advancement of Science 
wrote: "If we may use the great names of Kepler and 
Newton to signify stages in the progress of human discov- 
ery, it is not too much to say that without the treatises of 
the Greek geometers on the conic sections there could 
have been no Kepler, without Kepler no Newton, and 
without Newton no science in the modern sense of the 
term. . . ." 

The earth follows a nearly elliptical orbit around the 
sun," projectiles approximate parabolic paths, a shaded 

light illuminates a hyperbolic arch. If it required the 
Delian problem to reveal these common curves to eyes 
blinded by circles, the problem would have done more 
than its turn for mathematics. But, indirectly, there were 
more great gifts to come. 

It was some seventeen hundred years after Menaech- 
mus, Apollonius and Pappus that an arrogant young 
Frenchman published a short mathematical treatise as a 
supplement to a larger philosophical work which he ex- 
pected to ensure his immortality. This mathematical 
treatise, entitled La Geometrie, began with one of the 
most important sentences in the history of mathematics: 

"Any problem in geometry," wrote Rene Descartes, 
"can easily be reduced to such terms that a knowledge of 
the lengths of certain lines is sufficient for its construction." 

We have already seen how the discovery that the diag- 
onal of the square is incommensurable with the side had 
driven the Greeks into a geometry without number and a 
theory of numbers expressed in the terms of geometry. 
"This great step backwards/' it is regretfully called by 
Morris Kline in his Mathematics and Western Culture, 
Descartes, by applying the concept of the variable, which 
he took from algebra (the great Eastern contribution to 
mathematics),* to the ancient method of mapping by 
coordinates, which was known to the Babylonians and the 
Egyptians, reversed this step into a giant stride forward 
a stride in fact into modern mathematics. 

* The historical development of algebra has been characterized 
in three stages: (1) rhetorical algebra, in which problems were 
solved by a process of logical reasoning but were not expressed in 
abbreviations or symbols; (2) syncopated algebra, in which ab- 
breviations and symbols were used for certain quantities and 
operations occurring most frequently; and (3) symbolic algebra, 
in which completely arbitrary symbols are used for all forms and 
operations a development of the period immediately before and 
after Descartes. 


It does not matter that in doing so, lie was as much 
under the spell of utterly impractical geometric construc- 
tion problems as were the ancient Greeks. The application 
of the method of coordinate mapping to geometry and 
algebra, which is called in the history of mathematics the 
invention of analytic geometry and credited to the young 
Frenchman, was one of those innovations which, as soon 
as they are finally made, seem as if they had always been 
inevitable. It freed both subjects from bonds which until 
then had appeared inherent in them. Geometrical figures 
were transformed into algebraic equations and equations 
into figures. Problems which had eluded the genius of the 
Greeks dropped into the hands of schoolboys. 

In this chapter we shall offer a glimpse of this new tool 
by examining it in relation to the line and the circle of 
Euclid and the conic sections of Apollonius in short, the 
curves known and studied geometrically by the ancient 

We begin with the selection on the plane of a point- 
it may be any point we care to choose and we label this 
point "O" for origin. It is at this point that we set out on a 
very different mathematical road from that traveled by the 

Through the point O we draw a line which extends 
indefinitely to the right and to the left of O. On it, we mark 
off units very much as years are marked off from an origin 
point which is the birth of Christ. The units after the birth 
of Christ, or to the right of O, are labeled with a plus; the 
units before, or to the left of O, with a minus. We then 
draw perpendicular to our first line another which passes 
through O and extends indefinitely above and below it. 
On this line we mark off units as degrees of temperature, 
for instance, are marked off on the thermometer: the units 
"above" O being labeled with a plus and the units "below" 
with a minus. We call our original horizontal line the *-axis 

and this new vertical line which is perpendicular to it, the 

On the plane which we have marked off we can now 
locate uniquely any point by stating its position on the 
plane in the terms of its coordinates on the x and y axes. 
One number (x) tells its distance from the y-axis; another 
(y), its distance from the x-axis. This x, y pair is the ad- 
dress, as it were, of the point on the plane: 

It is, of course, essential that the "addresses" of these 
points be stated in the proper order. Just as 70 Twenty- 
sixth Street is not the same address as 26 Seventieth Street, 
the number pair 26, 70 does not locate on the plane the 
same point that the number pair 70, 26 does. ( If the reader 
will transpose the numbers in each pair located on the dia- 
gram above, he will find that he has located three entirely 
different points, one [ 3, 3] remaining the same point 
even when the coordinates are transposed. ) 

The concept of points as such ordered number pairs is 
the key to analytic geometry. For this reason the coor- 
dinates of a point are always given in order: the x 
coordinate first, then the y coordinate. Our new definition 
of a point is a long way from Euclid's "A point is that which 
has no part": 

A point is an ordered pair (x, t/) of real numbers. 

Not only can we think of these points ( every point on 
tlie plane ) as number pairs, but we can also handle them 
mathematically as number pairs. They are no longer 
geometrical points; they are things of arithmetic. This new 
approach works both ways, for we can also think of any 
pair of numbers as a unique point on the plane. The brief- 
est glance tells us in which quadrant it belongs, the 
slightest effort places it exactly, no longer a number pair 
but a point, again a thing of geometry. 

A point, we say then, is an ordered pair of real num- 
bers (x, y). When we know the values for x and y, we 
know where the point lies on the plane. 

But what happens to the values of x and y when our 
point moves about the plane leaving in its path a trail 
which we call a line? 

If we take the point ( 3, 3), which we located on 
the diagram above, and proceed to move it so that its path 
is a straight line toward and through O, the origin, we find 
that the values of x and y change continuously. But one 
tiling about them does not change: the x coordinate re- 
mains always the same as the y coordinate, just as 3 is 
the same as 3. If we move the point so that its path is a 
straight vertical line, the x coordinate remains the same 
( 3), but the y coordinate changes constantly. Moving 
the point on a horizontal line, we find that the reverse is 
true: the x coordinate changes constantly while the y co- 
ordinate remains the same ( 3). We have here three 


distinct lines, all passing through the point ( 3, 3). 
When we further consider the number pair ( 3., 3 ) , we 
are immediately aware that the x and y coordinates add 
up to 6. What happens when we mark on the plane near 
it the other points the coordinates of which also add up to 
6? We find that we are mapping yet another straight 
line which also passes through the point ( 3, 3 ) : 

Like points, all of the lines we have mapped can be 
uniquely identified in the terms of their x and y coor- 

But we may object we call these "paths of points" 
or "lines" in geometry, but aren't they just equations in 
algebra? Quite right. In analytic geometry we find that just 
as points on the plane can always be expressed as ordered 


pairs of real numbers (x, y), straight lines can always be 
expressed as equations of the first degree in two unknowns: 

ax + by + c = 0* 

Although we have not done so above, we can express 
all of the equations we have listed in this standard form. 
The last line we graphed, for example, has the equation 

Since we have geometrized algebra at the same time 
we have algebrized geometry, such equations are now 
known as linear equations. 

We have seen how the point of Euclid's geometry has 
become an ordered pair of real numbers and the line, the 
graph of an equation in the first degree with two un- 
knowns. Now we must see what has become of Euclid's 
circle in this new number-based geometry. 

On the Cartesian plane we draw a circle of unit radius 
with its center at the origin: 

* It was Descartes who originated this form of expressing any 
equation so that the right-hand side is 0. He also started the custom 
of using the last letters of the alphabet for the unknowns and the 
first for the constants 


What can we say about the points on the line, the circum- 
ference of the circle, which will identify it algebraically 
as we have already identified straight lines? We see that 
the four points of the circumference which fall upon the 
x and y axes can be easily identified as (0, 1), (1, 0), 
( 0, 1 ) and ( 1, ) . What have these four pairs of num- 
bers in common which might enable us to formulate a gen- 
eral rule for finding any point on the circumference of the 
circle we have drawn? Only that in each case the x and y 
coordinates add up to 1. Is this the general rule we are 
looking for? No, for when we locate the point ( H, % ) , the 
coordinates of which also add up to 1, we find that this 
point falls inside the circle and not on the circumference. 
We are not so far off, though, as we appear to be. 

The golden thread of the theorem of Pythagoras runs 
through our new algebrized geometry. If we draw a right 
triangle on the plane using the radius of our circle as the 
hypotenuse of the triangle, we can see by the Pythagorean 
theorem that r 2 , or in this case I 2 , must be equal to the 
sum of the squares of the other two sides: 

In the case of this particular triangle, the two sides are 
equal to the % and y coordinates of the point where the 
hypotenuse of the triangle cuts the circumference of the 
circle. We can say, therefore, that 

or, for this particular circle: 

Every point on the circumference of the circle we have 
drawn must be a number pair such that the sum of the 
squares of the two coordinates is 1 : 

1 V15\ il V8\ /I V3^ 
* 4 M3' 3]'l2' 2J"" 

The way in which we have obtained this formula is in 
one sense a long way from the Greeks; yet in another 
sense it is as old as geometry itself. Regardless of where 
our circle lies on the Cartesian plane, we can by means of 
the Pythagorean theorem, with only a simple variation on 
the method above, express it as a similar equation of the 
second degree in two unknowns. This "distance formula" 
of analytic geometry, which is just as valid in three di- 
mensions as in two, in seventeen as in three, runs con- 
currently with the geometry of Descartes as it is extended 
from two dimensions to three to four. . . .' But that is a 
story for another chapter. In the meantime we can see 
that just as the equation of the circle is derived by means 
of the distance formula, so can the equation for any and 
all of the conic sections. For they, like the circle, depend 
essentially on a distance ratio. It can be proved that the 
curve of any conic is the graph of an equation of the 
second degree. It can also be proved that, conversely, any 


curve defined by an equation of the second degree 

is one of the conies. 

Although we have gained only the threshold of the 
new world of curves and their equations which was 
opened up by the invention of analytic geometry, let us 
turn back now to the Delian problem, which started us on 
our journey. The oracle had advised doubling the cubical 
altar of Apollo to appease the gods. The Greeks had as- 
sumed that the construction must be made only by the 
instruments of pure geometry; but, failing to solve the 
problem with straightedge and compass alone, they had 
toyed with certain other solutions mainly effected by 
devices for drawing one or more of the conic sections. We 
recall that they could not even state the problem with the 
vividness and suggestiveness which algebraic notation 
gives us, nor could they express the relatively simple 
characteristic properties of the sections with any degree 
of economy. Paragraphs of cumbersome technical vocab- 
ulary led to the enunciation of truths which can be ex- 
pressed today with half a dozen letters, plus and minus, 
and die equals sign. And yet Menaechmus solved the 
problem of doubling the cube by the intersection of conic 

Let us look at the solution of this same problem, using 
the powerful new tool which Descartes put in the hands 
of the mathematicians of the Renaissance when he in- 
vented analytic geometry. 

We first graph the equation 

** = y 

which is an equation for a parabola, and then 

xy = 2 


which is an equation for a hyperbola: 

We have already said that the problem of doubling the 
unit cube is in modern notation the problem of solving 
for x the equation 

Now if we consider the coordinates of the point at which 
the two curves graphed above intersect, we shall see that 
by the formula for the hyperbola the product of the x and 
y coordinates at this point must be 2; but by the formula 
for the parabola the y coordinate must also be equal to 
the square of the x coordinate. If we substitute the x 2 
value for y (in the first equation) for the y in the second 
equation, we obtain 

The point of intersection of these two curves is then the 
solution of the Delian problem. If we take the length from 
the origin to the x coordinate of the point of intersection, 


we shall have the necessary length for the side of our new 
cube, which will be twice the volume of the unit cube! 

The Delian oracle is long silent, the original altar is in 
dust, the plague has been replaced by a thousand other 
plagues. To solve a problem made difficult by purely 
arbitrary restrictions, the conic sections have been dis- 
covered, analytic geometry has been invented. We are 
able to illustrate in this book the length of side of a cube 
which will be twice the size of the unit cube. But still the 
gods would not be satisfied, for the size of the new altar 
must be determined by straight line and circle alone! 


The reader may enjoy graphing the following equa- 
tions, the first few points of which are already indicated: 

x t/ = 2(2,0), (3,1),... 
4x + 3t/=18(0,6),(l, *%),... 

t, = **-2(0, -2), (!,-!),... 
^ =4 (1,4), (-y 2 ,-8),... 

=4 (0, 2), (1, V3), - . . 



How Big? 
How Steep? 
How Fast? 


Among these three apparently un- 
related questions there exists a deep 
and unexpected point of contact which 
can serve us as an introduction to the 
calculus, one of the most powerful 
tools of mathematics. It is by means 
of the calculus that mathematics has 
been able to make an effective attack 
on those problems which in earlier 
times admitted only of approximate 

Invented in the seventeenth cen- 
tury by Sir Isaac Newton (1642-1727) 
and Gottfried Wilhelm von Leibniz 
(1646-1716), who worked independ- 
ently, the calculus had had its be- 
ginnings long before in two purely 
geometrical problems : how to compute 
an area bounded by a curve and how 
to draw a tangent to a curve at any 
point. More than nineteen hundred 
years before either of the inventors of 
the calculus was born, these two prob- 
lems were solved (for special types of 
curves) by Archimedes (B.C. 287?- 
212), who used what were essentially 
the methods of the calculus of New- 
ton and Leibniz. 

How big? 

The question How big? was one of 
the first to which mathematics sought 
an answer, and one of the first to which 
it found one, although not for all cases. 


Given a rectangular area, it is a simple matter to compute 
that area as a sum of unit squares or, as we more often 
express it, the product of length and width. Given a tri- 
angular area, it can be shown that the area of a triangle 
is half that of a rectangle with the same base (width) and 
height (length) or, again, as we more often express it, 
one-half the product of base and height. Since any 
straight-edged surface, no matter how irregular its bound- 
ary, can be subdivided into triangles, the only remaining 
problem is to find the area bounded in part or in whole 
by a curve. 

One method of doing so is to divide the area insofar 
as possible into rectangles and add together the areas of 
these. In the first figure following, it is clear that the sum 
of the areas of the rectangles, which we can compute 
exactly, gives us a fair approximation. In the second, it is 


clear that more rectangles give an even more accurate 
approximation of the area which lies unoler the curve. We 
can continue, indefinitely, dividing the area into more and 
more rectangles and including as a result more and more 
of the total area under the curve. When we say that we 
can continue indefinitely, this is just what we mean : there 
is no limit to the number of rectangles into which we can 
divide the area the number can "approach infinity." 
There is, however, a very real limit to the sum of the 
areas, no matter how many rectangles we use: for the 
sum can never exceed the area under the curve. 

This limit provides us with a mathematically precise 
definition of what we mean by the area under the curve. 
It is the limiting value of the sum of the areas of the rec- 
tangles as the number of rectangles becomes indefinitely 

Is this a satisfactorily accurate method of determining 
area? It is indeed. How very accurate it is can best be 
seen by applying it, not to a curved figure, the exact area 
of which we do not already know, but to a straight-edged 
figure like a triangle, the area of which we know is one- 
half the product of base and height. By this formula the 

area of the triangle opposite is exactly %. Since the y co- 
ordinate of any point on the hypotenuse has the same 
value as the x coordinate, we can easily determine the 
dimensions of each rectangle. In Fig. A, where we have 
divided the triangle into five intervals, the width of each 
being % of the base, we get the following sum when we 
add the areas together: 

I 5,1 11 21 31 4_10 

5' 5 + 5" 5 + 5' 5 + 5'5 + 5"5~25 

But in Fig. B, where we have divided the base into tenths, 
we get a sum which is closer to %, the true area of the 

JL JLj_I. -!_. -1 A i .!_ 9 - 45 
10 ' 10 + 10 ' 10 + 10 ' 10 + * ' ' + 10 ' 10 ~~ 100 

By increasing the number of intervals from 5 to 10, we 
have brought our approximation from .40 to .45. The area 
with 50 intervals would be .49; with 100 intervals, .495. 
If we take n as the number of rectangular intervals into 
which we divide the triangle, we obtain the following 
general formula for the sum of the areas of n rectangles: 

If the reader, using this formula, will compute the sum of 
the areas of five hundred and one thousand rectangular 
intervals, he will find that these and any other higher n he 
chooses to compute will yield sums between .495 and .50. 


Under no circumstances will the sum of the rectangles 
into which he divides the triangle be more than .50. That 
this is true is intuitively clear when we look at the triangle 
being subdivided and note the tiny triangles above the 
tops of the rectangles which can never be included in the 
sum of the areas. It is also clear when we further simplify 
our general formula for the sum of the areas: 

As n gets larger (i.e., we cut our triangle into more and 
finer rectangles), 1/n gets smaller. As this happens, the 
value of 

will approach H, the actual area.* 

This method of determining area was called by Ar- 
chimedes the method of "exhaustion" and by Newton and 
Leibniz, "integration." The latter two were fortunate in 
having at their disposal a tool which was not available to 
Archimedes. This was the analytic geometry of Descartes, 
with which as has been frequently pointed out a mod- 
erately intelligent boy of seventeen can solve problems 
which baffled the greatest of the Greeks. This statement 
is made, not to discredit Archimedes, whose place with 
Newton and Gauss in the pantheon of mathematics is uni- 

* We can achieve the same result by circumscribing our 
rectangles so that they include more than the area of the triangle. 
As the number of rectangles gets larger, the sum will approach, 
from above, the limit which is the area of the triangle. 


versally acknowledged, but only to emphasize the power 
of the method of analytic geometry. 

When we can place our curves and figures on the 
plane formed by the x and y axes, we have a great ad- 
vantage over Archimedes. Curves, as we have already 
seen, are no longer merely beautiful lines but definite 
relationships among numbers which can be expressed in a 
most general form for the whole extent of the curve by 
algebraic formulas. The straight line, or "curve," which 
forms the hypotenuse of the right triangle on the lower 
part of page 80 is determined by the algebraic equation 
y = x. When we say this, we mean that the numerical 
value of the y coordinate at any point on the curve is the 
same as the numerical value of the x coordinate at that 
point. If we are given x = 9 at a given point, we know 
that y = 9; if x 21, y = 21; and so on. The curve on the 
lower part of page 79 is determined by the equation 
y = x 2 . On this curve the numerical value of the ij coordi- 
nate is always the square of the value of the x coordinate: 
if x = 3, y 9; if x = 9, y 81; and so on. The reader 
will recognize the equation for the parabola we used to 
solve the Delian problem in Chapter 5. 

This method of analytic geometry is even more useful 
in answering our second question than it was in answering 
the first. 

How steep? 

The question How steep?, like the question How big?, 
is simple enough to answer when only straight lines like 
y = x are involved. If we look at the line below, we see 
that one measure of its steepness is the angle it makes 
with the x-axis and another is the ratio between the two 
coordinates x and ij. If we take y/x as a measure of steep- 
ness, we see from the second figure that the greater y is in 
proportion to x, the steeper the line. 

_ __ x jjjsj| 

Neither method appears to be available to us when we 
want to determine the steepness of the parabola, or the 
curve represented by the equation y = x' 2 . Yet if we could 
draw a line which would have the same slope as the curve 
at some particular point, the same two methods of meas- 
uring steepness would serve. 

Although the problem of determining such a line was 
solved by Archimedes in the special case of the spiral, it 
was not solved generally for all curves until, in the cen- 
tury before Newton and Leibniz, Fermat developed a 
general method of drawing a line (called a tangent) 
which touches a curve at only one point and hence has 
the same slope as the curve at that point. 

When our curve is the arc of a circle, a line erected 
perpendicular to the radius at the point where it cuts the 
circumference will be tangent to the circle at that point. 
If we place the circle on the Cartesian plane with its cen- 


;/ .^Sy'MiJSSaSSSCif ?,! 

ter at the origin, the lines constructed perpendicular to 
the y-axis at the points where it cuts the circumference 
will be parallel to the x-axis and will represent the high- 
est and lowest points (or extrema) of the curve. The de- 
termination of such high and low points for any curve was 

l * / \ A- f ifei 7 ^ / 4 " 

%/ / ^^^ 

fa-jrr^, k \ -9-, ^-* 1 i *,f / J % r=^-^5 f 




the particular problem which interested Fermat and for 
which he created a general method for drawing tangents. 
To draw a line tangent to the point P in the figure 
below, according to Format's method, we mark on the 
curve in the neighborhood of P another point Q and draw 
a line from P to Q. As we slide the point on the line now 
marked Q along the curve toward P, always keeping the 
line PQ going through P, the closer Q gets to P, the more 
nearly will the line PQ represent the slope of the curve at 
P. In the language of the calculus, as Q is allowed to 
approach P, the line PQ will approach a limiting position 
which is the desired tangent to the curve at P. 

These two geometrical problems, computing the area 
bounded by a curve and finding the slope of a curve at a 
given point, are at the very foundations of the calculus. 
The first is the fundamental problem of the integral cal- 
culus; the second, of the differential calculus. Both, as we 
have seen, were recognized from antiquity, tackled and 
partially solved long before the invention of the calculus 
in the seventeenth century. Newton and Leibniz were the 
first to recognize that these two problems were but facets 
of one and the same problem, and that the integral and 
the differential calculus were essentially one the calcu- 
lus. The theorem which states this truly deep relationship 
was discovered independently by both of them. It is the 
Fundamental Theorem of the Calculus. 

Although the theorem cannot be stated or understood 
without some grasp of the technicalities of the calculus, 
the glimpse it can give us of this mighty tool in action is 
well worth the effort required to follow unfamiliar sym- 

bols and concepts. Already we have gained some idea of 
the two main concepts, those of limit and of function. 
These are basic to much of mathematics .beyond the cal- 
culus, and mathematicians can (and must) go on for 
pages defining precisely what they mean by limit and 
function. We, however, can make do with very little of 
this. We have seen that the area under the curve is de- 
fined as a limiting sum and the tangent to the curve as a 
Limiting position. These give us an intuitive, if not too 
precise, idea of a limit. We have dealt with the curves of 
two functions so far, although we have never referred to 
them as functions. For our purposes, the simplest and 
most easily grasped definition of a function is a strictly 
mathematical one. A function is a rule by which y is de- 
termined as soon as x is given. If we apply this definition 
to the straight line determined by the equation y = x and 
to the curve determined by y = x 2 , we have no trouble in 
recognizing that both of these equations identify functions. 
To express this concept of function there is a very 
simple and useful notation, f(x)> which is read "/ of x" 
or "function of x." In the first of the examples we have 
given, /(x) = x; in the second, f(x) = x 2 . Since any curve 
represents a value y determined by a value x at each point 
of the curve, we can identify any curve in a general way 


as /(x), or as a function of x, even though we may not 
know the particular f(x) that determines the curve. 

Sometimes we are concerned not with the curve as a 
whole but with a particular point on the curve. Knowing 
that the x coordinate of the point is, say, 2, we can then 
write of y that y = /(2). Whether y necessarily equals 2 
depends solely upon the particular f(x) which determines 
the curve as a whole. When the curve is determined by 
f(x) = x,y = /(2) = 2; but when the f(x) of the curve is 
f(x) = x 2 , then y = /(2) = 4. 

Unfortunately, without understanding this much of 
the notion of function, we cannot possibly follow even the 
simplest applications of the calculus. At the end of this 
chapter, therefore, are a few problems which will enable 
the interested reader to clarify and make firm his own 

With such a general notion of limit and function, we 
now need an understanding of the concept of an incre- 
ment if we are to follow the Fundamental Theorem of the 
Calculus. The technique of the calculus depends essen- 
tially upon this concept. An increment is an arbitrarily 
small increase in xoff(x) which, since- y = f(x), results 
in a corresponding (though not necessarily the same) 
arbitrarily small increase in the value of y. We symbolize 
the increment added to x by A% and the corresponding 
increment in y by Ay, and write 

where A is read "delta." To express what we have done in 
this general way, we do not have to know what x is, what 
die arbitrarily small increase in x is, what f(x) is, or what 
the corresponding small increase in y is. We can even pro- 
ceed, still not knowing the value of any of our terms, to 

express At/, or the increase in y, solely in the terms of x. 

Once we have expressed At/ in terms of a;, we can express 
the ratio A yj A% in terms of x. 

Ay f(x+ Ax) ~f(x) 
Ax Ax 

Perhaps we appear to be getting nowhere fast? 

But it is one of the marvels of mathematics that such 
apparently pointless manipulation of symbols should be 
the source of the power of the calculus, one of the most 
practical of the many tools with which mathematics has 
outfitted modern science! Appearances to the contrary, 
we are getting somewhere but fast. To see that we are, 
let us return to the curve of the parabola, which is repre- 
sented by the equation y = x 2 . We learned earlier how to 
determine the slope of such a curve at any given point, 
but now let us consider a less geometrical and more gen- 
eral question. What is the rate of change represented by 
this curve? How fast is y changing with respect to x? 
Actually, although these two questions sound quite dif- 
ferent, they are the same as the question How steep? 

Since in the case of this curve, f(x) = x 2 , we know 
that the value of y is increasing as the square of the value 

x 1 2 3 4 5 
y 1 4 9 16 25 

Obviously y is increasing much faster than x. Between 
and 1, both x and y increased by 1; but between 6 and 7, 


x still increased by only 1 but y increased by 13. Between 
and 7, x has gained 7 points while y has gained 49. The 
average gain of y in proportion to that of a is 7 to 1. But 
how fast is y gaining on x? 

Let us apply the method of the calculus to this prob- 
lem: a method which appeared a few pages back as a 
meaningless manipulation of symbols. We begin by add- 
ing an arbitrarily small amount to x in f(x) so that we 
have instead of /(x), f(x -f Ax). Since y = f(x), the new 
value of y is y + At/ = f(x -f- Ax). Now let us substi- 
tute for f(x) in its general form the specific function x 2 
with which we are dealing. We begin with 

After we add the increment to x 2 , we have 

When we express At/ in terms of x, we get 

At/ (x-f- Ax) 2 2 = x 2 H-2x-Ax + (Ax) 2 x 2 

If we now express the ratio between Ay and Ax in the 
terms of x and then cancel out identical terms in numer- 
ator and denominator, we arrive at 

Recalling that when we first added Ax to x in /(x), we 
defined it as "an arbitrarily small increase," we realize that 
as we choose smaller and smaller amounts for Ax, i.e., Ax 
approaches 0, the limiting value of the ratio At// Ax will 
be 2x. This is the rate of change of y with respect to x 
when /(x) = x 2 . 

We can see that 2x actually is the rate of change, or, 


to express it in a different way, the slope of the curve at 
a given point. We plot the parabola and then at any point 
draw a line the slope of which is equal to twice the value 
of the x coordinate of the point. For instance, at x = 1 the 
slope should be 2; so we line up our straightedge with a 
point 1 unit over and 2 units up from our given point on 
the curve. The slope of the line we draw will then be 2, 

and we can see that this line does represent the slope (or 
rate of change ) of the curve at this point. 

How big? How steep? How -fast? We have said that 
there is a fundamental point of contact among these three 
questions. We have shown that the answers to the last 
two are essentially the same. How steep? = How fast? 
Now we shall show the relation of the first to these two. 
That these three questions are so related has been called 
"one of the most astonishing things a mathematician ever 

We begin by taking the area under a curve which we 


can identify in a general way as f(x). We have seen that 
a curve is a function of x since each x coordinate deter- 
mines a y coordinate and hence the curve itself. The area 
under a curve is also a function of x but in a somewhat 
different sense. It is clear from the diagram below that if 
we take a as the x coordinate of the left-hand boundary 
of the area we wish to compute, and b as the x coordinate 
of the right-hand boundary, moving b to the right on the 
x-axis will increase the area. In this sense the area under 
a curve is a function of (i.e., is determined by) the value 
of the x coordinate at its right-hand boundary. 

' ' 

Since, although the area is also a function of x, it is not 
the same function as that which determines the curve 
above it, we represent the curve by f(x) and the area by 
F(x). This can be easily seen in the curves below. On 
the left we have a triangle under the curve f(x) = x, the 
value of each y coordinate being the same as that of the x 
coordinate of any point on the curve. If we compute 
the area of this triangle at each x coordinate as one-half of 
x 2 (or half the base times the height), we find that the 
curve representing the area as a function of x, or F(ac), is 

x 2 

an entirely different curve, F(x) = : 




-ft. 'VW />.'/ 


' i ... A=^ < <r,i ' >j / <>/ .// > \m 

* ' >j(fmK 

'.-I ' , f* 


4 f^f *l 


Now let us return to our main problem. 

To determine the area under the curve between a and 
b we proceed in the by now somewhat familiar method of 
the calculus. We go a little farther to the right on the oc-axis 
and add to x (represented on the diagram by b) an arbi- 
trarily small distance which we call Ax. This results in an 
appropriately small increase in the area under the curve, 
which we call AA. 

Instead of A = F(x) we now have 

and by subtracting the original area from the enlarged 
area we can determine the value of AA. 


If we look at our diagram we can see by inspection that 
AA, as well as having the value given above in terms of x, 
has also the approximate value of A times /(#), which 
would be the area of the largest rectangle we could in- 
scribe in. A A. The ratio A A/ As is then approximately 

lim AA_ lim Aa>/(x) _ . 
A*-*0 ~Kx~~~ Ax-*0 Ax ~~ * (X ' 

From the above we see that the area under the curve 
f(x) is determined by a function F(x) which has the 
property that its rate of change, or derivative as it is tech- 
nically called, is /(#)! 

Since F(x) answers the question How big? and f(x) 
answers How steep? and How fast?, we find all three 
inextricably bound together. This is the fundamental re- 
lationship of the calculus "one of the most astonishing 
things a mathematician ever discovered"! 

With a brief explanation of two notations which we 
have not already met, we are now ready to state and fol- 
low the Fundamental Theorem of the Calculus. For the 
derivative of F(x), we shall use the notation F'(ac); and 
for the area under f(x) between x = a and x = b, the 
notation below. 

J f(x)dx 

The Fundamental Theorem, discovered independently 
by Newton and Leibniz, states: 

If f(x) is continuous and F'(x) = f(x), then 



Let us apply this formula to the area under the line 
y = x between and 1, which we know is %, and the area 
under the curve y = x 2 between and 1, which we do not 
know. In the first case we must have a function of x, the 
derivative (or rate of change) of which is x. Since we 
earlier determined the rate of change of x 2 as 2x (on page 
90), we can surmise that the derivative of &t 2 is x. In 
the second case, the reader may be interested in working 
out (as on the same page) that the derivative of Ysx 3 is 

In the case of the triangle we know that the area is indeed 
%, which the Fundamental Theorem gives us as the limit. 
In the case of the area under the parabola, we did not 
know but now we know that the area under the curve, 
defined as the limit, is %. 


Thus the Fundamental Theorem of the Calculus 
brings together the answers to the three questions we 
asked about curves and the areas which He under them. 
How steep? has the same answer as How fast?, and the 
answer to How big? is the inverse of the other two. 

It was because they perceived this underlying unity 
that Newton and Leibniz, who were by no means the first 
to use the methods of the calculus, are given the full credit 
for its invention. 


The test below will enable the reader to make sure 
that he has a clear, if simple, notion of a function. 

1. Iff(x)=x,solvey = /(5)fory. 

2. Iff(x)=x 2 ,solvey=/(S)fory. 

3. If f (x) = x 2 , what are the y coordinates for 
a =1,2, 3? 

4. If f ( x ) = x f what are the y coordinates for 

5. Hf(x)= 

6. E/(x)= 

7. If f ( x ) = 1 /x, what is the value for y when x = 7? 

8. If f(x ) = 1 x, what is the value for y when x = 1? 

9. If /(*)= x s , solve y=f(2) fort/. 

10. If f (x) = x + 3, what is the value of y for x = 7? 

Ql = fi -oi $ = / '6 ?0 = / * '8 ' L A = fi 'I -9 = 
91 = fi '9 -9 '9 'f = fi 'f f 6 > 'I = ^ 'e -SS = fi 'Z f S = 



How Many 
Are Enough? 


1, 2, 3, ... are enough numbers to 
count the objects before us; yet when 
we encounter the ordinary operations 
of arithmetic for the first time, we find 
that they are not nearly enough. We 
can subtract only when the number 
being subtracted is smaller than the 
number it is subtracted from; we can 
divide only when the number being 
divided is a multiple of the number 
being divided into it; we can extract a 
square root only when the number 
from which we are extracting it is a 
perfect square. For these simple opera- 
tions to be always possible, we must 
have more numbers than 1, 2, 3, .... 

We have seen in an earlier chapter 
how the necessity for a number for 
every point on the line resulted in the 
development of the concept of the 
arithmetic continuum. Now we shall 
see how the necessity for an "answer" 
to every problem in arithmetic resulted 
in a parallel development that went 
one step beyond the concept of a 
unique number for every point on the 
line to a unique number for every 
point on the plane! 

The necessary extension was a mat- 
ter of centuries. Although we shall 
follow it in a more or less logical order, 
it was neither orderly nor logical. 

Numbers began as a way of count- 


ing. It seemed natural that a number should correspond 
to each thing counted, so in later times when all sorts of 
curious quantities were being used as if they were num- 
bers, these original counting numbers came to be thought 
of as the natural numbers. "God made the integers," thun- 
dered a mathematician of the nineteenth century.* "All 
else is the work of man." 

Numbers other than the original natural numbers 
turned up in the process o solving problems: first prob- 
ably in making accurate measurements, later in finding 
the roots to equations. Even mathematicians had curious 
attitudes toward them. The Greek mathematicians used 
rational quantities, or fractions, but refused to call them 
numbers; and the beautiful theory of numbers which they 
created deals, to this day, only with whole numbers. The 
Indian mathematicians did not consider the negative 
solutions to equations as solutions "because people do not 
approve of negative roots." The mathematicians of the 
Renaissance, who solved otherwise unsolvable equations 
by acting as if 1 had a square root, uneasily dismissed 
V 1 (after they had used it) as "imaginary." 

But in spite of the fact that the mathematicians did 
not really believe that anything other than 1, 2, 3, ... was 
a number, they ended up by justifying their use of these 
quantities as numbers by the fact that they used them in 
the same way they used 1, 2, 3, . . . , adding, subtracting, 
multiplying and dividing them according to what they 
considered the natural laws of arithmetic. 

Although very few of us could state these Laws of 
Arithmetic on a quiz program, we obey them almost un- 
thinkingly. The Associative Laws, the Commutative Laws 
and the Distributive Law, as they are called, are no more 
than the formal statements of how the natural numbers 

* Leopold Kronecker (1823-1891). 

behave under the operations of addition and multiplica- 
tion and, by implication, subtraction and division. 

The Associative Law of Addition, for instance, tell us 
that when adding 1 and 2 and 3, we can perform the 
operation in several different ways and still get the same 
answer; and the Associative Law of Multiplication tells 
us the same thing in regard to multiplying: 

1+2+3=6 1X2X3 = 6 

(1 + 2) +3, or 3 + 3 = 6 ( 1 X 2) X 3, or2 X 3 = 6 

1+ (2 + 3), or 1 + 5 = 6 IX (2 X3),orl X 6 = 6 

It is important to note that the Associative Laws do not 
tell us that we can change the order of 1, 2 and 3 when 
we add or multiply them and still get the same sum or 
product. That is reserved for the Commutative Laws. 

