.

A convenient way of writing the sequence of fractions that results is

1 + 1/(1 + 1/(2 + 1/(1 + 1/(2 + ...,

or

/-\ /-\
| | | |
| | | |
| | | |
| | | |
| | \-/
| | -------------------
| | /-\
| | | | 1
| | | | ---------
| | | | 1
| | | | 2 -----
\-/ \-/ 1 1/2...



125

These are called continued fractions. The most useful thing to do is
to start with the iteration of

y = 1 + 1/(2 + 1/y).

Taking y = 1 as a beginning, we get successively y = 1 1/3, y = 1 4/11,
y = 1 15/41, ..., and the corresponding values of x are x = 1 3/4, 1 11/15,
1 41/56, .... Note that

(1 41/56)^2 = 3 1/9409.

The sequence for (root)2 + 1 is (2, 2, 2, ...); for (pi) it is

(3, 7, 15, 1, 292, 1, 1, 1, 2, ...).

This explains why it is often stated that (pi) = 3 1/7.





2. The incommensurable of the incommensurable


Let us begin to consider an event which may or may not have hap-
pened to the reader in the course of his life to date, in a semblance
more or less as is to be described here; and one which may--or
possibly may not--befall him in the future. Most of us have a sort of
routine by which we trace out the day. A man will go to his daily
place of occupation, where the surroundings change little, or change
in an expected way; where he finds tbe people seem to understand
what he says with a fair degree of accuracy, and to show little sur-
prise at the man's own commonplace gestures and remarks. The sort
of thing that has just been described is not unusual. It may not hap-
pen all the time, but it happens at least a good part of the time; the
number of days that vary from the pattern are few, if any, and they
do not vary much from the pattern. The psychological effect of this
monotony is to produce a commonsense outlook, a reasonable,
matter-of-fact way of looking at the world.

Then WHAMMO! The illusion is suddenly shattered. You are creep-
ing, let us say, down the queue in the refectory at lunch-time, just as
you do every day. Nothing at all out of the ordinary has happened.

126

The level of hunger is normal for such a situation. There is a choice
of meat pie or spaghetti; you must speak.

'Meat pie, please.'

'Sorry, sir, no cheese today. Only meat pie or spaghetti.'

'I'll have meat pie, please.'

'I told you, sir. You'll have to take what we've got.'

'All right.' By now you are a wee bit puzzled, and mere acqui-
escence seems an easy way out of the difficulty.

'We only feed that to the rats, sir. If you'd like to come around the
back I'll see what I can do.' The assistant is plainly puzzled by your
obstinacy, but means to be cooperative. She leads the way behind the
counters, the ovens, into a corner, through a door, into a broom
closet. The door of the closet is shut on you. It is dark. A muffled
voice is heard outside.

'The spiders don't often come before teatime, sir. Shall I slide a
few lollies under the door? A nice lolly will help to keep off the peck-
ish feeling, you know.'

'No! Why am I in here? Let me out!'

'Oh, well. I'll have to ask the butcher. Are you sure you can't
wait? The knives are all dull, and there's still a lot of meat in the
freezer. Most of the customers prefer spiders if the knives are dull.'

'Help! Police!'

The door opens, and you are on a vast plain of tall grass. Nothing
is visible except a caravan of Tartars approaching in the distance. As
their voices grow nearer and nearer, the words of their chant become
distinctly audible:

'Meat pie! Meat pie! Meat pie! ...'

Something like this happens if you construct an isosceles right
triangle. There is a procession from rationality to irrationality. There
is an encounter, where by an encounter we mean something different
from a lunch-counter. There is a meeting-up with something that,
while perhaps not Wholly Other, is rather Other. There is the feeling
of the Absurd; there is the wish to cry, Help! Police! The encounter
may come when two circles meet. It may come in a variety of ways.
When it comes, we cross over from rationality to irrationality, and
enter a new domain of incommensurability.

The word 'surd' is popularly used to designate irrationality. This
usage has fallen out of favour with mathematicians, but is still found

127

among certain classes of pedagogues and is used in school text-
books. The basic meaning of the word is 'deaf', but it may mean
irrational or unresponsive. Some early etymologists derived the word
from 'sordid', probably incorrectly, because those whose ears are
dirty or sordid are unable to hear clearly. A surd is defined as

(1) any irrational solution of an equation

x^n = q,

where q is rational; or else

(2) any number obtained by a succession of rational operations
and extractions of roots, beginning with rational numbers; or

(3) any algebraic irrational.

Since surds were discussed before cases (2) and (3) were known to
be distinct, and before the question whether they might be distinct
had been much considered, it is not always clear which kind of surd
is meant. Here, as in many cases in mathematics, one has a usefully
expressive term which time and the development of the subject have
passed by. In the meaning (1), it sometimes says beautifully what one
would like to say about an irrational; namely, that nobody can give
a rational solution to the equation, so that in a sense the equation is
deaf and unhearing to the question we ask it, and also that we can
give approximate solutions, but that to do so is ugly and sordid. Just
as with prime numbers, one must not expect an ordinary dictionary
to tell the truth about surds. One dictionary [14] defines a surd num-
ber as one that cannot be squared. This would be ridiculous if it
meant what it seems to mean, namely that it is impossible to multiply
the number by itself, since every element of a ring may be multiplied
by every other. If the definition meant that, then the number 17
would be surd when we considered it as an element of the abelian
group Z, and rational as an element of the ring Z. That would be
absurd. In fact that is not what is meant, in this old-fashioned
terminology, when it is said that a surd is a number that cannot be
squared. What is meant is that a surd is a number q such that the
equation

x^2 = q

has no rational solution. This differs in only two respects from the
defintion (1): we must replace 2 by n, where n is an integer at least 2;

128

and we must call x, not q, the surd. This use of the word 'square' to
mean 'square root' derives from geometry.



3. The inconstructible of the inconstructible


Once it became thoroughly well known that surds were quite surd,
and that it was absurd to keep on asking them, 'Who are you?
Please answer in the form of the quotient of two integers (the second
being non-zero),' people cast about for other methods of obtaining
information. Certain surds, it was found, would respond if subjected
to a very tiresome, long-drawn-out process requiring lots of strong
light and the use of (1) a sharp pointed stick; (2) a pair of sharp
pointed sticks joined at the blunt ends and capable of being set at any
desired degree of opening at the angle thus formed; and (3) a stiff,
straight rod of suitable length. Numbers which became more com-
pliant under this mode of investigation were known as constructible
numbers. All this occurred in an age of the world when people were
far more cruel than they are now. In obtaining information of this
type, it was not thought to be particularly relentless or cruel to use
such tools, or to stretch the number out along a line segment and
so possibly to put it painfully out of shape.

It would perhaps not have been surprising, considering the lengths
to which people were prepared to go in representing surds geo-
metrically, or constructing them, as it was quaintly termed, if all the
surds had given way. But there were surds too obstinate to accept
such treatment. Not only would these surds not speak up and say
who they were, they would not even lie down meekly on a line seg-
ment with their feet (so to speak) up in the air. These brave numbers
won the grudging admiration of even the ruthless investigators them-
selves for their unflinchingly irrational behaviour under the most
brutal methods of the police state, which included bisection, in-
definite extension, rotation, translation, cutting by lines and circles--
but the reader must be spared the full force of this infamy. Suffice it
to say that some of those that withstood such treatment became
known as surds of the third degree. In the curious parlance of that
abominable era, they were inconstructible.

129

QUESTION 23. Whether a surd of the third degree is inconstructible?

Objection 1. We must never give up hope. Indeed, one hears daily
of someone who has found a new way of treating some surd--say
<3>(root)2--that seems, to the investigator at least, to promise success.
There is so much more time available now for scientific research than
ever before that it hardly seems plausible to suppose the problem will
not be broken in the near future.

Reply to Objection 1. It is not a problem any longer. It is quite
certain that surds of the third degree will never yield to the dreaded
ruler and compass. Time spent on them is time wasted utterly, and
beyond hope of return. No research of this kind can be called
scientific.

Objection 2. Clearly the methods used in the past for the treatment
of irrationality have been wrong; they are essentially those of the
snake-pit. As to the irrationals of the third degree, they have been
subjected to this sadistic cruelty for so long that they may quite
possibly be beyond help by now. But some of the higher algebraic
irrationals might be chosen for a more modern approach, such as
neo-Freudian analysis or prefrontal lobotomy.

Reply to Objection 2. Prefrontal lobotomy should be done only
under medical supervision, even as applied to irrationals. It, too,
involves the use of a pointed instrument. Under analysis, even
transcendental irrationals become relatively tractable, it is admitted.
It is quite true that irrationals of the third degree may become con-
structible if the methods of ruler-and-compass geometry are aban-
doned; if for instance, the geometer is allowed a device that draws
the curve

| X
| X
| X
| X
| XX
-----------X-X-+-X-X-----------
XX |
X |
X |
X | 3
X | y = x\/


130

tangent at its point of inflection to a given line at a given point.
It is difficult to understand why such a device is not provided in the
kit with the other supplies when young children first set out in
geometry.

Objection 3. It is easy to construct the cube root of a given number,
which is a surd of the third degree. Voila`:



C
/|\
/ |/\
/ | \
/ | \
/ | \
/ | \
/ | \
/\ | \
B --------+-------- D
\ |O
\ |
\ |
\ |
\ |
\ /|
\ |
A



Here AO = n (the given number), OD = 1, the triangles are similar,
and O is the centre of a circle with diameters AC and BD.

Reply to Objection 3. Unless n = 1, there is no such circle. If
n = 1, there is a simpler construction:



|---------|
O A



Here OA is the given length 1; it is also the cube root of that length,
since 1.1.1 = 1. But since 1 is rational, this is not a surd.

131

I ANSWER THAT it is impossible to construct any root of a cubic poly-
nomial

ax^3 + bx^2 + cx + d

unless one of the roots is rational. Thus no length that can be con-
structed by the methods of ruler-and-compass geometry can ever be
a solution of an equation

ax^3 + bx^2 + cx + d = 0,

where a <> 0 and where a, b, c, d are rational numbers; i.e., quotients
of integers.

In what ways is it possible to write the number 3 as the sum of a
finite series of positive integers? The following ways are certainly
possible ones:

3 = 3
3 = 2 + 1
3 = 1 + 1 + 1

If one requires further that the sequence be non-increasing, then it
can be shown that these possibilities are the only ones (this con-
dition implies, for example, the exclusion of '3 = 1 + 2', since 2 is
greater than 1 and that makes this an increasing series). Essentially
all that is required is the inequality

0 < 1 < 2 < 3,

together with the fact that no integer lies strictly between two succes-
sive members of this inequality.

From the fact just mentioned, it is possible to show that

ax^3 + bx^2 + cx + d

must satisfy one of the following conditions: either

(i) ax^3 + bx^2 + cx + d cannot be factored; i.e. cannot be
written as the product of polynomials of lower degree having rational
coefficients; or

(ii) this polynomial can be factored, and one of the factors is
linear; i.e.,

ax^3 + bx^2 + cx + d = a(x + t)(x^2 + ux + v)
or = a(x + r)(x + s)(x + t);

again, the coefficients are rational. Ths is clear because if the
factorisation is

132

k
(product) f[i](x)
i = 1

k
then (sum) deg(f[i]) = 3,
i = 1

and because by commutativity we can assume that the degrees do not
increase.

