LONG
WAY
FROM
EUCLID
by Constance Reid
$5.00
A
LONG
WAY
FROM
EUCLID
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
>0J>2.0 " ty jl
I,;'-! :" 1 ]]
r. ! JGi5 m
3 114
HA1
5AN1M979
AUG2713:
fllW
MAY 2 ? 1981
tt OCT291984
mw JUM04]
Held
A long way from Euclid
63-2^699
A LONG WAY FROM EUCLID
By the Author
From Zero to Infinity
Introduction to Higher Mathematics
A Long Way from Euclid
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.
CONTENTS
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
vii
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.
-EUCLID TO PTOLEMY I
D
Modern geometry is a royal road.
A
1
The
Golden Knot
in the
Golden Thread
IN ANCIENT GREECE, WHERE MODERN
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.
2
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
girl,
* 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-
gested:
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.
5
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
6
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
mathematics.
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
knot.
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
9
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
10
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.
11
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
or
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
even.
12
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
&lastiques.
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.
13
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
Number.
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."
FOR THE READER
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-
14
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
17
12 17
4r
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
V2.
OZ, s ! *&
Nothing,
Intricately
Drawn
Nowhere
"A POINT IS THAT WHICH HAS NO PAHT."
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
way.
16
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
17
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
work.
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
18
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.
19
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
20
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
4
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.
21
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
Theorems
Axioms
Definitions
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-
22
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
circle.
He clearly recognizes that he will never be able to pro-
23
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
Aristotle.
24
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
"importance."
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
25
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-
26
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,
AXIOMS AND POSTULATES OF EUCLID *
AXIOMS
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.
POSTULATES
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.
27
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.
FOR THE READER
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
28
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-
tions:
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
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
ANSWERS
66 >T 'T >
FORM AND NUMBER.
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
him.
"The higher arithmetic," wrote
30
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-
tact."
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
them.
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 :
31
4 00 9 000
00 000
000
6 000 10 00000
000 00000
8 0000 12 000000 0000
0000 000000 or 0000
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
32
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
33
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 :
000
0000
00000
000
0000
00000
000
0000
00000
1 + 3
0000
00000
= 4
1 + 3 + 5
00000
= 9
1+3+5+7
= 18
1+3+5+7+9
= 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-
34
THE FIRST FIFTY SQUARE NUMBERS
100
400
900
1600
1
121
441
961
1681
4
144
484
1024
1764
9
169
529
1089
1849
16
196
576
1156
1936
25
225
625
1225
2025
36
256
676
1296
2116
49
289
729
1369
2209
64
324
784
1444
2304
81
361
841
1521
2401
THE FIRST FIFTY PRIME NUMBERS
2
31
73
127
179
3
37
79
131
181
5
41
83
137
191
7
43
89
139
193
11
47
97
149
197
13
53
101
151
199
17
59
103
157
211
19
61
107
163
223
23
67
109
167
227
29
71
113
173
229
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
35
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.
36
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.
37
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.
38
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
39
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
squares").
* It was proved by Gauss.
40
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
41
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
42
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.
43
FOR THE READER
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?
45
4
A Number
for Every Point
on the Line
FROM WHOLE NUMBERS TO RATIONAL
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-
46
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-
bers?
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
47
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;
9
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
48
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
numbers.
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.
49
It was this new definition which Euclid used in the
Elements.
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
and
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."
50
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
and
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
follows:
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:
51
ma > nb and me > nd
or
ma nb and me nd
or
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. )
52
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,
53
that all repeating decimals are representations of rational
numbers.
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-
54
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,
55
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:
56
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.
57
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
58
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.
FOR THE READER
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:
I JL JL
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!
59
THE STRAIGHT LINE AND THE CIRCLE.
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
61
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^,
62
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!)
63
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
64
(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
line:
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.
67
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
Greeks.
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
Greeks.
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
z/-axis.
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
70
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-
dinates:
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
71
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
72
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
74
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
sections!
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
75
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,
76
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!
FOR THE READER
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), - . .
77
6
How Big?
How Steep?
How Fast?
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
answers.
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.
78
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
79
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
large.
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
triangle:
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.
81
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.