We are all familiar with the fact that if we take two 
of something like an apple and then three, we shall have 
as many apples as the person who first took three and then 
two. If we take two apples three different times, we shall 
have as many apples as the person who reached for the 
bowl only twice but took his apples three at a time. These 
simple facts of social life are formalized in the Laws of 
Arithmetic to the effect that addition and multiplication 
of the natural numbers are commutative operations: 

2 + 3 = 3 + 2 and 2X3 = 3X2 

The Distributive Law merely brings addition and 
multiplication together with the statement that 2 X ( 1 + 
3) is the same as (2 X 1) + (2 X 3). 

In the past, mathematicians firmly believed that these 
laws were as "natural" and God-given as the numbers to 
which they applied; yet all around them were "multiplica- 
tions" and "additions" not associative or commutative. 


We have actually seen that addition and multiplica- 
tion, when applied to apples, are commutative; but do we 
know that they are always commutative in respect to 
things other than apples? We, like the mathematicians of 
the past, probably think that we do; but let us look for a 
moment at baseball hits instead of apples. If our team gets 
a three-bagger and a home run, the total number of bases 
hit will be the same whether we add 


4B + 3B 

but there will be a considerable difference in the score 
depending on which hit was made first: 


If we buy an insurance policy after we have had an 
automobile accident, the result of the combination of 
accident and policy is quite different from what it would 
have been if the combination had been made in the re- 
verse order: 

Policy -f Accident = $1000 

Accident + Policy = $0 

There are many other examples in everyday Me where 
the order of combination changes the result of an opera- 
tion. We offer these only to show that while it may be 
impossible for us to think of 2 X 3 as not being equal to 
3X2, we can think of ab, under certain conditions, as 
not being equal to ba. 

Subtraction and division are neither associative nor 


commutative. But do multiplication and addition have to 
be associative and commutative? Does addition have to 
be distributive with respect to multiplication whenever 
we are dealing with quantities we choose to call "num- 
bers"? Up until a little more than a hundred years ago, it 
was thought by all mathematicians that they did; that, in 
fact, they must. The Laws of Arithmetic were considered 
a logical necessity of number. 

We have heard a great mathematician say that God 
made the integers and all else was the work of man. This 
was the attitude of mathematicians from the time of 
Pythagoras. To facilitate measurements and the solutions 
of equations, mathematicians might have to extend the 
concept of number to include quantities other than the 
integers, but they could at least see that, like the integers 
which God made, these followed the God-given Laws of 
Arithmetic. In all the extensions of the number concept 
which we shall describe in this chapter, this principle was 
followed. It was called the Principle of Permanence of 
Form; and it meant that the fundamental Laws of Arith- 
metic, which we have already examined, remained in 
force with the new numbers as well as with the old. This 
made everybody feel much better about using the strange 
new "numbers." 

To understand the extensions which were made, we 
shall begin with a picture of the natural numbers marked 
off, unit by unit, upon a straight line extending indefinitely 
to our right: 

Immediately we note a curious thing about this pic- 


ture. While 1 marks the distance 1 unit from the begin- 
ning of the line; 2, the distance 2 units from the beginning; 
3, 3 units, and so on there is no number among the 
original natural numbers which can mark the beginning 
of the line. Yet if we take away, or subtract, 4 units from 
the point marked 4, this beginning point is exactly where 
we obtain our answer. What is the answer to the question 
How many is 4 4? The answer is none at all or, nu- 
merically speaking, 0. So let us call a number, since it 
answers the question How many? just as the other count- 
ing numbers do, and then let us mark the beginning of the 
number line with 0.* 

Zero makes possible the subtraction of a number from 

But even with 0, subtraction is not always possible. 
We still cannot subtract a larger number from a smaller 
and get as our answer a number on the line above. When 
we take 6 from 5, we find that we are 1 unit short. In 
other words, we could perform the operation if we had 
one more unit to the left of 0. So, arbitrarily, we add it 
and an infinite number of such units. We extend the 

* This is not at all the way that was invented. It was in- 
vented, not as a number, but as a symbol to mark those columns 
in the representation of a number which contained no digits. The 
use of made possible the representation of all numbers with only 
ten different symbols and was probably one of the most important 
practical inventions in the history of the world. The idea of as a 
number (rather than merely a symbol) is not very important to 
anybody but a mathematician, to whom it is quite important. In 
the modern theory of numbers, is usually treated as one of the 
natural numbers. 


number line to the left of 0, and we mark it off in units just 
as we did the line to the right. Since these units are less 
than 0, we place a minus sign in front of them and call 
them negative. To be consistent, we must then place plus 
signs in front of what were once all the numbers, and call 
them positive. Zero has neither plus nor minus in front 
of it is neither positive nor negative. The extended num- 
ber line now looks like this: 

The negative numbers make subtraction always pos- 

Now we come to division and face to face with the 
unpleasant fact that most divisions do not come out even. 
If we are to perform the operation of division whenever it 
is indicated and get an answer among the numbers on our 
line, we must have parts of numbers, divide our units into 
sub-units, and allow these to be "answers" too. Unless we 
do so, we can divide a number only into a multiple of 

Although we shall indicate on our extended line just 
those sub-units obtained by dividing the unit in half and 
then in half again, we must understand that to make 
division always possible we have to include among our 
new numbers every quantity which can be represented by 
the ratio of two whole numbers. We give these new quan- 
tities the Greek name of the rational numbers. As a class, 
the rationals include the whole numbers, for these can 
always be expressed as the ratio of themselves over 1. 
With the extension of the number concept to include 
fractional parts of the unit, our line begins to look like 


this on the portion between 1 and +1: 

The rational numbers make division, except division 
by 0, always possible. 

Things are getting a little crowded even with only the 
few numbers we are indicating on the number line. It is 
now, in fact, what mathematicians characterize as dense, 
which means that between any two numbers there is al- 
ways another number. As we have seen, the Greeks at the 
time of Pythagoras, with far fewer numbers (for they had 
not extended their concept of number to include either 
or the negative integers), thought that they had quite 
enough for all practical purposes, including the measure- 
ment of the Universe. We have also seen how the most 
shattering discovery in the history of mathematics was the 
discovery that this beautiful array of whole numbers and 
their ratios was not enough to furnish an exact measure- 
ment of the diagonal of the unit square. The square root 
of 2 was a non-rational number. 

How many such non-rational, or irrational, numbers 
are there? Merely an infinite number and this in spite of 
the fact that, as we have seen, the rational numbers are 
dense upon the line. By multiplying by itself a rational 
number which is not a whole number, we can never get a 
whole number as our result. All numbers, therefore, which 
are not perfect squares, or generally perfect powers, of 
some other whole number must have as their roots irra- 
tional numbers. 


The irrational numbers make the extraction of roots of 
positive numbers always possible. 

Up to this point we have been writing of these suc- 
cessive extensions o the concept of number as if they 
were things that we would be unable to live without. Yet 
the majority of people in even the most civilized countries 
do not consider a number, but rather a symbol which is 
indispensable for the representation of numbers in the 
decimal system. It is most unusual to see the digits ar- 
ranged in their natural order 0, 1, 2, 3, . . . , usually 
being placed instead after 9. Only of late, with the "count- 
down," has been publicly recognized as a number, and 
then it is counted back to, rather than up from. Although 
we are all familiar with debts, losses, arrears and such 
unpleasant figures, we never put a minus sign in front of 
them in our accounts, but write them in red. We treat 
both profit and loss as positive quantities and subtract the 
smaller from the larger to find out whether we are ahead 
or behind, and how much. We would find it difficult to 
live without the rational numbers, since sub-units of the 
unit are necessary for even fairly approximate measure- 
ments; but considering the infinities of rational numbers- 
infinity upon infinity which are at our disposal, we use 
practically none of them. The ordinary foot ruler distin- 
guishes only to YIQ of an inch. Since we can place any 
irrational root to as many decimal points as we wish, and 
have the time and energy to compute, it is obviously of 
no great concern that we cannot pkce it exactly. 

The truth of the matter is that the successive exten- 
sions of the number system took place, not to make the 
ordinary operations of arithmetic always possible in 
everyday life, but to make them always possible in algebra. 
If we are to be generally effective in the solution of alge- 


braic equations, we must know before we start that there 
exists a number which will satisfy each respective un- 
known in an equation. This does not mean, even when we 
come now to the final extension to the so-called imaginary 
numbers, that the successive extensions of the number 
system have no practical value. Algebra is one of the most 
practical subjects in the world. Just ask any scientist! 

But let us imagine for a moment that we are limited in 
our algebra to solutions for x which are among the orig- 
inal natural numbers. Then let us try to solve the follow- 
ing simple equations by finding in each case a value for x: 

We can see by inspection that to solve these equa- 
tions, we must in each case extend the concept of number 
from the natural numbers to zero, to the negative num- 
bers, to the rational numbers, to the irrational numbers.* 
Not one of these equations would have a solution if we 
were limited in our algebra to the original natural num- 
bers! If this restriction had been placed on our solutions, 
we would have seen that we had to stop before we started. 
But in more complicated equations we cannot see so easily 
that there is no solution for x. If we are to proceed in our 
manipulations of the symbols with any assurance that 
these manipulations are not a waste of time, we must 
know before we start that in every case there exists a 
number for x. 

* In actuality we have added many numbers which we do not 
need for the solution of algebraic equations numbers, called 
"transcendental," which cannot be roots of algebraic equations. We 
shall hear more of these numbers later. 


Having extended the number system four times al- 
ready, we can now find roots for any of the equations 
above and for any similar but much more complicated 
equations. Yet, we are not through. There are still com- 
paratively simple equations for which we can find no 
roots at all among the numbers we already have. Such an 
equation is 

x 2 + I = 

It is obvious that if we are to add 1 to x 2 and obtain 
0, x 2 must have the value of 1. It is equally clear that 
x then must have the value of V 1. BUT, under the rules 
by which the negative numbers were allowed to be 
brought into our number system, it was implicitly stated 
that a negative number could not have a square root! 
Recall the rules for multiplying positive and negative 
numbers, which were necessary for maintaining the Prin- 
ciple of Permanence of Form. A positive number multi- 
plied by a positive number yields a positive number, as 
does a negative number multiplied by a negative number: 

Only when we multiply together a positive and a negative 
number do we get a negative product: 

(-2) X (+2)=-4 

We must remember that -j-2 and 2 are two different 
numbers, located at two entirely different points on our 
number line. But by definition a square is the product of 
a number multiplied by itself. Under this definition a 
negative number simply cannot be a square. Yet there is 
our equation 


If we cannot find a solution for this equation, we shall 
be severely handicapped in our algebra. We shall have 
failed in our avowed purpose of extending the number 
system so as to make the operations of arithmetic always 
possible. We shall have to concede that any equation for 
which x 2 1 has no root. 

Let us not give up too easily. In the first half of the 
sixteenth century Girolamo Cardano ( 1501-1576 ) , saying 
frankly that roots of negative numbers were "impossible" 
there could be no such roots! nevertheless began to use 
in solving otherwise unsolvable equations a symbol which 
he called the square root of 1. Since he did not consider 
this symbol a real number ( for he knew as well as anyone 
that there could not be a number which when multiplied 
by itself would produce 1) Cardano called his symbol 
an imaginary number. The strange thing was that by 
using such imaginary numbers when necessary, Cardano 
found that he could obtain very real, practical results 
with equations which otherwise he would not have been 
able to solve! 

But let us return for a moment to our own extension 
of the concept of number. How can we, refusing to have 
anything to do with Cardano's highhanded invention of a 
"number" for the square root of 1, go about finding such 
an impossible root for an equation like x 2 + 1 = in a 
logical and orderly extension of our concept of number? 
There is no root among the integers, the rationals or the 
irrationals. At this point we cannot change the rules under 
which we brought these quantities in as numbers. We 
cannot, for instance, say that a negative number multi- 
plied by a negative number yields a negative number, for 
that would involve us in impossible contradictions. It was 


to avoid the contradictions that we made the rules in the 
first place. There is only one thing we can do. Just like 
Cardano, we can make up another number. We can 
simply define it as V 1 and call it i (for Cardano's 
imaginary number). 

We have no everyday justification for what we are 
doing. We can compare the negative numbers to things 
like debts and temperatures below zero and the years be- 
fore the birth of Christ, but the number i we can compare 
to nothing in everyday life. It was for this reason that 
mathematicians, although they went right along using i 
to solve equations, felt a little guilty about what they were 
doing. God, they felt, had made the whole numbers. If 
He had wanted man to have them, He would have made 
negative numbers and given them square roots! 

Yet the extension of the concept of number to the 
imaginary numbers parallels in a logical and orderly way 
the extension to the negative numbers. The negative num- 
bers were invented to make subtraction always possible; 
the imaginary numbers were invented to make extraction 
of roots always possible. There was only one condition 
upon the admission of negative numbers to the number 
system: they must be used in accordance with the Laws 
of Arithmetic, the Principle of Permanence of Form must 
be maintained at all costs. This same condition was im- 
posed upon the imaginary numbers. They were just as 
much numbers, and every bit as "real," as the negative 
numbers. Unfortunately, in the beginning they were 
called "imaginary" by Cardano and the name has stayed 
with them and undoubtedly always will. 

Today the words "real" and "imaginary" are used to 
distinguish the two axes of a number plane which is as 
real as the plane of analytic geometry, and identical with 
it. Obviously i and its multiples 2t, 3i, . . . , cannot go on 


our number line, since all the points on the line are already 
accounted for by numbers. They can, however, have a 
line of their own the pure imaginary line which, like the 
y-axis of the Cartesian plane, is perpendicular to the real 
number line, or cc-ax s, at 0: 

~ ~ ~ TJT^ 


' I 

'W ''< 

With this geometric interpretation, we find that our 
seemingly "imaginary" numbers begin to assume an every- 
day reality. Like the reals they have a line of their own. 
Combined with the reals, they serve to locate uniquely 
each point on the plane. These new combinations which 
do the same job as Descartes' pairs of real numbers (x,y) 
represent, however, an important advance in our concept 
of number. While Descartes' real-number coordinates are 
"pairs" of numbers, these combinations of real and imag- 
inary coordinates are individual numbers. 

These new numbers of the form (x + yi) are called 
complex numbers because they have more than one part. 
They are represented abstractly as (x + yi) where x and 
y are real numbers and i is defined as V 1. When x has 
the value of 0, the "complex" number becomes a pure 


imaginary (0 -+- yi yi), while when y has a value of 
it becomes a real number (x +0i = x). The pure imag- 
inaries and the reals are, therefore, merely sub-classes of 
the complex numbers: 


Real Numbers ( y = ) Imaginary Numbers ( x ) 

Rationals Irrationals 

Integers Fractions 

The imaginaries make the extraction of the roots of 
negative numbers always possible. 

We have come a very long way in our extension of the 
concept of number. We began with the natural numbers, 
which could be paired in one-to-one correspondence with 
objects which were to be counted. By retaining the rules 
which we had made up for the behavior of these numbers, 
we were able to extend without logical difficulty our 
concept of number to the so-called "real" numbers, which 
could be paired in one-to-one correspondence with every 
point on a line. Still retaining the same rules, we further 
extended our concept of number to the complex numbers, 
half "real" and half "imaginary," which could be paired 
in one-to-one correspondence with every point on a plane. 
We have enlarged our number system, step by step, so 
that now for every operation of arithmetic we can obtain 
an answer within our number system. Just as with the 
original natural numbers we could always add or multiply, 
now with the complex numbers we can always add, sub- 
tract, multiply, divide and extract roots. 


But we are still troubled. 

We saw that the extensions to and the negative 
numbers made subtraction always possible; the rationals 
made division always possible; the irrationals made the 
extraction of roots of positive numbers always possible; 
the pure imaginaries and the number i that generates them 
made the extraction of roots of negative numbers always 
possible.* We now have a number for every point on the 
real axis and every point on the pure imaginary axis and, 
also, a number for every point on the plane. Surely these 
should be enough numbers to make the operations of 
arithmetic always possible and to provide every algebraic 
equation with a root! But what about an equation like 
this one? 

Won't we need to extend our number system once 
again, beyond i to V ? 

The answer to this question is a very simple one, which 
mathematics can offer with all the finality of mathematical 
proof. The answer is no. We have gone as far as we need 
to go. It can be shown and this is known as the Funda- 
mental Theorem of Algebra that any algebraic equation 
has a root within the system of complex numbers. 

To mathematicians i, the square root of 1, is the 
wonderful square root. In the satisfying language of 
mathematics it is both necessary and sufficient. 

That pesky equation? Don't we need a square root of 
i to get a root for that x? Oh no, 

Multiply it out, and see for yourself! 

* V 2 = iV2, and so on. 


One of the quickest ways to get rid of the idea that 
complex numbers are mere figments of our imagination is 
to pin them down geometrically. This is exactly what was 
done in the early nineteenth century when it was shown 
by Gauss and others that the domain of the complex 
numbers is mathematically equivalent to Cartesian plane 

Geometrically, a complex number (x + iy) is consid- 
ered as a composition of the two vectors of its real and 
imaginary (or x and y) coordinates. In non-mathematical 
language we can think of it as the diagonal formed when 
we complete a rectangle from these coordinates: 

This same idea is extended to the geometrical defini- 
tion of addition of two complex numbers (which are, of 
course, in themselves additions of real and imaginary 
parts). To add two complex numbers (x + iy) and 
(u + iv), we simply add real and imaginary parts sep- 


arately and obtain as our answer the complex number 

(* + *) 

Geometrically, we "complete the parallelogram" begun 
by the vectors of the two numbers, as below: 

The reader will find it interesting to add the complex 
numbers below by botb methods: 

(3 + 2i) + (4 + 50 
(2 + 50 + (6 + 20 



Where Thought 
Is Double 


of art. 

From the annexation of these two 
territories mathematics gained what 
has been called "an enchanted realm, 
where thought is double and flows in 
parallel streams." But such is the un- 
derlying unity of all mathematics that 
just as we found the lines and circles 
of Euclid and the conic sections of 
Apollonius in the graphs of algebraic 
equations on Descartes' numbered and 
coordinated plane, so we meet them 
again in this new domain. 

The "enchanted realm" is projec- 
tive geometry, the mathematics which 
was born in the struggles of the early 
Renaissance painters to transfer three 
dimensions to two without losing the 
appearance of reality. 

The approach of projective geom- 
etry to the familiar subject matter of 
geometry is synthetic we proceed by 
synthesis, or putting together, from 
the figures to the principles. The ap- 
proach of analytic geometry is, as its 
name tells us, the direct opposite. We 
proceed by analysis, or taking apart, 
from principles to figures. Yet, curi- 
ously, the small volumes which for- 
mally introduced these two new geom- 
etries to mathematics were published 
within a few years of one another! 
Although neither book was to have 


much influence on the immediate mathematics of its 
time, the very look of geometry had been irrevocably 
changed with their publication. 

In the geometry of Euclid we drew circular circles 
and square squares, equilateral triangles with three equal 
sides, and parallel lines which never met. But these are 
not what we see. To the human eye, circles are not gen- 
erally circular, squares are not square, equilateral triangles 
do not have equal sides or equal angles, and parallel lines 
approach one another. The only time we come even close 
to seeing these shapes in their pure Euclidean form is 
when we look at them head-on so that our eyes are more 
or less in line with the center of the figure. If we had just 
one eye, which could then be directly in line with the 
axis, we would be able to see them even more "accurately." 

Given the problem of drawing a three-dimensional 
cube (or a box or a house) on a two-dimensional piece 
of canvas, we are immediately confronted by a paradox. 
We know that each of the six sides of the cube is a 
square a quadrilateral with equal sides and equal angles 
and that all six squares are the same size. If we look at 
the cube head-on, we see just one square; but if we draw 
the cube as a square, it certainly will not look to the eye 
like a cube. Our eyes tell us that the square is a cube 
because, having two eyes, we have brought together two 
slightly different views of the face of the cube and thus 
obtained at least a sense of depth. The square on our 
canvas is a Cyclops view. It has no depth unless the 
square face of the cube is seen among other three-dimen- 
sional objects which are not drawn head-on. To make the 
cube by itself appear solid to the eye, we must draw it 
from an angle which shows more than one face; and when 
we try to do just this, we find that not one of the faces 
which we draw is still a square! 


We see now as we look at the cube from different 
angles that the faces change shape as we change our 
point of view. We may see one, two or three faces of the 
cube at once; but all will be different. We know, how- 
ever, that they, as well as the faces which we cannot see, 
are all the same. After all, what is a cube but a solid with 
six square faces! Even though they may not look as if they 
are, all lines and all angles must be equal. 

There is one thing about the faces of the cube which 
does not change. We may look at the cube from above, 
from below, from the left, from the right, the fact remains 
that every face we see is always a closed figure bounded 
by four straight lines a quadrilateral of Euclid's geometry. 

What rules determine these new quadrilaterals which 
are no longer squares? It was this question and similar 
ones about other geometrical shapes which led the paint- 
ers of the early Renaissance to investigate and formulate 
the principles of perspective. 

The word perspective in its original form means "to 
see through." It is an almost literal statement of what the 

painter conceived himself to be doing. From every visible 
point of the object which he saw before him, a ray of 
light entered his eye. If a pane of glass were to be placed 
between the objects and his eye, each of these rays would 
pierce it at a definite point. The painter then could con- 
ceive of his painting as an imaginary glass through which 
he saw the scene. By drawing on his canvas the outline of 
objects exactly as they would appear on the imaginary 
glass transposed between his eye and the objects them- 
selves, he could paint "what he saw." Some painters of 
the day actually used such mechanical aids. 

Since these men of the early Renaissance saw all ob- 
jects as essentially the shapes of classical geometry (for 
they believed with Plato that God eternally geometrizes ) , 
they recognized that the relationship between the shape 
of an object as it was and the shape as it appeared to the 
eye from varying angles of vision must be expressible in 
terms of mathematics. They worked out various inde- 
pendent and disconnected theorems of perspective and 
gave to the geometry that grew out of their work its two 
basic terms: projection and section, the latter referring 
to the point of view, or eye of the painter, from which an 
object or a group of objects is viewed; and the former, to 
an imaginary plane which intercepts or cuts that view, and 
is the picture itself. 

Mathematically, we express these same relationships 
in the following manner: 

From a point O, lines are drawn to every point of a 
geometric figure F; these lines issuing from O are cut 
by a plane w. . . . The set of lines joining a point O to 
the points of a figure F is called the projection of F 
from O, If a set of lines issuing from a point O is cut 
by a plane w, the set of points in which the plane w 


cuts the lines through O is called the section of the 
lines through O by the plane w. 

Let us observe now what this definition means in rela- 
tion to the projection and section of the circle. 

We begin by drawing a circle and selecting a point O 
in space above the circle and directly above the center. 
From every point on the circumference of the circle we 
conceive a line joining that point and O and continuing 
on past O. We now have in our mind a set of lines form- 
ing an infinite double cone. This is the projection from O 
of our original circle. If we conceive of a plane surface 
cutting this projection in various ways, we shall have a 
series of sections of the projection, A section which puts 
one portion of the cone will give us a picture of our circle 
as an ellipse. A section which is made parallel to any Line 
of the cone will give us a parabola; and a section which 
cuts both portions of the cone, a hyperbola (page 120), 

Here is a beautiful example of the elegant generality 
of this new geometrical way of thinking. Menaechmus 
and Apollonius studied the conies under the almost un- 
bearable weight of a cumbersome terminology. Descartes 
dealt with them as varying forms of the general equation 
of the second degree in two unknowns. Projective geom- 
etry now enables us to define the conic sections with even 
more stunning simplicity: 

The conic sections are simply the projections of a 

circle on a plane. 

The artist's yet unorganized mathematics of perspec- 
tive and the mathematician's yet undiscovered art of 
projective geometry met in a man named Gerard De- 
sargues (1593-1662), a self-educated architect and engi- 
neer. Desargues' interest in the subject was purely 
practical. "I freely confess," he wrote, "that I never had 


taste for study or research either in physics or geometry 
except in so far as they could serve as a means of arriving 
at some sort of knowledge of the proximate causes ... for 
the good and convenience of life, in maintaining health, in 
the practice of some art. . . ." He began by organizing 
numerous useful theorems and disseminating these 

through lectures and handbills. Later he wrote a pamphlet 
on perspective which attracted very little attention. His 
chief contribution, the foundation of projective geometry, 
appeared in 1639. With a few important exceptions, it 
was entirely ignored by his contemporaries; and every 
printed copy was lost. Only by the chance discovery of a 
manuscript copy, two hundred years later, was Desargues* 
original contribution to mathematics known at all. 

The difference between Desargues' geometry and 
Euclid's can be seen most vividly in the figures as they 
are presented by each man. Euclid is concerned with 
showing that two figures are congruent. This means that 
if we could slide a given figure by what a geometer calls 
"rigid motion" (which causes no change in the figure 
during the moving) along the plane to a second figure, the 
two would coincide. In the language that the new pro- 
jective geometry was to bring to mathematics, we say 
that Euclid was concerned with those characteristics of 
the figure which are invariant (do not change) under the 
transformation of rigid motion: 

Euclid was also concerned with those characteristics of 
the figure which are invariant under the transformations 
of uniform expansion or condensation: 

Euclid, of course, did not himself think of his concerns in 
these terms. Given two triangles, he took as his object to 
show that they were congruent. ( They were, for instance, 
if all three sides of the second were equal to the corre- 
sponding sides of the first.) If they were not congruent, 
his object was to show that they were similar. ( They were 
similar if all three corresponding angles were equal. ) 

The one theorem in projective geometry which bears 
the name of Desargues shows immediately by its word- 
ing and the accompanying figure that Desargues was con- 
cerned with relationships between triangles quite different 
from those that had Euclid's attention: 

THEOREM: If in a plane two triangles ABC and A'B'C' are 
situated so that the straight lines joining corresponding 
vertices meet in a point O (in the language of art, are in 
perspective from O), then the corresponding sides, if ex- 
tended, will intersect in three collinear points QRP. 

The triangles in Desargues' theorem are neither con- 
gruent (sides not equal) nor similar (angles not equal); 
yet there exists between them a relationship, as stated by 
the theorem above, which does not change remains in- 
variantunder the transforming powers of projection. 
The reader can test this statement experimentally by 


drawing other figures to illustrate the theorem. With one 
exception, which we shall take up later in this chapter, he 
will find that the theorem always holds. 

Desargues' excitingly new theorem and his really 
revolutionary little book were taken seriously by few peo- 
ple. A mapmaker named Philippe de la Hire, who had 
been one of his pupils for a time, utilized the new ideas 
of projection in his work and made a careful manuscript 
copy of Desargues' work, which saved it for posterity. 
Another fellow countryman, a youthful genius named 
Blaise Pascal (1623-1662), using the method of projec- 
tion, proved what has been called "one of the most beauti- 
ful theorems in the whole range of geometry." (We shall 
state this theorem later, on page 124. ) With these two ex- 
ceptions, projective geometry, which was invented by 
Desargues in 1639, might just as well have not been in- 
vented until the beginning of the nineteenth century, 
when it was invented all over again! 

The story of the second invention of projective geom- 
etry is one of the most dramatic in the history of mathe- 
matics. During Napoleon's retreat from Moscow, a young 
officer of engineers named Jean Victor Poncelet (1788- 
1867 ) was left for dead on the battlefield. He was picked 
up by enemy soldiers only because they thought that 
being an officer he might be able to give useful informa- 
tion. As a prisoner of war, he was forced to march for 
nearly five months across frozen plains to his prison on 
the banks of the Volga. At first he was too exhausted, cold 
and hungry even to think; but when spring came ("the 
splendid April sun"), he resolved to utilize his time by 
recalling all he could of his mathematical education. 
Later he was to apologize that "deprived of books and 
comforts of all sorts, distressed above all by the misfor- 
tunes of my country and my own lot, I was not able to 


bring these studies to a proper perfection." Nevertheless, 
a year and a half later, he returned to his native France, 
carrying with him the notebooks which were to serve as 
a passport for all mathematicians to "the enchanted 
realm." He was twenty-four years old at the time. 

In his classic treatise on projective geometry, pub- 
lished in 1822, Poncelet introduced a convention which 
has been used in all textbooks on the subject since his 
time. This was a simple typographical arrangement which 
brings immediately to the eye "the enchanted realm, 
where thought is double." 

Point and line in plane projective geometry are called 
dual elements. Drawing a line through a point and mark- 
ing a point on a line are dual operations. Two figures are 
said to be dual if they can be obtained each from the 
other by replacing every element and operation by its 
dual element and operation. Two theorems are called 
duals if one becomes the other when all elements and 
operations are replaced by their duals. 

Poncelet emphasized this distinguishing duality of 
thought in projective geometry by displaying all theorems 
in pairs. Thus the beautiful theorem of Blaise Pascal 
which we mentioned earlier in this chapter is displayed 
beside its dual, a theorem proved much later by C. J. 
Brianchon (1785-1864):* 

If the vertices of a hexagon If the sides of a hexagon 
lie alternately on two pass alternately through 
straight lines, the points two points, the lines join- 
where opposite sides meet ing opposite vertices are 
are collinear. concurrent. 

* The hexagons referred to in these theorems are figures formed 
when any six points are joined serially. The reader may enjoy 
joining six straws of varying lengths and discovering the varied 
hexagons he will obtain that way. N 


The duality o the two theorems stated above becomes 
even more vivid when we list parallel terms in parallel 
columns : 


vertices sides 

lie alternately on lines pass alternately through 


points lines 

where opposite sides meet joining opposite vertices 

collinear concurrent 

These theorems do not show an obvious resemblance; 
yet they are as firmly linked as Siamese twins. 

Pascal's theorem was proved in about 1639, before 
his sixteenth birthday; Brianchon's was discovered- 
through the principle of duality while he was a student 
at the ficole Polytechnique, and was printed in the 
school Journal in 1806 when Brianchon was twenty-one. 

According to the Principle of Duality, the dual of any 
true theorem of protective geometry is likewise a true 
theorem of protective geometry. 

Projective geometry is indeed an enchanted realm a 
sort of Big Rock Candy Mountain of mathematics 
where every theorem yields a twin and the proof of the 
first provides, with the proper exchange of dual elements 
and operations, the proof of its twin. 'Thought is double 
and flows in parallel streams." 

There is, however, a truly marvelous paradox in this 
world of parallel thought. For the beautiful and com- 
pletely general duality of projective geometry depends 
upon the fact that in projective geometry there are no 
parallel lines. 

In the elimination of parallel lines, projective geom- 
etry makes a complete break with its parent art, perspec- 
tive. When we are drawing a scene, we draw those lines 


which are parallel to the frame ( the pillars at either side, 
the edge of the floor or the table top ) parallel. This is in 
spite of the fact that we never actually see these lines as 
truly parallel, for all parallel lines appear to the eye to be 
approaching each other. In the small area framed by the 
picture this optical illusion, however, is not usually 

The simple sketch below shows two different treat- 
ments of the parallel lines in the scene. The pillars at the 
sides are parallel to each other and to the sides of the 
frame. The horizontal lines of the tile floor are parallel to 
each other and to the upper and lower sides of the frame. 
All other "parallel" lines in the picture appear to be ap- 
proaching a single "vanishing point" on the horizon. 

In Iris great work on projective geometry, however, 
Poncelet proposed a convention which would do away 
entirely with parallel lines in the mathematics that was 
the child of perspective. What he proposed was very 
simply to expel parallel lines from projective geometry 
by fat. Although his principle as stated sounds as meta- 
physical as anything in mathematics, his purpose was 
merely the practical, down-to-earth one of eliminating 
bothersome exceptions always having to be made in 
theorems and proofs for the special case of parallel lines. 


Let us recall as an example Desargues' theorem, which 
we stated earlier on page 122. We illustrated this theorem 
with a figure similar to the one below, and we observed 
that it is indeed true that the extended corresponding 
sides of the two triangles meet in pairs in three collinear 
points. But now let us consider a slightly modified figure 
where one side of one triangle is parallel to the correspond- 
ing side of the other triangle. 


' ' w -^' 

Although we can never hope to examine these lines in 
their entirety, it is immediately clear to us that the lines 
BC and B'C' will never meet. We cannot make this state- 
ment on the evidence of our eyes, for if we extend the 
lines far enough our eyes will tell us that they are indeed 
approaching each other and must, therefore, eventually 
meet. We make this statement because we know that 
parallel lines will never meet, "because," we say, "that is 
what parallel lines are lines that never meet." 

But what shall we do about the statement of De- 
sargues' theorem that the corresponding sides of the 
triangles, if extended, will meet in three collinear points? 
We shall have to add a qualifying clause to the theorem, 
"unless the corresponding sides are parallel." 

It would be one thing if Desargues* theorem were the 


only one to which we had to add such a clause to cover 
the exceptional case of parallel lines; but it is not. Almost 
every principle, theorem and proof in projective geometry 
must be modified to cover the exceptional case of the 
parallel. Such modifications are repugnant to mathema- 
ticians. Economy is one of the prime requirements for 
beautiful, general and effective mathematics. A theorem 
which applies to one specific triangle only is no theorem 
at all. A theorem that applies to almost all triangles is an 
improvement. But a theorem that applies to all possible 
triangles without exception now there is a theorem to 
delight a mathematician! 

Obviously, life in the enchanted realm of projective 
geometry would be much better for mathematicians and 
for mathematics if there were no parallel lines! This is 
exactly what Poncelet proposed to accomplish. 

It is an axiom of ordinary geometry that any two 
straight lines ( except two parallel lines ) intersect at one, 
and only one, point. If we now postulate that any two 
parallel lines have one ideal point in common, then we 
can state this important axiom with even greater gen- 

Any two lines meet in one, and only one, point. 

In the case of non-parallel lines the intersection is a real 
point; in the case of parallel lines it is an ideal point. But 
this distinction is trivial compared to the fact that the 
axiom now applies to all lines without exception. 

Unfortunately, as we have seen in the extension of 
the concept of number, the use of such words as "real" 
and "imaginary" and now "ideal" is often, even with 
mathematicians, a great hindrance to the grasping of a 
new idea. If, as is often done today, we simply postulated 
in our mathematics the existence of two kinds of lines 


and two kinds of points, we might escape this language 
trap. Any pair of lines, we might say, meets in one and 
only one point. Whether this point is of Class A or Class B 
depends upon the class to which the pair of lines belongs. 
Removed from the crippling language of everyday life, we 
might pursue our object with logic alone. 

The language of everyday life, however, is not com- 
pletely crippling even to mathematics. In fact, to switch 
metaphors, it often provides us with a very useful crutch 
in developing new ideas. Because parallel lines appear to 
meet or to be approaching a meeting place at the horizon, 
we say in our mathematics that parallel lines meet "at 
infinity." Since all parallel lines with a common direction 
are conceived as having an ideal point in common, we 
also conceive of ah 1 these ideal points of all possible sets 
of parallel lines as being on an ideal line, "the Line at in- 
finity." The mathematician uses this language very much 
as a poet uses a metaphor. Although he can make this 
principle premise in an analytic basis, he finds the lan- 
guage of everyday life both simple and suggestive in 
handling these new ideas. 

Most of us have forgotten that we are doing very much 
the same thing when we talk about the "real" points of 
ordinary geometry. A "real" mathematical point is the 
idealization of a real everyday point made with a pencil 
or a pen. This point, no matter how carefully we make it, 
has of necessity dimension. In fact, it has three dimen- 
sionslength, breadth and a certain theoretically measur- 
able depth when it is made with a pencil or a pen. Our 
so-called "real" point of geometry has no dimension at all. 
Yet we easily conceive of a real line in our geometry, the 
length of which is composed of an infinite number of 
these dimensionless points! 