Thus, if the cubic equation has no rational root then the poly-
nomial is irreducible; henceforth, let

f(x) = ax^3 + bx^2 + cx + d

be irreducible. If p(x) is another polynomial, and if p(x) is not of the
form f(x)q(x) for some polynomial q(x)--all polynomials shall have
rational coefficients--then since the ring

Q[x]

of polynomials with rational coefficients in an indeterminate x is a
principal ideal domain, there exist polynomials a(x), b(x) such that

1 = a(x)f(x) + b(x)p(x);

in particular, this is true if p(x) is quadratic or linear. Obviously,
then, no root (alpha) of f(x) satisfies any quadratic equation with rational
coefficients. Moreover, it is clear that

Q[x]/(f(x))

is a field containing Q as a subfield. If Q[(alpha)) is the smallest subfield
of F that contains both Q and (alpha), where (alpha) is a root of f(x) in some
extension F of the rationals, then (f(x)) is in the kernel of

(psi): Q[x] -> Q[(alpha)]
x -> (alpha),

and the induced map

Q[x]
(phi): ------ -> Q[(alpha)]
(f(x))

is a surjective field homomorphism, and hence an isomorphism.
Hence (1, (alpha), (alpha)^2) is a basis of Q[(alpha)] as a vector space over
Q. But since

133

constructible numbers can be produced by a succession of rational
operations and square-root operations, the smallest subfield of F
containing such a number has even dimension over Q; we merely
use the fact that if each of the fields k~, l~, m~ extends the preceding and
is a finite-dimensional vector space over the preceding field, then the
dimension of m~ over k~ is the product of the dimension of m~ over l~ and
the dimension of l~ over k~. But 3 is not even.


How to solve a cubic equation

It is easy to solve cubic equations. Let the equation be

ax^ + bx^2 + cx + d = 0,

and suppose the roots are x[1], x[2], x[3], so that

ax^3 + bx^2 + cx + d = a(x - x[1])(x - x[2])(x - x[3]).

If (sigma)[1] = x[1] + x[2] + x[3],
(sigma)[2] = x[1]x[2] + x[2]x[3] + x[3]x[1],
(sigma)[3] = x[1]x[2]x[3]

are the elementary symmetric functions, then writing

f(z) = x[1] + x[2]z + x[3]z^2

we get

3x[1] = f(1) + f(w) + f(w^2),

where

f(w)^3 + f(w^2)^3 = 2(sigma)[1]^3 - 9((sigma)[1](sigma)[2]
- 3(sigma)[3]),
f(w)f(w^2) = (sigma)[1]^2 - 3(sigma)[2].

Hence a root is

1/(3a){-b + [-2b^3 + 9abc - 27(a^2)d + 3(root)(-3(a^2)(b^2)(c^2) +
+ 81(a^4)(d^2) - 54(a^3)bcd +
+ 12(a^2)(b^3)d + 12(a^3)(c^3))]^(1/3).2^(-1/3) + [-2b^3 + 9abc -
- 27(a^2)d -
- 3(root)(-3(a^2)(b^2)(c^2) + 81(a^4)(d^2) - 54(a^3)bcd +
+ 12(a^2)(b^3)d + 12(a^3)(c^3))]^(1/3).2^(-1/3)}.

This is known as the cubic formula; in taking the two cube roots that
are indicated in it, always remember that among the nine combin-
ations that result from choosing among three roots in each case, only
those three combinations are allowed for which the product of the
cube roots is 2^(1/3)a^(-2)(b^2 - 3ac).

134

EXAMPLE. A man wishes to build a belfry, which shall be square at
the base, and which shall be three dekacubits higher than it is wide.
The desired volume of the belfry is four cubic dekacubits. Please give
the required measurements.


Solution. Let x be the height of the belfry in dekacubits. Then the
width and thickness at the base are both (x - 3). The equation is
this

(x - 3)(x - 3)x = 4,

or

x^3 - 6x^2 + 9x - 4 = 0.

The formula gives

x = (1/3){6 + [432 - 486 + 108 + 3(root)(-8748 + 1296 - 11664 +
+ 10368 + 8748)]^(1/3).2^(-1/3) + [432 - 486 + 108 -
- 3(root)(-8748 + 1296 - 11664 + 10368 + 8748)]^(1/3).2^(-1/3)},

or 4. Thus, a belfry 4 dekacubits high, and square at the base with
breadth 1 dekacubit will have volume 4 cubic dekacubits. From this
it should be easy to compute the volume in decibels when the bells
are installed. The bells are to be installed, of course, on decibelisation
day, and special decibel coins will be issued by the vicar of the
parish to all those who attend the ringing-in. On the great day when
for the first time all measurements will be in cubic units, it was thought
especially fitting to offer a seemly entertainment centred on the
church and capable of competing, noisewise, with the most deafening
discotheque. Afterwards there will be a talk on cubic measure-
ments, the cubit equation and the cubit formula; it is hoped that
no-one present will be able to hear what it said. Here is a run-down
on cubit measurements; note that no conversion factor from
standard British measurements is necessary:

1000 cubic millicubits = 1 cubic centicubit
1000 cubic centicubits = 1 cubic decicubit
1000 cubic decicubits = 1 cubic cubit
1000 cubic cubits = 1 cubic dekacubit
1000 cubic dekacubits = 1 cubic hectacubit
1000 cubic hectacubits = 1 cubic kilocubit
1000 cubic kilocubits = 1 cubic myriacubit.

135

The cubic cubit is established as the volume of water displaced by
the forearm of the person doing the measuring. This may be estab-
lished by the method of the great Greek mathematician, philosopher,
and physicist Eureka, who when he had put his whole forearm into
the bathtub to test the temperature of the water shouted 'Archi-
medes!', or as we would say in English, 'Decibelisation!' Regarding
the above table, it is necessary to remark that the number 1000
appears only as a rough guide, which is valid in almost all cases,
and that the general rule in the case of people with missing fingers
or parts of fingers as well as people with more than ten fingers is to
use the cube of the actual number of fingers present at the time when
the measuring is done. The joints of the fingers are usually taken as
dividing the fingers into thirds, and if extra joints are present the
finger is counted as more than one whole finger. Thus if you have 1 1/3
fingers missing from working in a sawmill, and if you were born with
11 fingers, then you must use the conversion factor 903 8/27, so that in
your case that number of cubic kilocubits make a cubic myriacubit.

The linear cubit, or measurement of length after decibelisation day,
is defined as the length of any edge of a cube of water at a comfortable
temperature that has been displaced by your forearm. The cube of
water should of course be in Euclidean 3-space, and should not be
frozen to facilitate measuring the edge as that would change the
volume somewhat and hence also the length of the edge. The square
cubit is similarly the area of a face of such a cube of water.

Ordinarily, a linear cubit is called simply a lineit, and a square
cubit is a quadratit. A measurement of 'hypervolume' in four-
dimensional space is defined in a completely similar way, and is
termed a biquadratit, or more freely, a quartit. This is sometimes
shortened to quart; other abbreviations are frowned on, and it is
especially hoped that the unpleasant dratit will not replace the more
seemly and correct quadratit.

In getting your own cubit, use the arm that seems more natural.
It is quite unnecessary to sever the limb at the elbow; simply hold the
arm in the water so that the elbow line lies in the surface of the water.
Artificial limbs having well-defined elbows may be used exactly as if
they were real ones.

136

4. Pain de maison


Poor M. Boulangiaire, the French baker, remained long in a state of
hypostatic tension over his quadratic equations. The trouble was
about those imaginary roots. He knew they would not come up in his
own bakery, because of certain facts relating to the coefficients. No-
one in the village was more sensitive to the good man's anxieties than
his widowed sister Gasparde Mange. Indeed, she felt the weight of it
more than he, being one of those women who live on anxiety. It
could almost be said that Gasparde got fat on anxiety; at least, she
tended to eat when she worried, and she worried prodigiously. Her
brother, Cre'tain the baker, was the focus around which her worries
orbited. The year after the invasion of the Ostrogoths, Gasparde
determined that her body and mind could use a course of slimming,
and took off for London. Her previous holiday, a couple of lustra
ago, had also been in London; Gasparde's life centred on Beaulieu
Derrie`re, and she could not bear to leave the village more frequently
than that. Long ago, she had learned that the food in London was
inedible, and that she had nothing to worry about in the land where
everyone does his duty.

Gasparde's sojourns in the homeland of the Anglo-Saxons were a
spasm of relief for her from the crowded activity of the village. She
usually avoided anything to do with home--she did not look at the
bucolic Butter Marketing Board posters, the vegetable shops, the
police horses, the 'Pinta' advertisements. (What a strange word for
milk that was! But everything about the English was strange.) On all
the other journeys, her eyes had never tarried on a baker's window;
that of all things was to be forgotten. Yet somehow, this time, she
could not help it. How queer the bread looked! She knew, of course,
that it was not bread; English bakers had a secret art of making the
appearance of bread out of air and water. But, it went without saying,
they could not make it quite like real bread if they chose those in-
gredients. Whatever their process might be, it certainly did not in-
volve baking. Why, it would all evaporate in the heat--and you could
see for yourself that there was no crust.

It must have been just then, when she was ruminating in front of
the bakery window, that the idea came to Gasparde. When the

137

assistant came out to ask her to leave (her immense form had been
blocking most of the light trying to come into the shop through the
plate glass) she had a question. 'Excusez-moi. Zat olmo^st spherical
wan wiz minimal superficial airia, wat is he call?'

'Cottage loaf,' the young person had said. By means of her pocket
dictionary, Gasparde resolved the cryptic name: bread of house, it
meant; a curious name. Probably because the shape of the loaf,
which had a little knob on top, could not be imitated and was a
speciality of this bakery. But it was the shape, all huddled together
on the inside with no outside surface to speak of, that interested
Gasparde. There was something about that that poor Cre'tain back
home in the village might find very interesting. Feeling her appetite
rise again at the thought of the serious business of the bakery at
home, Mme Mange hurried to the window of a tea-shop for the
only infallible remedy: the sight of Englishmen eating mashed
potatoes, tinned spaghetti and baked beans on toast. Hunger
vanished immediately, and soon the widow's mind returned to her
favourite pursuit on holiday--discovering new twists in the un-
fathomable ways of the islanders.

That winter, she recalled her moment of enlightenment for Cre'tain.
'You know, it would solve all your problems; only, you must not
make it like two balls smashed together. It is more like a chignon.'

'-- I could never, never compute the volume of a chignon; even
the sphere would introduce a difficulty, because the number (pi) on
my slide-rule is out by several thousandths.' When they spoke to
each other, Cre'tain and Gasparde always began the first sentence
with a long dash, as everyone does who speaks good conversational
French. It is the first indispensable rule; whoever ignores it is exposed
as a conversational barbarian before he has uttered a word.

'-- No, no, Cre'tain. Listen to me, I am telling you. You will not
make them like that. Do you even make the ordinary loaves like the
other bakers? Yours are pure rectangular parallelepipeds. You will
make them cubes, of course.'