82
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-
84
;/ .^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 "
%/ / ^^^
J
fa-jrr^, k \ -9-, ^-* 1 i *,f / J % r=^-^5 f
ii
.
85
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
87
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
understanding.
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
of*.-
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,
89
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,
90
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
discovered.''
We begin by taking the area under a curve which we
91
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) = :
2
92
-p,,_7~^r
-ft. 'VW />.'/
M
' i ... A=^ < <r,i ' >j / <>/ .// > \m
* ' >j(fmK
'.-I ' , f*
\,^'.
4 f^f *l
tWA
fair*
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.
93
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
J
94
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 %.
95
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.
FOR THE READER
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 =
96
7
How Many
Numbers
Are Enough?
HOW MANY NUMBERS ABE 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-
97
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.
99
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
3BH-4B
or
4B + 3B
but there will be a considerable difference in the score
depending on which hit was made first:
but
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
but
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
100
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-
101
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
itself.
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.
102
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-
sible.
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
itself.
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
103
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.
104
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-
105
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.
106
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
(+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
107
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
108
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
109
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^
/t
' 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
110
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:
COMPLEX NUMBERS (x + yi)
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.
Ill
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.
112
FOR THE READER
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
geometry.
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-
113
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
114
Enchanted
Realm,
Where Thought
Is Double
THE ART OF GEOMETRY. THE GEOMETRY
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
115
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!
116
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
118
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
119
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
120
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
122
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
123
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):*
PASCAL'S THEOREM BRIANCHON'S THEOREM
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
124
The duality o the two theorems stated above becomes
even more vivid when we list parallel terms in parallel
columns :
PASCAL'S THEOREM BRIANCHON'S THEOREM
vertices sides
lie alternately on lines pass alternately through
points
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
125
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
apparent.
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.
126
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.
y#'"
' ' 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
127
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-
erality:
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
128
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
129
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
mathematics.
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:
130
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
point!
131
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!
132
TO TRISECT A GIVEN ANGLE.
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
133
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.
134
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 1f-n.fi 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
135
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
equation
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
136
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
137
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:
138
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
139
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-
140
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
141
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 :
142
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.
143
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
cube.
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-
144
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
"comment."
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).
145
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."
FOR THE READER
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."
146
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.
147
THE SUBJECT OF GEOMETRY IS ALMOST
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
mathematics.
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-
148
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-
lates.
A straight line is the shortest distance between two
points.
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
149
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
150
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
151
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.
152
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
153
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.
154
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
line.
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
155
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
156
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.
157
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
158
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
159
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
161
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.
162
THE COMMON METHOD OF MAPPING BY
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
space?
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
163
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
164
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
165
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
167
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 y-ax.es. 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
168
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
identified.
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
169
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).
170
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-
171
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
4
For the volume of a sphere, V H- nr 3
o
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.
172
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!
173
12
Where Is In
and
Where Is Out?
WHEN, AS VERY YOUNG CHILDREN, WE
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
circle.
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
circle.
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
174
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
175
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
176
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.
177
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,
178
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."
179
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
180
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
181
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
182
change the fact that they have no edge. These, like the
characteristic of dividing space into two parts, are in-
variant.
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
below.
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
183
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
again.)
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
184
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
185
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
186
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.
187
IF EUCLID WERE TO RETURN TODAY,
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."
Invariant.
Group.
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-
ments.
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-
188
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
190
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.
191
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
192
REQUIREMENTS FOR GROUP
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
193
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-
194
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
195
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
196
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-
197
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.
198
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
Identity:
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
199
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:
200
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-
201
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-
formations.
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
202
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
similarities.
The changeless in a changing world!
FOR THE READER
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?
CLASS
OPERATION
Gi
1.
2.
All integers
All rationals
+
3.
All rationals
_
4.
All even numbers
+
5.
All even numbers
X
6.
All odd numbers
x
_
7.
1
X
8.
1, 1
+
9.
1, *
X
10.
1, 1
-r-
11.
1, 0, 1
X
_
12.
1, i, 1, i
X
ANSWERS
dnoi y -[ S) "9 dnoi y *
dnoiS y '01 s's't^ -9 dnoj y
dnojS y -Q S<T ) *g **8iQ -[
203
THE INFINITE PROBLEM . . . PARADOX
. . . 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
short!