In the familiar mathematics of everyday life we are 


working with ideas that are every bit as "far out" as the 
concept of parallel lines meeting in ideal points that lie 
on an ideal line at infinity. Although we are probably 
not aware of the fact, we have been forced to accept these 
ideas as logical necessities. Without the axiomatic state- 
ment that a line is composed of an infinite number of 
points and that there is a unique number for every point 
on a line, neither analytic geometry nor the calculus, to 
mention only two examples, would work as effectively as 
they do in practical problems as well as in higher 

It works. It is a logical extension of and logically con- 
sistent with our basic principles. And it works. This is 
justification enough for a mathematician to incorporate 
into projective geometry a postulate which eliminates the 
nagging exceptional case of the parallel. Just as we can 
postulate that any two non-parallel lines meet in one, and 
only one, point (in spite of the fact that the actual point 
in which they intersect covers an infinite number of 
mathematical points), in the same way we can postulate 
that any pair of parallel lines have in common one ideal 
point (in spite of the fact that they only appear to our 
eye to meet). 

We have already seen that when we accept this prin- 
ciple, the statement of Desargues' theorem no longer 
requires an exception for the special case where any pair 
of sides of the two triangles is parallel. But the simplest 
example of the way in which this principle allows mathe- 
maticians to unify and generalize projective geometry lies 
in the concept of projection itself. Originally it was neces- 
sary to distinguish between different types of projection, 
one in which the lines from the points of the figure meet 
in a single point and one in which the lines are parallel 
to one another: 


When we examine these projections in the figure ahove, 
they appear to be completely different. But if we conceive 
of O, the point of projection in the left-hand diagram, as 
moving away from the figure toward infinity, it is clear 
that as O approaches infinity the lines joining the points 
of the figure to O will become more and more "parallel." 

Utilizing our new concept of ideal points, we can say 
that both projections are from a point O. In the left-hand 
diagram above, O is a real point; and in the right-hand 
diagram, it is an ideal point. We can now discuss both 
projections without distinguishing between them as spe- 
cial cases, because they are both projections from a single 


Such elegant economy is a prized virtue in mathe- 
matics. It makes for practicality as well as for beauty. 

A realm where thought is double and flows in parallel 
streams where the statement of every theorem and its 
proof automatically yields another true theorem and 
another proof this is enchanting economy. A realm, how- 
ever, where exceptions must be made in every theorem 
and every proof is under an anti-mathematical spell. 
Mathematicians have been exorcising such spells for two 
thousand years, and will continue to do so! 



Mathematicians, amateur and pro- 
fessional alike, have struggled with 
this simple-sounding problem. Plato as 
well as Archimedes tried to trisect the 
angle. Another man, two thousand 

9 years later, wrote in his autobiography: 

"When I reached geometry, and be- 
came acquainted with the proposition 
the proof of which has been sought for 
centuries, I felt irresistibly impelled to 
try my powers at its discovery." Any 
The mathematically inclined person will 

Possibility recognize this response. We have all 

of Impossibility tried to trisect the angle. 

The trisection of the angle was one 
of the four great construction problems 
that the Greeks left to mathematics, 
the other three being the doubling of 
the cube, the squaring of the circle, 
and the construction of a polygon 
other than triangle and pentagon 
with a prime number of equal sides. 
From a practical point of view these 
constructions are not too difficult. With 
a protractor and a ruler we can draw 
what will appear to be a quite perfect 
regular heptagon. We can make a 
square having essentially the same 
area as a given circle and a cube 
having essentially twice the volume of 
a given cube. With protractor and 
ruler we can also divide any given 
angle into three "equal" parts parts 


which, for all practical purposes, will be quite equal. 

The protractor and ruler we use for these construc- 
tions have, however, one tiling in common which would 
make them repugnant to Greek eyes. They are both meas- 
uring devices. Tl\e protractor measures off the circular 
angle in degrees, minutes and seconds; and the ruler 
measures off lengths in units and parts of units. They are 
both very useful instruments, but there is a certain mean 
practicality about them. To the eye of man, constructions 
made with such instruments might appear accurate, but 
the gods would know different. The Greeks and the gods 
were not interested in the practical construction of 
squared circles, doubled cubes or trisected angles. They 
were interested in constructions which would in theory 
be absolutely exact even though in practice, because of 
limitations inherent in man and his instruments, they 
would be indistinguishable from the approximated con- 
structions by ruler and protractor. 

Although no mechanical device can possibly mark off 
on a line the exact point which is the irrational distance 
from the beginning represented by V2, we can in theory 
mark off the exact distance by constructing a right triangle 
of unit size and swinging across the number line an arc, 
the radius of which is the length of the hypotenuse, or 
V2. This arc actually marks \/2 no more exactly from a 
practical point of view than an ordinarily good ruler 
would; but in principle it is exact. If the number line 
could be represented by an infinite number of points and 
if the compass could trace the path of just one point at all 
positions the same distance ( V2) from the point on the 
line, this path would of necessity intersect with our right 
triangle at the point which is the vertex and the number 
line, or extended base of the right triangle, at the point 
which is the distance V2 from the origin. 


The reader will note that for this "theoretical" con- 
struction of V2, we have used no measuring device like 
protractor or ruler. We have assumed, with Euclid, that 
from a given point we can draw a straight to an- 
other point 1. (We do not have to measure this distance, 
since any distance we choose can serve as our unit.) We 
have also assumed, with Euclid again, that we can extend 
a given straight line and that we can draw a circle with 
given center and radius. The construction of the isosceles 
right triangle on a given base is Proposition 10 of Book I 
of the Elements. 

For our construction then we have used only an 
unmarked straightedge and a compass. These, being the 
mechanical manifestations of the straight line and the 
circle, were, as far as classical Greek mathematics was 
concerned, the only instruments which could be used in 
construction. The traditional problems thus were: 

To construct by straightedge and compass alone: 
A regular heptagon. 

A square equal in area to a given circle. 
A cube double the size of a given cube. 
An angle one-third of a given angle. 

It was the restriction to straightedge and compass alone 
which made these problems "problems." 

Even if we eliminate the crass idea of marked-off 
measure but allow an instrument other than straightedge 


and compass, we can make all of these constructions. As 
an example, we have already seen in Chapter 5 that the 
problem of doubling the cube, or the solution of the 

x = ^2 

can be determined by the intersection of conic sections 
which require only simple mechanical instruments for 
their construction. Without the restriction to straightedge 
and compass, there would have been no classic construc- 
tion problems. 

It is impossible at this date even to estimate the 
mathematical man-hours that have been devoted to the 
classic construction problems. For more than two thou- 
sand years every mathematician born in the Western 
world has had his turn at one or all of them. New mechan- 
ical devices have been invented, new curves have been 
discovered, new branches of mathematics have been 
developed, all in the course of efforts to solve these prob- 
lems. Yet on the eve of the eighteenth century all four of 
them still stood, absolutely undented. Their hour, how- 
ever, had at last arrived. 

In the long assault there had always been an unstated 
and equally unquestioned assumption on the part of the 
mathematicians who tackled the problems. Everybody 
assumed that it was possible to construct a regular hepta- 
gon, to square a circle, to double a cube, and to trisect an 
angle with straightedge and compass alone. In 1796 a 
young man, just nineteen, became the first person in the 
history of mathematics to question this age-old assump- 
tion. Karl Friedrich Gauss considered an entirely new 
idea: perhaps it is impossible to construct these figures 
under the classic restriction. 

The possibility of impossibility! 

It was a revolutionary idea. Up to the beginning of 


the nineteenth century, in the history of mathematics 
there had been only one other comparable thought. That 
was when the Pythagorean, pondering the diagonal of the 
unit square, considered the possibility that there might be 
no rational number which when multiplied by itself 
would produce 2. 

The young Gauss was particularly interested in just 
one of the classic problems, the construction of the regu- 
lar polygons. The Greeks, some two thousand years before 
him, had constructed within the circle the equilateral 
triangle, the square and the regular pentagon. From these 
basic figures they had gone on to construct the regular 
hexagon, octagon, decagon and 15-gon, the number of 
sides of which in each case is a product of the basic 2, 3 
and 5 of triangle, square and pentagon. It was clear that, 
by continuing to bisect the sides of these polygons, they 
could produce a 12-gon, 16-gon, 20-gon, 30-gon and so on. 
But could they produce a regular heptagon (7 sides) with 
straightedge and compass alone? This the Greeks left as 
an exercise for the future; and the future up until the 
time that the young Gauss entered it had produced 
neither a regular heptagon nor a single regular polygon 
the construction of which had not already been known to 
the Greeks. 

Gauss, however, began with great advantages over the 
ancients. He had a language, algebra, and a tool, analytic 
geometry, which allowed him to attack the problem in a 
much more general way than had been possible for them. 
Although all of the construction problems are presented 
differently some even, like the Delian problem, with a 
story to go with them they are, in the language of 
algebra, essentially the same: certain lengths are consid- 
ered to be given, and one or more lengths must be found. 
To solve a given problem, we must find a relation between 
the unknown quantities (x, y, z, . . . ) and the known 


quantities (a, b, c, . . . ). We must state this relation as 
an equation; and then and here is the crux of the matter 
we must determine whether the solution to this equation 
can be obtained by algebraic processes which are the 
equivalent of straightedge and compass constructions. 

At first we may be set back by the idea of algebraic 
processes as geometric constructions with straightedge 
and compass; but a moment's thought will assure us that 
we have thought for a long time in this manner. It is clear 
that, taking two segments of lengths a and b (in terms of 
a given unit segment), the solutions to such simplified 
equations as 

a-\-b = x or a b = x 
can be found with these traditional instruments: 

a-{-b a b 

It is not quite so immediately clear that we can also solve 
such equations as 

ab x 

or a-+-= 

with similar constructions. Yet these too are possible: 


Utilizing the fact that in both of these examples we have 
constructed similar triangles with one side of the smaller 
triangle as the unit, we can show that in the multiplica- 
tion problem illustrated above: 

_ ., 

c ~ ab 

the segment c being the desired segment ab. On the other 
hand, in the division problem, which is illustrated in the 
figure below, we can determine 

the segment c being the desired segment a/b, or a -4- &: 

From these simplified examples it is clear that the 
rational operations of algebra addition, subtraction, 
multiplication and division can all be performed by 
geometrical constructions which require only straightedge 
and compass. It follows that any equation which can be 
solved by any finite combination of one or more of these 
processes can also be constructed by straightedge and 
compass alone. (It must be a finite combination because 


obviously if the number of operations required were in- 
finite we would never be able to finish the construction. ) 
Besides the four basic operations of addition, subtrac- 
tion, multiplication and division, there is one other opera- 
tion in algebra which is the equivalent of a construction 
by straightedge and compass alone. That is the extraction 
of square root. Given the equation 

we can solve for x in the following manner. 

p~^_^^^ ^^^^^^^^^^ ^^ ^ ^ 

/; j f //", . 4 f r ^4^ 

^-r .'Vi^n^x' " \ 

j^^^^v^i^. /; 

'^/^^l^^^^S 1 ^-^''^ V , 
.\.j^4th?7rSrr,.rig3sgzl:.Mijg/T7^ t. ^ L .' 

After establishing the similarity of the triangles in this 
figure, we can conclude 

a _x 
x ~l 

It can be shown that the solutions for x which can be 
obtained by any finite number of additions, subtractions, 
multiplications, divisions and extractions of square root 
include all possible segments from a given set which can 
be constructed by straightedge and compass alone. There 
is nothing at all mysterious about this relationship between 
the solution of equations and the construction of geo- 

metric figures. We need recall only the fact that a straight 
line and a circle are represented in analytic geometry by 
equations of the first and second degree, respectively, and 
that the determination of circles with straight lines, or 
with other circles, leads analytically to the solution of 
equations which involve no irrational operations other 
than the extraction of square roots. 

Herein lies a method for establishing that a given 
construction problem is impossible if the tools of con- 
struction are restricted in the classic manner. All we have 
to do is to show that the problem requires the construc- 
tion of a segment which cannot be obtained from the 
measure of the given segments by straightedge and com- 
pass; i.e., the solution of an equation which cannot be 
obtained by the four basic operations and the extraction 
of square root. This is, naturally, not so easy as it sounds. 
Yet, one by one, the famous construction problems of 
antiquity, which withstood so firmly the full arsenal of 
two millenniums of mathematics, have fallen before this 
new approach, called the algebra of number fields. 

The first problem to be toppled by the young Gauss 
himself was that of constructing a regular heptagon with 
straightedge and compass alone. Such a construction, 
Gauss showed, is impossible because, unlike the pentagon, 
it results in a cubic equation the solution of which cannot 
be obtained by the four rational processes and the extrac- 
tion of square root. In the course of showing that the 
required construction of a regular heptagon is impossible, 
he established the fact that the only constructible regular 
polygons with a prime number of sides are those with p 
sides where p is a prime of the form 2 2n -f- 1. The first 
such constructible regular polygon after the triangle and 
the pentagon of the Greeks is the 17-gon (2 22 -f 1), 
Gauss's general proof, which established the conditions 


for constructibility of the regular polygons and provided 
a tool for attacking the other construction problems, was 
a magnificent achievement. Even Gauss himself was im- 
pressed by it. He had been torn between a career in 
philology and one in mathematics, but now he definitely 
decided in favor of mathematics. When the score is added 
for the classic construction problems time spent against 
the advantages accrued the recruitment of Gauss must 
weigh heavily. 

Last of the problems to topple was the famous ques- 
tion of squaring the circle. Almost a century after Gauss's 
solution of the problem of the regular polygons, Ferdinand 
Lindemann (1852-1939) succeeded in proving that rr can- 
not be the solution of an algebraic equation with rational 
coefficients. Since all constructions by straightedge and 
compass can be represented by equations with rational 
coefficients, this indirectly established the impossibility 
of squaring the circle, or solving the equation 

In the century between Gauss and Lindemann, the 
other two problems yielded almost automatically. Both 
are impossible under the classic restriction. We have 
already seen that the solution of the Delian equation, 
x 3 2, involves the extraction of a cube root; and we 
shall now examine the proof that in general the trisection 
of the angle is also impossible by means of straightedge 
and compass alone and for the same reason. 

We begin by inscribing on the complex plane a unit 
circle with center at O and an arbitrary angle with vertex 
at and one side lying along the real axis. The point 
where the arbitrary side of the angle cuts the unit circle 
is represented by the complex number : 


This complex number, as we recall from Chapter 7, is 
of the general form x -j- iy, where x and y are real num- 
bers and i = V 1. It is uniquely determined by its dis- 
tance from the point of origin and by its angle with the 
positive side of the real axis. These two characteristics 
are called, respectively, the absolute value and the argu- 
ment of the complex number : 

Absolute Value of Argument of 

In the geometric interpretation of complex numbers, 
multiplication of two complex numbers is defined as the 
product of the absolute values and the sum of the argu- 
ments. Since the absolute value of is 1, any root of 
will be a complex number on the circumference of the unit 
circle, all of which also have an absolute value of 1.* Its 
exact location on the circumference must be determined 

* This is easily established by the theorem of Pythagoras. 


by the argument, or size of the angle it makes with the 
real axis. The square root of , for example, will be that 
point, or complex number, where the bisected angle x 
cuts the circumference. This is an operation which we can 
perform with straightedge and compass alone: 


8flw%S^' ; '- 

The cube root of will also be a complex number on the 
circumference, one that makes an angle with the real axis 
equal to one-third of angle x, or a trisected angle x. We 
cannot, however, locate this number as we located the 
square root of because it is impossible to extract the 
cube root of a complex number by algebraic operations 
which correspond to construction by straightedge and 
compass alone. It is, therefore, impossible to trisect a 
given angle under the classic restriction which the Greeks 
pkced upon die problem, just as it is impossible to con- 
struct a regular heptagon, square a circle or double a 

That should settle the question for all time, but there 
is a psychological epilogue to the proofs that each of these 
famous problems is impossible. Mathematicians, amateur 
and professional alike, have shown a great reluctance to 
part with their old friends. Even the great Irish mathe- 
matician William Rowan Hamilton (1805-1865) wrote to 
De Morgan as late as 1852: "Are you sure that it is ira- 


possible to trisect the angle by Euclid [i.e., under the 
restriction}? I fancy that it is rather a tact, a feeling, than 
a proof, which makes us think that the thing cannot be 
done. But would Gauss's inscription of the regular poly- 
gon of seventeen sides have seemed, a century ago, much 
less an impossible thing, by line and circle?" 

This is curiously emotional language from a mathe- 
matician, especially when the essence of Gauss's proof is 
not the possibility of constructing a regular 17-gon but 
the impossibility of constructing a regular heptagon. Ap- 
parently the impossible is hard for any of us to accept. It 
seems almost a personal challenge, and this feeling is 
perhaps responsible for the fact that in spite of the finality 
of mathematical proof that the things cannot be done, 
would-be constructors of regular heptagons, squared 
circles, doubled cubes and trisected angles continue with 
us well into the twentieth century. Any statement in print 
that one of the problems is impossible invariably brings 
to the author a beautifully drawn construction, usually 
with protractor and ruler, with a modest request for 

Why have these famous problems captured the general 
imagination so permanently? Perhaps because, stated as 
they are in the language of construction, they have a 
practical sound which is refreshingly removed from the 
abstractions of most higher mathematics. This is ironic 
for in these problems no one, including the Athenians who 
consulted the oracle, was ever concerned with the actual 
construction of anything. Even Gauss's famous proof that 
it is possible to construct by straightedge and compass 
alone a 17-sided regular polygon did not show how to 
construct such a polygon.* The truth of the matter is that 

* A simple method of constructing the regular 17-gon is given 
by H. S. M. Coxeter in his Introduction to Geometry (New York: 
John Wiley and Sons, Inc., 1961). 


the construction problems, in spite of their practical 
sound, are as highly artificial as any mathematical prob- 
lems can be. 

It is indeed a curious thing that mathematics would 
hobble itself with an impossible restriction and then spend 
two thousand years trying to construct regular heptagons, 
squared circles, doubled cubes and trisected angles which 
could be constructed in a trice with a reasonably accurate 
protractor and ruler. But it was fun and mathematically 
speaking it was extremely profitable fun. Asked, "Was it 
worth it?" mathematics as a whole, inrmeasurably en- 
riched by the discovery of the conic sections, the inven- 
tion of analytic geometry, the winning over of Karl 
Friedrich Gauss to mathematics, the algebra of number 
fields, would echo with Hamilton: "I have not to lament 
a single hour thrown away on this attempt." 


It was Augustus De Morgan, the great mathematical 
writer of the last century, who mourned the Greek limita- 
tion to straight line and circle: 

"What distinguishes the straight line and circle more 
than anything else, and properly separates them for the 
purpose of elementary geometry? Their self-similarity. 
Every inch of a straight line coincides with every other 
inch, and of a circle with every other of the same circle. 
Where, then, did Euclid fail? In not introducing the third 
curve which has the same property the screw. The right 
line, the circle, the screw the representation of transla- 
tion, rotation, and the two combined ought to have been 
the instruments of geometry. With a screw we should 
never have heard of the impossibility of trisecting an 
angle, squaring a circle, etc." 


Let us take a moment to examine how De Morgan's 
proposed inclusion of the mathematical screw, or helix, as 
an instrument of construction would allow us to trisect an 
angle. Since the helix makes one complete turn in its 
length, the angle of the screw thread is proportional to 
the length of the shank; one-third of a complete turn of 
the screw would require one-third of the length of shank 
necessary for a full turn. The problem of trisecting any 
given angle would then be merely one of obtaining a seg- 
ment one-third of a given length of the shank. This we 
could easily do; for while we cannot trisect an angle with 
straightedge and compass alone, we can trisect a line. 

To divide a given segment into three parts, we con- 
struct an angle with the given segment as one side. We 
mark off the unit three times in succession on the other 
side. We join the point which marks the end of the third 
unit with the end of the given segment and join the ends 
of the other unit lengths to the given segment by parallel 
lines. In this way we have constructed three similar tri- 
angles, the corresponding sides of which are in the same 
ratio. Since the segment AC is divided into unit thirds, 

the given segment AB must also be divided into thirds. 

, ___ _ . _.__ _ _ 

Using this same method, the reader should try dividing 
an arbitrary segment into sevenths. 



synonymous with the name of Euclid. 
For this reason, when we first hear of 
something called non-Euclidean geom- 
etry, we feel that there is some misun- 
derstanding. Why, Euclid is geometry! 

A ^"""^ But our trouble is only in our tenses. 

j| m 1| Euclid was geometry for more than 

1 m m two ^ lousanc ^ y ears - He isn't any more. 

" ^^^^ The story of how Euclid was de- 

posed, and at the same time elevated, 
is one of the longest, in many ways the 
Euclid most ironic, and without question one 

Not Alone of the most important in the history of 


As we recall from Chapter 2, 
Euclid deduced aU of his theorems, or 
propositions as they were sometimes 
labeled, from a relatively small set of 
definitions and basic assumptions, 
called, more or less interchangeably, 
axioms or postulates. For a very long 
time it was believed that these assump- 
tions of Euclid's, which we have printed 
in full on page 27, were true, in the 
ordinary way of what we mean by 
"true"; and because they were true, 
the theorems which were logically de- 
duced from them were "true" in the 
same ordinary way. 

Yet geometry is a subject whose 
"truth" is immediately controverted by 
its very name. Geometry means earth- 
measurement, and that was an accu- 


rate name for the art which the Greeks learned from the 
Egyptians. On the small part of the earth which was 
flooded each year by the Nile, the Egyptians found it 
necessary to develop a system of measurement by which 
they could reestablish boundary 7 lines after each inunda- 
tion. But let us take a globe for the earth itself, as we 
shall see, is for various reasons too large for our purposes 
and let us take a few of the "truths" which the Egyp- 
tians arrived at from experience and which the Greeks 
deduced in logical fashion from their axioms and postu- 

A straight line is the shortest distance between two 

The sum of the angles of a triangle is 180". 
The circumference of a circle is 2nr. 

These ideas of straight lines, triangles and circles are 
almost as familiar as our own faces. We all know, for in- 
stance, what a straight line is. It is the shortest distance 
between two points, and it is, well, straight. But when we 
try to draw a straight line on the surface of the globe, it is 
immediately apparent that we can't draw any sort of line 
which even begins to meet our intuitive idea of what a 
straight line should be. Obviously (it is not at all obvious, 
but we think it is! ) , we can stretch a thread across the sur- 
face of the globe between any two points ( say, San Fran- 
cisco and London), and find the shortest distance between 
them. Since "the shortest distance between two points" 
satisfies part of our definition, we can call the line marked 
by the thread a straight line if we will just forget what we 
usually mean by straight. If we extend the line which 
marks the shortest distance between San Francisco and 
London all the way around the globe, we find that it di- 
vides the surface into two equal parts. In other words, it 


is a great circle. The great circle with which we are most 
familiar is the one we call the equator. Although arcs of 
these great circles are the straight line segments of our 
surface being the shortest distance between two points 
our idea of straightness is violated by calling them such, 
and so we caU them the geodesies of the surface. The 
geodesies of the Euclidean plane, or a perfectly flat sur- 
face like a floor, are what we call "straight" lines. 

Since we cannot draw "straight" lines on our globe, 
we cannot have straight-sided triangles. Our triangles will 
bulge on the sides and in the center. If we take one such 
triangle, flatten it with as little distortion as possible onto 
this page, and then join its vertices with straight lines, we 
see at a glance that if the sum of the angles of the interior 
triangle is 180, as we know by Euclidean geometry that 
it is, the sum of the angles of the spherical triangle must 
be more than 180. 

r >'V4 

& i ^^ J7mv j^rr frr ^^ f '' y V&L. <Vf / /%j 

- fe^>/l!l3^:/^l!^.frt/j 

We have seen that the shortest distance between two 
points on the globe is not a straight line, that the sum of 
the angles of a triangle on the globe is not 180. Now let 

us draw a circle on our globe. It meets exactly the Euclid- 
ean definition of a circle as "the locus of points equi- 
distant from a center," and we may jump to the conclusion 
that all we know about a Euclidean circle will also be true 
of such a circle. But the interior of this circle does not 
look like the interior of the circle we know about. It is 
two-dimensional, but it is not flat. It may look, depending 
on how large it is and how large is the globe upon which 
it is drawn, like a ball cut in half or a hub cap or merely 
a saucer. If we place it on this page and trace around its 
edge, we shall have a flat circle. Of this circle we know 
that the circumference is twice the product of TT and the 
radius. But obviously the curved circle drawn on the sur- 
face of the globe, which must have had the same circum- 
ference, cannot have had the same radius. Its radius must 
have been greater because of the curvature of the surface 
on which it was drawn. Its circumference, therefore, can- 
not be equal to 2m* . 

Although geometry means earth-measurement, it is 
apparent that the measurement of the earth has very little 
to do with the geometry of the Euclidean plane. This was 
not because the Greeks of Euclid's time (300 years before 
Christ) did not know that the earth was round. They had 
calculated that it was, from the fact that the North Star 
was higher in Greece than it was in Egypt. But the geo- 
metrical figures on which they based their geometry were 
drawn on only a small part of the surface of the earth, 
and that part, for all practical purposes, was flat. It would 
be more exact to say that they based their geometry on 
idealized figures on an ideal plane, and these were only 
represented by those which they drew on the earth. 
Euclid's geometry was indeed, as Edna St. Vincent Millay 
has written, "nothing, intricately drawn nowhere." 

Yet for two thousand years, in spite of the fact that 


the geometry of Euclid did not truly apply to the only 
large surface which man knew and had not constructed 
himself, it was felt that this geometry then the only 
geometry represented "truth," in so far as man could 
know it. One philosopher (Kant) called the ideas from 
which Euclid deduced his theorems "the immutable 
truths"; another (Mill) considered them "experimental 
facts." Mapmakers and sailors might struggle with the 
geometry of the boundless, finite surface that is our planet; 
but Euclid's geometry, extended to three dimensions and 
a space which was thought both boundless and infinite, 
was the geometry of God's mind. 

That the geometry of Euclid was not the only one 
possible, either physically or mathematically; that it was 
deduced not from self-evident truths but from arbitrarily 
chosen and unprovable assumptions; that another choice 
of assumptions could yield a geometry just as consistent, 
just as useful and just as true, never occurred to anyone 
for more than two thousand years unless, in a sense, it 
had occurred to Euclid himself when he set out the as- 
sumptions on which he based his geometry. For today it 
is clear that Euclid recognized what no other man be- 
tween his time and that of Gauss recognized: that his 
axioms and postulates were assumptions which could not 
be proved. 

The idea of those who followed Euclid and extolled 
him was that the axioms and postulates of his geometry 
did not have to be proved because they were self-evident. 
There was only one impediment to the full and complete 
acceptance of this point of view and that was the fifth 
postulate, which makes a statement very roughly equiva- 
lent to our common statement that parallel lines never 
meet. From the beginning, compared to the other axioms 
and postulates, this one did not seem quite self-evident 
enough,. even to the most devoted admirers of the master. 


The famous fifth postulate stated as follows: 

If a straight line -falling on two straight lines makes 
the interior angles on the same side less than two right 
angles, the two straight lines, if produced indefinitely, will 
meet on iliat side on which the angles are less than two 
right angles. 

As J. L. Coolidge has remarked in his History of Geo- 
metrical Methods, ". . . . whatever else this postulate may 
be, self-evident it is not." 

The fifth postulate makes a statement about the entire 
length of a straight line, a statement which can never by 
its nature be verified by experiment. To remove this flaw 
from the work of the master, generation after generation 
of mathematicians attempted to prove the statement about 
parallel lines from the other postulates. Time after time 
they failed: they were never able to prove the fifth postu- 
late without substituting for it still another postulate, 
which simply varied the problem rather than solving it. 

Among the last of die attempts to "free Euclid from 
every flaw" was one made by a Jesuit priest, Geronimo 
Saccheri ( 1667-1733). It was Saccheri's idea that although 
the parallel postulate did not, on the surface, seem as self- 
evident as the others, he could show that it was the only 
possible assumption because any other 'led to absurdity." 
This, as we have seen, is an ancient and honorable method 
of mathematical proof. We assume the falsity of that 
which we wish to prove true, or the truth of that which 
we wish to prove false, and then show that such an as- 
sumption is unfeasible because it leads us to a contradic- 
tion; hence follows the truth of whatever we were trying 
to prove in the first place. Saccheri's method was mathe- 
matically sound; the only thing which was not sound was 
his attitude. When he found that assumptions about par- 
allel lines quite different from the famous "fifth" did not 


lead Mm into the expected contradictions but into a 
strange and fantastic geometry which was nevertheless as 
consistent as Euclid's, he fell back upon his feelings instead 
of his brains and peppered the last pages of his work with 
such epithets of the logically defeated as "destroys itself,*' 
"absolutely false/' "repugnant." Great discoverers have 
made great mistakes. Columbus found the new world and 
at first thought that it was the old. Saccheri found a new 
world and refused to disembark because he thought he 
knew that there could be only one world. 

It was a century after Saccheri that three mathema- 
ticians in three different countries,* independently and 
apparently without knowledge of Saccheri's curious con- 
tact with non-Euclidean space, came to the conclusion 
that Euclid had known exactly what he was doing when 
he made his statement about parallels a postulate instead 
of a theorem. He had recognized what no else had recog- 
nized: that it was completely independent of the other 
postulates and therefore could not possibly be deduced 
from them. 

To prove this suspected independence of the "fifth," 
it was necessary only to substitute for Euclid's assump- 
tion about parallels a contrary assumption and then to 
show that the geometry deduced from it, in conjunction 
with the other postulates and axioms of Euclid, was as 
consistent as Euclidean geometry itself. 

The first of the non-Euclidean geometries was, in the 
relation its axioms bore to those of Euclid, the simplest 
possible. All the axioms were exactly the same except one, 
the famous "flaw," the long-worked-over statement about 
parallels. We have noted that this parallel postulate may 
be stated in various ways, all equivalent in the sense that 
the same set of theorems can be deduced from any of the 

* Nikolai Ivanovich Lobachevski, Russia; Janos Bolyai, Hun- 
gary; and Karl Friedrich Gauss, Germany. 


various versions. The statement which appears in the set 
of axioms on page 27 is the earliest known; but since even 
the oldest manuscripts we have of the Elements date from 
a time nearly a thousand years after the death of Euclid, 
the master himself may have stated the parallel postulate 
in a somewhat different form. It is clear from the theorems, 
however, that some statement of like nature must have 
existed among the original set of axioms. The most easily 
grasped statement is a later one, known as the Postulate 
of the Unique Parallel: 

Through any point not on a given line, one and only 
one line can be drawn winch will never meet the given 

Now let us make a contrary assumption and let us 
change the postulate to read: 

Through any point not on a given line, infinitely many 
lines can be drawn which will never meet the given line. 

Before our intuition objects to the postulate in this 
new form, let us recall that on the globe, where the 
equivalent of a straight line is a great circle, it is impossi- 
ble to draw through a given point even one line which 
will never meet a given line, since every great circle in- 
tersects every other great circle. A word of caution, 
though. We mention the contrary example that on a sphere 
every straight line or geodesic of the surface intersects 
every other straight line, only to put intuition in its proper 
place. Mathematically, it has nothing whatsoever to do 
with whether the alternate above is a proper postulate. 

When a set of axioms more or less agrees with our 
idea of reality, we will deduce from that set of axioms a 
geometry which also agrees pretty well with the same 
idea of reality. This does not mean that our idea of reality 
is right, but only that our axioms agree well enough with 


whatever reality there is so that the geometry deduced 
from them works. 

We have seen that die earth is not the infinite plane of 
Euclidean geometry; yet small parts of it are, for all prac- 
tical purposes, very much like small parts of the plane; 
and so for building pyramids and supermarkets it works 
very well indeed. But we shall also see that the non- 
Euclidean geometries, which attempted to show only from 
an intellectual point of view that it was possible to deduce 
geometries as consistent as Euclid's from a different set 
of assumptions, turned out to have quite a bit to do with 
reality, too. 

The first non-Euclidean geometry, based on the same 
set of assumptions as the old (except for the new Postu- 
late of Infinitely Many Parallels for the old Postulate of 
the Unique Parallel), applies to a surface which is the 
direct opposite of the surface of any part of the sphere. 
The surface of the sphere is what we intuitively think of 
as "evenly curved"; in mathematics this is more precisely 
defined as "constant positive curvature." The surface to 
which our first invented non-Euclidean geometry applies 
is one of "constant negative curvature.'* It is not (prob- 
ably fortunately) a very common one in the physical 
world; but we can find examples of such a surface : a sad- 
dle, for instance, or a mountain pass or the surface 
around the hole of a doughnut. In these, however, the 
negative curvature is only local. For a surface of constant 
negative curvature, we can look ahead to the illustration 
on page 158. 

If we place a plane tangent to a single point on a sur- 
face of constant negative curvature, like a portion of a 
saddle, we find that it cuts the rest of the surface in two 
hyperbolas. For this reason the earliest non-Euclidean 
geometry, which applies to such a surface of negative 
curvature, is called hyperbolic geometry. If we place a 


plane tangent to a single point on a surface of constant 
positive curvature, like a portion of the sphere, and then 
shift the plane ever so slightly so that it is parallel to its 
original tangent position, we find that it cuts the surface 
in the shape of an ellipse. (In the special case of the 
sphere, it will cut a circle, which is the limiting form of 
an ellipse.) For this reason a later non-Euclidean geom- 
etry, which applies to such a surface of positive curvature, 
is called elliptic geometry. It substitutes for the Postulate 
of die Unique Parallel the following statement: 

Through any point not on a given line, no line can be 
drawn which will not intersect ihe given line. 

From our earlier experiments with our globe, we rec- 
ognize that on the surface of a sphere, where a straight 
line is a great circle, the above postulate holds. For our 
purposes in this chapter, a sphere can serve as an exam- 
ple of a surface of elliptic geometry. Actually it is what is 
called "locally elliptic." To make the entire surface ellip- 
tic, a curious change must be made. As we recall, the 
purpose of non-Euclidean geometries is to establish the 
fact that geometries as consistent as Euclid's can be de- 
duced with a different parallel postulate, the others re- 
maining the same. It is an axiom of Euclidean geometry 
that two straight lines can intersect at only one point, but 
on the sphere two great circles always intersect at two 
points. To get around this difficulty, in elliptic geometry 
we identify the two points of intersection as one point. 
Although in this respect the geometry of the surface of 
the sphere as a whole is not technically elliptic and non- 
Euclidean, it is locally; and we can take a sphere as our 
sample elliptic surface. 

The true surface of hyperbolic geometry not just a 
portion but an entire surfaceis what is called the pseudo- 
sphere, a world of two unending trumpets. 


Let us now compare in a few simple respects the 
"truth" of certain geometrical statements in relation to the 
plane, the sphere and this pseudosphere. Straight lines, 
which are "straight" on the plane, follow the surface and 
therefore curve out on the sphere, curve in on the pseudo- 
sphere. Triangles on the sphere curve out; on the pseudo- 
sphere, in; and circles appear, depending on the surface, 
like saucers or limp watches. What happens to geometric 
"truths"? They are no longer true-false statements, but 
multiple-choice. The sum of the angles of a triangle is 
(equal to, more than, less than) 180. The circumference 
of a circle is ( equal to, more than, less than ) 2nr. Through 
a point not on a given line (one, none, infinitely many) 
lines can be drawn which will never meet the given line. 
Which is "true"? 