'-- What an idea! The only baker in France to make a cubical
loaf. You cannot imagine the joy of it! And the mathematical impli-
cations... Now I am glad that I could never find you a husband;
you can never know how you have eased my mind. Because of
course... '

138

'-- There is always at least one real root. It does not depend on
the supply of dough, on the orders... '

'-- Wait, Gasparde. What if no-one orders the variable cube? I
will be the laughing stock of the village--there would then be a
quadratic equation, not a cubic.'

'-- Do you think I have not thought of it? Do you still think, my
brother, that because I am a fat old woman I am stupid? Have I not
been thinking of it since the autumn? I promise always to eat the
large cube. I will place an order. I myself. Only because I am your
sister--who else would eat a huge lump of uncooked dough every
day? And you are wrong about being the only baker of cubes. That
you are already; only they are the small cubes with edge of length 1.
There are others besides you who bake by the quadratic equation.
But you will bake by the cubic equation; that will be your unique
distinction!'

Immediately, M. Boulangiaire reached for a pencil that adhered
to his left ear, and wrote on the surface of the table:

ax^3 + bx^2 + fx + c = p,

where the coefficients are a for the Anglo-Saxon loaves required,
b for the flat, square loaves preferred by the Goths (M. Boulangiaire
always called them Boches), f for the ordinary French loaves, c for
the little cubic loaves whose side was constant, and p for the total
amount of dough for which there was provision. Yes, it was going
to be a grand game. The villagers would not know what to make
of it. To questions about the new loaf he would only reply, 'It is
special--you would not like it.'



QUESTION 24. Whether the cubic equation will always provide a real
root, and in fact a positive one, so that the baker can begin to bake?

Objection 1. The coefficient a is the number of Anglo-Saxon
cubical loaves, preferably of large size so as to be soggy inside. This is
certain to be at least 1, since Gasparde has promised to consume one
loaf daily, and she of all women can always be relied on to eat what-
ever she says she will eat. No doubt she will want several. Moreover,

139

the supply p always provides for more of the little buns of unit side
than the requirement c, since the villagers are never that keen to have
them. Hence if f is the polynomial function whose value at x is the
left-hand side of the equation,

f(0) < p

and

lim f(x) = (infinity).
x -> (infinity)

Taking N very large and positive, and applying the Intermediate
Value Theorem to f on the interval from 0 to N, we see that

f(0) < p < f(N)

and hence there exists a real number (phi) such that 0 < (phi) < N and such
that f((phi)) = p. This is then a positive root.

Reply to Objection 1. This solution of the problem has already
occurred to M. Boulangiaire, but has been rejected by him for the
very good reason that it requires the Intermediate Value Theorem,
of which he was not given a rigorous proof at the E'cole Paranormale.
We, of course, know of a rigorous proof of the Intermediate Value
Theorem, but it is one that uses the idea of the existence of a univer-
sally attracting object in the category of archimedean fields, and the
fact that such a universally attracting object must necessarily contain
the least upper bound, or supremum, of a set of numbers that is not
empty and that possesses an upper bound. Not only would M.
Boulangiaire be unable at his present age to assimilate all the neces-
sary concepts to follow a proof, but he would be justifiably annoyed
at the introduction of analytical concepts in order to solve an
algebraic problem. 'You are employing a blast furnace in order to
toast marshmallows,' he would say in his shrewdly practical way.

_______
Objection 2. Replacing any root (alpha) by its complex conjugate (alpha),
we get

_______ _______ _______
a(alpha)^3 + b(alpha)^2 + f(alpha) + c
________________________________________ _
= (a(alpha)^3 + b(alpha)^2 + f(alpha) + c) = p,

_______
so that (alpha) is also a root. Hence the roots occur in conjugate pairs,
and since 3 is an odd number and there are three roots in all, one of
them must be equal to its complex conjugate and hence real.

Reply to Objection 2. In order to do the problem this way, it is
necessary to construct the complex number field, or some other field

140

containing all the roots of the equation, and possessing an alter-
_ _
nating automorphism k -> k; i.e., one such that k = k; the alter-
nation must have the property that the subfield it leaves invariant is a
real field. The study of automorphisms of extension fields and the
subfield that they leave invariant is Galois theory, and the study of
the complex numbers is analysis; hence, these methods are too high-
powered.


I ANSWER THAT there exists a real root of the cubic equation under
consideration, and that it is positive. This means that the baker will
always be capable of baking bread in such a way as to solve the
equation and just use up his dough--although he will have to
approximate and so may be a little bit out, the error and hence the
amount of dough left over or not available for finishing the last loaf
can be made as small as desired. Unfortunately, he will not in general
be able to construct x, the length of the French loaf and the edge of
the Gothic and Anglo-Saxon loaves, by ruler and compasses as he
used to do with the old method. When he had only a quadratic
equation to solve, it was possible to obtain the required dimension by
ruler-and-compass construction, and the baker greatly enjoyed the
process. This recreation will now have to be abandoned. A number
of good methods nevertheless exist by which the approximation can
be refined within acceptable bakery tolerances.

The point is to have a system of numbers in which it is possible to
do addition, multiplication, subtraction, and division, in which it is
possible to say whether a number is positive or negative in a way
consistent with the arithmetic of the system of numbers; the system
must be generous enough to include a positive number that corres-
ponds to the desired dimension x, where x is the solution of the
bread equation given by the orders of the various customers. That
will all be the case if there is a real extension field of the field of
rational numbers containing a root of the equation.

If the equation is reducible, this is obvious. Taking g(X) to be

aX^3 + bX^2 + fX + c - p,

and (alpha) to be a root of this polynomial, suppose -1 is a sum of squares

141

in Q((alpha)). We get the equation

n
-1 = (sum) p[i](X)^2 + k(X)g(X),
i = 1

where of course the p[i](X) have degree less than 3. Since k(X) there-
fore necessarily has degree 1, or is the zero polynomial, there exists
a rational number q such that k(q) = 0. Hence, -1 is the sum of
squares of a finite sequence of rational numbers. This is impossible,
and hence a real root exists.

The question remains whether any of the roots is positive. This
is clear if all roots lie in a real field, since their product is (p - c)/a,
which is a positive rational. Otherwise, the polynomial is reducible,
so that

aX^3 + bX^2 + fX + c - p = a(X - (alpha))(X^2 + BX + C).

Since B^2 >= 4C would imply that all roots of the equation lie in a real
field, it may be assumed that B^2 < 4C. Since c - p is negative, so
must be -(alpha)C, and it follows that (alpha) is positive.



How to find cube roots

In solving a cubic equation by the cubic formula, one is obliged to
take cube roots. This may be done by the method of approximation
by continued fractions to within any required degree of accuracy.
For example, to find <3>(root)2 we note that x^3 - 2 changes sign between
1 and 2; this leads to

x = 1 + 1/y,

and on substitution we find

y^3 - 3y^2 - 3y - 1 = 0.

The left-hand side changes sign between 3 and 4, which leads to

y = 3 + 1/z.

Similarly we get z = 1 + 1/w; continuing this we get

<3>(root)2 = 1 + 1/(3 + 1/(1 + 1/(5 + ...;

the computation taken thus far gives 29/23, the cube of which is

2 55/12167.

142

5. The Rule of Three


If a man and a half, all of whom speak Scots Gaelic as well as
English and have had 5 years of schooling, dig a hole and a half
while the temperature is 35 degrees Fahrenheit, smoking 22 cigars, eating
4 lb of preserved goose, reciting a Shakesperean sonnet, and taking
a day and a half to do the job, how long will it take one man, speak-
ing only Ancient Egyptian, Sumerian, and Chinook Trade Jargon
and having 3 years of schooling, to dig a single hole while the tem-
perature is 3 degrees Centigrade, smoking 11 cigars and eating 3 kilos of
moules marinie`re while saying off from memory all the poetic con-
tents of the Egyptian edition of The Lord of the Rings? [33]

The above is an example of a question in the Rule of Three. It is a
moderately complicated example of the genus, and the fact that it
can be solved at all is a tribute to the immense versatility of our con-
venient algebraic notation. A simpler problem is this: if John gets
10 hectathrills from accompanying to a ball a well-proportioned lady
of 25 years who is 5 feet tall, how many thrills does John get from
accompanying to the same ball a well-proportioned lady of 50 years
who is 10 feet tall? (The thrill is the basic measurement of joy; a
hectathrill is equal to 100 thrills.) Let us simplify still further, by sup-
posing that not only the gentleman, John, but also the age of his
young lady remains constant--whether she is 25 or 50 need not con-
cern us. The only thing that changes is her height. Then we note that
the problem is stated as a question. This is very unpleasant, because
it is not a proposition unless it is in the indicative mood. Hence we
make it a statement, using a letter (say O) to stand for the unknown
number of thrills. It now takes the form:

O thrills of John are to 10 feet of lady
as
100 thrills of John are to 5 feet of lady.

Here we have a problem in the standard rule of three, so called
because it has as its aim the solution of all problems in which one is
to find the fourth, unknown, term in a proposition in which three
terms are known. The more complicated examples, such as the one

143

previously mentioned involving a choice of cigars, languages,
delicatessen, etc., can only be treated once the main points of the
simple, classical case are fully understood. Before considering these,
it would be desirable to examine first why there is no such thing as a
Rule of Two, or a Rule of One. In a sense, of course, these things do
exist. A Rule of One might be thought of as a way of finding a single
unknown from a single given quantity, and a Rule of Two as a func-
tion of two variables. But these ideas do not fully mirror the structure
of the Rule of Three, which is not merely a matter of a function of
three variables. It is more nearly a matter of a class of functions
indexed by a parameter; two of the given numbers suffice to deter-
mine the parameter, and the third determines the particular value of
the function associated with the value thus determined for the para-
meter. Thus, no fewer than three given numbers can produce a
sufficiently rich structure. Rules associated with larger odd numbers
are then a matter of families of functions of several variables; i.e. the
Rule of 2n + 1 is simply produced by considering an index set A and
a function

(f[a]) a (epsilon) A: A -> Y,

where

n
X = X~ X[i]
i = 1

Hence, there can be no Rule of Two since 2 is even, and there can be
no Rule of One since n = 0 gives a Cartesian product over an empty
set of indices (from 1 to 0), so that X is a singleton {x}. Necessarily
the given term in the proposition is f[a](x), and this is just what we are
required to find--hardly an arduous task requiring a special rule for
its accomplishment.

In the case of John and the expandible-contractible ladies, it is
clearly presupposed that once we know how John reacts to a little
thing of 5 feet we can say just how a 10-foot version of the same
delicious morsel will hit him. Now various men react to different
women in various ways. For some, the thrill function reaches its
maximum value at 5 feet 2 inches. Clearly, what we must know from

144

the given information is John's thrill function. In other words,
the function

X -> Y

induced by A -> Y actually takes its values in the subset of Y
consisting of injections A -> Y. Thus, if two different men (John and
Xerxes, let us say) receive exactly the same thrillage from the com-
pany of a 5-foot woman, then they must also receive exactly the same
thrillage from a 10-foot woman. Otherwise, it is impossible to answer
the problem as set.