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?
204
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
birds.
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.
205
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.
COLLECTIONS WHICH CAN
BE PLACED IN ONE-TO-ONE
CORRESPONDENCE WITH
MODEL COLLECTION MODEL COLLECTION
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
206
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
207
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
208
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
square.
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:
ALL ODD ODD 4n -f- 1 SQUARE
NUMBERS NUMBERS PRIMES PRIMES NUMBERS
01350
135 1
257 4
37 9
4 9
5
6
7
8
9
* Galileo spoke here through the character of Salviatus in his
Mathematical Discourses and Demonstrations.
209
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?
ALL
ALL
ALL
ALL
ALL
ODD
ODD
4n + l
SQUARE
NUMBERS
NUMBERS
PRIMES
PRIMES
NUMBERS
I
3
5
1
3
5
13
1
2
5
7
17
4
3
7
11
29
9
4
9
13
37
16
5
11
17
41
25
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.
210
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-
finite.
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-
211
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
212
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
So?
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
213
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-
214
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
215
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!
216
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-
217
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.
O.OOOOOOOOOOOOOOOOOOOOO^ . . .
but
O.OOOOOOOOOOOOOW . . .
Have we proved, then, that it is Impossible to arrange the
real numbers from to 1 in such a way that they can be
218
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
placed.
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.
219
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
220
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.
, *?',
imj^W^''t4
:5. iawaii-sL USE i. ^ ^ Ji.z,
221
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
222
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
223
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.
224
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."
225
15
A Most
Ingenious
Paradox
SEGMENTS OF LINES HAVE LENGTH. SUR-
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
together.
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
"untrue."
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,
226
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
mathematics.
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
227
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
er+o'
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-
cumference
229
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
circumference
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-
230
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.
231
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-
merable.
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.
232
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-
nates.
3. The real numbers, which are the rational numbers
plus the irrational numbers, are a non-denumerable in-
finity.
4. The rational real numbers are a denumerable in-
finity.
5. The irrational real numbers are a non-denumerable
infinity.
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
part.
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.
233
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
234
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,
like
V2
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
236
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
point.)
237
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
238
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.
239
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-
240
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
241
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,
242
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
243
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.
244
FOR OVER TWO THOUSAND YEARS THE
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
245
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
chair.
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
needed.
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
246
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
247
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.
248
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."
249
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
250
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:
EUCLID HILBERT
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, . . . ;
251
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, . . . ;
itself.
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
itself.
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:
252
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
253
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
254
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.
255
"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
256
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-
lished.
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.
257
THE LADDER TO TRUTH WHICH EUCLID
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
258
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
259
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
concept.
' (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."
260
-> (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
if/'
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.
261
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
other.
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
262
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
right?
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-
263
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.
264
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
^p
2. Snow is white and all men are mortal. T F
pAq
3. Snow is white or all men are mortal. T F
4. If snow is white, then all men are T F
mortal.
p-*q
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,
265
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
T
T
T
T
T
T
T
F
F
T
F
F
F
T
F
T
T
F
F
F
F
F
T
T
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
266
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
267
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
__ _ __ _
FTP T
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
tautology.
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.
FOR THE READER
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
FT T F
F F T T
p q p-*q (p-qf)-*p [(p-* <?) ~ p] - p
T T T T
T F
F T
F F
269
puo
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.
snrnoreo
-U9S 9U} JO MV\ B 9JOJOJ9lp ptEB XO|O)l"lBq. B p99p
-TIT SI
-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
270
18
Mathematics,
the
Inexhaustible
AT THE MID-POINT OF THE TWENTIETH
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.
271
TARSKI*S ELEMENTARY PLANE GEOMETRY, OR Ea
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-
etry.
272
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
mathematics.
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
method?"
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
273
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
274
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
275
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-
structible.
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
276
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
machine.
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
277
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
278
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.
279
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
problems.
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
question
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.
280
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
century:
Not only are the problems of mathematics infinite
and hence inexhaustible., but mathematics itself is inex-
haustible.
FOR THE READER
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
281
A8 [Euclid's Axiom].
A txyzuVvw[/3(xut)
P(xzv)
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
interchanged.
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!