When we compare the geometries of these three very 
different surfaces, we see that the geometry of one surface 
cannot be applied to another. We see also that of these 
three, the surface of the sphere is the one which we can 
say with greatest accuracy "exists" for us. Yet portions, if 
not too large, of the imperfect sphere on which we live 
are more like portions of the Euclidean plane. On the 
Pacific Ocean we might choose the geometry of the sphere, 
but in our own backyard we'll take Euclid. So far no one 


in everyday life has found the geometry of the pseudo- 
sphere indispensable; nevertheless, logically it is one with 
the others. 

It is interesting to note at this point that if we did not 
know the nature of the surface of our "earth" we could 
discover whether the curvature was positive or negative- 
always provided that it was not too large in several dif- 
ferent ways. Perhaps the simplest would be by adding up 
the angles of a fairly large triangle. If they added up to 
definitely more than 180 we would know that we were 
living on a surface of positive curvative; if to definitely 
less than 180, that we had our existence on a surface of 
negative curvature. But it would be practically impossible 
to determine with finality that our "earth" was a bound- 
less, endless Euclidean plane. We could never go far 
enough out so that we could state that the plane was in- 
finite, and we could not even say definitely that it was a 
plane, or a surface of curvature 0. Whether the total de- 
grees of the three angles of a triangle was exactly 180, 
slightly more or slightly less, the range of experimental 
error would prevent our knowing for sure that it was flat. 
If, however, our surface is sufficiently large, whether the 
curvature as a whole is positive, negative or exactly zero, 
we will find Euclidean geometry most practical because 
any portion of the surface with which we are concerned 
will seem, for all practical purposes, fiat. 

Non-Euclidean geometries were invented not to pro- 
vide geometries for unusual surfaces but to show that 
from assumptions other than Euclid's (specifically, a dif- 
ferent postulate about parallels ) equally consistent geom- 
etries could be deduced. One of the ways of establishing 
this consistency is by identifying the objects and relations 
of Euclidean geometry with certain other objects and 
relations which result in a non-Euclidean geometry. All 
of the facts of Euclidean geometry then apply to the model 


of the non-Euclidean geometry with the exception of the 
Postulate of the Unique Parallel which, in the case of 
hyperbolic non-Euclidean geometry, is replaced by the 
Postulate of Infinitely Many Parallels. It follows, there- 
fore, from the model that the non-Euclidean is as con- 
sistent at least as Euclidean geometry. 

One of the best-known models of hyperbolic non- 
Euclidean geometry is that of Felix Klein (1849-1925). 
In this model the plane of Euclidean geometry is defined 
as the points of the interior of a circle. Each of these 
points is defined as a non-Euclidean point, and the chords 
of the circle are defined as non-Euclidean straight lines. 
Other definitions are made, but these three will be suf- 
ficient to explain the model below where, as we can see, 
through a given point P not on a straight line AB, in- 
finitely many straight lines can be drawn which will never 
intersect the given line. 

The invention of non-Euclidean geometry freed math- 
ematics from the tyranny of the "obvious," the "self- 
evident" and the "true," and in so doing served to reveal 
the nature of mathematics as well as the nature of geom- 
etry. With the invention of non-Euclidean geometry, it 
was recognized for the first time that the theorems of a 
geometry are logically deduced from a set of arbitrarily 
chosen assumptions. The truth of the geometry is deter- 
mined within this framework and has nothing to do with 
the "truth" ( as judged by external facts ) of the assump- 
tions from which it is deduced. 

We are inclined to think of a geometry as being 
tailored, as it were, to fit a particular surface; but actually 
geometries are rather like ready-made suits. They can be 
used if they fit. Euclidean geometry fits portions of the 
earth very well, although the idealized type of surface 
which is implicit in the geometry apparently does not 
exist at all "nothing, intricately drawn nowhere." The 
surface of elliptic non-Euclidean geometry on which we 
go halfway around and come back to our starting point 
and the surface of hyperbolic non-Euclidean geometry on 
which the "ends" of the world become smaller and smaller 
as they approach infinity are as non-existent as the Euclid- 
ean plane. The fact has nothing to do with their mathe- 
matical importance. They were not invented to be useful. 

It is important that we clearly understand this point, 
for something happened sometime after their invention 
which gave to these non-Euclidean geometries the same 
kind of physical importance that was for so many cen- 
turies the unique possession of Euclidean geometry: the 
geometry for relativity was discovered in a non-Euclidean 
geometry of boundless, finite, "curved" space. In such a 
space the geodesies are paths of light waves, which are 
deflected in varying degrees from their "straight" course 
by the various masses in space. It is easy to glimpse from 


just the brief examination we have made of the geodesies 
of plane, sphere and pseudosphere the implications such 
deflection would have for any geometry of space. 

Mathematically, the usefulness of non-Euclidean 
geometry was a bonus over and above its mathematical 
usefulness, which was, as we have seen, the freeing of 
mathematics from its ancient bonds. 

The new freedom, which included freedom from the 
axioms of Euclid, did not, however, include dispensing 
with Euclid's axiomatic method. This had been the ideal 
of all mathematicians since his time. Yet actually it had 
been hobbled by the definition of an axiom as a self- 
evident truth. When this definition was dispensed with 
and an axiom recognized as simply an arbitrary assump- 
tion, the axiomatic method became infinitely more valu- 
able to mathematics. 

So it is that while Euclid is no longer all geometry, 
he is the axiomatic method the logical ideal and aim of 
mathematics and of all science and the "flaw" which so 
many generations of mathematicians labored to remove 
from the work of the master is seen as no flaw at all, but 
the hallmark of his genius. 



coordinates, which enables us to find 
our way about an unfamiliar part of 
our city, enables mathematicians to 
move mentally in a world of n dimen- 
sions with as much freedom as they 
Umove physically in a world of three. 
The idea of dimensionality has 
been with mathematics since the time 
of the Greeks. The lengthless breadth- 
less point traced out a line, which had 
one dimension. The line traced out a 
Worlds plane, which then had two dimen- 

We Make sions; the plane traced out a space, 

which then had three dimensions. No 
one with a human desire for consist- 
ency could follow this process and fail 
to ask the next question. Why not a 
figure, a sort of hypersolid, traced out 
by a solid moving in a 4-dimensional 

But a 4-dimensional space! What 
could it possibly be like? 

Although we cannot visualize a 4- 
dimensional space, we can visualize 
the effect that "going into the fourth 
dimension," as science fiction writers 
say, would have on an object from 
space of three dimensions. This we can 
come to by a logical extension of what 
we can actually see of the relation be- 
tween space and the plane. Let us take 
a piece of paper, trace out die soles of 
our shoes, and cut them out. We have 


a right sole and a left sole, mirror images of each other. 
If we limit ourselves to sliding them around in the plane, 
on a table top, for instance, we can never make them 
both left soles. But if we lift the right sole off the table 
( out of the plane and into space ) , we can turn it over so 
that it is a left sole when we return it to its mate. Now 
let us take the shoes, one right and one left, from which 
we traced the soles. These are 3-dimensional objects in 
3-dimensional space. We know from experience that we 
can never turn the right shoe into a left shoe. But if we 
were able to lift it out of our space and into a 4-dimen- 
sional space, turn it over and return it, what would have 
happened to it? 

There is yet another way by which we can get a visual 
idea of 4-dimensional space. This too is by a logical exten- 
sion from the three dimensions with which we are familiar. 
Let us take the simplest figures in each dimension: 

A line segment is bounded by two points. 
A triangle is bounded by three line segments. 
A tetrahedron is bounded by four triangles. 

Should there not be, in a 4-dimensional space, a figure 
bounded by five tetrahedra? This logical extension of the 
first three figures we call a pentahedroid. When the five 
tetrahedra are regular, the pentahedroid (it can be 
proved) is one of the six regular bodies possible in a 4- 
dimensional space. 

What does the pentahedroid look like? Well, it is a 
figure bounded by five tetrahedra. Although we are some- 
what like the Lady of Shalott in that we cannot turn and 
see it and live, we can look at it in several of the ways in 
which we usually look at 3-dimensional figures. 

We can "see" a hypersolid in a manner similar to the 


one in which we are accustomed to seeing 3-dimensional 
solids in two dimensions. We are all familiar with the real 
appearance of these objects in photographs and paintings. 
Of course, it is our actual experience with the objects in 
three dimensions which gives for us a reality to their rep- 
resentation in two, and this actual experience is not possi- 
ble with 4-dimensional objects. Nevertheless, we can 
construct a perspective model in three dimensions of a 
never-seen and never-to-be-seen but logically thought 
out figure in four dimensions. We can give an example 
which is so simple as to be trivial and yet illustrates 
exactly the relationship. If we are drawing a tetrahedron 
and are looking at one of its triangular sides from a posi- 
tion directly in front of it and level and parallel with it, 
we see only and draw only the triangular face of the 
side of the tetrahedron which is toward us. In this par- 
ticular case, a triangle is a 2-dimensional representation 
of the 3-dimensional tetrahedron. In the equivalent 3- 
dimensional representation of a 4-dimensional pentahe- 
droid we have before us because our projection of the 
pentahedroid into our space is "head-on" a tetrahedron, 
which is one of the faces of the pentahedroid. In the com- 
parable projection of the tetrahedron, the other three 
faces have been projected into the three straight lines 
bounding the triangular face we saw. When we look at 
the tetrahedron which is the head-on projection of the 
pentahedroid, each face of the tetrahedron which we see 
is a projection of a bounding tetrahedron, comparable to 
the projection of planes into lines in the projection of a 
tetrahedron into a triangle. 

We can also make 3-dimensional patterns of 4- 
dimensional hypersolids almost as easily as we can make 
2-dimensional patterns of 3-dimensional solids. To make 


a plane pattern of the solid tetrahedron which we have 
represented below 

we simply open it up and flatten it out on the page: 

An equivalent, but of course 3-dimensional, pattern for a 
pentahedroid would involve spreading out the hypersolid 

in space. The resulting pattern would be a tetrahedron 
with a tetrahedron upon each face. 

Just as one of our 3-dimensional children would have 
no trouble folding the 2-dimensional pattern back into a 
3-dimensional tetrahedron, a 4-dimensional child would 
make quick work of folding the 3-dimensional pattern 
back into a 4-dimensional pentahedroid! 

We can also dismantle a pentahedroid, as if it were a 
Tibetan monastery being prepared for shipment to the 
home of an American millionaire, the pieces carefully 
labeled so that they can be put together again in another 
land. Then we should actually have seen a 4-dimensional 
body in pieces! 

It is, of course, impossible to construct an actual 
model of a 4-dimensional figure, but mentally we are not 


so limited. If we do not insist upon an answer to our very 
human question, "But what does it actually look like?" we 
can think freely of objects in space of any number of 
dimensions or, as the mathematicians say, n-dimensional 
space where n is any number greater than 3. 

After our excursion through what might be called the 
sideshows of dimension theory, let us go back to the sys- 
tem of coordinate axes by which we find our way about 
an unfamiliar part of our city. These can be the same axes 
by which we map the points, lines, figures and surfaces of 
2-dimensional space in analytic geometry. We saw that 
any point on the plane could be uniquely located by a 
number pair (ac, !/); now we see that any point in three 
dimensions could be uniquely located if only we had a 
third axis. This, the 2-axis, we erect at the origin perpen- 
dicular to the plane formed by the ac and Now 
instead of two coordinates, x and y, to locate a point, we 
need a third, z. 

To see how this extension of the system of coordinate 
mapping works, let us consider the points in the illustra- 
tion on page 69. On the plane they are uniquely identified 


by their jc and y coordinates as (2, 1), ( 4, 2), ( 3, 3), 
(4, 2). If we raise the first two points one unit above 
the plane and lower the last two points one unit below the 
plane, we get (2, 1, 1 ) and (4, 2, 1) above the plane and 
(3, 3, 1) and (4, 2, 1) below the plane. If we 
raise each of the four original points a different amount, 
the first one unit, the second two units, and so on, we get 
the points (2, 1, 1), (4, 2, 2), (3, 3, 3) and (4, 
2, 4 ) , each one a unique point and each one uniquely 

Following the general method we have already out- 
lined, we can locate points, lines, plane figures and solids 
in 3-dimensional space. The only difference is that instead 
of expressing these by equations of two variables we shall 
need equations of three variables. The equation ax -f- by 
+ cz + d = represents a plane in 3-dimensional space 
just as the equation ax + by -f- c = represents a line in 
2-dimensional space. 

It is only natural at this point that we ask what is rep- 
resented by an equation in four variables? We do not 
have to be mathematicians to guess the answer to this one. 
If an equation of the first degree in two variables repre- 
sents a line in 2-dimensional space, an equation in three 
variables represents a plane in 3-dimensional space, then 
an equation in four variables represents a space ( or hyper- 
plane) in 4-dimensional space, and so on. 

The reason that we are able to move so freely in n- 
dimensional space is that, thanks to analytic geometry, we 
no longer have any need to visualize what we are talking 
about. We are just talking about algebraic equations. But 
do not make the mistake of triirildng that the geometry of 
ft dimensions is all algebra after n = 3. There is a division 
of labor. Algebra does the work, and geometry suggests the 
ideas. If, for instance, in 2-dimensional space we have a 


number pair (x, y) and another pair (x f , if), geometry 
suggests that we can use in our algebra the concept of 
"the distance" between (or, y) and (x', if), since any given 
number pair can always be represented as a unique point 
in the plane. The way in which we do this is as old as 
geometry itself. If we draw a line from (x, y) paraUel to 
the y-axis and a line from (*',#') parallel to the x-axis, 
the two lines will intersect. When we join (x, y) and 
(x',1/) we have a familiar fig 

By the Pythagorean theorem we know that the hypotenuse 
of the right triangle, which is also the distance between 
our two points, is the square root of the sum of the squares 
of the two sides. We say, then, that our distance formula 
for two ordered pairs of numbers (x, y) and (yf,i/) is the 
one below.* 

In the specific case of the two points above (3, 4) and 
(7, 1), the formula gives us (7 3) 2 + (4 I) 2 = 25. 

* Note that the result of squaring (x a') is the same as the 
result of squaring (x' x). 


The square root of 25 being 5, this is the desired distance 
between the two points. 

When we put our points in a 3-dimensional space as 
number triples (x, y,z) and (x 1 , \f,tf), we have the same 
formula for the distance between the two points except 
that we have a third variable, z and ^. 

We can apply this formula in the following concrete prob- 
lem. We wish to determine the distance from the back 
right-hand corner of the top of our desk to the bottom of 
the front left-hand leg. To do this, we determine first the 
length of the diagonal of the top of the desk. Then, with 
this as one side of our right triangle and the front left-hand 
leg as the other, we determine the length of the hypot- 
enuse, which is the desired distance. Try it some time 
with a desk. 

It is not just the abstract concepts of geometry like 
that of distancewhich suggest ideas to algebra. Even the 
geometric figures of space of four dimensions, which we 
found impossible to visualize a few pages back, become 
mere formulas and lead us to extensions of themselves in 
higher and higher dimensions. We are all familiar with 
the circle and its extension into three dimensions, the 
sphere. If we map a circle on the plane with its center at 
trie origin, the formula for its radius is 

x 2 + y* = R 2 

and this means simply that the square of the radius is the 
sum of the squares of the x and y coordinates of any point 
on the circumference. 

Just as we extended the distance formula into dimen- 
sions higher than 2, we can extend the formula for the ra- 


dius of a circle to the radius of a sphere, a hypersphere, 
and so on. 

x 2 + y 2 = R 2 

= R 2 

= R 2 

We must not think that the mathematics of n dimen- 
sions is nothing more than adding another letter for each 
dimension that we add. Things that are mathematically in- 
teresting begin almost as soon as we add that next letter, 
and they are not at all predictable. If they were, the mathe- 
matics of n dimensions might be very useful which it is 
but it would not be very interesting and it is. 

Although the extension of the formula for the radius 
of the circle into three and four dimensions was made in 
routine fashion, the extension of the formula for the area of 
the circle into higher dimensions is not nearly so routine: 

For the area of a circle, A irr 2 


For the volume of a sphere, V H- nr 3 


For the hypervolume of a hypersphere, H ^ n 2 r 4 

Here we have a very interesting and unexpected relation- 
ship. Two generalizations are involved and they alternate, 
depending upon whether the dimensionality of the figure 
is even or odd. If the number of dimensions is even, n = 
2fc, we have 

but if the number of dimensions is odd, n = 2k ~f 1, the 
general expression is quite different. 


As we go further into the geometry of n dimensions, 
we find that we never know at just what n our extension 
may become suddenly more difficult. Consider the problem 
of packing spheres into space so that in some regular pat- 
tern we can fit in the greatest number of spheres. For n = 2 
we get the most circles on the plane by alternating stag- 
gered rows. 


For n = 3 we arrange each layer of spheres in the same 
way that we arranged the circles but stagger the alternate 
layers. We can continue in similar ways, although it is not 
at all easy to prove, through n 8. At n 9 3 the problem 
inexplicably takes a more difficult turn. At the present time 
there is no one who can tell us how to pack 9-dimensional 
spheres in 9-dimensional space! 

The geometry of n dimensions might just as well be 
called the algebra of n variables, but either way the intel- 
lectual journey which begins at O on the Cartesian plane 
takes us through fascinating if purely mental country, and 
never ends! 



Where Is In 


Where Is Out? 

are told to copy a drawing of a triangle, 
we produce a blob. If we are then given 
a neat little square to copy, we produce 
a brotherly blob. A long thin rectangle 
is transformed into a blob, and so is a 

As far as we are concerned, the blob 
is a reasonable reproduction of any 
number of simple geometric figures. It 
is generally admitted that we do not 
draw very well; yet we have perceived 
the essential likeness of all the figures 
we have been given to copy, a likeness 
which will escape us in later Me when 
a rectangle, for instance, will seem like 
something entirely different from a 

The fundamental similarity of tri- 
angle, square, rectangle and circle is 
that they all divide the plane (or the 
piece of paper on which they are 
drawn) into two distinct and mutually 
exclusive parts: that part A, which is 
inside the boundary, and that part B, 
which is outside. A point C which is in 
A cannot simultaneously be in B. For C 
to move from A to B, it must cross the 
boundary of the figure we have drawn, 
whether it be triangle, square, rectangle 
or circle. If we think of each of these 
figures as drawn on a thin sheet of rub- 
ber, we can see that no matter how we 
pull the sheet about, so long as we do 


not cut or tear it, we shall never be able to affect in any 
way this basic and common characteristic. 

If, however, we take certain figures like those below 
which divide the plane, or the paper on which they are 
drawn, into more than two parts, we shall find that no 
amount of stretching will turn them into the figures we 
were first concerned with, 


Yet, although we cannot reduce any of these figures to our 
first simple blobs, we can reduce each of them to a blob 
with a blob cut out of it; and this is the way, as children, 
we would have drawn any one of them. 

Recalling the straightedge and compass of Euclid's 
geometry, the protractor in its envelope at the back of the 
text, the painstaking care with which we drew each figure 
and lettered the appropriate points, we find it hard to be- 
lieve that tins casual approach to figures can be geometry 
too. Yet it is. Topology, as this geometry is called, is one 


of the newest, the most all-inclusive and the most abstruse 
branches of mathematics. It concerns itself with the truly 
fundamental properties of geometrical figures, surfaces 
and spaces. Most of its problems are so removed from our 
everyday experience that it is impossible for us even to 
glimpse them, let alone grasp them; yet, as in the higher 
arithmetic, some of its most difficult problems can be 
stated in the language of a child. 

This is not as surprising as it might at first seem. In 
an article entitled "How Children Form Mathematical 
Concepts" (Scientific American), Jean Piaget has written: 

A child's order of development in geometry seems 
to reverse the order of historical discovery. Scientific 
geometry began with the Euclidean system (con- 
cerned with figures, angles and so on), developed in 
the 17th century the so-called projective geometry 
(dealing with problems of perspective), and finally 
came in the 19th century to topology (describing 
spatial relationships in a general qualitative way for 
instance, the distinction between open and closed 
structures, interiority and exteriority, proximity and 
separation). A child begins with the last: his first 
geometrical discoveries are topological. At the age of 
three he readily distinguishes between open and closed 
figures: if you ask him to copy a square or a triangle, 
he draws a closed circle; he draws a cross with, two 
separate lines. If you show him a drawing of a large 
circle with a small circle inside, he is quite capable of 
reproducing this relationship, and he can also draw a 
small circle outside or attached to the edge of the 
large one. All this he can do before he can draw a 
rectangle . . . Not until a considerable time after he 
has mastered topological relationships does he begin 


to develop his notions of Euclidean and projective 
geometry. Then lie builds those simultaneously. 

Yet the only formal geometries with which most adults 
are familiar are these last two! 

In that with which we are most familiar the Euclidean 
geometry we were taught in high school we studied and 
proved statements which established the likenesses among 
different types of figures triangles, for instance. We were 
especially fond of the right triangle. Following in the foot- 
steps of Pythagoras, we found that the square constructed 
on the hypotenuse of the right triangle was equal to the 
sum of the squares on the other two sides and that all right 
triangles, regardless of their sizes and shapes, were alike 
in this respect. (We have seen how this ancient theorem 
runs through all mathematics: arithmetic, algebra and 
analysis as well as geometry we even meet it, in a modi- 
fied form, in the mathematics of relativity; but one place 
we never meet it is in topology! ) 

The other geometry with which we may also have be- 
come f amiliar in high school, in the art course, since it is 
not taught as mathematics at that level, is projective geom- 
etry. ( It was Cayley who exclaimed, "Projective geometry 
is all geometry!" but it is not topology.) Here, when we 
attempted to draw the comer of a room, we discovered a 
curious thing. The corner was formed by the meeting of 
three right angles and we knew by Euclidean geometry 
that a right angle is 90 and that the sum of three right 
angles must be 270; but when we drew the corner on 
paper, so that it looked to the eye exactly like the corner 
we saw, the sum of the three right angles was always 
360! * 

* This, of course, is because the comer when projected to the 
plane on which we are drawing it must fill an entire circle, or 360. 


Invariants under rigid motion length, angle, area- 
are the subject of Euclidean geometry. Invariants under 
projection point, line, incidence, cross-ratioare the sub- 
ject of projective geometry. ( Rigid motions are technically 
a class of projections.) No matter how we slide a right 
triangle about on the plane, we never affect its "triangle- 
ness" nor its "lightness"; but when we draw it from differ- 
ing points of view, although we retain its "triangleness," 
we lose its "rightness." The transformations of topology, 
which include rigid motions and projections as special 
classes, are in general much more drastic. Under the par- 
ticular group known as the deformations, a right triangle 
can be transformed into any other type of triangle, a poly- 
gon of any number of sides more than three, an ellipse, a 
circle and so on. Yet, through all these changes the char- 
acteristic which we perceived when we drew our first 
triangle as a blob will remain invariant: it will divide the 
plane into two distinct and mutually exclusive parts, an 
inside and an outside. This characteristic is invariant 
under deformation for any figure like the triangle which 
topologists classify as a simple closed curve. 

Although intuitively we have an idea of what we mean 
by a simple closed curve, let us arm ourselves with a more 
precise definition. When we tliink of a curve we probably 
think of something the opposite of sharp, angular, straight; 
but in mathematics the sharp, the angular and the straight 
may all be curves. The ancient definition of a curve is that 
it is the path traced by a moving point. In the spirit of this 
definition, a closed curve is one whose end point is the 
same as its beginning point; and a simple curve is one 
which does not pass through the same point more than 
once. It is obvious from this definition that circles, tri- 
angles, rectangles and higher polygons, as well as blobs, 


are all simple closed curves. It is not quite so obvious that 
the figure below is a simple closed curve. 

What we perceived so early in Me about simple closed 
curves that they divide the piece of paper on which they 
are drawn into an inside and an outside is one of the 
fundamental theorems of topology. 

THEOREM: A simple closed curve in the plane divides the 
plane into exactly two domains. 

There are many mathematical theorems which, in the 
course of this book, we will receive with puzzled frowns 
or raised eyebrows; but the Jordan Curve Theorem, as the 
above is known, is not one of them. This theorem was first 
stated by Camille Jordan (1838-1922). Besides being a 
mathematician of the first order, Jordan was a great 
teacher and the author of a textbook, Cours d'analyse, 
which is an acknowledged masterpiece. In A Mathema- 
tician's Apology Hardy has stated his own debt to Jordan 
and to Ms book as follows: "I shall never forget the 
astonishment with which I read that remarkable work, the 
first inspiration for so many mathematicians of my genera- 
tion, and learnt for the first tune as I read it what mathe- 
matics really meant. From that time onwards I was in my 
way a real mathematician, with sound mathematical am- 
bitions and a genuine passion for mathematics." 


We have included this testimonial from Hardy to make 
clear that Jordan was a mathematician of stature and in- 
fluence. If Jordan was interested in the fact that a simple 
closed curve divides the plane into two domains, it must 
be more interesting and less obvious than the observation 
of a three-year-old would lead us to believe. (Actually 
modern mathematicians have a considerable respect for 
the obvious. They have found that quite often what ap- 
pears obvious is not at all; in fact, quite often it is not 
even true. They have also found that even when it is true, 
it is often almost impossible to prove that it is true.) 
Jordan experienced considerable difficulty in trying to 
prove the obvious theorem which bears his name, so much 
difficulty that his proof did not meet the rigorous stand- 
ards which he himself had set up in his Cours d'analyse. 
Time and effort on the part of other mathematicians finally 
filled the logical gaps in his reasoning. When at last it was 
completely acceptable from the rigorous point of view, the 
proof of this "obvious" theorem was nothing for children. 
It was so extremely technical that even mathematicians 
found difficulty in following it. 

Why should it be so difficult to prove what we have 
shown is readily apparent even to a three-year-old? 

The answer to this question lies in the complete gen- 
erality of Jordan's theorem. It is simple (relatively) to 
prove it for any special case of curve. For instance, we 
can give a simple method for determining whether a given 
point is inside or outside the labyrinthine "simple closed 
curve" that we drew on page 179. Incidentally, the reader 
can first determine that this is, indeed, a simple closed 
curve by tracing it. He will find that without lifting his 
pencil and without crossing a line he can go around the 
entire curve and return to his starting point. It is a little 


harder to determine whether a given point is inside or 
outside. To do so, we take a direction which is not parallel 
to any side of the figure. Although sometimes difficult, 
this is not impossible, since any straight-edged closed 
curve has only a finite number of sides and hence of direc- 
tions. To determine whether a given point is inside or out- 
side the curve, we direct a "ray" in the chosen direction 
from the point and past the curve. If the ray crosses the 
boundary an even number of times, the point is outside; 
if an odd number of times, inside. Below we have applied 
this method to a fairly simple figure, but the reader should 
also apply it to the figure on page 179. 

The general problem in other words, the proof in re- 
spect to all simple closed curves presents difficulties 
which do not occur in the special case of straight-edged 
closed curves. All simple closed curves include, in addi- 
tion to the various examples we have already mentioned > 
such curiosities as curves which have area, curves to which 
no tangent can be drawn, curves which cross and recross 
a straight line infinitely many times within an arbitrarily 
small distance. Although these are contrary to all we think 
we know about curves, they too may be simple closed 
curves; and when we make a statement about simple 
closed curves, as we do in the Jordan Curve Theorem, we 


are making a statement which must be shown to apply 
also to such curious curves! 

The greatest difficulty of all in proving this theorem is 
one which seems at first preposterous. Where is in and 
where is out? It is very easy to show that there exists at 
least one point which is outside the curve. Knowing that 
the plane is infinite in extent, we select a point sufficiently 
far away from the boundary so that it is unquestionably 
outside. But how do we go about showing that there is 
at least one point which is inside the curve? In the case 
of the ordinary everyday simple closed curve, the land 
which makes the Jordan Curve Theorem seem so obvious, 
we find our inside point by selecting one which is on the 
other side of, and an arbitrarily small distance from, the 
boundary. Even mathematicians agree that such a point is 
inside. But this method will not be of any use to us when, 
in going even an arbitrarily small distance across the 
boundary we shall have already crossed and recrossed the 
curve an infinite number of times. Such problems, not 
obvious at all, made the general Jordan Curve Theorem 
so difficult to prove. Today, proved at last with full rigor 
and generality for all possible simple closed curves in the 
plane, the theorem has been extended for their equiv- 
alents in space. These are the simple closed k surf aces like 
the sphere and the polyhedra which divide space into two 
distinct and mutually exclusive parts, that which is inside 
them and that which is outside. 

We again imagine these figures to be made of rubber, 
thin enough to be stretched at will into any topologically 
equivalent shape we choose yet strong enough to hold a 
shape. As we pull them about, what other characteristics 
about them remain invariant? No matter how we stretch 
these surfaces, we cannot change the fact that each has 
two sides, an inner side and an outer side. We also cannot 


change the fact that they have no edge. These, like the 
characteristic of dividing space into two parts, are in- 

If we puncture our general balloon-like surface and 
carefully stretch it out flat, we get a surface which we can 
call a disk. This disk, which we can say is the topological 
equivalent of a sphere with one hole in it, does not of 
course divide space into two parts because it encloses no 
space. It is not unbounded, as the sphere is, and therefore 
it has an edge where the sphere has none. It has, however, 
one characteristic of the sphere. It has two sides. Unless 
we are somewhat informed on the subject of topological 
curiosities, we may think that all surfaces have two sides, 
and if this is ah 1 the sphere and the disk have in common, 
it isn't much. However, although it is impossible to have a 
three-sided surface, it is perfectly possible to have a sur- 
face with only one side. 

We can take our disk, with its two sides and its one 
edge, and stretch it out into a long thin strip like the one 

Let us paint one side of this strip red, and one side green. 
Then let us pick it up and join the two ends so that red 
meets red and green, green. We have a band which is red 
on one side and green on the other. Like the strip ( from 
the disk) with which it was formed, it has two sides; but 
unlike the strip, it has not one edge but two. The original 
strip was the topological equivalent of a sphere with one 
hole in it; the band is the topological equivalent of a 
sphere with two holes in it. 

Now let us take another similar but unpainted strip 


and give it a half twist before we join the ends together. 
We do not have a band, but something quite different 
topologically. Where the band has two sides and two 
edges, the Mobius strip, as it is called, after A. F. Mobius 
(1790-1868), has only one side and one edge. If we at- 
tempt to paint one side red, we shah 1 never find a place to 
stop until we get back to where we started, and by then 
the entire strip will have been painted red! 

The Mobius strip and the band were both made from 
a strip which was a stretched-out disk; yet no amount of 
stretching will enable us to make a Mobius strip into a 
band or a band into a Mobius strip. What happens, 
though, when we perform a similar operation upon all 
three? We cut them down the length. The original strip 
falls into two strips; the band falls into two bands; but the 
Mobius strip remains in a single piece! (Try cutting it 

Topologists, it is clear, look at things and see them dif- 
ferently from the way most of us do. Where we see a 
circle or a triangle or a square, a topologist sees a simple 


closed curve; where we see a knot just a knota topolo- 
gist sees many different kinds of knots, and lie is fas- 
cinated by them. 

By a knot, a topologist means nothing so simple as 
even the most complicated knot that Boy Scout, first-aid 
instructor, cowboy or sailor can tie. A knot which is tied 
can be untied. It is, therefore, topologically equivalent to 
the piece of string or rope out of which it was tied, a hue 
segment or a simple open curve. A topologist is interested 
in knots which are not tied and therefore cannot be un- 
tied. Such knots are essentially loops or circles, simple 
closed curves in space, but with a difference. 

The most famous of these is probably the trefoil, or 
clover leaf, knot pictured below in two different forms. 

No amount of stretching or pulling or clever weaving 
can transform one of these knots into the other. Yet both 
(in fact, any knot) can be mapped upon a simple closed 
curve a rubber band, for instance. We put the band and 
the string out of which the knot is made together at one 
point and then keep them together at each point as we 
move around the rubber band. Eventually we come back 
to where we started, never having had to separate string 
and rubber band at any point. In this respect a knot is 


equivalent to a simple closed curve; yet no amount of 
stretching, nothing short of cutting the knot and rejoining 
the ends, can make a knot into a simple closed curve, for it 
is embedded in three-dimensional space in a different way. 
We may think of knots only as pleasant puzzles, yet 
they present topology with one of its greatest unsolved 
problems: that of classifying different kinds of knots ac- 
cording to their invariants. One method which works very 
well for the great majority of knots is that of associating 
each one with a certain surface, the edges of which can 
be arranged so that they trace out that particular knot. A 
Mobius strip with three half-twists instead of the usual 
one, for instance, will trace out in the path of its edge a 

trefoil knot. But a general method of classification which 
would cover all cases has not yet been discovered. 

Here, as in the proving of the Jordan Curve Theorem, 
the difficulty lies in the complete generality of the prob- 
lem; yet if a general method of classifying knots can be 
found, much in related topological fields will fall auto- 
matically into place, like minor candidates riding into 
office on the leading candidate's coattails. 

Perhaps the solution to this problem lies within the 


future grasp of some chubby hand drawing circles and 
triangles as indistinguishable blobs. 

"One thing seems certain," wrote E. T. Bell, in The 
Development of Mathematics: "to think topologically, the 
thinker must begin young. The cradle with its enchained 
teething rings may be a little too early; but the education 
of a prospective topologist should not in any case be de- 
ferred beyond the third year. Chinese and Japanese puz- 
zles of the most exasperating kind, also the most devilish 
meshes of intertwisted wires to be taken apart without a 
single false move, should be the only toys allowed after 
the young topologist has learned to walk." 

Topology is one of the youngest branches of an ancient 
subject, and much of its strength has come from the 
youthfulness with which it has looked at age-old figures. 
It has seen what was always there but never seen before 
by grownups. 



seeking news of what lie loved best, he 

might be surprised to find in the 

schools only his own geometry his own 

theorems. He might well wonder if 

nothing at all had happened to mathe- 

^j .^""""""'^^ matics in some twenty centuries. But, 

I ^r * n *^ e urn * vers iti es > ne would find out 

1 ^ """^fc what had happened. He would be 

m ^^^.^^ confronted by not one but many, many 

different geometries, of which his own 

was only the most elementary. "A 

What a geometry," he would be told in 

Geometry IS strangely unfamiliar terms, "is the 

study of those properties of figures 

which remain invariant under a given 

group of transformations." 



Transformation . 

Even in Greek, these words would 
have no mathematical meaning for 
Euclid. Yet with the help of the con- 
cepts which they represent, mathe- 
matics has been able to bring together 
into one unified whole all the very dif- 
ferent geometries which have been 
developed in the twenty-three hundred 
years since Euclid composed his Ele- 

The key word of the three is 
"group," a concept which has been 
called the unifying principle of mod- 
ern science. For simplicity's sake, how- 

ever, we shall begin our examination of "what a geometry 
is" with the two less complicated concepts of "invariance" 
and "transformation," which we have "already met in sev- 
eral earlier chapters. 

These two ideas are in a sense diametrically opposed. 
The concept of transformation represents change: invari- 
ance represents changelessness. When we combine the 
two, we are concerned with that which is changeless 
under change. 