Let us suppose, in order to be able to do the problem, that a lady
h feet high produces in a man a number of thrills equal to

a sin(((pi)/10)h)

where a is the parameter. We easily determine in the present case that
for John the parameter a takes the value 100; hence it is easily com-
puted that he gets exactly no thrills from taking out a 10-foot lady-
friend.



6. Polynomials


In treating problems related to the Rule of Three, it is often assumed
for simplicity that the functions f[a] in the problem are all of the form

f[a](x[1], x[2], ..., x[n]) = a(x[1]^k[1])(x[2]^k[2])...(x[n]^k[n])

where the k[i] are integers, and where A, X[1], X[2], ..., X[n] are all the
same field. If we restrict this further to the case where the k[i] are
natural numbers, we reach a kind of function closely related to the
polynomial. Polynomials are sometimes divided into monomials (pro-
perly spoken, these should be mononomials), binomials, trinomials,
.... There seems to be no special name for polynomials that have no
terms at all, but otherwise these are just fancy ways of incorporating
into the name exactly the number of terms that exist in the poly-
nomial. Of these, only the first type, the monomial, deserves separate
study. Hence we begin the study of polynomials with that of mono-
nomials, or what might almost be called terms.

145

An example of a monomial is

(tau)(A^a)(B^b)(C^c)(D^d)(E^e)(F^f)...(Z^z)

where (tau) is likely to be an element of a ring, and where the exponents
a, b, c, ..., z are natural numbers. The letters A, B, C, ..., Z are
called letters or indeterminates. It can easily be shown that the
monomial just mentioned is a monomial in (at least) 26 letters; in
order to simplify slightly the questions which arise, we shall begin by
considering monomials in just one letter.

An example of a monomial in just one letter is aX. From this
monomial we can make a function, in a way which will be more fully
described below. If we assume that the thrill function with parameter
is obtained from this monomial, rather than from the function sin,
then we shall be able to compute that if John gets 100 thrills from
dancing with a lady 5 feet tall, then

100 = a.5,

so that a = 20. This shows that if John dances with a lady 10 feet
tall then he gets 200 thrills. (This approach is called linearising the
problem, and shows one of the many charming applications of this
special kind of problem.)

Another kind of monomial is X^a. Using this monomial we produce
a parametrically indexed family of functions, and the same problem
as before--if the reader will forgive the intrusion of one more worked
example of the Rule of Three--can again be worked on the hypoth-
esis that the thrill reaction of a man dancing with a lady depends on
the height of the lady according to the law

h -> h^a,

where h is her height and where a is a parameter which has to be
determined in the case of every individual man, since men do not all
react to ladies in the same way. In the present case, we get

100 = 5^a,

and since a is a natural number, this is impossible, since it leads to

5^a = 5^3 - 5^2,

which can be shown by computation to be false if a = 0, 1, 2, or 3,
and which is false for a >= 3 because

Z^(a-2) - Z + 1 = 0

146

has as its only rational roots a subset of {-1, 1}. Why can the prob-
lem not be done? There are two explanations: one, that the word
'lady' no longer has a definite meaning. In 1840, a self-respecting
servant girl could say, 'Why, if I were a lady, I should be delighted
to be the object of Captain Ainstruther's affectionate interest.' Then,
it was a matter of fact whether a woman was a lady or not. At
present, many women who are certainly not ladies are most insulted
if they are called women, and speakers of the language are divided
between those who call all women ladies, and those who call no
women ladies. Now in a mathematical exercise, it is of crucial import-
ance that all the words used have a definite, clearly defined meaning.
It is no good applying mathematics to a mixture of vague impres-
sions, compounded of the notions 'lady', 'gentleman', 'well-propor-
tioned', etc. All the concepts must rest on a firm logical basis, as
ball:

{x (epsilon) R<3>: |x| <= 1}

--one may gather that it is the 3-ball from the fact that the people
who are going to the ball are in a problem illustrating the Rule of
Three, but if it were some other odd number, one would simply take
the ball in the Euclidean space of the corresponding number of
dimensions. That is one reason why we cannot expect very startling
success in determining the thrillage. The second reason is that in this
case the image of 5 under X -> Y is not a surjection A -> Y, and
that this is just as necessary as that the image be an injection.



QUESTION 25. Whether a polynomial is a kind of function, or not?

Objection 1. A polynomial is a kind of function. In fact, if you
start with a variable x, by which is meant a number that can be any-
thing it likes, and if there are some constants, by which are meant
some numbers that are stuck and cannot change at all, no matter how
hard they try, and if addition, subtraction, and multiplication are
allowed but not division, then one will get an expression of the form

a[n]x^n + a[n-1]x^(n-1) + ... + a[0].

Now if you let the variable x vary--which is to say, change all over

147

the place--then this quantity will also vary. As x varies independently
of any control, the quantity formed in the way described will vary in
a way that depends on the way x varies. Hence x is called the in-
dependent variable and y is the dependent variable. But where a
dependent variable depends on an independent variable, there one
has a function; and functions constructed according to this formula
are called polynomials.

Reply to Objection 1. It is bad form to mention variables; not only
that, but it is very imprecise to talk about numbers changing all over
the place, moving and heaving about like quicksand. It would be
incorrect to say that the thing presented here is a polynomial, and
just as imprecise to call it a function. It is by no means clear just
what it is.


I ANSWER THAT polynomials are not functions. The reason is that
polynomials can be used to obtain functions, but that two poly-
nomials may produce the same function without being the same
polynomial. Before explaining function, we must of course go back
to monomials, and indeed we must begin by considering monomials
without coefficients. There will be a set of indeterminates, or letters,
which are to be used; let this be S. Of course, if M is any monoid
then M|S| has the obvious monoid structure, and it is called the set of
monomials in the letters of S without coefficients, and with exponents
in M|S|. One ordinarily takes M to be the natural numbers N[0]. In terms
of notation, there is a trick to be learnt; to indicate the element of
M|S| that sends A[i] to n[i] for i and integer such that 1 <= i <= k, where
S = {A[1], A[2], ..., A[k]} and k is a positive integer, we write

(A[1]^n[1])(A[2]^n[2])...(A[k]^n[k])

Moreover, if A (epsilon) S and if A is sent to 1, it is permissible not to
write the 1; and if A is sent to 0 it is permissible to write neither the A nor
the 0; but this does not apply if S = {A}.

The obvious structure on the monoid N[0]|S| then makes it obvious
that (taking S = {X}) we have

(X^m)(X^n) = X^(m + n);

it is impossible to exaggerate the importance of this identity in com-
putations with polynomials.

148

Unfortunately, the above considerations of notation have the
effect of importing a certain difficulty into the study of the monomial.
Taking again the case S = {X} for simplicity, it is clear that the
expression X^n can mean two separate things: firstly, the function
S -> N[0] that sends X to n; secondly, the nth power of X = X^1,
which is the function S -> N[0] sending X to 1. The solution is not far
to seek--we merely map N[0] homomorphically to N[0]|S| by

n -> X^n,

which means two homomorphisms by the two interpretations of
'X^n', and observe that the two homomorphisms are not distinct,
making one homomorphism and hence one interpretation of 'X^n'.

By the same token, we see that there is an a priori danger of con-
fusion between AB on the one hand and AB on the other; or slightly
more generally between (X^m)(Y^n) and (X^m)(Y^n). It would of course be
somewhat discomfiting for the theory of polynomials if these were
really as distinct as one must suppose, without proof, they may be.
Therefore let us attempt to show, if possible, that they remain the
same, always restricting ourselves for simplicity to the case AB
versus AB.



QUESTION 26. Whether one may be certain that AB is the same as
AB?

Objection 1. This, surely, of all the ridiculous quibbles, of all the
effete nonsense, of all the crabbed contortions found in the present
compendium of cant, is the silliest and most abominably absurd! Is
a thing the same thing as itself, when it is mentioned under the same
name? Flogging is the answer to those who permit themselves to ask
such questions, if no more ignominious punishment is available.
And a remedial period of good, hard work could without detriment
follow, to be continued until a more socially oriented behaviour
pattern develops. Apiary work would do. The patient might be re-
quired to repeat, whenever stung, 'A bee is a bee is a bee.'

Reply to Objection 1. It may possibly seem, at first sight, that when
two things are mentioned under the same name they are the same

149

thing. If one is set the problem: Prove that Oberon is the King of the
Fairies, it is not surprising that there is some work to do--one may
have to find a fairy and say, 'Take me to your leader!' But on being
asked to prove that Oberon is Oberon, one reacts differently. Since
it is not clear how to proceed, even though the task hardly promises
to be one of extreme difficulty, one may feel inclined to bite one's
nails or to call for the police. It is not unnatural to react in this way.

The present case is in fact slightly deceptive. The conventions of
notation have produced two expressions that were intended to say
different things, and that were arrived at by different roads; and by
pure chance the two expressions have the same form. It is rather like
the two English sentences

'Fruit flies like a banana.'

and

'Fruit flies like a banana.'

The first of these sentences means that certain small flies, some of
which belong to the species Drosophila, and all of which feed on
fruit when they are in the larval stage, if they can get it, have among
the fruits for which they feel a preferential inclination a yellow,
elongated fruit of the genus and species Musa paradisiaca sapientum.
In other words, those fruit flies really go for a banana. The second of
the two sentences means something quite different; to wit, that
vegetable produce succeeding the flower passes through the air in a
manner resembling that in which a banana would pass through the
air. The first sentence could be tested as to veracity by presenting
some fruit flies with various kinds of food: a measuring tape, some
beefsteak tartare--the list can vary without restriction, except that
it must contain a banana. That way, you could tell if fruit flies like a
banana. To test the second proposition, one might use a wind tunnel,
or a large catapult, in order to study the aerobatic properties of
mangoes, lichees, pears, and (just possibly) aubergines and horse-
chestnuts, and to compare these with the aerobatic properties of a
banana. 'Fruit flies like a banana' is a proposition in entomological
gastronomy, whereas 'Fruit flies like a banana' is a proposition in
horticultural aerobatics. These two subjects, while they are important
intellectual disciplines demanding our utmost respect, are still in
their infancy, and unfortunately we cannot safely rely on them for
more than tentative, if hopeful, guesses as to whether fruit flies do

150

like a banana, or as to whether fruit does fly like a banana. The im-
portant point is that the two fields have as yet no common ground;
at present, no reputable interdisciplinary work has been done in
both at once; no joint degree has been taken in entomological
gastronomy as applied to horticultural aerobatics; no lecture en-
titled 'Fruit flight and the diet of Drosophila: bananism versus
pomegranitry' has been read. There is no connection between the
two sentences. If we are to prove that AB, understood in one sense,
is the same as AB, understood in a quite different sense, we shall not
do it just by remarking that they look the same.


I ANSWER THAT without doubt AB = AB. Consider the three
monomials involved in the equation: A, B, and AB. These are
monomials in a certain set of letters S, where S certainly contains A
and B--considered as letters, not as monomials--and where S may
possibly contain any further finite set of letters--say, all Russian
minuscules that normally precede the letter 'e' in a native Russian
common noun, all Hebrew letters that take daghesh forte but not
daghesh lene, and the Devanagari voiced aspirates together with
Latin majuscules not congruent to 3 modulo 4.