282
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,
214-217
algebraic processes as geometri-
cal constructions (see also
construction problems) ,
138-148
analysis, 43-44, 115, 177
analysis situs (see also topol-
ogy), 191
analytic geometry, 7, 67-77, 82-
83, 115-116, 130, 137, 141,
168-169
"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-
147,225
Apollo, 61, 75
Apollonius, 18, 63, 66, 67, 68,
115, 119
283
Arabs, 17
Archimedes, 18, 63, 82, 83, 84,
133
area (see also calculus), 78-82,
178, 226, 243
argument, 143
Aristotle, 23, 24
arithmetic (see also numbers,
theory of), 49, 70, 177,
255-257
elementary, 278
Fundamental Theorem of, 33-
34
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,
271
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-
282
Banach, Stefan, 241
Banach-Tarski Paradox, 241-243
284
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,
94-96
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,
142
class, 195, 198-200
Common Notions, see axioms
Commutative Laws (see also
Arithmetic, Laws of)^ 98-
99
complex numbers, 110-112, 113-
114, 142-144, 200, 215,
217
composite numbers, 31, 33-34
computers, electronic, 37, 42
cone, 64-65, 119
congruence, 121-122, 233-236,
253
conic sections, 60-67, 74-75,
115, 119, 146
conjunction, see "and"
consistency, 16, 25, 254-257,
274
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,
190
decimals, 43, 53-55, 59, 218-220
decision theory, 272-281
Dedekind, Richard, 55, 56, 58,
59
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,
146-147
dense, 104, 230-231
Desargues, Gerard, 119-120,
122, 123, 127, 130
Desargues' Theorem, 122-123,
127-128
Descartes, Rene, 67, 72, 75, 110,
115, 119, 227
Development of Mathematics,
The, 187
dimensionality (see also geom-
etry, Ti-dimensional), 129,
163
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,
271
elements of Hubert's geometry,
251-252
ellipse (see also conic sections),
64,66
ellipsis, 63
Encyclopaedia Britannica, 18
Epimenides, 257
"equal," 209-211, 217, 220
equation of fifth degree, 201
equations:
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,
48
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
286
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,
281-282
foundations, crisis in, 245-246,
254, 256
Foundations of Geometry, The,
250
Fourier, Jean Baptiste Joseph,
243-244
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,
192
analytic, 7, 67-77, 82-83, 115-
116, 130, 137, 141, 168-
169
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-
197
group property, 193-194
groups, theory of, 188-203
Hamilton, William Rowan, 144,
146
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-
246
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-
251
287
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,
87
Lindemann, Ferdinand, 142
Littlewood, J. E., 66
Lobachevski, Nikolai Ivanovich,
154
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
288
Mobius strip, 184, 186
models of 4-dimensional objects,
164-167
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,
227
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,
273-274
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-
132
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,
190
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,
255
points, ideal, 128-130, 131
point sets, theory of, 226-244
points of line, 46, 56-57, 97,
204, 221-222, 227, 228,
231
points of plane, 22, 228-231,
234-235
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,
248
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-
276
prime factorization, see Arith-
metic, Fundamental The-
orem of
289
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,
5
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
290
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,
258
reciprocals, 197
rectangle, 174, 178
reductio ad absurdum, 11, 153,
218
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),
13
rotation, 190, 242
ruler and protractor, 133-134,
135
Russell, Bertrand, 239
Saccheri, Geronimo, 153-154
Schopenhauer, Arthur, 20
screw, see helix
section, 118-119
segment (see also straight line),
254
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,
43
space, 163, 164, 168-169, 172-
173, 182, 252
Spain, 19
sphere, 157, 158, 183, 241-243
square, 23, 116, 117, 133, 137,
174
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,
277-278
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,
202
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),
184
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
291
truth, geometrical, 148-149, variable, 67
152, 161 Vatican, 19
truth or falsity of sentences, volume, 226, 243
261-262
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
292
(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.
THOMAS Y. CROWELL COMPANY
2O I PAKK AVENUE SOUTH
NEW YORK 3
ESTABLISHED 1834
Live Music Archive
Librivox Free Audio
Metropolitan Museum
Cleveland Museum of Art
Internet Arcade
Console Living Room
Books to Borrow
Open Library
TV News
Understanding 9/11