Let us take a very simple geometrical example, re- 
membering as we do so that these concepts can be applied 
to much more than geometry, to much more in fact than 
mathematics. We pick up a right triangle (A) and move 

/ ' . ^"""*^ / *# 7>/> * f ; i 

' " r , '," /^ -f 1 /^', 

*' * "* I ^l^^/^^ 1 

it by what geometers call "rigid motion" from one place 
to another (B). We find that certain of its geometric 
properties change but others do not. Its position changes, 
for instance, but its size and shape do not. If, however, we 
proceed to expand it in a uniform manner (C), its shape 
does not change but its size does. We now move it in 
space so that it is "in perspective" (D) with the position 
it originally held. It remains a triangle, but it is no longer 
a right triangle. Now we drape it over a globe (), allow- 
ing the sides to fall along the shortest distances between 
the vertices. We find that we still have a three-sided fig- 
ure, but the sides are not straight lines and the angles, 
unlike the angles of A, B, C and D, add up to more than 
180. We take up our triangle and s-t-r-e-t-c-h it out be- 
tween our fingers (F). It remains, like our original tri- 
angle, a simple closed curve, dividing the surface on 
which it lies into two distinct parts, that which is outside 
the curve and that which is inside but everything else 
about the original triangle has changed. It is no longer 
even a triangle! 

We have subjected a given right triangle to four dif- 
ferent changes, or transformations, and in each case at 
least one of the properties of our original figure has re- 
mained invariant under that particular kind of transforma- 
tion. We recognize that each of the transformations has 
given us a figure characteristic of one of the geometries 
which we have already examined in this book: Euclidean 
geometry, projective geometry, elliptic non-Euclidean 
geometry and topology, or "rubber-sheet" geometry. Yet 
we have touched on only a few of the more obvious 
transformations to which we can subject a right triangle. 
We are reminded among other things of reflections, 
translations, dilations, inversions and rotations. With each 


of these transformations we can find ourselves with a dif- 
ferent geometry! t 

Even a hundred years ago, the garden of mathematics 
seemed rankly overgrown with geometries. Projective 
geometry threatened to take over the place. "Projective 
geometry is all geometry!" one enchanted mathematician 
was heard to exclaim. Yet among the neatly tended rows 
of Euclidean geometry all sorts of non-Euclidean geome- 
tries were springing up. Topology showed a tentative 
blade of green as analysis situs. It was obvious that a 
period of wild growth now needed to be followed by some 
attention to pruning. 


It was at tills time that young Felix Klein, whose 
model of non-Euclidean geometry we have akeady met in 
Chapter 10, made a speech at Erlangen University which 
offered the very tool needed for the pruning. Because of 
the location where the speech was made, Klein's proposal 
has come to be known in the history of mathematics as the 
Erlangen Program. 

Klein suggested that under an entirely new definition 
of geometry, all of the many apparently disconnected 
geometries could be brought together, classified and uni- 
fied. Once more, geometry would be one great subject of 
study instead of many smaller subjects. The new defini- 
tion which he proposed is the one we have akeady met: 

"A geometry/' Felix Klein suggested, "is the study of 
those properties of figures which remain invariant under 
a given group of transformations." 

The concept of a group, upon which Klein's Erlangen 
Program depended, had been used earlier in connection 
with the solvability of algebraic equations. We shall see 
now how it was used to unify and define the many 
branches of geometry. 

Group is one of those everyday words which a mod- 
ern mathematician uses in a very precise sense. By it he 
means nothing so vague as the "assemblage" of Webster. 
A mathematical group must satisfy four specific require- 
ments, which are labeled in the way of mathematics Gi, 
Go, Gs, d and are listed on the next page. Without any 
further explanation, these very abstract requirements 
would probably seem to the reader entirely removed 
from the world he knows; yet the group concept is one 
with which we live and work every day as we use the 
ordinary operations of arithmetic. The rational numbers, 
for instance, constitute a group (G) with respect to the 
operation (o) of addition; and the non-zero rational 



Gi If A and B are in G, then AoB is in G. 

G 2 If A, B, C are elements of G, the result of 
operating upon the elements A and BoC, in 
the order named, is the same as the result of 
operating upon AoB and C, in the order 
named, or Ao(BoC) = (AoB)oC. 

G 3 There exists in G an element I such that 
Aol = A for every A. 

G4 There exists in G, corresponding to an ele- 
ment A, another A', such that AoA' = I for 
every A. 

numbers, a group with respect to the operation of multi- 
plication. The positive integers, on the other hand, do not 
constitute a group under either operation. 

Let us, therefore, approach the group concept through 
the positive integers. 

We can begin with something as simple as 2 + 2 = 4. 

This we know. All our lives we feel that it is some- 
thing we can depend upon. It is our symbol for what is 
changeless in a changing world and, curiously enough, we 
are not so wrong about that. For the fact in which we 
have such confidence is a specific example of a most gen- 
eral property: a property which has provided mathe- 
matics and, through mathematics, the physical sciences 
with a scalpel for laying bare the very bones of structure. 

How is it possible that something as simple as adding 
together two numbers and obtaining a third of the same 
kind can lead us to such a unifying concept? To answer 
this question, we must begin by abstracting from the 


statement that 2 + 2 = 4 the general property of which 
it is a specific example. Let us call our 2's A and B and 
say nothing more about them other than that they are 
members of the same class. (As A and B, they may be 
either the same or different members of the class.) Let 
us then call the addition represented by + an operation, 
or rule of combination, and designate it o. Instead of 
2 -f- 2 = 4, we now say that when A and B are members 
of a class, AoB (the result of combining A and B by the 
operation o) is also a member of the class. In the same 
way 2 and 3 are positive integers, and 5 the result of 
combining them by the operation of addition is also a 
positive integer. The property exhibited by A and B in 
respect to o, and 2 and 3 in respect to -f, we call the 
group property. We have already met it as Gi, the first 
of our requirements for a group. 

Gi If A and B are in G, then AoB is in G. 

This group property is the first in a succession of ab- 
stractions which has made the branch of mathematics 
called "the theory of groups" something especially abstract 
even in a subject as abstract as higher mathematics. 

To be sure that we thoroughly understand this first 
abstraction upon which all the others will rest, let us 
translate it back into the concrete. If instead of A and B, 
2 and 3 are members of the class of positive integers; and 
if we consider in order the common operations of addi- 
tion, multiplication, subtraction and division, we find that 
the results of certain operations (2+3), or 5, and 
(2 X 3), or 6, are also positive integers; but the results of 
other operations, (2 3), or 1, and (2 ~ 3), or %, 
are not. We say, then, that the positive integers exhibit 
the group property or the first requirement of a group- 
under the operations of addition and multiplication, but 
not under subtraction and division. 

So far we have been using the words class and opera- 


tion on the assumption that we know well enough what 
we mean by them; but before we continue we should do 
well to pause and define our terms in a more mathe- 
matically approved manner. We say that a class of objects 
is defined whenever a rule or condition is given whereby 
we can tell whether an object belongs or does not belong 
to the class. If we say "all positive integers," we have de- 
fined a class which does not include 0; but when we say 
"all non-negative integers," we have included in the 
class which we have defined. We say that an operation 
upon the elements A and B of a class is defined if, cor- 
responding to those elements, there exists a third thing 
called C, the result. In this general definition of an opera- 
tion nothing is said about C's being an element of the 
same class as A and B. When it is an element, as in addi- 
tion and multiplication of the positive integers and only 
then we can say that the class has the group property 
under that particular operation. 

We have seen that Gi of the four requirements for a 
group is merely the statement that a class which consti- 
tutes a group must possess the group property. We also 
recognize Gz now as the abstraction of the familiar fact 
that 1 -h (2 + 3) = (1 + 2) + 3 and that 1 X (2 X 3) 
= (1 X 2) X 3, what the textbooks call the Associative 
Laws of Arithmetic. 

Gi If A, B, C are elements of G, the result of 
operating upon the elements A and BoC, in 
the order named, is the same as the result of 
operating upon AoB and C, in the order 
named, or Ao(BoC) = (AoB)oC. 

We already know that among the positive integers the 
operations of addition and multiplication are associative 
( although of course subtraction and division are not ) , so 
we can move to the third and fourth requirements. 

G 3 and G 4 require that in a group there must be two 


elements which have very specific functions. The first, 
called the Identity, is an element which when combined 
with A always gives the result A. The second, called the 
Inverse, is an element which when combined with A al- 
ways gives the Identity as the result of combination. 

Gs There exists in G an element I such that 
Aol = A for every A. 

G-i There exists in G, corresponding to an ele- 
ment A, another A', such that AoA' = I for 
every A. 

Let us consider now whether the positive integers 
1, 2, 3, . . . , which exhibit the group property ( Gi ) and 
observe the associative requirement (G2) with respect to 
addition and multiplication, also meet the third require- 
ment for a group by possessing among their elements 
both an Identity and an Inverse. Gs requires for the op- 
eration of addition among the positive integers an I such 
that A + I = A. Since is the only number which can be 
added to an integer without changing its value (A + 
= A) and since is not included in the class of positive 
integers, we have to conclude that the positive integers 
do not constitute a group with respect to addition. In 
respect to multiplication, however, there is a number, the 
number 1, by which any integer can be multiplied with- 
out changing its value (A X I = A). So, with respect to 
multiplication, the positive integers do meet the first 
three requirements for a group. 

If they then meet the requirement of G^, they consti- 
tute a group. But G^ postulates the existence of an ele- 
ment for every member which when multiplied by that 
member will yield the Identity in this case, the number 
1. There are no such numbers among the positive integers. 
To meet the requirements of G^ we must enlarge our 


class to include the reciprocals of all the positive integers: 
%, %, }4, . . . Then A X I/A = 1, the Identity. 

If, however, we conclude that the positive integers 
and their reciprocals form a group with respect to multi- 
plication, we shall have fallen into error. Our enlarged 
class no longer exhibits the group property, although our 
original class of the positive integers did. When we mul- 
tiply integers and reciprocals., we get results which are 
neither integers nor reciprocals, and therefore not mem- 
bers of our class: 

Doggedly, we enlarge our class once again to include 
all the positive rational numbers. And now, at last, we 
have a group! 

But we have seen that the technical requirements for 
a group, although they are only four in number, can be 
slippery things indeed. To discover whether he has them 
firmly in mind, particularly the requirements for the 
Identity and the Inverse, the reader should take the simple 
test at the end of this chapter. 

In spite of the fact that there exist infinitely many 
groups, our chances that a particular class will meet the 
requirements for a group are relatively slim just as our 
chances that a particular number will be a prime are slim, 
although the number of primes is infinite. For this reason 
we say that, in spite of the fact that the number of groups 
is infinite, almost all classes with respect to a particular 
operation are not groups. 

Up to this point we have been thinking exclusively of 
groups in which members of a class (like numbers) are 
combined by a certain operation (like addition or multi- 


plication). We can also think, however, of a group as 
a class of operations which can be performed one after 
another ( the rule of combination, in this case ) to yield a 
result which could have been achieved by a single opera- 
tion. This is the same as getting "an answer" which is in 
the class when we combine two members of a class. For 
example, in the class of whole numbers, the two opera- 
tions (add 2) and (add +5) when performed in suc- 
cession yield a result which could have been achieved by 
the single operation (add -j-3). 

This concept of a group as a class of operations can be 
better understood when we examine a class of actual 
physical operations. Consider, for instance, the rotations in 
the plane which will turn a square, placed with center at 
the origin, into itself. The members of this class are four 
in number, the rotations of 0, 90, 180 and 270: 

When we subject this class of four rotations to the re- 
quirements for a group, where our "operation*' is perform- 
ing one rotation after another, we find that it meets all 
four requirements, as listed below. 

GI Any two rotations when performed in suc- 
cession are the equivalent of performing just 
one rotation: 

The rotation of 90, for example, fol- 
lowed by the rotation of 180 is the 
equivalent of the single rotation of 270. 


G2 The order of combination of the rotations 
does not affect the result. 

Gs There is an Identity element the rotation of 
which does not change the effect of any 
rotation with which it is combined. 

G-i There is for each rotation another, an Inverse 
element, which when combined with it re- 
turns the square to the starting point and 
is the equivalent of a rotation of 0, the 

A rotation of 270 followed by a rotation 
of 90 is the equivalent of a rotation of 
0, since it returns the square to its start- 
ing point. 

The group of four rotations which will turn a square 
upon itself is not only a finite group, but a very small 
finite group. Yet from it we can get a glimpse of the great 
power of the group concept. 

By working out the various possible combinations of 
our four rotations, we can construct a "multiplication 
table" for our group, where I, A, B, C are rotations 
through 0, 90, 180 and 270, respectively: 

_ I A B C 
IC = C BC = A B 2 = I A 

I A B C 

A B C 1 

B C I A 

C I A B 

This same multiplication table will work for other groups 
which do not, at first glance, appear to have any con- 
nection whatsoever with the four rotations in the plane 


which turn a square at the origin upon itself. If, for in- 
stance, we take the numbers 1, i, 1 and i and label 
them in order I, A, JB, C, we shall find that their multipli- 
cation table is the same as that of the four rotations: 

IA = A, or 1 X i = i 

AB = C, or i X 1 = 

1C C,orl X 1 = i BC = A,orl X~~i=i 

I 2 = I, or I 2 = 1 
A 2 = B,ori 2 = 1 
B 2 =l,or( l) 2 = l 
C 2 = B,or(i) 2 =~l 

This should not surprise us when we recall our interpreta- 
tion in Chapter 7 of the complex number plane as 
formed by two axes, of the real and imaginary numbers, 
placed perpendicular to one another. If we concentrate 
upon that portion of the real axis which is to the right of 
the origin ( the positive reals ) , we can see that successive 
rotations of the number plane through 0, 90, 180 and 
270 are the equivalent of multiplying the positive reals 
by 1, i, 1 and i, respectively: 


The multiplication table for a group reveals to us what 
is called its abstract group. We have seen that the four 
rotations in the plane which turn a square into itself and 
the four roots of unity have the same multiplication table. 
We know, therefore, that they have the same abstract 
group, and we can now concentrate upon one group in- 
stead of two. What we learn about the abstract group we 
can apply to the group of four rotations and to the group 
of four roots of unity as well as to any group of four ele- 
ments generated by the powers of one element. This 
means, among other things, that when in the investigation 
of some phenomenon we come upon the hitherto-unsus- 
pected pattern of our abstract group, the mathematics 
is already there and waiting for us. 

The recognition that several apparently disparate 
theories have the same abstract group may also result in 
the discovery of significant and previously undetected 
relationships among them. Consider the case of a group 
of rotations somewhat similar to our group of four. This 
is the group of all those rotations in space which turn a 
20-sided regular solid, or icosahedron, upon itself so that 
after each rotation it occupies the same volume it did 
before the rotation. The abstract group of these rotations 
is also the abstract group of certain permutations which 
we come up against when we attempt to solve the general 
equation of the fifth degree; the same group occurs in the 
theory of elliptic functions. The relationship? It turns out 
that the general equation of the fifth degree, which can- 
not be solved algebraically, can be solved by means of 
elliptic functions. Such is the power of the group concept 
to uncover similarities among apparent dissimilarities! 

With the concepts of invariance and transformation 
added to the basic concepts of group and abstract group, 
mathematics has an unbelievably powerful tool for strip- 


ping away the externals and revealing the essentials of 
structure in the physical world as well as in the mathe- 
matical. This tool is not limited in any way. It is a method 
o looking at any class of any thing under any operation 
which combines any two members of the class. It is not 
limited to infinite classes or even to very large classes. It 
is not limited to classes whose individual members have 
gaps between them but may be exhibited by classes in 
which the individual members are, practically speaking, 
indistinguishable from one another. It is not limited to 
classes in which all of the elements are essentially the 
same or in which the same operation is performed upon 
every pair of elements. We have seen that in mathematics 
the group concept is not limited to numbers. The idea of 
groups was first used in connection with the solvability 
of algebraic equations. Yet it was basic to a program 
which unified and defined the many branches of geometry. 

By utilizing the concepts of invariance, group and 
transformation, Felix Klein was able in his Erlangen 
Program to propose a criterion for determining whether 
a given discipline, perhaps as far removed from the 
geometry of Euclid as topology, is "a geometry." Under 
this great unifying principle we are able to classify some 
of the varied geometries we have already met in the fol- 
lowing manner: 

Euclidean geometry is concerned with those proper- 
ties of geometric figures which are invariant under the 
group of similarity transformations, while topology is 
concerned with those properties of geometric figures 
which are invariant under the group of continuous trans- 

But the group concept, applying equally to algebra 
and geometry, is not limited even to mathematics. It ex- 
hibits itself in the structure of the atom and the structure 


of the universe. Wherever we can apply the theory of 
groups, we are able to ignore the bewildering variety, to 
see among similarities differences and among differences 

The changeless in a changing world! 


Keeping in mind the four requirements for a group, 
which are listed on page 193, try to determine which, if 
any, of the four requirements are met by each of the fol- 
lowing classes. Which are groups? 






All integers 
All rationals 



All rationals 



All even numbers 



All even numbers 



All odd numbers 







1, 1 



1, * 



1, 1 



1, 0, 1 




1, i, 1, i 



dnoi y -[ S) "9 dnoi y * 

dnoiS y '01 s's't^ -9 dnoj y 

dnojS y -Q S<T ) *g **8iQ -[ 



. . . and paradise has been with, math- 
ematics since its beginnings. It lies, 
unstated, in the assumption upon which 
Euclid's geometry rests. It is implicit 
in the first numbers with which we 

M begin to count. 
0, 1, 2, 3, ... * 
The three dots after these first few 
numbers indicate to us that they are 
enough for counting: that we shall 
never run out of numbers to count 
Counting with, for there is no last number. The 

the Infinite counting numbers are infinite. They 

are also enough to count the infinite, 
provided it is not too large. They are 
not, however, enough to count the 
points on any line, no matter how 

Before we can understand these 
paradoxical statements about counting 
the infinite, we shall have to revise 
our ideas about several things: about 
"counting," for one, and about "the 
infinite," for another. 

It is quite possible to count with- 
out 0, 1, 2, 3, .... A bird that can tell 
when one of four eggs has been re- 
moved from her nest probably has a 

* The reader may find it difficult to ac- 
cept as one of the counting numbers, but 
with what other number will he "count" the 
unicorns in his living room? 


mental picture of the eggs in the nest with which she can 
"count" the eggs upon her return. Man's first numbers 
apparently consisted of such grouping pictures man him- 
self, bird wings, clover leaves, legs of a beast, fingers on 
his hand with which other groups could be compared 
and "counted." If there were as many birds as fingers on 
his hand, and as many arrows as fingers, then he knew 
there were "as many'* birds as arrows, and an arrow for 
every bird. 

Formally we call what he was doing "counting by 
one-to-one correspondence" and we probably think of it 
as a rather inferior trick compared to counting with num- 
bers. Yet what we are doing with our numbers is essen- 
tially the same tiling. Say that we have a bowl of apples 
and a party of children. We count the apples and find 
that we have 7; then we count the children and find that 
we have 7. We have the same number of apples and chil- 
dren, so we have an apple for every child. We could also 
have handed an apple to each child and when we came 
out even we would have known, without knowing the 
number of children and apples, that we had "as many" 
apples as children. When we diagram what we have done, 
we see that in both cases we were counting by one-to-one 
correspondence very much like man with arrows and 

apple < > child apple < > 1 < > child 

apple < > child apple < > 2 < > child 

apple < > child apple < 3 < child * 

Counting by one-to-one correspondence is the most 

* We have followed here the conventional method of begin- 
ning to count with 1; but is logically one of the counting numbers 
and we can count just as well by beginning with 0. When we do, 
the answer to the question "How many?" is the successor of the 
last number which we paired with the last member of the collection. 


primitive and also, as we shall see, the most sophisticated 
method of counting. 

The ancient method of directly comparing two col- 
lections to determine the number of members is the logical 
basis for a definition of what we mean by "number" 
which can be extended to infinite as well as finite collec- 
tions. Let us firmly banish 0, 1, 2, 3, ... from our minds 
for a moment and think instead of all the finite collections 
we might possibly want to "count" being grouped in such 
a way that all those which can be placed in one-to-one 
correspondence with each other all the collections of a 
dozen members, for example are in the same group. 
These groups do not need to be arranged in order of the 
size of their respective collections. For the moment it is 
sufficient for our purposes that they have been grouped. 
We have all those collections whose members can be 
placed in one-to-one correspondence with a dozen eggs, 
all those whose single member can be placed in one-to-one 
correspondence with the sun, and so on. 



Day, Night eyes, antlers, wings, man 

and woman, good and 
evil, . . . 

Breakfast, Lunch, Dinner ears and mouth, clover 
leaves, man-woman-child, 
stars in Orion's belt, . . . 

Sun head, self, earth, moon, 

god, . . . 

Now, instead of having to keep in mind the specific 
collections we are using for our models, we can substitute 


an X for each member so that we have XX, XXX, and X. 
We can then easily arrange these new model collections 
in the order of their increasing size and, if we want, can 
give them names. We are now ready to define A, or what- 
ever name we have given the model collection X, as the 
cardinal number of any class whose members can be 
placed in one-to-one correspondence with X, or the Sun. 
If someone objects and says that all we have done is to 
define the number 1, why we shall be generous and call 
A "1." Then we shall call our next krgest model collection 
"2" and define it as the cardinal number of any class 
whose members can be placed in one-to-one correspond- 
ence with XX, or Day and Night; and so on, to infinity. 

The number of cardinal numbers we can define in this 
way is infinite, but the members of each collection in the 
classes so defined will be finite. The number of members 
in each collection may be very large: all those collections 
whose members can be placed in one-to-one correspond- 
ence with all the stars in the Milky Way, all those whose 
members can be placed in one-to-one correspondence 
with all the grains of sand on the earth, all those whose 
members can be placed in one-to-one correspondence 
with all the electrons in the universe. It may be personally 
impossible for us to count all the members of a particular 
model collection, but they are "countable" in the sense in 
which we commonly use the word. The cardinal numbers 
which we have defined are finite cardinal numbers. 

But is there any reason why in this same way we can- 
not define transfinite cardinal numbers for classes which 
contain an infinite number of members? 

It is at this point that we must change our idea of 
"the infinite." For instance, instead of thinking of the 
counting numbers 0, 1, 2, 3, ... as an ever-growing pile 
filling room, world, universe, . . . , we must think of them 
stuffed, as it were, into the metaphorical suitcase of their 


class. In short, we must think of them not primarily as 
infinite in number but as an infinite class, something 
which we can handle as a unit, just as we handle finite 
classes, but something which is still different from a finite 
class because of the fact that it is infinite. This was not 
an easy idea, even for mathematicians, to accept. Yet once 
we accept it, we have something "capable not only of 
mathematical formulation, but of definition by number." 
These are the words of the man who, almost singlehanded, 
corralled the infinite for mathematics. 

Georg Cantor, whom we met before as one of the 
authors of the Cantor-Dedekind axiom, was one of those 
rare people who are able to look at the familiar as if they 
have never seen it before and thus become the first to see 
it. How revolutionary was his idea of the infinite, as 
something consummated, is shown by his own words in 
presenting it to his mathematical colleagues: "This con- 
ception of the infinite is opposed to traditions which have 
grown dear to me, and it is much against my own will 
that I have been forced to accept this view. But many 
years of scientific speculation and trial point to these con- 
clusions as a logical necessity." 

Once we have recognized counting as matching one 
class to another in one-to-one correspondence and an in- 
finite number as something consummated an infinite 
class we are ready to take the next step, which is count- 
ing the infinite by placing one infinite class in one-to-one 
correspondence with another! Doing this, and even the 
specific way of doing it, was not original with Georg 
Cantor, living and creating in nineteenth-century Ger- 
many and fighting an abstractly bloody battle not only 
with his colleagues but also with a mathematical tradi- 
tion of the infinite which went back to the Greeks. 

Three hundred years before Cantor, in the Italy of the 
Inquisition, Galileo had pointed out that the infinite class 


of squares can be placed in one-to-one correspondence 
with the infinite class of natural numbers: that there are 
fully "as many" squares as there are natural numbers, 
since every number when multiplied by itself produces a 

Unfortunately, Galileo, with Cantor's theory of the in- 
finite in his palm three hundred years before Georg 
Cantor was even born, dismissed it: "So far as I see, we 
can only infer that the number of squares is infinite and 
the number of their roots is infinite; neither is the num- 
ber of squares less than the totality of all numbers, nor 
the latter greater than the former; and finally the attri- 
butes equal, greater, and less are not applicable to in- 
finite, but only to finite quantities." * 

What Georg Cantor did three hundred years after 
Galileo was to take the attributes of equal, greater and 
less and apply them to infinite quantities. 

When we take the first few numbers and set them off 
according to some of the various classifications which have 
been made, we come out with something like this: 



135 1 

257 4 

37 9 

4 9 






* Galileo spoke here through the character of Salviatus in his 
Mathematical Discourses and Demonstrations. 


If we total these various classifications, we find that 
among the first ten numbers we have five odd numbers, 
four squares, three odd primes, and only one prime of the 
form 4n + 1. We have no trouble in determining that the 
class of numbers from through 9 is greater than any of 
these sub-classes, that the odd numbers and the even 
numbers are equal, and that the class of primes of the 
form 4n + 1 is less than any of the other classes. If we 
attempt to place any of these sub-classes in one-to-one 
correspondence with the numbers from through 9, we 
shall have at least five numbers left over. But what hap- 
pens if, in following the same system, we take, instead of 
the first ten, all of the natural numbers and all of the 
members of the same sub-classes? 








4n + l 



































It is already apparent. The three dots at the end of 
each column indicate that each class of numbers is in- 
finite; in spite of the fact that we appear to be exhausting 
some of the classes, like the 4n + I primes, more quickly 
than the others, we only appear to be doing so. We can 
never exhaust an infinite class. When we consider a finite 
class of whatever size we please, the natural numbers in 
the chosen ckss will far outnumber any one of the sub- 
classes; but when we take all of them, they are equal to 
any one of the equal sub-classes. 


Galileo said that they were neither more nor less, and 
that the attribute of equal was not applicable to infinite 
quantities. Cantor said that infinite quantities are equal 
when they can be placed in one-to-one correspondence 
with each other; they have the same cardinal number! 

Just as we said that all classes which could be placed 
in one-to-one correspondence with the class of the Sun, or 
X, had the same cardinal number, which we call 1, Cantor 
said that all classes which can be placed in one-to-one 
correspondence with the natural numbers have the same 
cardinal number, which he called aleph-zero or KO- It is 
different from the finite cardinals only in that it is trans- 

We have already seen how sub-classes of the class of 
natural numbers can be placed in one-to-one correspond- 
ence with the whole of which they are a part; but so curi- 
ous are the workings of infinite classes, as opposed to 
finite classes, that we can also do our pairing the other 
way around. We can set off in one-to-one correspondence 
with the natural numbers a class of numbers of which they 
themselves are a sub-class. The class of all integers has one 
peculiarity which its sub-class, the natural numbers, does 
not have: it has neither a last nor a first member. How, 
then, can we pair it off with the natural numbers? This is 
not so difficult as it might seem. It is merely a matter of 
ordering the integers in such a way that they can, as it 
were, stand up and be counted. With no beginning, we 
begin right in the middle at and then count each pair of 
integers, positive and negative, in turn. 

1 2 3 4 5 6 7 8... 

I 1 I 1 I I 1 I I 

+1 1 +2 2 +3 3 +4 4 ... 
There is no particular trick to pairing the natural 
numbers with the integers, which include them as a sub- 


class; but such a pairing does serve to show an important 
technique in counting the infinite. A class of numbers 
which may not appear to be countable ( in the case of the 
integers, because there is no first number) can often be 
rearranged in such a way that it can be counted. Consider 
the class of all positive rational numbers. These are num- 
bers of the form a/b where a and b are both integers. 
When a is smaller than b, we have what we called in 
grammar school a "proper" fraction; when b is smaller, 
an "improper" one. The class of all positive rational num- 
bers is no straightforward sort of infinity like the class of 
integer squares where we have just one member of the 
class for each integer. Just one small sub-class, a/b where 
a is 1, is infinite in number. Since a may take any integer 
value and for every a, b may take any integer value, we 
appear to have among these numbers infinity upon in- 
finity, an infinite number of infinities. 

If we take the positive rationals in what might be 
called their natural order, omitting those with common 
factors since they are already represented, we find that 
placing them in one-to-one correspondence with the nat- 
ural numbers is impossible. Not only is there no "smallest" 
fraction, but also there is no "next largest" fraction. Be- 
tween any two a/b and c/d an infinity of fractions larger 
than ajb and smaller than cjd spring up to vex us. Ob- 
viously it is impossible for us to pair off with the natural 
numbers a class of numbers which behave in this fantastic 
fashion. We have sown dragon teeth on the number line. 

But remember, we have said nothing about the 
rational numbers having to be paired off in their natural 
order only that they must be paired in such a way that 
we can see that we are going to be able to count them 
with the natural numbers. So let us rearrange the rational 
numbers. Let us organize them into battalions: the first 
battalion consisting of all those rational numbers whose 


numerator is 1, the second battalion consisting of all those 
whose numerator is 2; and so on. 

This arrangement is reminiscent of one of those parades 
during which we wait restlessly for the band while an 
apparently endless procession of foot soldiers goes by. 
The only difference between our parade and the actual 
parade is that it is not just seemingly endless; it is end- 
less. The band, or even the second battalion, can never 
pass by. Obviously, again it is impossible to count off by 
placing in one-to-one correspondence with the natural 
numbers a set of numbers which behave in this fashion; 
for although in counting the primes, for instance, we 
would never finish, we would always be able to count as 
far as any prime we might care to choose. With this ar- 
rangement of the rational numbers, not only could we 
never get to the end, but we could never get to %! Have 
we then come at last upon an infinity which is impossible 
to pair with the natural numbers, an infinity whose car- 
dinal number is different from and perhaps larger than 

No, we have not. 

The simple method by which Georg Cantor ordered 
the positive rational numbers so that they can be placed 
in one-to-one correspondence with the natural numbers 
has the quality of genius. All he did was to take the group- 
ings which we have called battalions and arrange them 
in rows instead of in one long straight line. 

M % y* y* y> VG ... 

% % % % % MI ... 

At this point we might stop for a moment and see, with 


this much of a hint, whether we can now order the 
rationals in such a way that every one will be paired with 
a unique natural number and whether we will be able to 
count with the natural numbers to any rational we choose, 
such as "%. . . . 

Cantor's way was to order them diagonally, begin- 
ning in the upper left-hand corner with M . 

Thus we have all the rationals placed in one-to-one cor- 
respondence * with the natural numbers and we quite 
promptly get to %. 

012 3456789... 

4'4'4' * v 4 1 4' J' * NT 

M % y 2 % % % # % % % ... 

This is mere child's play compared to the task of 
arranging the algebraic numbers so that they too can be 
placed in one-to-one correspondence with the natural 
numbers. The algebraic numbers are all those numbers 
which are roots of algebraic equations of the form 

aox n + ax n -i + . . . -f- dn-ix + On = Q 

in which the coefficients do, a-i, . , a are all integers. 
This is nothing more than the general expression for the 
algebraic equations with which we are familiar where n 
has a value of 1 or 2. When n = 1, we have a simple equa- 


tion like 2x 1 = 0, where we can see at a glance that 
the root, or value of x, must be %. When n = 2, we have 
a familiar quadratic equation like 3x 2 + 4x + 1 = 0, 
where the roots, or values of x, are 1 and %. The essen- 
tial thing for us to remember is that when such an alge- 
braic equation has whole-number coefficients, as in our 
examples, it always has a root among the complex num- 
bers. (This is the Fundamental Theorem of Algebra, 
proved by Gauss. ) Those complex numbers which can be 
roots of such algebraic equations are called the algebraic 
numbers. They are not, as we shall see, all of the complex 
numbers by any means. 

Cantor's proof that these algebraic numbers can be 
placed in one-to-one correspondence with the natural 
numbers has been called "a triumph of ingenuity"; yet it 
is essentially as simple as the alphabetization of the tele- 
phone book. The crux of the method is what Cantor called 
the height of an algebraic equation. This is the sum of the 
absolute values of the coefficients plus the degree of the 
equation less 1. (The absolute values are the numerical 
values of the coefficients with no attention paid to whether 
they are positive or negative; the degree is the highest 
power of the unknown x, or the value for n in the general 
expression as given above.) Thus the equation of the 
third degree 

5 = 

has a height of 19, since 3 + 4 + 5 + 5+ (3 1)= 19. 
Having assigned for every algebraic equation a 
method of determining its height as an integer, Cantor 
proved that for any integer there is only a finite number 
of equations which have that particular integer for their 
height. From this point on, the method of the phone book 
comes in handy. When we have ordered all algebraic 


equations according to their height, we find that in most 
cases we have more than one equation of a particular 
height. Undaunted, we arrange the equations of the same 
height according to the value of their first coefficient and, 
where the first coefficient is the same, according to the 
second, and so on. Since there is only a finite number of 
equations with the same height, and since no two equa- 
tions can have exactly the same coefficients, we have 
assigned every algebraic equation to a unique position in 
an order arrangement. 

Our purpose, however, is not to order the equations 
but to order the numbers which can be their roots the 
algebraic numbers so that they can be placed in one-to- 
one correspondence with the natural numbers. So we 
continue by taking the roots of the ordered equations, 
which may be more than one but are never more than the 
degree of the equation, and arranging them according to 
their increasing value, first according to the value of the 
real part and then, where several numbers have the same 
real part., according to the value of the imaginary part. 
By agreement, as in the case of the rational numbers, we 
throw out those which are repetitions. We now have a 
method by which every number which can be the root of 
an algebraic equation can be paired with one of the nat- 
ural numbers this in spite of the fact that we have not 
actually written down the roots of a single equation! 

Cantor's "triumph of ingenuity" can be best appreci- 
ated when we recall our diagram of the complex number 
plane as formed by axes of the pure imaginary and of the 
real numbers and recall that, although the algebraic num- 
bers are not all the numbers upon the plane, they are 
everywhere dense upon it, while the natural numbers 
mark only the units on one-half of the real-number axis! 


Yet these two seemingly unequal classes have the same 
cardinal number, Ho- 
ls KO the only transfinite cardinal? 
We are beginning to suspect that perhaps it is. We 
have examined many infinite classes of numbers which 
represent certain specific points upon the complex number 
plane. All of them are, of course, sub-classes of the com- 
plex numbers. Some are sub-classes of the natural num- 
bers as well, and some include the natural numbers as one 
of their sub-classes. Yet always we have found (with 
Cantor ) that the classes we have examined can be ordered 
in sucb a way that they can be pkced in one-to-one cor- 
respondence with the natural numbers and, therefore, 
have the same transfinite cardinal, KO- 

[ 4n + 1 primes | 
| odd primes | 

[ odd numbers | 
| natural numbers [ 

| integers [ 

I rational numbers I 

I algebraic numbers I 

Although we can define infinite classes as being equal, 
it seems that we cannot define them as being greater or 
less. Perhaps we were right to begin with: an infinite 
number is just an infinite number. Fortunately, we were 
wrong. If we were right, the infinite would be an Infinitely 
less interesting subject than it is. There is a transfinite 
cardinal greater than KO there is, in fact, an infinite 
number of greater transfinite cardinals! But at the mo- 


ment we shall be satisfied with only one. We can find an 
infinity which is greater than the infinity of natural num- 
bers on a very small part of the real-number line: the 
segment between and 1. 