Now as a monomial,

A: A -> 1

and if (mu) (epsilon) S (without) {A} then

A: (mu) -> 0.

Displaying in a table this information for the monomial A with
similar information for the monomials B, AB,

| (mu) = A | (mu) = B | (mu) (epsilon) S (without) {A,B}
------------+------------+------------+----------------------------------
A: (mu) -> | 1 | 0 | 0
B: (mu) -> | 0 | 1 | 0
AB: (mu) -> | 1 | 1 | 0

we get that A~ A((mu)) + B((mu)) = AB((mu)).
(mu) (epsilon) S


Those who have mastered the art of forming monomials may, if
they are good, be admitted to the secrets of polynomials. These are

151

among the most important weapons in the arsenal of the mathe-
maticians, so that familiarity and skill in the handling of polynomial
manipulations is absolutely essential for passing examinations. The
most important skill of all, the sine qua non of the would-be alge-
braist, is the ability to recognise a polynomial on sight. This is not
as simple as it sounds. Recognising polynomiaIs on sight can be done
on various levels. At a rather lowish level, one may be shown this:

a[0] + a[1]X + a[2]X^2 + ... + a[n]X^n,

and one may be asked, 'Well, how about it? Is it a polynomial, or
isn't it?' The low-level answer is (depending on the dialect one
speaks),

'Yeah, sure I guess. Looks like one. Yeah, sure, that's a poly-
nomial, all right.'

Now, such an answer is neither too extremely bad nor on the other
hand is it as good as it might be. The worst answer (excluding some
very improbable sorts of answers, since there is not room here to
consider any but the most common possible responses) would be a
flat, definite No. After that, the next to the worst answer is a flat,
definite Yes. The present answer, a hesitant, fumbling Yes, has many
advantages. Hesitation is often a symptom of superior knowledge; in
fact, one may have rarely had the opportunity to see polynomials, so
that one is just barely able to tell when something looks like a poly-
nomial. In order to hide one's ignorance, one is inclined to be daring
and come out with a flat answer. But then one realises that experts
ofen hesitate. Experts have all sorts of reasons to think two ways on
a question, because they realise all the fearful complexities of a sub-
ject. By hesitating, one may get oneself mistaken for an expert. The
finishing touch on this approach is to use the all-important phrase,
'It all depends'. (For added effect, learn to say this in several foreign
languages and say it as if quoting from an eminent algebraist who
spoke the appropriate language, as 'Das kommt darauf an, as Gauss
might have said'. Gauss may be used for almost any subject.)

Rather better than this last answer is one that shows, if not any
great familiarity with polynomials, at least some idea about mathe-
matics; one may say, 'That all depends. Do you treat it as a poly-
nomial? Do you operate on it as you operate on a polynomial? If so,
it is a polynomial, as near as makes no difference. If not, then it is

152

certainly not a polynomial.' This could be extended further by, 'Is it
part of a system isomorphic to a system composed of polynomials?'
and words to that effect.

Of course if one can define a polynomial, it is a good thing to do so;
and in that way one can of course give the correct answer to the
question. Knowing what a polynomial is comes before all other
knowledge in the field of polynomials as an essential prerequisite.
Nothing could be simpler. First, polynomials have got monomial
expressions, and these to begin with are pure and sans coefficients.
Then, they have got to have a supply of coefficients from somewhere.
Since the coefficients are going to have to be added and multiplied
at some point, they are usually taken to come from a ring. Thus we
have the pure-monomial-expression monoid N[0]|S| and the ring (sigma).
The most obvious thing to do with these is to form the monoid
algebra of N[0]|S| over (sigma), which may be denoted (sigma)N[0]|S|, or for
simplicity (sigma)[S]. This consists of all functions in

(sigma)N[0]|S|

that take the value 0 everywhere except on a finite subset of N[0]|S|.
These functions are added in the obvious way, and are multipied by
convolution or Faltung. This means that if p and q are polynomials,
their product r has the property that

A~ r((mu)) = (sum) p(x)q((tau))
(mu) (epsilon) N[0]|S| x(tau) = (mu)

An obvious change of notation produces the more usual form; thus
when S = {X} we can write the polynomial in the form

a[0] + a[1]X + ... + a[n]X^n.

The reader will note that this change of notation bears an obvious
similarity to integral transforms. The verification that the sum of X
and Y, considered as polynomials in the letters X and Y, is exactly
the polynomial X + Y need not detain us; it is no harder than the
proof that the product of X and Y is XY, which has already been
explained.





153

Polynomial identities

One of the most basic identities for practical purposes in connection
with computations is

(X - Y)(X + Y) = X^2 - Y^2.

This is sometimes expressed in words: the sum times the difference
is the difference of the squares. Since we know how to multiply
polynomials (previous section), this should not offer any great
difficulty. Every term in the polynomial (X - Y) has degree 1; in
other words, the value of the function at an element of the monoid of
pure monomials is 0, unless the element of the monoid of pure
monomials is (X^h)(Y^i) with h + i = 1. Note that for simplicity it has
been taken for granted that all the polynomials are polynomials in
the letters X and Y, and that if this supposition is not made then the
last statement must be modified. Similarly, the polynomial (X + Y)
is a function taking the value 0 on the pure monomial (X^j)(Y^k) unless
j + k = 1. Note further that we have no idea what is meant by the
statement 'the function takes the value 0 at a monomial', since we
have no idea what this 0 is and are even unaware what ring 0 is the
zero element of; and note also that this makes no difference to the
argument. Putting

h + j = m
i + k = n,

we see that the product of the two polynomials has degree 2; that is,
their product is a function that must take the value 0 at (X^m)(Y^n)
except in case m + n = 2. This is because m + n = (h + j) +
(i + k) = (h + i) + (j + k) = 2 whenever even one term in the con-
volution sum is non-zero. Hence we know that the product takes the
form

aX^2 + bXY + cY^2,

and it remains only to compute the actual values of a, b, and c. In
order to compute a, it is necessary to observe only that if

/- -\ /- -\ /- -\
| 1 0 1 0 | | h | | 2 |
| 0 1 0 1 | | i | = | 0 |
| 1 1 0 0 | | j | | 1 |
| 0 0 1 1 | | k | | 1 |
\- -/ \- -/ \- -/

154

then (h, i, j, k) must differ from (1, 0, 1, 0) by an element of the
kernel of the homomorphism Z<4> -> Z<4> determined by the matrix;
hence is of the form

(1 + w, -w, 1 - w, +w),

where w is an integer. Since it is further required that h, i, j, k be
natural numbers, we must necessarily have

w = 0.

Hence the coefficient of X^2 is the product of the coefficient of X in
(X - Y) and the coefficient of X in (X + Y), or 1.1, giving

a = 1.

In a similar way, if (X^h)(Y^i) and (X^j)(Y^k) are to make a contribution to
the coefficient b of XY then we must have

(h, i, j, k) = (1 + w, -w, -w, 1 + w).

There are seen to be two solutions in naturals, namely (1, 0, 0, 1) and
(0, 1, 1, 0), and we get b = 0. Similarly c = -1 and we are done.

Note that the above product X^2 - Y^2 becomes X^2 + Y^2 over
rings of characteristic 2, where -1 = 1.

An exactly similar method may be used to show that many other
familiar polynomial identities hold. Alternatively, of course, one
may verify that the monoid algebra (sigma)[S] satisfies the ring axioms;
i.e., one may verify that the convolution product is associative:

(sum) ((sum) p(x)q((tau)))r((mu)) = (sum) p(x)
(delta) = i(mu) i = x(tau) xi = (delta)
((sum) q((tau))r((mu)))
(tau)(mu) = i

and that the 'constant polynomial' taking the value 1 on the neutral
monomial and taking the value 0 elsewhere is neutral for con-
volution. Distributivity over addition must also be verified.




7. What are brackets?


Nothing is so important in mathematics as knowing what brackets
are. In America and elsewhere abroad, brackets are parentheses; at
least the people in those places think that brackets are parentheses,

155

and call them parentheses. Doubtless this is but one instance of the
fact that, broadly speaking, the people abroad speak broadly.
Americans, especially speak so broadly they speak expansively; they
expand the word 'lift', a box for hauling people up and down in, into
'elevator', and they have applied the same principle to the word
'bracket'. This habit is known as American largesse.

The division of brackets is twofold. First, they are divided into
species, which are the round, the square, and the curly. Secondly, each
species is itself divided into two sexes, the left-hand (or sinister) and
the right-hand (or dexter). For example, ( is a left-hand bracket, and )
is a right-hand one.


A parenthesis on the sex life of brackets

Human beings live in a very different world from that inhabited by
brackets. Our world (the author is himself a human being) is a three-
dimensional one; that is, disregarding time and speaking locally, a
human being who refrains from extended space travel spends most
of his time in a homeomorph of R<3>. In a similar sense, we may call a
bracket a two-dimensionalite; he (or she) lives in R<2>. Just as we can
hardly imagine the life of a being in four-space, so can brackets only
dimly guess at us. A thoughtful bracket could certainly form the
abstract idea of the space we live in. An imaginative bracket could
perhaps people such a figmentary space with putative beings. A lucky
bracket might guess something about the division of our race into
sexes, and the shapes of our bodies. But surely no being so alien to
us could ever suppose our complex customs of marriage and court-
ship to be such as they are. Brackets, on those rare occasions when
they think of us at all, must picture us as some kind of huge, curly
bracket.

But are we in a better state with regard to our little fellow creatures?
Most of us see them often enough; how often do we try to think
what life is really like for them? Yet the rules of their game, the sys-
tem sanctioned by nature and society by which they regulate their
lives, is perhaps as complex for brackets as our own system is for us.

One misconception must be scotched at the start. Just because the
left- and the right-hand brackets look rather alike, people sometimes
think that no polarity exists between them. This is not true, as we shall
see. An especially dangerous idea is that a left bracket can, by leaping

156

upward on the page and turning a somersault, become a right
bracket--in short, that sex-change is possible for brackets. Again,
not true at all. The bracket remains on the line, and never tilts for-
ward or backward; so that he (or she) can never change sex. Motion
exists, but it is always a motion along the line of print.

Brackets marry. Unmarried brackets are extremely rare; indeed,
they are almost non-existent. (There is an exception in France.)
Brackets marry brackets of the opposite sex, but of the same species.
Again, one would like to say that this rule is universally true, but
there are (alas!) exceptions. These are as follows, and readers may
well wish to close their eyes: square and round brackets of opposite
sexes do marry. Unions of male or left-handed squares with female
rounds, and unions of female squares with male rounds are about
equally frequent; the couples are never very deliriously happy,
bcing separated by an interval; e.g., [a, b) on the one hand and (a, b]
on the other. No such mixed matings are ever indulged in by the
curlies. The name of marriage is too holy to dignify the unnatural
liaisons that occur chiefly in France, to the shame of that proud
nation: such monstrosities as [a, b[ and ]a, b] in which two brackets
of the same sex join horribly. Even harder for virtuous nations to
understand, perhaps, is the ridiculous sort of marriage that occurs
there between two brackets who, although they are of the same species
and of opposite sexes, have got their ro^les reversed: ]a, b[. Even in
France, it is only the square brackets that commit these abnormali-
ties; thus curly brackets are the purest of all in their marriage
behaviour.