To show that these real numbers, which are the equiv- 
alent of all the points on the segment, cannot be placed 
in one-to-one correspondence with the natural numbers, 
Cantor began by assuming that they could be. This is a 
method of mathematical proof as old as Euclid, who used 
it to show that the number of primes is infinite. It was 
also used by Fermat to show that all primes of the form 
4n + 1 can be expressed as the sum of two squares. In this 
case, to prove that the placing of the real numbers in one- 
to-one correspondence with the natural numbers is im- 
possible, Cantor risked assuming that such a pairing was 
indeed possible. 

As we saw in Chapter 4, all numbers on the real- 
number line between and 1 can be represented as never- 
ending decimal fractions, and this is the way in which 
Cantor chose to represent them. If, however, we start to 
write down the actual decimals, we immediately become 
involved in all sorts of difficulties. The first would be 
0.000000000 . . . with the O's continuing to infinity; but 
what would be the second decimal? No matter how many 
O's we place between our decimal point and our first posi- 
tive place value, we can always construct a smaller deci- 
mal by inserting one more and moving our first positive 
place value over one more place to the right. 




Have we proved, then, that it is Impossible to arrange the 
real numbers from to 1 in such a way that they can be 


placed in one-to-one correspondence with the natural 
numbers? No. We have proved nothing of the kind. Only 
that we have not been able to find a way of doing what 
we want to do. The question then becomes, not whether 
we can find a way, but whether there is a way. 

To prove that there isn't a way, we begin by assuming 
that there is. We solve the problem of determining the 
second decimal and all succeeding decimals by assuming 
that they have been determined. We then think of them 
abstractly as expressions like ^.a^a^a^a^a^a^a-: . . with 
each On denoting the particular value (0, 1, 2, 3, 4, 5, 6, 7, 
8 or 9) of each place in the decimal; and we place them 
in one-to-one correspondence with the natural numbers, 
in accordance with our assumption that they can be so 

Cantor showed that such an assumption was false 
because, even assuming that all decimals could be and had 
been placed in one-to-one correspondence with the nat- 
ural numbers, he could construct a decimal which had 
not been included in the class of "all" decimals so ordered. 
This decimal he indicated by 

mi being any digit (except 9)* other than the digit rep- 

* Since terminating decimals like .25 can be represented as 
non-terminating decimals in two ways: either as .250000 ... or as 
.249999 . . . , we exclude 9 to avoid having our new decimal a 
different representation of a number which has already, in a dif- 
ferent form, been included in the class of "all" decimals. 


resented by ai in the first decimal; m 2 being any digit 
(except 9) other than the digit represented by b 2 in the 
second decimal; and so on. This new decimal would be 
one not included in the original class of "all" decimals 
because it would differ from every included decimal in at 
least one place: from the first in at least its first place, 
from the second in at least its second place, and so on. 

We can see a little more vividly what Cantor did if 
we take a concrete set of decimals and then by following 
his method construct a decimal not in our set. 

0.02468 ... To get a decimal not in the 

\ set, we make the first place of 

0.13579 . . . our new decimal different from 

\ 0; the second, from 3; the third, 

0.23571 . . . from 5; and so on. It will differ 

\ in at least one place from any 

0.35712 . . . decimal in the set: 0.14623 . . . 

\ is not included, and there are 

0.49012 . . . many other possibilities. 

It is almost impossible to overestimate the importance 
of this achievement. Already Cantor had shown that the 
attribute equal was applicable to infinities; now he showed 
that the attributes greater and less were also applicable. 
The new cardinal number, which is easily shown to be 
larger than xo, the cardinal number of a "countable" in- 
finity, is C (pronounced like "c"), the number of what 
Cantor called the continuum an "uncountable" infinity! 

What other infinities have this same t as their car- 
dinal number? 

The answer to this question is completely contrary to 
intuition. We have noted that the real numbers from to 
1 are equivalent to the points on the segment of the real 
number line from to 1, just as all the real numbers are 


the equivalent of all the points on the line. Our intuition 
tells us that the infinity of real numbers must be greater 
than the infinity of real numbers between and 1, just as 
the infinity of points on the line must be greater than the 
infinity of points on the line segment between and 1. 
Yet it is very easy to prove that for every point on the 
long line there is a point on the short line and that, there- 
fore, there are as many real numbers between and 1 as 
there are in all the length of the real-number line! 

To prove this statement, we shall take two lines (one 
short, which we shall call AB, and one somewhat longer, 
which we shall call CD) and place them parallel to one 
another. We shall then construct one line which passes 

through A and C and another line which passes through 

___ ^ 

B and D. The intersection of these two lines we shall call 
O. It is clear that we can draw a line from O to any point 
Q which we choose on line CD, and that this line OQ 
will of necessity intersect line AB at some point P. 

, *?', 


:5. iawaii-sL USE i. ^ ^ Ji.z, 

For every Q on the longer line there will be a unique 
point P on the shorter line which can be placed in one-to- 
one correspondence with it. 

It is also possible to prove, although not so easily, that all 
the points on the plane can be placed in one-to-one corre- 
spondence with the points on a line segment of any finite 
length. All of these infinities of points have the same car- 
dinal number, c. Since the real numbers represent all the 
points on the line, and the complex numbers all the points 
on the plane, they also have c as their cardinal number. 

Now that we have distinguished between two types 
of infinities, those which, like the natural numbers, are 
"countable** and those which, like the real numbers, are 
"uncountable," we might think that we were finished with 
the subject of the infinite. But the infinite is not so easily 
disposed of. 

There are an infinite number of transfinite cardinals 
which are greater than c, which is greater than K o- 

This important fact in the arithmetic of the infinite is 
stated by a very simple theorem to the effect that 

2 n is always greater than n 

and supported by a very simple proof. If we consider this 
theorem when n is a finite cardinal number, we can see 
that it is true. We take n blocks n in this case being 
equal to 3 and paint each block either blue or red. The 
number of possible color schemes will equal 2 n , or 2 3 = 8 
in this case. 

B B B 

B B R 

B R B 

B R R 

R B B 

R B R 

R R B 

R R R 

As here, when n is a finite cardinal, we can actually count 
the color schemes and can actually see that we have ex- 
hausted the possibilities: no one can turn up with another 
color scheme for three blocks painted either red or blue 
which is not already among the color schemes we have. 
But now let us take n = xo- Let us take as many 
blocks as we have positive integers. Again, let us paint 
each block either blue or red. How many possible color 
schemes can we have? Certainly an infinite number. For 
instance, we could in each case paint the nth block blue 
and all the others red. 

1 . First block blue and all others red. 

2. Second block blue and all others red. 

3. Third block blue and all others red. 

Obviously this is too easy. After we have paired a 


unique color scheme with every one of the positive inte- 
gers, we can think up one or an infinity more of schemes 
which we have not included. For instance, we could paint 
the nth and the (n + l)th block blue and all the others 
red, and this would give us a completely different set of 
color schemes which could also be placed in one-to-one 
correspondence with the positive integers. But remember 
that even another infinity of color schemes does not prove 
that both sets of color schemes could not be placed in 
one-to-one correspondence with the positive integers, or 
even all possible color schemes! 

So let us assume that by some method we have de- 
termined all possible color schemes and to each block we 
have attached one of the color schemes. Now can we 
come up with a color scheme which is not among those 
attached to the blocks? We can and we do using the 
same method by which we constructed a decimal which 
was not in our original set of "all" decimals. We pick up 
the first block and note from the color scheme attached 
to it what color it is to be painted in that particular 
scheme. Then we paint it a different color, red if it was 
blue on the list, blue if it was red. The color scheme which 
results from our newly painted blocks cannot possibly be 
one of those already attachedor paired in one-to-one 
correspondence to the blocks. It will differ, for at least 
one block, from each of the color schemes we already 
have. The cardinal number, then, of all possible color 
schemes is greater than the cardinal number of the blocks 
because the color schemes cannot be placed in one-to-one 
correspondence with the blocks. Our theorem 2 n is 
greater than n is true whether n is finite or transfinite. 

It follows, therefore, that for any transfinite number 
there is always another and greater transfinite number. 


There is no last transfinite number. The number of trans- 
finite cardinals is infinite! 

Of this infinitude of transfinite numbers NO, as its sub- 
script indicates, is the first. What is K i ? The cardinal num- 
ber of the continuum, c, is larger than K O . There is no 
known transfinite number that is smaller than t and larger 
than So- But is C the second transfinite number? Is it Si? * 

In modern mathematics this problem holds the place 
that the problem of the trisection of the angle held in an- 
cient mathematics. We have indeed counted the infinite, 
but we are not done with it! 

* That c is x x is the famous "continuum hypothesis." 



A Most 



faces have area. Solids have volume. 
The measure assigned to a figure- 
length, area or volume, as the case may 
be is unaffected by rigid motion of the 
figure. The whole is greater than any 
part, and is the sum of all the parts 

These statements are as ancient as 
Euclid and at the same time so com- 
monplace that we cannot conceive of 
their being controverted. Yet in the 
theory of point sets, a branch of mathe- 
matics in which the paradoxes are al- 
most as numerous as the points (and 
the points are very numerous indeed), 
we are forced to the conclusion that 
under certain conditions, involving the 
most familiar figures of geometry, some 
of the statements we have made are 

To understand the necessity for 
this conclusion, we must go back to 
that unfortunate Pythagorean who dis- 
covered that there can be no rational 
number for the point on the measuring 
stick which coincides with the diagonal 
of the unit square, and perished at sea 
for his pains. From this point, quite 
literally, we are logically committed to 
the theory of point sets, although the 
theory itself was not founded until 
some twenty-five hundred years later. 
When, toward the end of this chapter, 


we find ourselves balking at some of the conclusions at 
which we arrive, we must remember that here at the be- 
ginning we easily accept in fact, insist upon the assump- 
tion from which the conclusions will necessarily follow. 
Who among us would now renounce the idea that for 
every length there is such a unique measure as V2 for the 
diagonal of the unit square? 

The logical consequences of this concept of a number 
for every point on the line, or the theory of point sets, will 
be the subject of this chapter. In the course of it we shall 
find ourselves juggling infinities and distinguishing pre- 
cisely between those which are non-denumerable and 
those which are denumerable; transforming by rigid mo- 
tion whole infinities of points; selecting single points from 
infinities. Unfortunately, this is not material that can be 
skimmed. We can only remind the reader that there is no 
royal road to even the faintest understanding of the con- 
cept of point sets, and assure him that if he follows the 
rocky road of reasoning he may be more than repaid by 
the satisfaction he gets from a personal contact with pure 

We must begin by considering what we mean by "a 
point." When we take a pencil and make with it on paper 
what we call a point, we have what for all practical pur- 
poses is a point. But a point (mathematicians agreed 
about the time of the Pythagorean) is that which has 
position but no magnitude. Since any representation of 
a point must have magnitude, it cannot be a point. More 
recently, since the time of Descartes, mathematicians 
have based their definition of a point on its representa- 
tion by numerical coordinates. A point on the line is a real 
number. A point in the plane they define as an ordered 
pair of real numbers; a point in space, as an ordered triple 
of real numbers; and so on. It is from this definition of a 


point as a number, and a number as a point, that the great 
paradoxes of point-set theory develop. 

When we start to think of points as numbers, we gain 
an advantage in handling them. Each one becomes an 
individual, easily distinguishable from ah 1 the others. We 
can divide an infinity of points into mutually exclusive 
sets and have no trouble at all in determining whether 
a given point belongs in a set. All the points on the line, 
for instance, can be divided into those which represent 
a real number less than and those which represent a real 
number greater than 0; while a third set, the single point 
0, serves as the boundary between the other two sets. 

We can make a similar division of the points on the plane 
by including in one set all those the x-coordinate of which 
is less than and in the other, greater than 0. Here the 
boundary set will contain not just one point but all those 
points with x 0, or the f/-axis itself. 

If we inscribe a figure on the plane let us say a circle 
of radius 1 about the origin we can distinguish the points 
which are on its circumference from all the other points in 
the plane. Physically, this is impossible; for our drawing, 
no matter how finely done, must add magnitude to the 
position of the points. Mentally, though, such a selection 
is perfectly possible. 

The equation for the given circle is 

since, by the Pythagorean theorem, the sum of the squares 
of the x and y coordinates at any point on the circum- 
ference will give us the square of the hypotenuse, which 
is also the square of the radius of the circle, in this case 1. 
There are various sets of points which we can represent 
by means of this knowledge. The equation itself is the 
equivalent of the statement "all the points x, y for which 
the equation holds." If we take at random two points, say, 
(4, 3) and (%, %), we find that 


where the symbols >, < are read as "is greater than" and 
"is less than," respectively. It is clear that (4, 3) and 
(%, % ) are not among the points on the circumference of 
our circle. If, in fact, we locate them on the plane pictured 
on page 69, we can actually see that (4, 3) would fall out- 
side of a circle of radius 1 about the origin while (%, %) 
would fall inside. Thus, with the equation for the circle 
already given and various related equalities, we are able 
to divide the points on the plane into various sets: 

A. x 2 + y 2 = 1 the set of points on the cir- 


B. x 2 -f y 2 < 1 the set of points interior to 

the circle 

C. x 2 + y 2 > 1 the set of points exterior to 

the circle 

D. x 2 -f- y 2 ^ 1 the set of points on the cir- 

cumference and the inte- 
rior of the circle 

E. x 2 -f- y 2 ^ 1 the set of points on the cir- 

cumference and the exte- 
rior of the circle 

F. x 2 + y 2 = 1 the set of all points not on the 


Certain pairs of these sets, when combined, will include 
all of the points in the plane and yet will have no points 
in common: (A) and (F), (B) and (E), (C) and (D). 
These are called complementary sets. 

When we divide the entire plane into such parts, even 
though we cannot physically represent some of them, like 
the points on the circumference or the interior of the circle 
without the circumference, we are still dealing with the 
concept of the whole and its parts in the traditional man- 
ner. The plane is the sum of its sub-sets (A), (B) and 
(C); each occupies a "different" portion of the plane. Yet 
with point sets it is possible to divide the plane into var- 
ious pairs of complementary sets in such a way that each 
set of the pair by itself is everywhere dense upon the 
plane. Such a pair would be the set of all points in the 
plane which have rational coordinates; and its comple- 
ment, the set of all points which have at least one irra- 
tional coordinate. Together, they include all the points in 
the plane, which are everywhere dense. Yet, when we 
remove either set of points, the points remaining are still 
everywhere dense in the plane. This curious situation 
arises from the fact that the rational numbers are every- 


where dense (i.e., between any two rational numbers 
there is always another rational number) and that the 
same characteristic is exhibited by the irrational numbers. 

There is yet another unconventional way in which we 
can divide the whole point set into parts, or sub-sets of 
points, a way which is not available to us when we are 
dealing with geometrical figures in the traditional manner. 
As we have seen, we can divide a point set into a finite 
number of complementary sets, or parts; but we can also 
divide it into an infinity of such parts. The number of 
points on a line, in a plane or in a space is always the 
same: a non-denumerable infinity. If we divide any one 
of these point sets into sub-sets, each of which contains 
but a single point, we have divided the whole into a non- 
denumerable infinity of parts. 

Such a non-denumerable infinity is infinitely more 
numerous than a denumerable infinity; yet we can also 
divide a point set which contains a non-denumerable in- 
finity of points into a denumerable infinity of sub-sets. 
Later we shall see that this is sometimes a rather com- 
plicated procedure, but now we shall merely divide the 
real-number line into a denumerable infinity of parts. This 
is child's play in the theory of point sets. By defining each 
sub-set as all the real numbers equal to and greater than 
a given integer n but less than the next largest integer, 
or n + 1, we have solved the problem. The integers, 
a denumerable infinity themselves, divide the non- 
denumerable infinity of real numbers, which represent 
all the points on the line, into a denumerable infinity of 
sub-sets, each of which of course contains in turn a non- 
denumerable infinity of points.* 

* The reader is reminded of the proof on page 218 and the 
following pages that the real numbers between and 1 are a non- 
denumerable infinity, and of the proof on page 221 that the num- 
ber of points on any portion of the line is equal to the number of 
points on the entire length of the line. 


The distinction between non-denumerable and denu- 
merable infinities, as confusing as it may be to us at first, 
is essential to our gaining even a glimpse of the reasoning 
which leads to the paradoxes of point-set theory and their 
implications for the theory of measure. We must, there- 
fore, make sure that we have it clearly in mind before we 
go any further in this chapter. We recall from Chapter 
14 that a denumerable or countable infinity (the "small- 
est" of all infinities ) is one whose members can be placed 
in one-to-one correspondence with the integers, and thus 
in the sense that there is an ordered pairing between its 
members and the integers can be counted. Such count- 
able infinities include the integers themselves; such sub- 
sets as the natural numbers, the even numbers, the primes, 
and so on; and, what is particularly important to us in 
point-set theory, the rational numbers. A non-denumerable 
infinity, as we saw in the same chapter, is more numerous 
than the integers, cannot be arranged in any way so that 
its members can be paired with them, and hence cannot 
be "counted" in the same sense that a denumerable in- 
finity can be counted. Such uncountable or non-denu- 
merable infinities include the real numbers which singly, 
in pairs, or in triples can be placed in one-to-one corre- 
spondence with the points of line, plane and space, re- 
spectively. They also include a non-denumerable sub-set 
of the reals which is particularly important for point-set 
theory the irrational numbers. It is essential that we keep 
in mind the fact that while the rationals and the irrationals 
are complementary sub-sets of the real numbers, the ra- 
tionals are denumerable and the irrationals are non-denu- 

In brief summary: 

1. Each of the geometrical figures, plane and solid, 
with which we shall deal in the next few pages contains a 
non-denumerable infinity of points. 


2. Each and every one of such a non-denumerable in- 
finity of points can be handled as an individual because 
it can be uniquely defined by ordered real-number coordi- 

3. The real numbers, which are the rational numbers 
plus the irrational numbers, are a non-denumerable in- 

4. The rational real numbers are a denumerable in- 

5. The irrational real numbers are a non-denumerable 

We are now prepared to follow the reasoning which 
will lead us to a fundamental paradox of point-set theory: 

The whole is not necessarily greater than one of its 
proper parts, but on the contrary can be congruent to that 

The word congruent here means "equal" in that special 
sense in which we use it in the geometry with which we 
are all familiar. In point sets we always use it in this sense. 
As a specific example, we say that the triangles A and B 
below are congruent if, without lifting the left-hand tri- 
angle out of the plane, we can, by rigid motion alone 
(sliding along the page in this case), superpose it upon 
the right-hand triangle so that the two occupy exactly the 
same position and there is a one-to-one correspondence 
between their points. The triangle C, as can be seen, is a 
proper part of A; but since A can never, by rigid motion 
alone, be superposed on C, they are not congruent. 


In point-set theory the meaning of the word congruent 
is exactly the same as it is in traditional geometry super- 
position and one-to-one correspondence achieved by rigid 
motion alone. But here th6 resemblance stops. For in tra- 
ditional geometry we never find, as we do in point sets, 
that the whole can be congruent to its proper part. We can 
never superpose A in the figure above upon C, its proper 
part; but we can superpose the whole right-hand half of 
the plane, or the set of all points such that x > 0, upon a 
proper part, the set of all points with x > 1: 

It is "obvious" to us that the entire right-hand half 
of the plane (x > 0) is "larger" than that "part" of it 
(x > 1) which lies to the right of 1, "larger" in the same 
way that triangle A is larger than triangle C. Yet, recalling 
Cantor's theory of the infinite, we know that it is perfectly 
possible for an infinite set (such as the integers) to be 
equal (because placed in one-to-one correspondence with 

it) to a proper part (such as the even numbers) . It is only 
a step to the recognition that the half -plane of points can 
be superposed on its proper part because the points of 
each can be placed in one-to-one correspondence merely 
by sliding the whole onto its part. Since such superposi- 
tion achieved by rigid motion is the accepted definition of 
congruence, we can say in this situation that the whole is 
congruent to its proper part. 

In point-set theory this same notion of congruence is 
found in sets much more complicated than the points of 
the half -plane. For an example of such a set, we begin by 
marking off on a circle an angle which is an irrational 
multiple of one complete rotation of the circle, or 360. 
If we were to make our angle a rational multiple (for 
instance, 90 or one-fourth of a complete rotation), we 
would find that after we had marked off four angles our 
next would coincide exactly with one which we had pre- 
viously marked off. When, however, our angle is irrational, 


no matter how many times we go around the circle we 
shall never mark off an angle which coincides exactly 
with one which we have previously marked off. 

We thus divide the circle into an. infinite number of line 
segments in this case a denumerable infinity, since each 
segment can be paired in order with an integer, the first 
with 1, the second with 2, and so on. The point set which 
we extract in this way from all the points of the circle con- 
sists of a denumerable infinity of line segments, each of 
which contains a non-denumerable infinity of points. Now 
this point set can be shown to be congruent to a proper 
part. By rigid motion in this case a rotation of the whole 
point set through the distance of our chosen angle we 
bring each segment into the position originally occupied 
by the next segment in the construction. Since, however, 
no segment can have been brought into the position orig- 
inally occupied by the first segment, we have shown that 
the whale is congruent to its proper part. 

The same result could have been achieved by con- 
sidering as our point set the end points of the segments 
which lie on the circumference of the circle. Then by 
rotating the circumference we would place the first point 
on the second, the second on the third, and so on. A nu- 
merical equivalent would be the placing of the positive 
integers in one-to-one correspondence with the positive 
integers greater than 1: 

1 2 3 4 5 6 7 ... 

4- J/ 4- ! \l- Np ^ 

2 3 4 5 6 7 8 ... 

For our next example, instead of extracting a denu- 
merable infinity of points from the circumference, we shall 
divide the entire circumference into a denumerable in- 
finity of congruent pieces. This is not, by any manner of 
means, child's play. The difficulty lies in the phrase "a 
denumerable infinity .** It is no problem at all to divide 


tlie circumference of any circle into a finite number of 
congruent pieces. We can, for instance, use the length 
of the radius to mark off six arcs on the circumference, 
any one of which can be superposed by the rigid motion 
of rotation on any one of the others.* Nor is it a problem, 
as we have already seen in connection with the real-num- 
ber line, to divide the circumference into a non-denumer- 
able infinity of congruent pieces, since when each piece 
consists of just one of the non-denumerable infinity of 
points on the circumference, all of them together are a 
non-denumerable infinity of congruent pieces. Here, how- 
ever, the method we used to divide the real-number line 
into a denumerable infinity of pieces, which were integer 
intervals, will not work, since the real-number line is in- 
finite in length while the circumference of the circle is 
finite. To solve our problem we must resort to a much 
deeper type of reasoning. 

We begin with a circle the circumference of which is 
one unit in length. The circumference can then be thought 
of as the portion of the real-number line between and 1. 
Except for the fact that and 1 are the same point, all the 
other points on the circumference are uniquely identifiable 
as the real numbers from to 1. 

To divide this non-denumerable infinity of points on 
the circumference into a denumerable infinity of sub-sets, 
we first gather together a set ( or what we shall call a fam- 
ily, to distinguish it from the other types of sets) which 
consists of all those points that differ from some point on 
the circumference by a rational number, or distance. The 
first family consists of all those points which are a rational 

* Even in such a simple problem as this, in point sets we have 
to decide how to distribute the end points of the arcs, since each 
shares its end points with the adjacent arcs. (We usually go around 
the circle counterclockwise and assign to each arc its first end 


distance from the point 0. This family, we can see, will in- 
clude all the rational points on the circumference since 
-h % gives us the rational point %; + H gives us the 
rational point %, and so on. We do not need to bother with 
selecting points a rational distance from any rational point 
other than 0, for the points so selected would necessarily 
be duplicates of those already included in the first family 
(the sum of rational numbers being always a rational num- 
ber). We turn our attention to selecting families of points 
which are a rational distance from each irrational point 
in turn. One of these families, for instance, will be all 
those points a rational distance from the irrational point 
1/V2. Since we are choosing a different family of points 
for each irrational point and since there is a non-denumer- 
able infinity of such irrational points between and 1, we 
shall divide the non-denumerable mfinity of points on the 
circumference into a non-denumerable infinity of families, 
or point sets. How many points in each of these families? 
Only a denumerable infinity, for there is just one point for 
each rational distance and the rationals themselves are a 
denumerable infinity. 

From the families, we now gather together a new kind 
of point set which we can call, to distinguish it from a 
family, a set of representatives from each family. The first 
set of such representatives is obtained by choosing from 
each of the non-denumerable infinity of families a single 
point, and will thus contain a non-denumerable infinity of 
points.* The next set of such representatives is obtained 

* After all that we have accepted so far, we probably have no 
difficulty in accepting the idea that we can choose from each of 
a non-denumerable infinity a single point. Yet this statement- 
known as the Axiom of Choice-has been one of the most con- 
troversial in modern mathematics. It is easy to see that if we have 
a finite number of sets, no two of which have a common member, 
we can in a finite number of operations choose a member from 

by rotating the entire circle a given rational distance and 
taking from each family a second point which is that 
rational distance from the first. To obtain yet another set 
of representatives we again rotate the circle a different 
rational distance; we continue in this manner until we 
have a set of representatives for each of the denumerable 
infinity of rational distances on the circumference. 

It is logically clear (although it may take a moment 
for one unused to juggling infinities to see that it is ) that 
we will end up with a denumerable infinity of sets of rep- 
resentatives, for there will be one for each rational dis- 
tancea denumerable infinity. Each set of representatives, 
however, will contain a non-denumerable infinity of 
points, one from each of the non-denumerable infinities 
of families we first selected. 

None of the sets of representatives can have a point in 
common with any other set because each rotation gave us 
a choice which, by its nature, could not include any of the 
points selected by previous rotations. Since the points 

each set so that we have a new set which has just one member in 
common with each of the original sets. If, however, we have an 
infinite number of sets to choose from, we cannot choose the new 
set in a finite number of operations unless we have some way of 
automatically distinguishing the member to be chosen. This dif- 
ficulty is illustrated in Bertrand Russell's story of the infinitely rich 
man with infinitely many pairs of shoes and socks. He can easily 
form a set which has one member in common with each pair of 
shoes. The rule for membership in this set can be that each mem- 
ber must be a left shoe. With the statement of this rule, the set is 
automatically chosen. In the case of the socks, however, no such 
rule is possible. A sock must actually be chosen from each pair 
since socks, unlike shoes, are not automatically distinguishable as 
'left" or "right." Since even our infinitely rich man could never 
complete this infinite task, the set containing one sock from each 
pair could never be chosen. Mathematicians usually overcome this 
difficulty with the Axiom of Choice by means of which they simply 
assume, as an axiom, that it is always possible to choose one mem- 
ber from each of an infinite number of sets. 


which we are choosing constitute a denumerable infinity, 
as does each of the families from which they are being 
chosen, every point on the circumference will be included 
in some set of representatives. We have, therefore, by 
dividing the circumference into mutually exclusive sets 
including every point, divided the circumference into a 
denumerable infinity of pieces. These pieces are con- 
gruent in the sense of elementary geometry, for all were 
obtained by the rigid motion of rotation. We have solved 
the given problem : to divide the circumference of a circle 
into a denumerable infinity of congruent pieces. 

The significance of what we have done may not be im- 
mediately apparent to the reader whose head is still rock- 
ing with non-denumerable and denumerable infinities; but 
let us consider for a moment the problem of assigning a 
measure, or a length, to these pieces of the circumference. 
Among them are included all the points on the circum- 
ference. By everyday standards they are the parts of the 
circumference and the circumference is the whole, so the 
sum of their lengths should be the length of the circum- 
ference. But by everyday standards they are also con- 
gruent, or equal. If, in an everyday sense, any measure is 
assigned to the pieces, the same measure must be assigned 
to each one of them. There are two possibilities: either a 
measure of for each piece or a positive measure. The 
circumference of the circle is one unit, and the pieces into 
which we have divided it must, if they are to have any 
length, add up to 1. Yet the sums of the only measures 
we can possibly assign to them are zero or infinity. We 
are forced to the necessary conclusion that these pieces 
the congruent point sets into which we have divided the 
entire circumference Jo no* have a length. 

The problem which we have just detailed rather com- 


pletely is an example of the type of reasoning, although 
much less deep, which led to the most famous paradox of 
point-set theory and an implication in regard to everyday 
ideas of measure much more startling than the one above. 
The Banach-Tarski paradox was propounded in 1924 by 
Stefan Banach (1892-1945) and Alfred Tarski (1901- ). 
These two mathematicians proved that it is possible to 
disassemble a solid unit sphere into a finite number of 
pieces in such a way that the pieces could be reassembled 
into two spheres the same size as the original sphere! 

Mathematically, the most unusual thing about the 
Banach-Tarski work was that its paradox of measure 
rested, not upon an infinity of pieces, as in the case of the 
problem we have just finished examining, but on a down- 
to-earth finite number of pieces. How many pieces? They 
did not say. A very large number of pieces? They did not 
say. Merely a finite number of pieces. That in itself was 
sufficiently startling. 

The exact number of pieces necessary was given, some 
twenty years later, by R. M. Robinson (1911- ), and 
it was very small. Working with only five pieces, Robinson 
showed it is possible to disassemble a solid unit sphere 
(point by point, of course) and reassemble it into two 
spheres the same size as the original. The reasoning which 
led Robinson to this conclusion was very complex, but 
basically similar to that which we followed in dividing the 
circumference of a circle into a denumerable infinity of 
congruent pieces to which no length could be assigned. 

In determining the smallest finite number of pieces 
into which the solid sphere can be divided for the Banach- 
Tarski paradox, Robinson began with the simpler problem 
of determining the number of pieces into which the sur- 
face of such a sphere or a hollow sphere must be divided 


so that it could be reassembled into two spheres the same 
size as the original. He showed how it was possible to 
divide the point set of the surface into four sub-sets 
A, B, C and D which exhibit a truly remarkable property. 
The sub-sets A and B are congruent to each other; and 
each of them is also congruent to the sum of A and B. In 
the same way C and D are congruent to each other and 
each of them, to the sum of C and D. Thus by rotating 
A into A -f B and C into C + D, we are able to form Si, 
a sphere which is exactly like our original sphere. We then 
rotate B into A + B and D into C + D to form S 2 , a 
second sphere exactly like Si and hence exactly like our 
original sphere. Thus four pieces were shown to be suf- 
ficient for reassembling a hollow sphere into two spheres 
the same size as the original. 

The solution of the problem for the solid sphere was 
then shown by Robinson to be essentially the same as that 
for the hollow sphere. Yet there was a difficulty. We can 
of course extend the four pieces of the surface A, B, C and 
D into the center of the sphere, but which piece will then 
include the point which is the center? If we are willing to 
simply assign the point to one of the four pieces so that 
it has one more point than the others, then we can re- 
assemble A, B, C and D into two solid spheres exactly like 
the original except for the fact that one of the new spheres 
will not have a point at its center. Most of us would be 
satisfied with tin's solution, but a mathematician will go 
to considerable trouble to get a center for that other 
sphere. Having found a point by a method too devious 
to record here, Robinson' brought it to the center of the 
sphere by translation* (all the other rigid motions in- 

* Translation is distinguished from rotation in that, under the 
rigid motion of translation, all the points are moving in the same 
direction at the same time, 


volved in the solution being rotations about the origin 
which of course could not produce the needed copy of the 
origin). Five, then, was determined as the necessary and 
sufficient number of pieces for the Banach-Tarski paradox. 

The significance of this paradox for the theory of 
measure is immediately apparent. When we consider geo- 
metric figures as point sets in 3-dimensional space and we 
do nothing more to them than what we do to the usual 
run of geometric figures with which we are familiar, we 
are forced to the conclusion that we cannot assign to them 
a measure of either area or volume. If the four pieces into 
which the surface of our sphere was divided had an area, 
their sum would be both the area of the original sphere 
and twice the area. If the five pieces of the solid sphere 
each had a volume, their sum would be both the volume 
of the sphere and twice the volume. In these particular 
situations the sum of the parts is not the whole, but twice 
the whole! 

A conclusion like this completely contrary to every- 
thing our intuition tells us, to what we have always known 
with confidence that we knew, and to what we feel is 
true separates the mathematical minds from the inher- 
ently non-mathematical. For there are always those who 
want to go back to the beginning, change the rules, forbid 
such exceptions, refuse such conclusions. The man who 
was the founder of point-set theory was not one of these. 

Georg Cantor came to the theory of point sets because 
he was forced this was his own word for it by logic. He 
did not invent his theory, arbitrarily, to confound intuition 
and experience. It is indeed one of the neatest ironies of 
mathematics that this theory, which seems as completely 
removed from the practical world as do the dreamy specu- 
lations of Laputan philosophers, grew out of the work of 
Jean Baptiste Joseph Fourier ( 1768-1830) , a physicist who 


expressed Ms opinion frequently and positively that math- 
ematics justified itself only by the help it gave to the 
solution of physical problems. (Fourier's own consider- 
able contributions to mathematics were in the theory of 
functions, and resulted from his researches in the conduc- 
tion of heat.) Although the line from Fourier to Cantor 
is a direct one, it is not the whole line. The theory of 
point sets is more truly a modern step on a logical path 
to which mathematics committed itself when it accepted 
the idea that there is a measure for every length a real 
number, rational or irrational, for every point on the 
number line. 

Georg Cantor followed this path where it logically led 
and drew the necessary conclusions although they were 
contrary to his own intuition, training and desire, and 
made him the object of an attack which had been un- 
equaled, in mathematics, since the Pythagorean who dis- 
covered the irrationality of V2 perished, mysteriously, 
at sea. 



Elements of Euclid commanded the 
almost unqualified admiration of man- 
kind. It could be said and was: 

". . . from its completeness, uni- 
formity and faultlessness, from its ar- 
rangement and progressive character, 
anc ^ fr m th 6 universal adoption of the 
completest and best line of argument, 
Euclid's Elements stands preeminently 
at the head of all human productions." 

It could be added and was: 

The "For upward of two thousand years 

New Euclid it has commanded the admiration of 

mankind, and that period has sug- 
gested little toward its improvement." 

At the beginning of the twentieth 
century, however, the suggestion box 
was open. 

This was the period known in the 
history of mathematics as "the crisis in 
foundations." A quarter of a century 
had elapsed since Georg Cantor had 
presented his theory of the infinite; 
and mathematics, somewhat like a man 
with a new living-room chair, had at 
last settled back comfortably with the 
once revolutionary idea of the infinite 
as something consummated. This was 
the moment that the Italian mathema- 
tician C. Burali-Forti (1861-1931) 
chose to produce by using exactly the 
type of reasoning that Cantor had used 
to establish his theory of infinite sets 


a flagrant contradiction which, at least for the moment, 
virtually invalidated Cantor's entire theory. The new chair 
collapsed; and, of course, like any normal man, mathe- 
matics in general now refused to use any chair but that 

The effort to set logically aright the foundations of 
mathematics and yet retain Cantor's new theory of the 
infinite ("No one shall expel us from this paradise which 
Cantor has created for us!") was led by a great German 
mathematician who now occupied Gauss's old place at 
Gottingen. David Hilbert (1862-1943) was actually the 
greatest mathematician in the world during the time that 
the newspapers and the man on the street thought un- 
questioningly it was Einstein (who was not a mathema- 
tician but a physicist). Besides the notable work which 
Hilbert accomplished in several fields, he offered mathe- 
matics leadership at a time when it was desperately 

Faced with the crisis hi foundations, Hilbert led his 
followers back to the Greeks, back to Euclid, to begin in 
an almost literal sense at the foundations themselves and 
re-erect the edifice of mathematics, block by block, with 
modern rigor. 

While the Elements of Euclid had served as the model 
of logical thought since antiquity, it had been observed 
by various mathematicians during that time that there 
were, nevertheless, certain logical lapses in the logical 
model. In the very first proposition of the very first book 
one such flaw is immediately apparent to the rigorous eye. 
Euclid lays the first block of the edifice of elementary 
geometry by attempting to show that (relying only on the 
previously stated definitions and axioms) it is possible 
"on a given finite straight line to construct an equilateral 
triangle." In his proof, invoking Postulate 3, he inscribes 
a circle with center A and radius AB on the given segment 


AB and another circle of the same radius with center B. 
He then proceeds with his proof from the point C "in 
which the circles cut one another." 

"It is a commonplace," says Sir Thomas Heath, rather 
tiredly, in his English edition of the Elements, "that 
Euclid has no right to assume, without premising some 
postulate, that the two circles will meet in the point C" 

Like Euclid, we think that we know for haven't we 
drawn this same figure many times to find the center of a 
given line? that the circumferences will intersect at a 
point equidistant from A and B. We cannot know this 
from experience, for we cannot have drawn all possible 
circles; we can only assume from what experience we 
have had that such pairs of circles will always meet and 
that they will always meet in just one point above the line. 
This, then, is an assumption upon which our geometry is 
based and, as such, it should be stated with the other 
assumptions. Because such assumptions of intersection 
are not explicitly stated in the Elements, it is possible by 
using only Euclid's stated definitions and axioms to 
"prove" such paradoxical propositions as "Every triangle 


is an isosceles triangle." (The "proof" is given in J. W. 
Young's excellent little book Fundamental Concepts of 
Algebra and Geometry, Macmillan, 1930.) 

Unstated assumptions of intersection pervade the Ele- 
ments; for Euclid uses whenever possible the method of 
actual construction for proving the existence of figures 
having certain properties. Constructions are made on the 
basis of Postulates 1-3, with straight line and circle alone. 
What enables Euclid to build with these straight lines 
and circles is the fact that they determine by their inter- 
section other points in addition to those originally given 
in the problem, and that these points can then be used 
to determine new lines, and so on. The method of the 
Elements logically demands that the existence of such 
points of intersection be either proved or postulated in 
the same way that the existence of the lines which pro- 
duce them is postulated by 1-3.* The ladder to truth rests 
on the idea that after the original assumptions are granted, 
no other assumptions will ever be required. 

Hilbert's problem in subjecting the Elements to true 
modern rigor was, not only that Euclid often assumed 
assumptions which he had not made, but that he relied 
upon definitions which did not actually define. We are all 
familiar with this problem of definition. We define an 
orange with Webster as "the nearly globose fruit, botan- 
ically a berry, of an evergreen rutaceous tree (genus 
Citrus)"; and immediately find ourselves involved in a 
multiplying set of definitions: What is a fruit? What is a 
globose fruit? What is a nearly globose fruit? It is obvious 
that unless we can begin with the assumption that there 
are certain terms which everybody intuitively "knows," 
we shall have to give up our project of a dictionary. 

* The only statement of intersection in the axioms is the 
negative one of the Parallel Postulate, where it is stated that under 
certain conditions two lines will never meet. 


Euclid faced the same problem as the dictionary 
maker when he went to compile the Elements. He began 
in traditional style with first things first: twenty-three 
definitions at the beginning of Book I, ranging from "A 
point is that which has no part" to a breakdown of such 
quadrilateral figures as the square, the oblong, the 
rhombus and the rhomboid. "And let quadrilaterals other 
than these be called trapezia/' he concluded. Although 
Euclid grouped these definitions together, he made in 
the Elements a distinction, never stated but clearly im- 
plied. The existence of points, lines and circles had to be 
assumed by the reader; existence could not be proved for 
any of these. But after the existence of these geometrical 
objects was assumed, such a figure as met the require- 
ments of a rhomboid, for instance, could be constructed 
and displayed to the reader, its "existence" established 
by proof. 

Although a rhomboid is perhaps a vague figure to the 
ordinary person,* points, straight lines and circles are not. 
Everybody who has seen a point on paper, for instance, 
knows intuitively what Euclid meant by his geometric 
point "that has no part/' 

The ancients argued quite extensively about the 
proper definition of a point: 

A point is an extremity of a line. 

A point is that which is indivisible. 

A point is that with position only. 

A point is an extremity which has no dimension. 

A point is the indivisible beginning of all magnitudes. 

Yet all of their definitions were attempts to express what 
they thought they already "knew" a point was. 

* What is it? A rhomboid, according to Euclid, is "that which 
has its opposite sides and angles equal to one another but is neither 
equilateral nor right-angled." 


Actually the definitions were not really necessary at 
all. Anyone who had never seen a point on paper would, 
by the time he had completed the propositions of the 
Elements, have a thoroughly accurate idea of the geo- 
metric point as a result of the various statements which 
are made about it in the propositions. 

From what we have said about the problem of defini- 
tion, three main ideas emerge as pertinent for modern 
mathematicians as for Euclid: 

1. Unless we are to continue defining indefinitely, 
some terms in our geometry will have to be ac- 
cepted as undefined, or primitive, terms. 

2. Their existence will have to be assumed just as 
the statements made in the axioms have to be as- 
sumed because they cannot be proved without 
bringing in other axioms which cannot be proved 
without bringing in other axioms, and so on. 

3. What these undefined or primitive terms stand for 
will, however, become increasingly clear as using 
only these terms, the axioms, and the rules of 
reasoning we make and prove more and more 
statements about them. 

Now let us return to Hilbert and his effort to place 
at least one domain of mathematics the geometry of 
Euclid on a thoroughly sound logical basis. (This effort, 
massive though it was, was presented in a tiny book 
which is available in English as The Foundations of 
Geometry [Open Court, La Salle, Illinois, 1938].) 

As the epigraph of his work, Hilbert took a quotation 
from the great German philosopher Immanuel Kant (1724- 
1804). It was Kant whose often quoted attitude toward the 


axioms of the original Euclid was that they were "a priori 
synthetic judgments imposed upon the mind, without 
which no consistent or accurate reasoning would be 
possible/' Since the time when Kant made this statement, 
it had become increasingly clear in mathematics, if not in 
philosophy that Euclid's assumptions were somewhat 
arbitrary and that other assumptions and other geometries 
were just as possible and just as "true." The quotation 
from Kant which Hilbert chose for his epigraph was not, 
therefore, this most famous one. He chose instead a state- 
ment to, emphasize the relation between the intuitive roots 
of mathematics and its abstract flowering: 

"All human knowledge begins with intuitions, thence 
passes to concepts and ends with ideas." 

The intuitions with which geometry begins, in both a 
literal and a figurative sense, are those of points and lines 
and the surfaces on which points and lines exist. These, 
then, are the ideas with which both Euclid and David 
Hilbert begin. But every one of the twenty-three hun- 
dred years which lie between the Greek and the German 
lie between their opening treatments of these ideas. To 
emphasize the contrast, we shall present their two be- 
ginnings in parallel columns: 


Definitions The Elements of Geometry 

Let us consider three dis- 
tinct systems of things. 

A point is that which has no The things composing the 

part. first system, we will call 

points and designate them 

by the letters A, B, C, . . . ; 


EUCLID (cont.) HILBERT (cont.) 

A line is breadthless length, those of the second, we will 

The extremities of a line are call straight lines and des- 

points. A straight line lies ignate them by the letters 

evenly with the points on a, b, c, . . . ; 

A surface is that which has and those of the third sys- 
length and breadth only, tern, we will call planes and 
The extremities of a surface designate them by the 
are lines. A plane surface is Greek letters <*, /?,-/,.... 
a surface which lies evenly 
with all the straight lines on 

The points are called the 
elements of linear geom- 
etry; the points and straight 
lines, the elements of plane 
geometry; and the points, 
lines and planes, the ele- 
ments of geometry of space 
or the elements of space. 

Euclid continues to a total of twenty-three defini- 
tions at the beginning of Book I. Hilbert is through with 
the definition of terms. 

But where the new Euclid is more concise in his 
definitions than the old, he finds a need for many more 
assumptions than the five Common Notions and five 
Postulates of the original Elements. In Hilbert's Euclid 
there are twice as many axioms, and the relationship be- 
tween the axioms and the three undefined "systems'* listed 
above is explicitly stated: 


We think of these points, straight lines and planes 
as having certain mutual relations, which we indicate 
by means of such words as "are situated," "between," 
"parallel," "congruent," "continuous," etc. The com- 
plete and exact description of these relations follows 
as a consequence of the axioms of geometry. Each of 
these groups expresses, by itself, certain related funda- 
mental facts of our intuition. We will name these 
groups as follows: 

I. 1-7. Axioms of connection. 
II. 1-5. Axioms of order. 

III. Axiom of parallels, 

IV. 1-6. Axioms of congruence. 
V. Axiom of continuity. 

The relationship between the three "elements" of his 
geometry and the twenty "axioms" is precisely stated by 
Hilbert within the paragraph quoted above: "The com- 
plete and exact description of these relations [between 
the elements] follows as a consequence [my italics] of the 
axioms of geometry'' 

For example, in the first group of axioms which have 
to do with the intuitive idea of "connection/* he begins 
with the assumption that "two distinct points A and B 
always completely determine a straight line a." He then 
points out that instead of the word "determine/* we may 
also employ other forms of expression: "For example, we 
may say A lies upon' a, A 'is a point of 0, a 'goes through* 
A 'and through* B, a pins* A 'and* or 'with* JB, etc." 

Within the body of the work, Hilbert places various 
necessary "definitions** as the need for them occurs. These 
are made in terms of the previously stated but undefined 
elements, point, straight line and plane, and the descrip- 
tion of their relations which follow as a consequence of 


the axioms. Such a definition is that of segment: 

DEFINITION. We will call the system of two points A 
and B, lying upon a straight line, a segment and de- 
note it by AB or BA. 

In the preceding paragraphs we have given a few 
quotations directly from Hilbert's work on the founda- 
tions of geometry so that the reader can feel at first hand 
at least the faint breath of the new spirit of rigor which 
entered mathematics at the beginning of the twentieth 
century. It was by such rigor no attempt made to define 
the undefinable, but every attempt made to state explicitly 
every necessary assumption that Hilbert attempted to 
resolve "the crisis" by setting the most ancient branch of 
mathematics upon a logically sound foundation. 

The crisis had been precipitated by the discovery 
that the type of reasoning and the assumptions Georg 
Cantor had used in developing the theory of sets could 
lead to contradiction. To resolve the crisis, Hilbert had 
returned to Euclid and attempted to place elementary 
geometry on a completely rigorous foundation which 
would eliminate such contradictions as the paradoxical 
proposition that all triangles are isosceles. He then at- 
tempted to do something which would never have oc- 
curred to Euclid. He set out to prove that reasoning with 
his assumptions could not possibly lead to such contradic- 
tionsthat the axioms of elementary geometry as he had 
restated them were now absolutely consistent! 

How is it possible to prove that a set of assumptions 
is consistent? How can we know before we start that we 
will never find ourselves in the position of having proved 
that A is equal to B and, also, that A is not equal to B? 

At the present time the only way of doing this is to 


match our abstract theory primitive terms and assump- 
tionswith some concrete representation of it, which is 
already granted to be consistent. For instance, we can 
take as our model of consistency the arithmetic of real 
numbers, since reasoning according to the rules has never 
yet brought us to the contradictory position of having 
proved that 2 -f 2 = 5 as well as 4! 

This is what Hilbert did, although taking an even 
smaller domain of arithmetic than that of the real num- 
bers. This was the domain X "consisting of all those alge- 
braic numbers which may be obtained by beginning with 
the number one and applying to it a finite number of 
times the four arithmetic operations (addition, subtrac- 
tion, multiplication and division) and the operation 
y 1 _j_ ttf 2 , where w represents a number arising from the 
five operations already given." The reader will recognize 
this domain as that of the constructible numbers, which 
we met in Chapter 9. 

In the terms of the arithmetic of this domain, Hilbert 
defined his primitive terms, point, straight line and plane. 
A pair of numbers (x, t/), for instance, became a point and 
the ratio of three such numbers (:i?:u?) where u and f 
are not both equal to 0, became a straight line. The exist- 
ence of the equation 

ux 4- vy + ? = 

was defined to express the condition that the point (x,y) 
lies on the straight line (u:v,w). He then showed how 
the various groups of axioms could be interpreted in the 
terms of the arithmetic of domain X. In this way he was 
able to establish that the arithmetic of domain X could 
be considered a concrete representation of his abstract 
geometry of three "systems" and twenty "axioms. 


"From these considerations," he concluded, "it follows 
that every contradiction from our system of axioms must 
also appear in the arithmetic related to the domain X." 

We may still ask, "But how did Hilbert know that the 
arithmetic of real numbers is consistent?" The answer is 
that he did not know. No one knows. The arithmetic of 
real numbers is considered to be consistent only because 
of the absence of any known contradiction. That it is in 
actuality consistent was an assumption that Hilbert made. 
He was aware that it was an assumption. Since the con- 
sistency of his geometry depended upon the consistency 
of the arithmetic of real numbers, only an absolute proof 
that the arithmetic is consistent would establish the abso- 
lute consistency of his geometry. Although there had not 
yet been such an absolute proof when Hilbert published 
his work, one was generally assumed to be possible. 

This was as far as David Hilbert could go in his effort 
to resolve the crisis in foundations by establishing the 
logical consistency of elementary geometry. At the time 
he wrote the words above, he was almost forty, the twen- 
tieth century was in its first year, and not yet born was 
the young man who would reveal the hopelessness of 
Hilbert's dream by demonstrating with finality that estab- 
lishing the absolute consistency of any such set of axioms 
is impossible. 

In 1931, at the age of twenty-five, Kurt Godel pub- 
lished a paper entitled "On Formally Undecidable Prop- 
ositions of Principia Mathematica and Other Related 
Systems." When, many years later, Harvard University 
awarded him an honorary degree for this work, the cita- 
tion referred to him as "discoverer of the most significant 
truth of this century, incomprehensible to laymen, revo- 
lutionary for philosophers and logicians." 

For the moment we shall not be concerned with these 


even wider implications of Godel's 1931 paper, but only 
with its primary subject. This was the demolishing of the 
hope that the absolute consistency of any mathematical 
system (including ordinary arithmetic) could be estab- 

Although Godel's is as complex a piece of reasoning as 
mathematics is ever likely to see, it depends upon a varia- 
tion of an ancient brain teaser with which we are all 
familiar. This is the statement of Epimenides, who was a 
Cretan, that all Cretans are liars. Was Epimenides a liar? 

In his epochal proof Godel showed that in any suffi- 
ciently strong mathematical system it is possible to con- 
struct a statement which asserts its own unprovability in 
that system. The consistency of the system cannot then 
be established within the system itself but must be re- 
ferred to a stronger system, where of course the same 
thing can be shown to be true, so that the consistency of 
that system must be referred to a still stronger system, 
and so on. 

With the kind of finality which is possible only in 
mathematics, Godel demolished Hilbert's project. There 
can be no proof of the absolute consistency of the founda- 
tions of mathematics. We must live and work on assump- 
tions of consistency. 

David Hilbert died in 1943 at the age of eighty-one. 
The problem which he had put for himself was one that 
would never have occurred to Euclid. The answer which 
young Kurt Godel established in his epochal paper of 
1931 was one that had never occurred to Hilbert. 



erected in the Elements consisted of 
the rungs of definitions, axioms, theo- 
rems and proofs suspended between 
strong side supports of logic. These 
supports were formulated in Euclid's 
^ ijj^^^j day as the laws of reason and they 

F Jr were formulated in words, for they 

J were not part of mathematics but of 
. , m logic. Today these laws are still the 

strong and indispensable supports of 

the ladder to truth, but today they are 

Of expressed in mathematical symbols; 

Truth Tables and any proof which utilizes a combi- 

and Truths nation of these laws can be tested for 

error by a mathematical methodthe 

method of the truth tables. 

The truth tables are a develop- 
ment of the sentential, or propositional, 
calculus. The sentential calculus, in 
spite of its formidable name, has a 
vocabulary which consists in its en- 
tirety of the small words, and, or, not, 
if, then, only, and the one relatively 
big word, sentence. It is a fragment 
and we must admit the most element- 
ary fragment of a great and modern 
mathematical study symbolic logic 
which subjects logic to the symbols 
and procedures of mathematics. 

The basic logical concepts of the 
sentential calculus are things which 
every mathematically minded person 
knows and uses intuitively. They 


sound, therefore, too obvious to bother with. But as 
mathematical sentences (or propositions) become longer 
and more complicated, intuition is not sufficient to deter- 
mine with finality their logical truth or falsity. Then a 
method is needed which is completely formal, and this 
method is furnished to mathematics by the sentential 
calculus under the slightly sinister title of truth tables. 

We shall, in the course of our exploration of the sen- 
tential calculus, use the method of truth tables to test the 
logical truth of certain statements; but before we can do 
so we must examine in some detail the meaning of its 
vocabulary and familiarize ourselves with the five symbols 
with which it conducts its business. The reader is strongly 
urged to do the simple problems as they occur, covering 
the answers with his hand and testing his memory of what 
has been explained, translating language into logical 
symbolism and logical symbolism back into language, and 
taking pencil in hand and determining for himself the 
truth of given sentences. It is guaranteed that he will be 
pleasantly surprised at the enjoyment he will get out of 
actually using truth tables. 

The vocabulary of the sentential calculus is, as we 
have said, limited to very simple and common words. 
These words are used, however, in a precise way which, 
in every case, seems different either to a large or small 
degree from the way in which we ordinarily use them. 
Because we use the words of the sentential calculus all 
the time, we have a tendency to feel that, like Humpty 
Dumpty, we have as much right as anybody to say what 
they mean. We are inclined to object to the meanings 
which the logicians assign to them. (Even logicians have 
this same feeling about the words and argue quite a bit 
among themselves. ) But if we are to understand, we must 
make a definite effort to erase from our minds our own 


personal meanings of the words which compose the vocab- 
ulary. We must consider these words as technical terms 
to which the logician, like any scientist, assigns the un- 
ambiguous definitions which are necessary for the func- 
tioning of his science. 

The most straightforward way of getting rid of the 
ordinary meanings of the words is to eliminate the words 
themselves from our preliminary discussion. So let us be- 
gin by giving our attention to the five symbols of the 
sentential calculus, each of which represents a logical 

' (Negation). This symbol, when placed before a 
sentence, or a letter which we take to represent a sen- 
tence, denies whatever follows it. If we represent a 
sentence by a variable p, then -p stands for "not p." If 
the sentence p is "Snow is black," then <~-p is "Snow is not 
black." We can call this logical concept "not." 

A (Conjunction). This second symbol indicates the 
joining together of the expressions on either side of it. If 
these expressions are variables p and q, then p A q indi- 
cates "p and q" If p is used as above and q in place of an- 
other sentence, such as "All men are mortal," then p A q 
is the sentence what we call grammatically a compound, 
sentence "Snow is black, and all men are mortal." We can 
call this logical concept "and." 

V (Disjunction), The third symbol represents a join- 
ing which nevertheless leaves the joined, expressions some- 
what separated. This is sometimes called an alternation. 
If V is used to join our two variables p and q, the resulting 
expression p V 9 is the equivalent of "p or q" In the case 
of the meanings we have been assigning to the variables, 
the expression can be translated as "Snow is black, or all 
men are mortal." This is called the logical concept "or." 


-> (Implication). With this fourth symbol we have 
what is grammatically called a conditional sentence. The 
expression p - q is read "If p, then q," or, "If snow is 
black, then all men are mortal." This logical concept is 
called "if, then." 

< > ( Equivalence ) . The relationship represented by 
our fifth and last symbol is biconditional. The expression 
p < > q is read "p if, and only if, q" "Snow is black if, 
and only if, all men are mortal." Here we have what is 
known in mathematics as "a necessary and sufficient con- 
dition" and we can call the logical concept "if, and only 

As we read over these definitions and the examples 
given for the relation between p and q as expressed by 
each of the symbols, we are naturally troubled by the 
fact that they do not seem, according to our understand- 
ing of the word, very logical. Snow is not black and what 
does all men being mortal have to do with snow, anyway? 
Surely the sentential calculus does not concern itself with 
such inanities! 

Let us consider these objections in order. 

First: the appropriateness of the examples. In the sen- 
tential calculus, p and q, or whatever other variables we 
use, stand for mathematical propositions. These proposi- 
tions may be true (All men are mortal), or they may he 
false (Snow is black). We are not concerned with their 
truth or falsity except as it affects the soundness (or 
logical truth) of the reasoning which follows from them. 
This important fact is emphasized when the propositions 
are selected outside the subject matter of mathematics. 

Let us take, as an example, one of the simplest and 
most obvious of the laws of the sentential calculus the 
Law of Identity. 

p > p or If p, then p. 


If we substitute for the variable p, the "false" statement 
"Snow is black," we then get the logically true statement: 
"If snow is black, then snow is black." This is just as sound 
reasoning as that represented by "If all men are mortal, 
then all men are mortal." A logically false, or unsound, 
statement is equally false whether p and q are themselves 
true or false. If, instead of p - p, we take 
pM _ p 

we find that it is as logically false when p stands for "Snow 
is black," which is false, as it is when p stands for "All 
men are mortal," which is true. "Snow is black if, and only 
if, snow is not black." "All men are mortal if, and only if, 
all men are not mortal." Both are unsound reasonings. 

The first hurdle we must overcome is this: We must 
understand that the truth or falsity of p and q does not 
directly determine the truth or falsity of the reasoning 
which is based upon them. The second hurdle is much 
more difficult. 

We were originally bothered by the statement that 
snow is black, but we were much more bothered about 
the fact that a statement about snow and one about mor- 
tality were combined. Snow and mortality, we objected, 
have nothing to do with each other; it isn't logical to com- 
bine them in one statement! We shall not at this point 
bring up the common poetic symbolism of winter and 
death, but shall content ourselves with the comment that 
it is quite difficult to determine with finality whether two 
ideas have or do not have something to do with each 

A simple example will serve. A says, "B attended the 
University of X and he is a Communist." Obviously, A 
considers these two ideas related. In the newspaper he 
has noted that a couple of people recently revealed as 
Communists attended the University of X. Some of those 

crackpot professors, he thinks, must be turning the kids 
into Commies! He connects the two facts that B is a Com- 
munist and attended the University of X connects them 
both in his mind and in his sentence. C, who is an alum- 
nus of the University of X, objects. There is no connection 
between the two facts. They do not belong in the same 
sentence. It is not logical to put them together! Who is 

If such are the difficulties of detennining "relationship'* 
in everyday Me, how can we hope to make such a concept 
precise? The logician answers this question and solves this 
problem by announcing in a firm voice that, for his pur- 
poses, it doesn't matter whether two sentences joined by a 
symbol of the sentential calculus are, or are not, related. 
A conjunction p A q will be true if p and q are both true, 
"Snow is white and all men are mortal" is a completely 
acceptable sentence from the point of view of the logician. 
Before we object (we who use "and** too and feel that we 
have as much right as he to express our opinion), let us 
remember that the logician does not even suggest that we 
be governed by the same rule when we use "and." He only 
says that, for the purpose of developing a calculus with 
which he can test the logical soundness of mathematical 
propositions, he must have an unambiguous rule for join- 
ing two sentences with "and." As an alumnus of the Uni- 
versity of X he would probably argue heatedly with the 
rest of us about the "logic** of the compound sentence 
which joins "B attended the University of JT* and "B is a 
Communist.** As a logician, examining the proposition, he 
will say that A*s statement is logically sound if it is true 
that B attended the University of X and if it is also true 
that B is a Communist. 

In the sentential calculus we are concerned with the 
truth of certain combinations of sentences effected by "not," 
"and," "or,** "if, then,** and "if, and only if.** We ignore com- 


pletely any questions of subjective relationship, like Should 
these two ideas be put together in the same sentence? 
Instead, we concentrate upon the objective relationship. 
When we put ' ' in front of p, the resulting sentence ^p 
can be true only if p is false. When we put A between p 
and q, the resulting sentence p A q can be true only when 
p and q are both true. Once we accept the idea that p and 
q do not have to "belong" in the same sentence, we have no 
objection to these rules. 

There are similar arbitrary rules for determining the 
truth of combinations made with the other symbols. These 
five symbols, and the logical concepts which they express, 
are no longer common expressions of everyday discourse, 
but the technical terms of the sentential calculus: 

Not. The sentence ^p is true only when p is false. 

And. The sentence p A q is true only when p and q 
are both true. 

Or. The sentence p V q is true if either p or q is true. 

If, then. The sentence p - q is always true except 
when q is false and p is true. 

If, and only if. The sentence p < > q is true only 
when p and q are both true or both false. 

These definitions of the conditions under which 'p, 
p h q, pVq, p -^ q and p < > q are true certainly ignore 
our everyday insistence upon a relationship between two 
sentences which are joined as one. To determine the log- 
ical truth of a combination, we do not even have to know 
what sentences the variables p and q represent. Given 
that p is true and q is false, we know that 

p is false while ~>q is true; 
p A q is false, but p V q is true; 
p - q and p < > q are both false. 


To test his understanding of these rules, the reader 
might like to mark the sentences below "true" or "false" 
from the point of view of a logician. 

p = Snow is white. 

q All men are mortal. 

1. Snow is not white. T F 

2. Snow is white and all men are mortal. T F 

3. Snow is white or all men are mortal. T F 

4. If snow is white, then all men are T F 


5. Snow is white if, and only if, all men T F 
are mortal. 

p= 2 + 2=5 
q= 2X3 = 4 

6. 2 + 2 ^ 5. T F 

7. 2 + 2 = 5and2X3 = 4. T F 

8. 2 + 2 = 5 or 2X3 = 4. TF 

9. If2 + 2 = 5,then2X3 = 4. T F 
10. 2 + 2 = 5 if, and only if, 2X3 = 4. T F 

True Sentences: 2, 3, 4, 5, 6, 9, 10. 

Note that in Sentences 1-5, p and q were both 

true while in 6-10 they were both false. 

For every p and q, we have four possible situations: 
the sentences which p and q represent can be both true, 


both false, p can be true and q false, or q can be true and 
p false. As we saw from our examples above, each of these 
situations may result in a change in the truth or falsity of 
the combination of p and q effected by a logical symbol. 
These various possibilities can be stated most simply in 
the form of a table. In the first column we list by T and F 
the different possible situations in regard to the truth or 
falsity of the sentences represented by p and q. The re- 
maining columns are allotted to the different logical rela- 
tionships; for each we indicate the truth or falsity of 
that particular combination under the situation regarding 
p and q as indicated in the first column. 

Since the table for the combination effected by ---', or 
"not," is much simpler than that for the others, we shall 
give it separately and first. 

T F 

F T 

In the following table for the four other combinations, 
the jTs and F's in the first and fourth rows across give us 
the correct answers to sentences 2-5 and 7-10 in our test 
on p. 265. 

p q pt\q pVq p-^q p^-^q 

























It is important for us to note that in each of the col- 
umns representing a combination of p and q by one of our 
symbols, we have at least one F. This means that for at 
least one of the possible situations regarding the truth or 


falsity of p and q their combination into one statement 
cannot be regarded as a "true" or logically sound state- 
ment. When, however, we construct the same type of 
table for what in the sentential calculus is called the Law 
of Identity, or p-* p, which we mentioned earlier, we find 
that regardless of the truth or falsity of p the combina- 
tion p p is always true. 

T T 

F T 

Since p - p is always true, we say that it is a true sen- 
tence. All such true sentences are laws of the sentential 
calculus and, as we have seen, this is the Law of Identity. 
We cannot be blamed if we are not too impressed with 
the Law of Identity. If p, then p. So p implies p. We are 
reminded of the word tautology. Our Law of Identity is 
certainly tautological. Webster says, "With needless repe- 
tition, as visible to the eye, audible to the ear." Logicians 
say, "A tautology is a true sentence, or law, of the senten- 
tial calculus." 

The most profound mathematical truths are as tauto- 
logical as p -> p, but because of their complexity we do 
not so immediately or intuitively recognize the quality in 
them. This is where the sentential calculus is indispensa- 
ble. By means of its so-called truth tables there is a general 
method for determining whether any statement (no mat- 
ter how extensive or complicated) is a tautologyin other 
words, a logically true statement. 

The table which we constructed f or p -> p is the sim- 
plest possible example of a truth table. As our sentences 
to be tested increase in the number of their relationships 
and the number of variables involved, so do their truth 


tables increase in complexity. Let us take a statement a 
little more complicated than the Law of Identity and by 
constructing its truth table determine whether it, too, is 
a law of the sentential calculus: 

(r~~p - p ) - p, or "If not p implies p, then p." 

The method which we follow to test this statement is the 
same one which we will follow for more complicated 
statements. We take the sentence, beginning most simply, 
combination by combination. 

1. Against the possible truth or falsity of p, we test 
p in column 2. 

2. Against the respective possibilities for p and r y 3 
we test the combination - ^? - p in column 3. 

3. Against the respective possibilities for ~p - p in 
column 3 and p in column 1, we test the entire sentence 
( p > p} -^ p in. column 4. 

p ^p ^p^p (,~p_p)-p 
__ _ __ _ 


Since, whether p is true or is false, the statement 
( p p) ~ p is always true (as we see in column 4), 
we know that it is a law of the sentential calculus, or a 

Since any sentence of the calculus can be tested for 
truth or falsity by means of truth tables, the sentential cal- 
culus is one of the few branches of mathematics which 
has a general method for solving all Us problems. This 
almost unique quality of the sentential calculus is ex- 
tremely significant when we realize that almost all scien- 
tific reasoning is based either directly or indirectly upon 
its laws. We are then, in the words of Tarski, able to dis- 

sect even the most complicated mental processes by "such 
simple activities as attentive observation of statements 
previously accepted as true, the perception of structural, 
purely external connections among J;hese statements, and 
the execution of mechanical transformations as prescribed 
by the rules of inference. It is obvious that, in view of 
such a procedure, the possibility of committing mistakes 
in a proof is reduced to a minimum." 

This achievement of the sentential calculus is all the 
more impressive when we consider the simplicity of the 
tools with which it works half a dozen concepts ex- 
pressed by some of the simplest words in the language. It 
is an achievement that Euclid would have appreciated. 


Following the method of the truth tables which we 
have detailed in this chapter, the reader can now deter- 
mine for each of the following two sentences whether it 
is a true sentence in the sentential calculus. (One is and 
one isn't.) 

p q p-*q q-*p (p-* q) < (q-*p) 

T T T T 

T F F T 


F F T T 

p q p-*q (p-qf)-*p [(p-* <?) ~ p] - p 

T T T T 
T F 
F T 
F F 



j: UQ 

The reader can now by the same method construct a 
truth table for a fairly complicated statement: 

If p implies q and q implies r, then p implies r 

When we transcribe this sentence into the symbolism of 
the sentential calculus, we get the statement below. 

[(p-><?) A (9->r)]-> (p->r) 

To construct a truth table for this sentence, we must first 
list the possibilities in regard to the truth or falsity of the 
three variables, p, q and r. We then check off against 
these possibilities the truth or falsity of the logical com- 
binations of the variables in the following somewhat 
nested order: 

We leave it to the reader to determine whether this is a 
law of the sentential calculus. 


-U9S 9U} JO MV\ B 9JOJOJ9lp ptEB XO|O)l"lBq. B p99p 

-joui 9JB UQui ^y w 'b joj puB ,/^OBjq ST AYoug^ 'd 
joj S9:m}psqns 9i{ jj -umnioa ^SB^ 9ip m 9OBds 
Xj9A9 in j; B tpiAV ^no 9UIOO pjnoqs j9pB9J 9qjL 






century, more than two thousand years 
after Euclid compiled his Elements, 
the axiomatic method the method 
which is synonymous with the Ele- 
ments of Euclid was the subject of an 
international symposium of mathe- 
maticians and scientists, the primary 
purpose of which was to determine 
the extent to which this classic method 
of mathematics could and should be 
further utilized by the physical sci- 
ences. The discussions, the concepts, 
and the vocabulary were a long long 
way from Euclid and they were per- 
haps farthest in the discussions of 
elementary geometry! 

"What is elementary geometry?" 
asked Alfred Tarski, the famed logi- 
cian, and answered as follows : * 

"We regard as elementary that part 
of Euclidean geometry which can be 
formulated and established without 
the help of any set-theoretical devices." 

Tarski then continued with a more 
precise statement of his view of ele- 
mentary plane geometry (or 2 ) as 
formulated in the terms of first-order 
predicate calculus, which is printed in 
full on page 272. 

* The following quotation and the axi- 
oms of elementary geometry appear in The 
Axiomatic Method, Leon Henlcin, Patrick 
Suppes and Alfred Tarski, editors, North- 
Holland Publishing Company, Amsterdam, 
The Netherlands, 1959. 



All the variables x,y,z f ... occurring in this theory 
are assumed to range over elements of a fixed set; the 
elements are referred to as points, and the set as the 
space. The logical constants of the theory are (i) 
the sentential connectives the negation symbol - ', the 
implication symbol -, the disjunction symbol V, and 
the conjunction symbol A; (ii) the quantifiers the 
universal quantifier A and the existential quantifier 
V; * and (iii) two special binary predicates the 
identity symbol = and the diversity symbol 9^. As 
non-logical constants (primitive symbols of the 
theory) we could choose any predicates denoting 
certain relations among points in terms of which all 
geometrical notions are known to be definable. Actu- 
ally we pick two predicates for this purpose: the 
ternary predicate ft used to denote the betweenness 
relation and the quaternary predicate B used to 
denote the equidistance relation; the formula P(xyz) 
is read y lies between x and z, (the case when y 
coincides with x and z not being excluded), while 
(xyzu) is read x is distant from y as z is from u. f 

Strange though the language of Tarskfs twentieth- 
century geometry might seem to Euclid, it would not be 
so far removed from the Greek as the simple statement 
of Tarskfs Theorem 3: 

THEOREM 3: The theory E 2 is decidable. 

* The universal quantifier A stands for "for every" and the 
existential quantifier v, for "there exists." 

f Using the vocabulary which is given here, the reader may 
enjoy translating into words Tarskfs axioms for elementary geom- 


This five-word theorem embodies that aspect of mod- 
ern mathematics which is undoubtedly farthest from 
Euclid what has been called "the most significant truth" 
of the twentieth century! 

In this chapter we shall try to give the reader a glimpse 
of this truth by clarifying the meaning of that deceptively 
simple word, decidable, in the statement of the theorem 
above. In general terms, the statement that the theory 2 
is decidable means that for elementary plane geometry, as 
formulated by Tarski in the paragraphs above, there ex- 
ists a method for solving all possible problems. 