Courtship as such is hardly known among brackets. Brackets are
mature almost immediately after 'birth', quite contrary to the rule
that long-lived creatures generally take a long time to grow up. They
marry just as quickly. There is almost never any doubt as to who is
married to whom. Thus in (a(bc)) the inner pair are married, and the
outer pair are married. The absolute fixity which reigns in the
relations of brackets might lead some human students to suppose
that love among the brackets is a very humdrum affair. As we all
know, courtship by the very uncertainty of its result adds a dimension
of suspense and anticipation to love. Many romantic novels play on
the theme of courtship; few carry on the story into married bliss. If
we examine our hearts, we see that we should not be satisfied if the

157

course of love were perfectly smooth. When we examine the life
story of a bracket, perhaps we ask, 'How can it interest them?' It
seems to us like utter monotony.

Nothing, in actual fact, could be further from the truth. The love
life of brackets is not static bliss, but dynamic anticipation. The male
and female brackets forever strive to come together, and forever
they are prevented. There is always something in the way. + Often one
sees couples for whom the gleam of hope must appear infinitesimally
tiny and distant, like a faint ray of light to a lost explorer of under-
ground caverns:

(x[1] + y[1] + z[1], x[2] + y[2] + z[2], ..., x[q] + y[q] + z[q]).

At other times, especially in these latter days, couples almost in
each other's arms:

f(.).

Indeed, a male bracket must feel much as did a knight in the days of
chivalry, who had pledged himself beyond recall to a lady far above
his station, to whom he could never hope to attain.

So all-pervasive is this feature of parenthetic love life that we
ought perhaps to speak of betrothal rather than marriage. Many
bracket couples not only cannot contrive to come together, but are
prevented even from seeing one another by older couples who stand
between the lovesick male and female bracket; while these older
couples may themselves be the objects of a similar chaperoning
operation performed by yet other couples:

((ab)((c(de))e))f.

Here the couple (de) separate another pair in (c(de)) and these
separate yet others, and so on. Only (ab) and (de) can see each other.

Looking on with pitying eye, if we are kind we may be seized with
a desire to interfere in the ro^le of Cupid. It would be possible for us,
from our god-like vantage, to unite the pairs, to bring to fulfilment
loves otherwise unrequited. We can twine two brackets in a true
lovers' knot, and from (ab) make

/----\
| ab |
\----/
--------------------------------------------------------------------------
+ See p. 61.

158

or from ((ab)((c(de))e))f make


/----------------\
| /----------\ |
/----\ | | /----\ | |
| ab | | | c | de | | e | f.
\----/ | | \----/ | |
| \----------/ |
\----------------/


Thus the two are made one, and the he-bracket may say to his she-
bracket, 'Borne of my borne, and fle`che of my fle`che!' It is possible
for us to do the thing; but is it wise? For what we have joined can no
longer be put asunder. What is written, is written, like the laws of
the Medes and the Persians. And it may begin to pall on the brackets.
Too much of a good thing may well be worse than none at all. Had
we not better leave well enough alone? For after all is said and done,
human knowledge of the love life of brackets is meagre.


The most important use of brackets is to show the order in which
operations must be performed. Thus, the above may indicate

F(F(F(a, b), F(F(c, F(d, e)), e)), f).

The function F is suppressed as being too obvious for mention, and
the simpler form involving only brackets results. There is a direct
approach which is used in Poland sometimes, whereby instead of
suppressing the functions one suppresses the brackets; for example,
one writes

FFFabFFcFdeef.

This way of writing things is said to be easy for machines to read.
One may also replace brackets by a numeral or mark representing the
depth of the deepest bracket at the point, as

ab///c/de//e////f.

However we choose to indicate the order of operations, we must
make it clear that one must perform either ab or de first, and that the
last step is the one that uses f.

It is clear that when brackets are used in this way, certain rules
must be obeyed if the result is to make sense. If the rules are not
followed a very unhappy situation develops in which brackets are

159

not married, or are not sure to whom they are married. A sensitive
reader soon notices when this has happened--we do not know quite
how; perhaps it is because of a kind of thought transfer from bracket
to man. For example, in writing

(ab)c)d)((ef)

the author has started some very sad brackets on the road of life.
In ((ef) we have, on the left, a sort of eternal best man who can never
be married and who is doomed to be part of an eternal triangle,
hanging about on the outskirts of the married couple (ef). It cannot
be good for any of them, although ) may be flattered by the extra
attention she gets. There are other heartrending situations present in
this example, and the author would never have created this woeful
little microcosm but for his desire to warn others not to make the
same mistake.

If we are to avoid such errors, we need a rule explaining when ex-
pressions obey the social rules so necessary for the smooth functioning
of domestic life among the brackets, and when the expressions
introduce friction into the well-lubricated workings of the bracket
household. We need to have a procedure for writing bracketed
expressions that will produce all the right expressions, and none of
the wrong ones. Since a bracketed expression can be obtained from
two smaller expressions by writing one down after the other, and then
surrounding the whole mess by a pair of brackets, we may write

S = A + (SS),

where A = a + b + c + ... + z, unless we wish to use further
letters; e.g., the 5000 most common characters of standard literary
Chinese. If we wish to use these they must be added into the sum for
A. Note also that the brackets are not brackets here, but generators
of a free semigroup on 28 generators (or 5028 generators if you
include the Chinese characters). It is 28 and not 26 because of the
two brackets. It is clear that all bracketed expressions are generated,
and only such expressions. Moreover, it is clear that no expression is
generated ambiguously (use induction on the number of letters in the
expression, not counting the brackets). It is extremely difficult to
expand S as a series in the semigroup on 28 generators with integer
coefficients, but as we have said the coefficients in such an expansion

160

are all 1. If, however, we delete ( and ) and replace A by a, we get

S = a + SS,

and if S = k[0] + k[1]a + k[2]aa + ... we see that this is equivalent to

(k[i]) * (k[i]) = ((delta)[i]<1>) + (k[i])
i (epsilon) N[0] i (epsilon) N[0] i (epsilon) N[0] i (epsilon) N[0]<9>

where * is the convolution product and where ((delta)[i]<1>) i (epsilon) N[0]
is (0, 1, 0, 0, 0, ...), the sequence that is 1 in the first place and 0 in
the zeroth and all other places. With q[i] = k[i] + (delta)[i]<0>, we get

(q[i]) * (q[i]) = 1/4 - a,

which gives us

2(2i - 3)!
k[i] = -------------
i~)2~(!~ - i!

by the binomial theorem. Thus there are 429 ways of inserting brackets
in an expression of length 8: for instance, one way of inserting
brackets in abcdefgh is

(a(b(c(d(e(f(gh))))))),

and there are 428 other ways.


















161

4

TOPOLOGY AND GEOMETRY


1. Into the interior


Topology begins at Ujiji, on the continent of Africa, where for the
first time the interior was penetrated by the explorer Stanley. If it
were not for Stanley, we should be as ignorant of the interior as we
were then, and topology would be dark to us--as dark as the thickest
jungle of the dark continent.

Though many people are acquainted with the outlines of the story,
not so many know the important details. We give here only as much
of it as is necessary for our purposes. [34] Stanley penetrated so deeply
and became so lost that an American reporter, Dr L. I. Presume,
was sent to find him. The American, as he approached Stanley,
extended his hand stiffly according to the custom of the time--people
in those days were still rather Victorian. There were no herald and
no butler to announce his arrival. After all, one was in the bush.
The inhabitants of Ujiji (called Bajiji) spoke Mujiji, not English.
Having no-one to announce his name, he announced it himself: 'Dr
Livingstone I. Presume!' This piece of American resourcefulness has
gone down in history, and has become a famous quotation.

But it is not this for which we mathematicians remember Dr
Presume. Indeed, we break off the history of exploration and dis-
covery at this point, not even pausing to recount the later death of
Stanley at Ujiji, and his mummification in the rays of the all-purging

163

sun; not dwelling on the forwarding of his remains to the isle of
Britain, where the skies are continually overcast with cloud, or on
their consequent putrefaction and interment. These things need not
detain us.

Dr Presume, being American, is usually called by his first name
Livingstone. It is not generally known, except to topologists, that
Livingstone approached Stanley along a net. A naked white man,
speaking only a variety of Swahili picked up from the apes with whom
this strange being consorted, had at first attempted to teach Living-
stone to approach Ujiji along a sequence, swinging from point to
point on a vine. This was the white apeman's own customary mode
of locomotion, acquired also from his simian companions. But
Livingstone found that his ten-gallon Stetson hat kept falling off in
the breeze. That was when the birthday boy (whose name was
Tarson, and who is thought to be related to the Tar-baby of American
Negro folklore) strung a net over the trees for Livingstone to ap-
proach Ujiji along. As he dangled from the net, swinging along
by his hands like a circus performer, Livingstone stretched, and was
taller when he arrived at Ujiji than when he had set out. There was
more Livingstone than before. For this reason convergence along a
net is sometimes called Moore-Smith convergence. +

The concept of the interior, and how a point of the interior may be
approached, is one of the great starting-points of topology. We hold
these truths to be self-evident:

(i) The interior of everything is everything;
(ii) The interior of the interior is the interior;
(iii) The interior of anything is a part of that thing;
(iv) The interior of the intersection is the intersection of the
interiors.

Perhaps these wlll be made more clear by a word of explanation.
Number (ii) says that if one considers, say, Africa, and then the
interior of Africa, and finally the interior of the interior of Africa,
the last two are the same place. Number (iii) says that the interior
of Africa is entirely African, so that the Spitsbergen archipelago is
not in the interior of Africa. To understand (iv), think of the United
--------------------------------------------------------------------------
+ Taking Smith as a sort of generalized name for Livingstone.

164

Kingdom and also of the island of Hibernia. These intersect in a
place called Northern Ireland. Its interior is the interior of Northern
Ireland. The interior of the United Kingdom and the interior of the
island of Hibernia then must also intersect in the interior of Northern
Ireland. The conditions may be expressed by writing

(i) X = X
(ii) A = A
(iii) A (subset) A
(iv) (A (intersection) B)
= A (intersection) B;

if we write AB for A (intersection) B and 1 for X we see that these conditions
make sense on any monoid, and that they say that A -> A is an
idempotent endomorphism of the monoid such that AA = A. In
order that (subset) should be a partial order, however, it is reasonable to
restrict attention to commutative monoids in which every element is
idempotent. For technical reasons, it is sometimes convenient to
consider the case in which the monoid M satisfies the further con-
dition that if (a[i]) i (epsilon) I is a family of elements then there exists
an element a (epsilon) M such that

A~ a[i]a = a[i]
i (epsilon) I

and such that ax = a whenever

A~ a[i]x = a[i].

We write a = V a[i], and note that if it exists, a is unique. In this
i (epsilon) I
case, it is obvious that the submonoid T of M consisting of elements
invariant under a -> a is itself closed under V, and that of course
the restriction of to this submonoid is the identity. One calls T
the topology of M. Very often, one takes an even more special case: the
set of all subsets of a set X is a commutative monoid in which every
element is idempotent (the operation being (intersection) and the neutral
element of course X), and so all the above remarks may be applied
to it. If we take A to be (psi) if a <> X, we get the indiscrete
topology ((psi), X). A space in the indiscrete topology is like an enormous
room. Everyone is lumped together, and nobody is housed off. In fact, the
indiscrete topology is the most un-housed-off topology there is. There
is no separation, no seclusion, and so if one is living in such a space

165

it is impossible to do anything discreetly. Lovers find indiscrete
spaces very tiresome indeed. But they are a paradise for exhibitionists.

Considering our own terrestrial orb for the moment as X, we
see that Ujiji is in the interior of Africa, but that Africa is not the
whole earth. Hence our world is not indiscrete. This will be no sur-
prise to lovers, who find this world a very pleasant place indeed,
especially in the spring. Nor (to leap ahead for a moment) is it
discrete--and none but hermits would have it so.



QUESTION 27. Whether it is possible to use nets in order to approach a
point in a topological space?

Objection 1. Nets are merely glorified sequences. But approaching
a point along a sequence is like marching on a pogo stick--you may
never get near your objective. Suppose topological space to consist
of the ordinal numbers up to, and including, the first uncountable
ordinal (omega). The open sets are the intervals [(alpha), (omega)] with
(alpha) < (omega); alternatively, the interior of A is [(alpha), (omega)] if
(alpha) is the first ordinal less than (omega) such that [(alpha), (omega)]
(subset) A, and is (psi) if no such (alpha) exists. If a sequence
((alpha)[n]) approaches (omega) it is clear that

(alpha)[n] = (omega)

from some point onwards. In other words, it is impossible to ap-
proach (omega) along a sequence except by being at (omega) almost from the
start. But that would be cheating. Hence, sequences are inadequate
for a proper stalking exercise in this kind of country, and no ex-
perienced ordinal-hunter would use them.

Reply to Objection 1. It is true that nets are glorified sequences,
and that sequences are woefully insufficient in many spaces. One
weeps to think how hard it would be to approach one's objective if
there were nothing better than sequences to use. It would be like
lion-hunting Masai style, with spear and leather shield. Fortunately,
weapons have been devised that make lion-hunting as easy as picking
up a sleeping pussy cat; they are, in essence, merely glorified spears.
The projectile is thrown more forcefully and accurately, and from
a greater distance. Similarly, nets can do what sequences cannot,
despite the fact that they possess the same basic principle of design.

166

Note carefully that the use of nets in approaching or converging
to a point in a topological space has nothing at all to do with the
use of nets in Roman circuses by fighters known as retiarii. These
did not hunt lions primarily, as is commonly thought, but were pitted
against an armoured swordsmen with a fish on his hat called a
murmillo. The retiarii carried tridents, which are of almost no use in
topology. The use of nets for catching armoured swordsmen is by
now almost a lost art, whereas thousands of undergraduates an-
nually engage in contests to determine which of them can best stalk
a point in a topological space using nets. The losers are fed to the
lions, in a quaint survival of the ancient life of the arena.

Objection 2. To approach a point along a net, the net must be
finer than the neighbourhood filter at the point. When Livingstone
set off for Ujiji on the net which his friend Tarson had provided for
him, he must have realised that even in the neighbourhood of that
savage place, the neighborhood filter could not be so coarse as his
net. Of course in the nineteenth century it was difficult to obtain
really fine filters in such heathen places (that was the main reason
why missionaries were sent out, bearing the message of modern
civilisation and comfort); but even so, it was inherently unlikely
that the wildest cannibal could drink his coffee without removing
the grounds with any device more effective than a fish-net.

Reply to Objection 2. It is an abuse of language to say that a net
is finer than a filter. Only another filter can be finer than a filter.
What is actually meant by the phrase is that a filter associated
with, or generated by, the net in question is finer than a certain filter.
It is also a mistake to talk about nets and filters as if they had holes
in them; or as if nets and filters were distinguished according to
whether the holes were larger or smaller than the holes in cheese-
cloth. But it is true that filters are a more refined concept than nets.

By the way, another possible conclusion may be avoided if it is
pointed out quite sharply that filters, in our sense, have nothing
whatever to do with philtres; thus, when Tarson is feeling very melan-
choly he may make use of philtres in wooing his girl friend Jane, but
he never lends philtres to American reporters who wish to approach
her. Philtres are not very fashionable in topology, unlike filters. The
two must never be confused. Putting a philtre through your girl friend
is very different from putting your girl friend through a filter. Like

167

any other substance, when she passes through a filter she comes out
in a refined state; whereas when a philtre passes through her the
probable effect on her behavior is not likely to be one that could
be described in terms of greater refinement.


I ANSWER THAT approaching points in a topological space along a
net is indeed perfectly possible.

Sometimes it is even possible to approach a point along a sequence.
We all know about the thirsty frog who wished to jump down the
well, but found that his strength was fast failing him. The well was
two feet away. His first jump took him one foot, but his second jump
carried him only six inches. Indeed, he found that after each jump,
he had only enough power left to jump half as far. Thus the total
distance he had jumped after n jumps was

2 - (1/2)^(n-1) feet.

By taking more jumps than a certain minimal number he could
ensure that he would come as close as desired to the well, and
that he would ever afterward remain that close to the well. (He
could never actually reach the well; but if we are to do experiments
on animals we must first harden our hearts to the promptings of pity.
It is of no interest to the investigating scientist if the frog feels
terribly dry, if his skin and throat become parched and his muscles
fatigued. No doubt frogs cannot really feel pain. In any case, if the
frog fell in the well he would surely drown.)

In a more general topological space than the real numbers, our
frog might find it necessary to proceed along a net indexed by a
more general ordered set than the set of natural numbers. So as to
avoid needless complexities, we may assume that the supremum of
any two elements in this ordered set always exists. We say that the
net (x[i]) i (epsilon) I converges to, or approaches the point x if for
every neighbourhood N of x one can always ensure that a term x[i] of the net is
in the neighbourhood N by requiring merely that i is greater than a
certain suitable element i[N] of the ordered set I, which must depend
on N only. If by a tail of the net we mean a net (x[i]) i (epsilon) I' obtained
by restricting the net to a subset I' of I of the form I' = {i (epsilon) I:
i >= j}, where of course j may be any element of I, then the net converges to
x if and only if any filter that contains the images of all the tails of

168

the net must necessarily contain the neighbourhood filter. If (x[N])
N (epsilon) N~ is a choice function for the neighbourhood filter N~ at x,
and if this is ordered in the natural way, then this net approaches x.



QUESTION 28. Whether Africa is connected to America?

Objection 1. It would appear obvious that America cannot be
connected to Africa. America is a continent; i.e., a vast extent of
land that one may run through without crossing the sea. [35] But, if
America and Africa were connected, then one could run about all
over America, cross over to Africa, and then run all over Africa. If
one could do that, then the union of both America and Africa,
together with the bit of land that one ran along, would amount to
a vast extent of land, and one would have run through it without
crossing the sea. That is, the union of these three land areas would
be a continent. But there are exactly seven continents, and no one of
them is a superset of two others. Hence America and Africa are
not connected.

Reply to Objection 1. According to the definition used, it is not
clear at all that America is a continent. The word 'vast' is defined
[14] as 'Wast, huge, hurly; wheady, wide, broad and large, misshapen,
illfavoured, ...'. It is perhaps doubtful if all these adjectives apply
to America. If they do apply, to what other parts of the earth do they
apply? Most countries are rather oddly shaped, and seem large if
one walks over them. But the condition about 'running through' the
vast extent of land is even harder to justify. The only sensible mean-
ing that can be given to this is that one is able to follow a path that
crosses every point of the land in question. It would not appear to be
sufficient that one could find a path that would pass within a short
distance of every point of America. The topological properties of
America are little known: is it, for instance, an open set? Is it
compact? Is it even a Borel set? And so it would appear difficult
to settle the existence of a continuous surjection [0, 1] -> America.

But let it be granted that America is a continent. Then Eurasia,
by the argument presented in the objection, is as much a continent as
any other. Hence a continent may indeed be a union of two others,

169

and the conventional total of seven continents is very wide of the
mark.

Objection 2. America is not even connected to itself, since 1914
when the Panama canal was completed, severing the North from the
South and putting asunder what was meant to stay together. How
then can America be connected to something else?

Reply to Objection 2. If X (union) Y and Y (union) Z are each connected, and
if Y (in this case, Africa) is not empty, then it is elementary that X (union)
Y (union) Z is connected, though X (union) Z may not be. But Africa has
at least one point; namely, Ujiji. Hence we cannot infer from the
argument in the objection that America and Africa are not connected.


I ANSWER THAT Africa and America are not connected. We may
safely take the surface of the earth to be a sphere S<2>. (This is known
to be only approximately true. We must first break down all natural
bridges, which of themselves make the earth a surface of high genus.
It is not known to the author how numerous these bridges are in
various parts of the earth, and they will present a difficulty if there
are any in certain parts of the planet, especially Greenland, Antarc-
tica, and the Eastern tip of Siberia, and certain small islands. The
restoration of the natural bridges, and the proof that even after
the restoration the Americas are still not connected to Africa, are
tasks left to the reader as exercises.) At least, we shall take the surface
of the planet to be a homeomorph of S<2>. Now it is well known that
neither the north pole v nor the south pole (sigma) lies in America or
Africa. Include the punctured sphere S<2> ~ {v, (sigma)} as the unit sphere in
R<3>; circumscribe about it the cylinder S<1> x (-1, 1) (subset) R<2> X R
= R<3>, which has the line through v = (0, 0, 1) and (sigma) = (0, 0, -1) as
its axis, and project the punctured sphere outward from the axis on to
the cylinder. The homeomorphism just described is, of course,
nothing but the usual way of defining the equiareal projection, a
kind of map. Mercator's projection would have done as well, but is
more difficult to describe. Now identify the punctured sphere with
the cylinder. Choose points (gamma) = Porto Praia in the islands of
Santiago, of the Capverdian chain, and (alpha) = Abkit or Anadyr (take
your choice). Then if (alpha)[1], (alpha)[2], (gamma)[1], (gamma)[2] are
points a little east and west of (alpha), (gamma), and if the projection on
S<1> sends these to a[1], a[2], c[1], c[2], a continuous function

170

S<1> -> [0, 1]

exists sending

a[1] -> 0
a[2] -> 1
c[1] -> 1
c[2] -> 0

and taking values that depend linearly on arc length over any interval
not containing any of these four points. (By an interval on S<1> we
mean a proper connected subset containing more than one point.)
This fact is not difficult to get, and can be obtained by noting that
S<1> is [0, 1] with 0 and 1 identified. It may be taken as a fact of
geography that by the composition of

S<2> ~ {v, (sigma)} -> S<1> X (-1, 1) -> S<1> -> [0, 1]

we have sent all of Africa to 0 and all of America to 1. Restricting
to Africa (union) America and corestricting to {0, 1}, we have a surjection.