What do we mean by a method of solving an infinite 
class of problems, such as all the problems of elementary 
plane geometry? This is a question to which Euclid's 
successors of the twentieth century have devoted con- 
siderable thought, and the answer they have at length 
come to is among the most significant in the history of 

Curiously, their interest in what they meant by a 
method developed from the consideration, suggested for 
the first time by Godel, that for some classes of mathe- 
matical problems there might be no method. This is un- 
derstandable. If someone comes to us and says, "I have a 
method of doing so and so," we do not stop him with, "See 
here. Just what do you mean by a method?" Instead we 
say, "What is it?" It is only when he comes and says, 
"There is no method of doing so and so," that we stop him 
with, "Just what do you mean when you say there is no 

This is essentially the situation that occurred in mathe- 
matics in 1931. In that year Godel, as we have told in 
Chapter 16, published a paper "On Formally Undecid- 
able Propositions of Printipia Mathematica and Other 
Related Systems." This was one of the great turning points 


in mathematical thought. Although the paper was con- 
cerned primarily with demolishing the idea that the 
absolute consistency of a mathematical system could be 
established within that system, implicit in it was the idea 
that for certain classes of problems (such as those encoun- 
tered in number theory), there can be no general method 
of solving all of the problems in the class. 

This truly monumental result started other mathema- 
ticians thinking for the first time upon the subject of 
methods in general. What did they mean by a method? 
Working more or less independently here and abroad, 
several of them formulated definitions of a method. Most 
definitions were extremely technical (one of the most 
important depending upon the idea of recursive func- 
tions ) j but there was one among them the mere name of 
which evokes a refreshingly non-mathematical image. 
This particular definition of a method was put forth by 
A. M. Turing ( 1912-1954) and is called a Turing machine. 

Since the mechanical way of thinking was almost as 
natural to Turing as the mathematical, it is not surprising 
that when he set out to define a method, he thought of it 
as something which could be performed by a machine. 
Said Turing: If a machine could be conceived of as solv- 
ing an arbitrarily chosen problem of an infinite class, then 
indeed we have a general method for that class of prob- 
lems. When we say there is no method of solving an 
infinite class of problems, we mean that it is impossible 
to conceive of such a machine. 

With a method, according to this definition, a machine 
could be given a set of specific instructions which it 
would follow for a finite length of time, depending upon 
the particular problem of the class that it was given; and 
eventually perhaps eons from now it would turn out an 
answer, the right answer, to that problem. Instructions 

for the machine would have to be absolutely determined 
in advance: do some specific thing until some other spe- 
cific thing happens and then do some specific other thing. 
The machine could ask no questions, exercise no judg- 
ments, make no innovations. Each problem would have to 
go in, and come out, with every step toward its solution 
automatically decided by the method alone. Otherwise, 
no method. 

Such a machine as Turing conceived is not even 
meant to be constructible. Conceptually, it is very like one 
of the great electronic computing machines which are in 
existence at the present time. In many ways it is conceived 
of as being less efficient than they, for its aim is not 
efficiency but simplicity. In other ways it is (quite lit- 
erally) infinitely more efficient. It is in the nature of the 
infinite classes of problems with which we are dealing 
that, while a computer may be in a sense "close" to a 
Turing machine, it can never in spite of all possible 
improvements in its efficiency be any "closer." This be- 
comes clear when we consider a specific and infinite class 
of problems for which a general method has been known 
since before the time of Euclid. Is a given number n a 
prime? Theoretically, we can solve this problem for any 
n by attempting to divide it by every prime which is 
smaller than (or equal to) Vw-; if none of these divides 
it, then n is a prime. Practically, though, we find very 
soon that n is too large for us to test by this method. 
Although mathematicians have devoted years to testing 
the primality of certain interesting numbers, life is literally 
too short to accomplish this, and they must yield to the 
electronic computing machines. But very soon n is too 
large for the machines. The largest number which has 
been tested and found prime is 2 9941 1. By everyday 
standards 2 9941 1 is quite a large number, being some 


3000 digits in length; yet among the primes it is a rela- 
tively small one. Since there are only a finite number of 
primes which are smaller than 2 0941 1 but an infinite 
number of primes which are larger, "almost all" primes 
are larger than the largest known prime. Obviously, an 
actual machine, because of the limits of time and storage, 
can never solve all or even certain specific problems of an 
infinite class. A Turing machine, being purely conceptual, 
has no such limits because it is conceived of as having an 
arbitrarily large amount of time and an arbitrarily large 
memory or storage as large as it needs for any given 
problem in a class. Only for this reason is it uncon- 

The mathematical point to the Turing machine is not 
whether there could or could not be such a machine. A 
Turing machine is simply a set of specifications, not for a 
machine, but for a method of solving an infinite class of 
mathematical problems. The limits imposed by the con- 
cept of the machine upon a method are as follows : 

The machine is allowed an arbitrarily large amount 
of time in which to solve a problem and an arbitrarily 
large amount of paper on which to do the work. A roll of 
tape keeps moving through it. This tape consists of a 
series of positions of rest which can be visualized simply 
as squares. At any particular instant only one of these 
squares is being scanned by the machine. How the ma- 
chine reacts is determined by (1) the contents of the 
square and (2) the internal state of the machine. The 
square contains one of a finite number of symbols and the 
machine is in one of a finite number of internal states. On 
the basis of these two factors, in the time interval allowed, 
the machine can change the contents of the square, 
change its position by no more than one square and /or 
change its internal state. It can have no choice, in the 


usual sense; what it does is absolutely determined by the 
method. Also included is a way of feeding problems to 
the machine and of recognizing when the machine has 
finished a problem. 

Such is the conceptual blueprint for a Turing machine. 
If what we call a method for solving an infinite class of 
problems (like determining whether or not n is prime) 
can be used within these limitations to solve any arbitrary 
problem of the class, then we have a method. When we 
say that there is no method for solving such an infinite 
class of problems, we mean that the class includes prob- 
lems which cannot by their nature be solved by such a 

By a method we mean a machine. 

Perhaps this does not sound like what w.e usually con- 
sider a precise definition; yet when we begin to apply it, 
we find that it does define what we mean by a method, 
and very precisely. The method for determining whether 
or not a given n is prime is a method in this sense; for, as 
we have seen, determining primality by machine is com- 
mon practice and limited only by physical considerations 
of time and storage. 

In the preceding chapter we described the method of 
truth tables by which it is possible to determine whether 
any sentence of the sentential calculus is a true sentence 
and, therefore, a law of the calculus. It is easily seen that 
this, too, is a general method according to our definition of 
a method as a machine. We can conceive of a Turing ma- 
chine which, using the method of truth tables, could solve 
any of the problems of the sentential calculus no matter 
how long and complicated the sentences involved might 
be. Since all of its problems are solvable by such a general 
method, we call the sentential calculus a decidable theory. 

Tarskfs Theorem 3, which we gave at the beginning of 


this chapter, tells us merely that elementary geometry is 
also a decidable theory. The more limited a class of prob- 
lems ( even though the class is infinite ) , the more likely it 
is that there exists a general method of solving all the 
problems in the class. The sentential calculus is the most 
fundamental and elementary theory of logic and is, as we 
have seen, a decidable theory. First-order predicate cal- 
culus, a step above it in complexity and importance, is an 
undecidable theory. The theory of numbers defined as all 
those problems which can be expressed in terms of the 
integers, the basic concepts of logic, and multiplication 
and addition is an undecidable theory, as Kurt Godel 
showed in 1931. When we take a more limited class of 
number problems, like those of elementary arithmetic, we 
find that we have a decidable theory. 

Sometimes, however, when we enlarge our definition, 
we get a decidable theory. When, as in the case of the 
problems of elementary algebra, we define our class in 
the same terms by which we define the problems of num- 
ber theory except for the fact that we substitute the real 
numbers for the integers, we find that we have a decid- 
able theory. Interestingly, Tarskfs proof that elementary 
geometry is a decidable theory follows from the proof 
( also his ) that elementary algebra is a decidable theory, 
elementary geometry and elementary algebra being both 
concrete representations of the same abstract theory. 

In the last quarter of a century, as a result of the pre- 
cise defining of method by Turing and others, modern 
mathematicians have been able to till a field which was 
undreamed of by their predecessors: the determination of 
undecidable theories, those classes of mathematical prob- 
lems for which there can be no general method. Just how 
undreamed-of this field is can best be illustrated by a 
famous problem proposed at the turn of the century by 


David Hilbert. As the leading mathematician of the day, 
he gave to his colleagues a list of problems which he felt 
needed to be solved. One of these was to determine a 
general method of solution for aU indeterminate, or 
Diophantine, equations. These, a sub-class of the prob- 
lems of number theory, take their name from Diophantus 
of Alexandria, who had a fondness for them. These are 
problems in two or more unknowns for which integer 
solutions are required. A simple example is x 2 y B = 17, 
which is one of an infinite class of problems represented 
by the equation x 2 y s = n, in turn a sub-class of the 
class of all Diophantine problems. 

When Hilbert, in 1900, proposed to his colleagues that 
they attempt to determine a general method for solving all 
Diophantine problems, he and his colleagues, as well as- 
sumed that such a general method existed. Today so great 
have been the recent developments in meta-mathematics * 
it is generally considered probable (although such has 
not yet been proved ) that there can be no general method 
for solving aU Diophantine problems: that it is an unde- 
cidable theory. Even its relatively small sub-class, men- 
tioned above, presents difficulties. It is not known whether 
there is a general method for solving the class of problems 
x 2 t/ 3 n. Such problems have only a finite number of 
solutions. This has been proved. For instance, the specific 
problem x 2 y 3 = 17, already mentioned, has the follow- 
ing solutions when x is positive: 

x 3, 4, 5, 9, 23, 282, 375, 378,661 
y 2, 1, 2, 4, 8, 43, 52, 5,234 

These solutions were obtained by a "method" which works 
in a great many cases in fact, has never failed to work in 
any case; yet it has never been shown in the sense of a 
* The study of the structure of mathematics. 


method such as that which can be performed by a machine 
that it will work in all cases. 

To show that the class of problems x 2 y 3 = n is de- 
cidable, someone must prove that this or some other 
method is a truly general method which could be used by 
a machine to solve any arbitrary problem of the class. To 
show that the class is undecidable, someone must estab- 
lish that in it there exist problems, x 2 y 3 = n, which by 
their nature cannot be solved by any general method. It 
is quite likely that this particular class of problems is 
decidable and that the known method is truly general. If, 
however, someone were to prove tomorrow that the class 
is undecidable, the result would have great significance: 
for, by establishing the undecidability of a sub-class of 
Diophantine problems, it would at the same time estab- 
lish the undecidability of the class of all Diophantine 

In such a way the determination of undecidable 
theories sub-classes in themselves of all mathematics- 
establishes, as well, a fact of overwhelming significance: 
that mathematics itself is undecidable. The answer to the 

Can there be a general method for solving all mathe- 
matical problems? 

is no! 

Perhaps, in a world of unsolved and apparently un- 
solvable problems, we would have thought that the desir- 
able answer to this question, from any point of view, 
would be yes. But from the point of view of mathema- 
ticians a yes would have been far less satisfying than a 
no is. Now it is established with all the certainty of 
logical proof that machines can never, even in theory, 
replace mathematicians. 


The language of twentieth-century elementary geom- 
etry, a curious combination of logic and letters, is a long 
way from Euclid. Decision theory was undreamed of in 
his mathematics; yet the conclusion to which mathematics 
has come as a result of GodeFs paper would be as satisfy- 
ing to Euclid as to any mathematician of the twentieth 

Not only are the problems of mathematics infinite 
and hence inexhaustible., but mathematics itself is inex- 


We have come a long way from Euclid, and perhaps 
how very far we have traveled is shown most vividly by 
a comparison of Euclid's axioms, which appear on page 27, 
and those of Tarskfs 2, which are printed in full below: 

Al [Identity Axiom for Betweenness] . 

A xy[fi(xyx)->(x = y)] 
A2 [Transitivity Axiom for Betweenness] . 

A xyzu[/3(xyu) KP(yzu) -* p(xyz)] 
A3 [ Connectivity Axiom for Betweenness ] . 

A xyzu[p(xyz) 

A4 [Reflexivity Axiom for Equidistance]. 

A xy[Z(xyyx)] 
A5 [ Identity Axiom for Equidistance] . 

A xyz[l(xyzz)-* (x = y)] 
A6 [Transitivity Axiom for Equidistance] . 

A xyzuvw[$(xyzu) A $(xyvw) - %(zuvw)] 
A7 [Pasch's Axiom], 

A txyzuVv[p(xtu) hft(yu&)-*fi(xoy) A 


A8 [Euclid's Axiom]. 

A txyzuVvw[/3(xut) 
A9 [Five Segment Axiom]. 

A xx'ijifzz ! uu'[%(xyx'ij} A o(yzy'z') A 

A o ( I/MI/ V ) A ft ( xijz ) A ( -ty z' ) 

A10 [Axiom of Segment Construction] . 

A xyiwV z[/3(xyz) A ^(z/sw;)] 
All [Lower Dimension Axiom] . 

V xyz[^j3(xyz) /\--~<j3(yzx) l\ 
A12 [ Upper Dimension Axiom ] . 

A xyzuv[o(xuxu) A (yuyv) A o(%w%u) A 
V P(yzx) V 

A13 [ Elementary Continuity Axioms ] . 
All sentences of the form 

where * stands for any formula in which the 
variables x,v,w,..., but neither y nor % nor u, 
occur free, and similarly for *, with x and y 

If the reader will take in hand pencil and paper and 
the vocabulary for E 2 from page 272, he will be sur- 
prised to find how easily he can translate some of these 
axioms into statements which will be meaningful for him. 
( The first few particularly! ) 

But it's a long way! 


absolute value, 143, 215 

addition (see also arithmetic, 
operations of) , 99-101, 
138-140, 193-196, 278 

aleph-one, 225 

aleph-zero (see dso infinite, the- 
ory of), 211, 217, 225 

Alexandria, 17, 18, 26, 279 

algebra, 67-68, 83, 106, 137, 

169, 171, 177 
elementary, 270 
Fundamental Theorem of, 

112, 215 

of n variables (see also geom- 
Index etr ^' "-dimensional), 173 

algebraic numbers (see also 
complex numbers; tran- 
scendental numbers), 42, 

algebraic processes as geometri- 
cal constructions (see also 
construction problems) , 

analysis, 43-44, 115, 177 

analysis situs (see also topol- 
ogy), 191 

analytic geometry, 7, 67-77, 82- 
83, 115-116, 130, 137, 141, 

"and" (see also calculus, senten- 
tial ) , 258, 260, 262-264, 266 

angle, 149-150, 158, 178, 190 
right, 23, 26, 177 
trisection of (see oho con- 
struction problems) ., 133- 
134, 135, 142-145, 146- 

Apollo, 61, 75 

Apollonius, 18, 63, 66, 67, 68, 
115, 119 


Arabs, 17 

Archimedes, 18, 63, 82, 83, 84, 

area (see also calculus), 78-82, 

178, 226, 243 
argument, 143 
Aristotle, 23, 24 
arithmetic (see also numbers, 

theory of), 49, 70, 177, 

elementary, 278 
Fundamental Theorem of, 33- 


Laws of, 98-99, 101, 109 
operations of, 55, 97, 105, 

108, 112, 193, 255 
Associative Laws (see also 

Arithmetic, Laws of), 98- 

99, 195 

assumptions, see axioms 
Athelhard, 19 
Athenians, 61, 145 
"at infinity," 129, 130 
Auden, W. H., 17 
axiomatic method, 22-23, 162, 


Axiomatic Method, The, 271 
Axiom of Choice, 238-241 
axioms (see also geometry, non- 
Euclidean), 17, 22-26, 28, 

128, 148-152, 155-156, 

246, 250, 252-253, 255, 

256, 258 

of Euclid, statement of, 27 
of Hilbert, 253 
of TarsH, statement of, 281- 


Banach, Stefan, 241 
Banach-Tarski Paradox, 241-243 


band, 183-184 
Barrow, Isaac, 52 
Bell, E. T., 7, 41, 187 
Berkeley, California, 28 
biquadratic reciprocity, 42 
Bolyai, Janos, 154 
Brianchon, C. J., 124, 125 
Brianchon's Theorem, 125-126 
British Association for Advance- 
ment of Science, 66 
Burali-Forti, C., 245 

calculus, 78-96, 130 
differential, 86 

first order predicate, 271-272 
Fundamental Theorem of, 86, 


integral, 86 

sentential, 258-270, 277, 278 
Cambridge University, 52, 66 
Cantor, Georg, 55, 59, 208-209, 
211, 213-214, 215-216, 
218-220, 234, 243-244, 
245-246, 254 

Cantor-Dedekind Axiom, 58, 208 
Cardano, Girolamo, 108-109 
cardinal numbers: 
finite, 207 
transfinite, 207, 211, 217-218, 

222, 224-225 
change, rate of, 89-91 
Cicero, Marcus Tullius, 19 
circle, 23, 24, 60-61, 72-74, 116, 
119, 133, 137, 149-151, 
158, 160, 171-173, 174, 
177, 178, 229-230, 235- 
240, 247-248 

squaring of (see also construc- 
tion problems), 133-135, 

class, 195, 198-200 
Common Notions, see axioms 
Commutative Laws (see also 

Arithmetic, Laws of)^ 98- 

complex numbers, 110-112, 113- 

114, 142-144, 200, 215, 

composite numbers, 31, 33-34 
computers, electronic, 37, 42 
cone, 64-65, 119 
congruence, 121-122, 233-236, 

conic sections, 60-67, 74-75, 

115, 119, 146 
conjunction, see "and" 
consistency, 16, 25, 254-257, 


construction problems, 133-147 

continuum, 46-59, 97, 220 
number of the, 220, 222, 225 

continuum hypothesis, 225 

counting, 97-98, 205-206 

Cours d'analyse, 179-180 

Coxeter, H. S. ML, 145 

cube, 21, 116-117 

doubling of (see also con- 
struction problems), 61-63, 
67, 75-77, 83, 133-136, 142 

cube root, 62, 144 

curvature, 156, 159 

curve (see also calculus), 178 
simple closed (see dso Jordan 
Curve Theorem), 178-182, 

decimals, 43, 53-55, 59, 218-220 
decision theory, 272-281 
Dedekind, Richard, 55, 56, 58, 

Dedekind cut, 55-58 

definitions (see also primitive 
terms), 22, 23, 248-250, 
251-254, 258, 264 

deformation, 178 

de la Hire, Philippe, 123 

Delian problem, see cube, 
doubling of 

De Morgan, Augustus, 144, 

dense, 104, 230-231 

Desargues, Gerard, 119-120, 
122, 123, 127, 130 

Desargues' Theorem, 122-123, 

Descartes, Rene, 67, 72, 75, 110, 
115, 119, 227 

Development of Mathematics, 
The, 187 

dimensionality (see also geom- 
etry, Ti-dimensional), 129, 

diophantine equations, 279-280 

Diophantus, 48, 279 

Dirichlet, P. G. Lejeune, 36 

Dirichlet's Theorem, 36 

disjunction., see "or" 

disk, 183, 184 

Disquisitiones Arithmeticae, 41 

distance formula, 74, 170-171 

Distributive Law (see also 
Arithmetic, Laws of), 99 

division (see also arithmetic, op- 
erations of), 29, 97, 99, 
100, 103-104, 138-140, 194 

dodecahedron, 22 

Duality, Principle of, 124-125 

eccentricity, 66 
edge, 182-184, 186 

Egyptians, 2, 67, 149 

Einstein, Albert, 8, 14, 246 

Elements of Euclid, 16-28, 30, 
31, 32, 46, 50, 52, 55, 58, 
135, 155, 245-254, 258, 

elements of Hubert's geometry, 

ellipse (see also conic sections), 

ellipsis, 63 

Encyclopaedia Britannica, 18 

Epimenides, 257 

"equal," 209-211, 217, 220 

equation of fifth degree, 201 

linear, 71-72 
solutions of, 97, 101, 105-112 

equivalence, see "if, and only if" 

Eratosthenes, 62, 64 

Erlangen Program, 192, 202 

Euclid, 14, 16-29, 47, 50, 52, 
55, 62, 66, 68, 70, 72, 115, 
121, 135, 148-162, 175, 
204, 218, 226, 245-254, 
257, 258, 269, 271, 272, 
273, 275, 281 

Euclid of Megara, 17 

Euclid's algorithm, 21, 28-29, 

Eudoxus, 17, 18, 20, 21, 46-53, 
55, 58, 62 

Eukleides, see Euclid 

Euler, Leonhard, 39 

even numbers, 31, 232, 234 

exhaustion, see integration 

extrema, 85-86 

Fermat, Pierre, 7, 38, 39, 40, 41, 
85, 86, 218 


Fermat's Last Theorem, 7 
form and number, 1, 30, 53 
For the Reader, 14-15, 28-29, 

45, 59, 77, 96, 113-114, 

146-147, 203, 269-270, 

foundations, crisis in, 245-246, 

254, 256 
Foundations of Geometry, The, 

Fourier, Jean Baptiste Joseph, 


Four Square Theorem, 38-40 
fractions, see rational numbers 
function, 87-88, 96 
functions, recursive, 274 

Galilei, Galileo, 208-209, 211 
Gauss, Carl Friedrich, 136-138, 

141, 142, 145, 146, 152, 

154, 215, 246 
Gem of Arithmetic (see also 

Quadratic Reciprocity, Law 

of), 40, 42 

geodesies, 149-150, 161-162 
Geometrie, La, 67 
geometry, 32, 49, 67, 116, 148- 

149, 169, 171, 188-189, 

analytic, 7, 67-77, 82-83, 115- 

116, 130, 137, 141, 168- 

elementary, 246, 271-273, 

278, 281-282 
Euclidean, 7, 16-28, 60, 72, 

116, 117, 121-122, 151- 

152, 156, 158-160, 175, 

176-177, 178, 188-190, 

202, 204, 271 
n-dimensional, 163-173 

non-Euclidean, 7, 148-162, 

190, 191 
protective, 115-132, 176-177, 

178, 190, 191 
rubber sheet, see topology 
Germain, Sophie, 13 
Gillies, D. B., 37 
Godel, Kurt, 11, 256-257, 273, 

278, 281 

great circle, 149-150, 155, 157 
"greater," 209-211, 217, 220 
Great Mathematicians, The, 17 
Greeks, 1-15, 16-29, 34, 45, 46, 

53, 60-61, 63, 66, 67, 68, 

98, 103, 104, 133-134, 137, 

149, 163, 208, 246 
group, requirements for, 192- 


group property, 193-194 
groups, theory of, 188-203 

Hamilton, William Rowan, 144, 

Hardy, G. H., 11, 66, 179, 180 

Harvard University, 256 

Heath, Thomas, Sir, 19, 20, 26, 
28, 58, 247 

helix, 146-147 

heptagon, 133, 137, 141 

Hilbert, David, 246-257, 279 

Hippocrates, 62, 63 

History of Geometrical Methods, 
A, 152-153 

"How Children Form Mathe- 
matical Concepts," 176 

hyperbola (see also conic sec- 
tions), 65, 66, 75-76 

hyperbole, 63 

hyperplane, 169 

hypersoHds, 163-168, 172 

icosahedron, 21-22, 201 

Identity, 196-197 

Identity, Law of, 260-263, 267 

"if, and only if' (see also calcu- 
lus, sentential), 258, 260, 
261, 264 

"if, then" (see also calculus, sen- 
tential), 258, 260, 264, 266 

imaginary numbers, 98, 106, 
108-112, 200, 216 

implication, see "if, then" 

incommensurable magnitudes, 
47-49, 51, 53 

increment, 88-89 

Indians, 98 

infinite, theory of, 204-225, 245- 

infinite descent, method of, 38 

integers (see also negative num- 
bers), 98, 108, 111, 195 
positive, 193-194, 195-197, 
211-212, 217, 231-232, 278 

integration, 82 

intersection, 247-248 

Introduction to Geometry, 145 

intuitions, 250, 253, 258-259 

invariance, 121, 122, 178, 188- 
190, 192 

inventum hecatomb dignum, 6-7 

Inverse, 196 

irrational numbers (see also 
square root of 2, irration- 
ality of), 53-58, 104-105, 
106, 108, 111, 230-233 

Jordan, Camffle, 178, 180 
Jordan Curve Theorem, 178- 
182, 186 

Kant, Immanuel, 16, 152, 250- 


Kepler, Johannes, 66 
Klein, Felix, 160, 192, 202 
Kline, Morris, 67 
knots, 185-186 
Kronecker, Leopold, 98 

Lagrange, Joseph Louis, 39 

Lehmer, D. H., 43 

Leibniz, Gottfried Wilhelm von, 

78, 82, 85, 86, 94, 96 
length, 67, 134, 138-139, 178, 

226, 240 

"less," 209-211, 217, 220 
limit (see also calculus), 80, 82, 


Lindemann, Ferdinand, 142 
Littlewood, J. E., 66 
Lobachevski, Nikolai Ivanovich, 

logic, see calculus, sentential; 

reasoning, rules of 

Mathematical Discourses and 
Demonstrations, 209 

Mathematicians Apology, A, 
11, 179 

Mathematics and Western Cul- 
ture, 67 

measure, 226, 240, 243 

measurement, 98, 101, 134, 149 

Memoire sur les surfaces elas- 
tiques, 13 

Menaechmus, 62, 63, 67, 75, 119 

Men of Mathematics, 41 

meta-mathematics, 279 

method (see also decision the- 
ory), 273-277 

Middle Ages, 20 

Mill, John Stuart, 152 

Milky, Edna St. Vincent, 151 

Mobius, A. F., 184 


Mobius strip, 184, 186 

models of 4-dimensional objects, 

multiplication (see also arith- 
metic, operations of), 45, 
99, 100, 101, 138-140, 193, 
194-196, 278 

natural numbers (see also num- 
bers, theory of), 44, 98, 
101, 103, 106, 111, 209- 
210, 211-214, 216, 217- 
218, 232 
classification of, 44 

negation, see "not" 

negative numbers, 98, 102-103, 
104, 105, 106, 107, 108, 
109, 111 

Newton, Isaac, Sir, 16, 52, 66, 
78, 82, 85, 86, 94, 96 

"not" (see also calculus, senten- 
tial), 258,260,264,266 

number for every point on line, 

numbers (see also algebraic 
numbers; cardinal numbers; 
complex numbers; compos- 
ite numbers; even numbers; 
imaginary numbers; inte- 
gers; irrational numbers; 
natural numbers; negative 
numbers; odd numbers; 
positive numbers; primes; 
Pythagorean numbers; ra- 
tional numbers; real num- 
bers; squares; transcen- 
dental numbers; whole 
numbers) : 
as points, 70 
theory of, 21, 30-45, 67, 278 

number system, extensions of, 
46-59, 97-112 

octahedron, 21 

odd numbers, 31, 34, 36, 209- 
210, 217 

one-to-one correspondence, 204- 
207, 208, 209-211, 217- 
218, 219, 233 

"On Formally Undecidable 
Propositions of Principia 
Mathematica and Other 
Related Systems," 256, 

operation, 195 

"or" (see also calculus, senten- 
tial), 258, 260, 264, 266 

origin, 68 

Pappus, 66, 67 

parable, 63 

parabola (see also conic sec- 
tions), 65, 66, 75, 83, 84, 
89-91, 95 

parallel lines (see also Postu- 
late, Fifth), 60, 116, 125- 

Parthenon, 60 

Pascal, Blaise, 123, 124, 125 

Pascal's Theorem, 123, 124-125 

pentagon, 133, 137, 141 

pentahedroid (see also hyper- 
solids), 164, 165, 166-167 

Permanence of Form, Principle 
of, 101, 107, 109 

perspective, 117-119, 126, 176, 

pi, 142 

Piaget, Jean, 176 

plane, 158, 159, 163, 164 

Plato, 1, 13, 47, 118, 133 

Platonic bodies, 21-22 

point (see also point sets, theory 
of), 16, 23-24, 27, 69-70, 
124, 129-130, 160, 163, 
168-169, 178, 227-228, 
248, 249-250, 251, 253, 

points, ideal, 128-130, 131 

point sets, theory of, 226-244 

points of line, 46, 56-57, 97, 
204, 221-222, 227, 228, 

points of plane, 22, 228-231, 

polygons, regular constractible 
(see also construction prob- 
lems), 133, 134, 135, 141- 
142, 145 

Poncelet, Jean Victor, 123-124, 
126, 128 

pons asinorum, 20, 27 

position, 190 

positive numbers (see also inte- 
gers, positive), 102-103, 
105, 107 

Postulate, Fifth, 60, 152-160, 

Postulate of Infinitely Many 
Parallels, 155, 156, 160 

Postulate of No Parallel, 157 

Postulate of Unique Parallel, see 
Postulate, Fifth 

postulates, see axioms 

prime, largest known, 37, 275- 

prime factorization, see Arith- 
metic, Fundamental The- 
orem of 


primes, 21, 31-41, 43, 133, 197, 
209-210, 213, 217, 218, 
232, 275-276, 277 
infinitude of, 32-33 
table of first fifty, 35 

primitive terms (see also defini- 
tions), 250, 255 

Prince of Mathematicians, 31 

Principia, 16 

projection, 118-119, 165 

proof (see also calculus, senten- 
tial), 17,22,28, 132 

proportion, theory of, 17, 20, 
46-53, 58, 62 

pseudosphere, 157, 158, 159 

Ptolemy I, 17, 22 

Pythagoras, 1, 3-4, 6, 48, 63, 
101, 104 

Pythagorean numbers, 8 

Pythagorean Proposition, The, 

Pythagoreans, 1-15, 43, 47, 48, 
55, 57, 59, 226, 227, 244 

Pythagorean Theorem, 1-14, 17, 
20-21, 27, 73-75, 143, 170- 
171, 177, 229 

Quadratic Reciprocity, Law of, 

40-42, 43 

quadrilaterals, 117, 248-249 
quantifier, existential, 272 
quantifier, universal, 272 

ratio (see also proportion, the- 
ory of ), 8-9, 47-52 

rational numbers, 8-15, 47-49- 
53-55, 56-57, 59, 98, 103- 
104, 105, 106, 108, 111, 
193, 197, 212-214, 230- 
231, 232, 233 


ray, 181 

real numbers (see also number 
system, extensions of; 
points of line), 55, 58, 109- 
111, 216, 218-222, 231, 
232, 233, 278 

reality, 155-156 

reasoning, rules of (see also cal- 
culus, sentential), 22, 250, 

reciprocals, 197 

rectangle, 174, 178 

reductio ad absurdum, 11, 153, 

relationship, 262-264 

relativity, 8, 177 

Renaissance, 115, 117, 118 

result, 195 

Rhind papyrus, 2 

rhomboid, 249 

rigid motion (see also congru- 
ence), 178, 190, 226, 233, 
235, 242 

rigor, 180, 246, 248, 254 

Robinson, R. M., 241-242 

Romans, 19 

roots, extraction of, 97, 105, 
109, 111, 140 

roots, irrational (see also square 
root of 2, irrationality of), 

rotation, 190, 242 

ruler and protractor, 133-134, 

Russell, Bertrand, 239 

Saccheri, Geronimo, 153-154 
Schopenhauer, Arthur, 20 
screw, see helix 
section, 118-119 

segment (see also straight line), 

sentence, 258, 259, 277 

17-gon (see also polygons, reg- 
ular constructible), 137, 
141, 145 

shape, 190 

sides, 183-184 

similarity, 121-122 

size, 190 

Smith, David Eugene, 43 

Source Book in Mathematics, A, 

space, 163, 164, 168-169, 172- 
173, 182, 252 

Spain, 19 

sphere, 157, 158, 183, 241-243 

square, 23, 116, 117, 133, 137, 

rotated into self, 198-200 

square root, 107, 144, 255 
of 1, see imaginary numbers 
of 2, approximation of, 14-15 
of 2, irrationality of, 1-2, 8-15, 
30, 46-47, 55, 104, 134- 
135, 137, 226-227, 244 

squares, 31, 34-41, 43, 44, 209- 

210, 218 
table of first fifty, 35 

squaring the circle, see circle, 
squaring of 

steepness, measure of, 84-86 

straightedge and compass (see 
also construction prob- 
lems), 60-61, 62, 75, 77, 
134-136, 138-141, 175 

straight line, 46, 60, 70-72, 84, 
128, 135, 149-150, 158, 
160, 190, 252, 253, 255 
and circle, 141, 146 

subtraction (see also arithmetic, 
operations of), 97, 99, 100, 
102-103, 109, 138-140, 194 

surface, simple closed, 182 

synthesis, 115 

tangent to curve, 78, 85-86, 87 

Tarski, Alfred, 241, 268-269, 
271-272, 273, 278 

Tarskfs Theorem 3, 272-273, 

tautology, 267, 268, 270 

tetrahedron, 21, 164, 165-167 

Thales, 3 

Theon, 18, 19 

theorems, 17, 22, 23, 25, 26, 28, 
148, 258 

theory of numbers, see numbers, 
theory of 

Three Square Theorem, 38-40 

topology, 174-187, 190, 191, 

transcendental numbers (see 
also pi), 106 

transfinite cardinals, see cardi- 
nal numbers, transfinite 

transformation, 121, 122, 178, 
188-190, 192 

translation, 190, 242 

trefoil knots (see also knots), 

triangle, 23, 79-82, 116, 122- 
123, 127, 133, 135, 137, 
139-140, 141, 149-150, 
158, 164, 165, 174, 177, 
178, 187, 189-190, 233, 
246-247, 254 

trigonometry, 7 

trisection of angle, see angle, 
trisection of 


truth, geometrical, 148-149, variable, 67 

152, 161 Vatican, 19 

truth or falsity of sentences, volume, 226, 243 


truth tables (see also calculus, whde numbers (seg ^ n&t _ 

Senten * al) ;5T^ o^o ural numbers) , 30, 48, 109 

Turing, A. M., 274, 275, 278 

Turing machine, 274-277 

Tumbull, H. W., 17 Yeats William Butler, 17 

Two Square Theorem, 37-38 

zero, 102-103, 104, 105, 106, 
unity, roots of, 201 204, 205 


(Continued from front flap) 

had failed them. They faced a problem they 
could not solve. They were so shocked, the story 
goes, that they "persuaded" the discoverer of 
this unhappy fact to drown himself. 

The author shows how this unsolvable prob- 
lem, and others that followed in later times, 
forced the invention of new concepts. The idea 
of number was successively broadened to in- 
clude zero, irrational numbers, negative num- 
bers, imaginary numbers, and infinite numbers. 
Geometry multiplied from one to many: projec- 
tive geometry, non-Euclidean geometries, the 
geometry of n-dimensions, topology or "rubber 
sheet" geometry until finally -what is meant by 
"a geometry" had to be defined in an entirely 
new way. 

In lively, understandable style, Constance 
Reid leads the reader to a new understanding 
of the abstract foundations of modern mathe- 
matics, the search for truly consistent assump- 
tions, the recognition that absolute consistency 
is unattainable, and the realization that same 
problems can never be solved. Mathematics 
itself is inexhaustible. 

Constance Keid grew up in San Diego, Cali- 
fornia, and attended San Diego State College 
and the University of California. Until her mar- 
riage in 1950 to Neil D. Reid, an attorney, Mrs. 
Reid taught in San Diego. For the past thirteen 
years she has devoted her time to her family 
and to the free-lance writing that produced her 
earlier books, From Zero to Infinity, and Intro- 
duction to Higher Mathematics. She has also 
been a contributor to Scientific American and 
numerous other journals and magazines. 
Constance Reid now lives in San Francisco.