2. The insides


People often talk as if they had insides. When a man is feeling queasy,
uneasy, under the weather, when he suffers from neuritis or neuralgia,
he may say mournfully, 'Oh! There is something the matter with my
inwards; I am not quite right inside; I am the victim of intestine
strife.' What rubbish! How does he know that he has any insides?
Just because human beings ordinarily possess livers, guts, gall-
bladders, and other such slimy organs, does not go any distance
toward proving that these things are inside the human beings. In
many fairy stories, people's insides are represented as being outside
the people. A little boy may be entrusted with the care and keeping
of his mother's heart, and may carry it about with him. How do we
know that this is merely an extraordinary fancy? Why do we so
easily suppose that it cannot happen to us? Perhaps everyone's
heart is outside him. Perhaps my heart is outside me, and my fingers

171

and eyes, the paper I am writing on, the reader and the book he is
reading are all inside me. Or perhaps the book and I are inside the
reader, and his stomach is outside him. Who can tell?

But even worse, why do we suppose that there is a distinction of
the world into two parts, one outside us and one inside? Ah, the
reader may say, now there you go too far. I am willing to entertain
the ridiculous notion that my insides are really outside me, and my
outsides are inside me. But now you ask me to think that there is no
difference between the inside and the outside; that my stomach and
the book I am reading are both on the same side, the only side;
that there are not two sides but one side. It is too much. I cannot
imagine it.

Yet, that is the situation we are faced wth. It is not asserted that a
human being has not got an inside and an outside; all that is asserted
is that there is a problem. And it is a topological problem. Now a
human being is topologically a difficult object to study. He is always
moving about; that is, he is constantly changing shape. A man with
his fingers in his ears has two more handles than a man standing
with arms and legs outstretched. If the alimentary passage is not
blocked, and if we make certain other simplifying assumptions, we
may consider the man to be composed of the skin and the lining of
the oesophagus, stomach, and intestines. It must be emphasised that
in this view, the bones, muscles, brains, etc. are not part of the man
thus considered. He is a surface; and it would help our deliberations
if he would be so kind as to keep open his mouth and his anus. The
breezes must pass freely through the tunnel. Then a man is a torus;
to be exact, he is a two-torus T<2> = S<1> X S<1>. He is the product of two
circles. More than that, he is a torus in three-space R<3>. Has he got an
inside and an outside? To put it another way, if we consider all the
rest of the universe besides the man, which consists of the guts, heart,
liver, brain, blood, etc. of the man together with the Eiffel Tower, the
Big Dipper or Ursa Major, Grant's tomb, and whatever else exists
besides the man himself--if we consider all that, is there a sensible
way of dividing it into two persons that shall be separated from one
another by the man? Is there a division of the waters under the firma-
ment from the waters above the firmament? Is the liver inside the
man and the Eiffel Tower outside--or vice versa? Later we shall
return to this question. Before we go on to simplify the problem

172

sufficiently to make it tractable, let us pause to consider a related
question, namely that of a man who has closed his mouth and his anus.
This type of man is a sphere, as we shall see later. What about him?
Has he got an inside and an outside?

Here is an easier problem:



QUESTION 29. Whether a circle has an inside and an outside?

Objection 1. Consider a hoop or loop floating in space after being
jettisoned by a lunar module. The hoop is a circle, but it is not a
subset of any particular surface. The remainder of the universe, after
the hoop has been deleted, is connected. Hence the circle has no
inside and outside.

Reply to Objection 1. It is necessary to stipulate that the circle lies
in some surface.

Objection 2. Consider a circle drawn on an inner tube, of the sort
used to inflate the tyres of automobiles. If the tube is cut along the
circle, it may or may not fall in two pieces. For a very small circle,
the tube will fall in two pieces (Figure 5a), but for a circle drawn
right round the tube, when the tube is cut it remains in one piece
(Figure 5b). The second kind of circle has no inside.


_________________ _________________
/ \ / / | \
/ _______ \ / \__|___ \
/ \/ \/ \ / \/ \/ \
\ \_______/ / \ \_______/ /
\ | A | B / \ /
\____|___|________/ \_________________/

(a) (b)



Figure 5



173

Reply to Objection 2. How true! It is clear that

(S<1> X S<1>) ~ (S<1> X {a}) = S<1> X (S<1> ~ {a}),

which is homeomorphic to S<1> X (0, 1), and the natural equivalence
of {0, 1} with ({0, 1}) proves (via ({0, 1}) = {0, 1} =
{0, 1}) that the product of connected spaces is connected. Hence if
a circle is to have an inside and an outside, it must not be drawn on
an inner tube.

Objection 3. The Equator is a circle running round the middle of
the earth. The Arctic Circle also runs round the earth, but not round
its middle. It is usually held that Santa Claus lives inside the Arctic
Circle, and that the Patagonians live outside the Arctic Circle. But
on which side of the equator does Santa Claus live? On the inside,
with the Patagonians outside, or is it they who live inside the
Equator?

Reply to Objection 1. There are many ways to answer the question.
Some people would say that if you walk round the Arctic Circle
going eastward then Santa Claus lives on the inside, but if you walk
round it going westward then he lives on the outside and the Pata-
gonians live on the inside. This is doubtless true, since anyone who
begins to walk round the Arctic Circle will certainly drown, freeze,
or be eaten by polar bears long before he finishes the journey. Other
experts hold that Santa Claus does not exist, but does live inside the
Arctic Circle. Still others would say that there are two sides of the
Arctic Circle, but that it is impossible to tell which of them is in and
which is out. This is the best answer, together with the stipulation
that the earth is a sphere [9] and that it is required to draw the
circle on a plane, and only on a plane. Note that the sphere, being a
closed, bounded subset of R<3>, cannot be a plane R<2>, since the latter
is not compact.



I ANSWER THAT a circle drawn in a plane has an inside and an outside
(Figure 6). This is sometimes hard for people to understand, for
various reasons. Some people think that if the circle is to have an
inside and and outside, there must be a door. Houses have insides, out-
sides, and doors; if the circle has an inside and an outside, where is
the door? Before we proceed to show that a circle has an inside and

174




______
/ \
/ \
< inside > outside
\ /
\______/

Figure 6




/
/----/ ---\ /-----------\
| | | |
| | | |
| | | |
| | | |
| | | |
\-----------/ \-----------/
(a) (b)

Figure 7



and outside, we had better dispense with this difficulty; otherwise, the
reader's mind may be insufficiently prepared for the truth, and unable
to assimilate it fully. In Figure 7 we have two houses. House (a) has
the door open; house (b) has the door shut. We are looking at a
floor plan. The reader should see immediately that house (a) has no
inside; it is porous, permeable, open; anybody can walk through the

175

door without committing a felony. There is no separation. The
walls and door of the house do not divide the world into an inside
and an outside. Now consider house (b). The door is shut, and that
is why it does not show in the figure. We shall return to this point
presently.

Now for simplicity in proving the circle has an inside, let it have
radius 1. Let P, P' be any two points of the plane, and let r, r' be the
distances |OP|, |OP'|. Since one side of a triangle is shorter than
the sum of two others, |r - r'| <= |PP'|, which shows that the
mapping

R<2> -> [0, (infinity))

defined by

P -> |OP|

is continuous. Since

| r r' |
| ----- - ------ | <= |r - r'|,
| 1 + r 1 + r' |

the mapping

f: [0, (infinity)) -> [0, 1]

defined by

r
r -> -----
1 + r

is continuous; notice that f is also strictly increasing, and that
f(1) = 1/2. If S<1> is our circle, we get a map

R<2> ~ S<1> -> [0, (infinity)) -> [0, 1].

By corestriction, we may get one

R<2> ~ S<1> -> [0, 1/2) (union) (1/2, 1]

and composing this with the obvious map

[0, 1/2) (union) (1/2, 1] -> {0, 1}

we are done with the major part of the problem; namely, to show
that the circle divides the plane into two pieces. Note that the same
argument works on the sphere, so that it is true that the Arctic
Circle (or the Equator) divides the surface of the globe into two parts.
This is because the sphere can be regarded as the one-point compacti-
fication of the plane, and all maps extend suitably. But beware! As

176

we saw, it is not enough in the case of the sphere to show that there
are two pieces, or components, because even so one is unable to
distinguish which of the pieces is the inside and which is the outside.
Can we do so in the present case?

Consider the two pieces. The piece containing O (the centre of the
circle) will be shown to be the inside, and the other piece will be
shown to be the outside. Thus, among other things we shall learn
that the centre of the circle is inside the circle. Write C[O] for the com-
ponent of the plane (with the circle taken out) that contains O, and
____
C[O] for the union of this set with the circle itself. Then it is to be
____
proved that C[O] consists of all the points either inside the circle or on
the circle: in any case, it is known that

____
C[O] = {P: |OP| <= 1}.

The other component forms in a similar way

_____________
C[(infinity)] = {P: |OP| >= 1}.

and it is to be shown that this is the set of all points either outside the
circle or on the circle.

If P is a point of the circle and if 0 <= r <= 1, there is exactly one
point P' on the line segment OP at a distance r from O; this provides
a mapping

____
S<1> X [0, 1] -> C[O]

which is easily verified to be a continuous surjection. By Tychonoff,
____ _______
it follows that C[O] is compact. For n (epsilon) N[0], let B[n](O) be the
closed ball of radius n at O. Then

____________ _______
C[(infinity)] (subset) R<2> = U~ {B[n](O): n (epsilon) N[0]},

_____________
but C[(infinity)] is not a subset of a union of any finite subcollection of
this collection. Hence there is an essential difference between the com-
ponent containing the centre of the circle and the other component.
It is therefore reasonable to call one of these components the inside
of the circle and the other the outside. For some reason, convention
has settled on the term 'inside' for the component of the plane, when
the circle has been cut out, that has the centre of the circle in it. Like
all conventions, this one is not easy to explain rationally. Why should
not the centre be said to be outside the circle, and why should not we

177

designate all the points of the plane very far from the centre as points
inside? In other words, why is the conventional terminology not
the one shown in Figure 8? There appears to be no satisfactory


______
/ \
/ \
< outside? > inside?
\ /
\______/


Figure 8


answer to this question. There is no real reason why one should
prefer to say that the centre of the circle is inside, and not outside.
However, if most people prefer to say that the centre is inside the
circle, it is good manners to acquiesce; nothing is ruder than to find
occasion for dispute in every inconsequential alternative. Really it
would be best if we could all agree to speak of the compactside and
the countercompactside of the circle. Thus, a man with indigestion
could say, 'I've got to see a doctor. There's something seriously
wrong with me compactside.' An economical housewife buying a
chicken might say to the butcher, 'And please could I have the com-
pactsides for the cat?' A prisoner in a maximum security institution
might mutter to his compactmates on the night before he expected
to get 'sprung': 'Youse won't see me tomorrow. Tomorrow I'm
gonna be countercompactside.'

But not so swiftly! So far it is only the circle which one knows to
have a compactside and a countercompactside (or an inside and
an outside, to use the terminology of stodgy convention). Prisons,
chickens, and men are much more complicated things, by their very
nature. It is now time to consider whether men have insides. Chickens
can be taken to be pretty much the same for our purposes as men;
certainly, in a mathematical discussion, a chicken is often as much

178

use as a man and makes comments of about equal intelligence. As
we know, there are really two very distinct types of human being:
mouth and anus open, and mouth and anus closed (Figure 9). (The
acute reader will note that it is possible, in general, to consider the
further cases that occur when one of the apertures of the alimentary


_______
/ o o \
/ @ * @ \
<={ --- }=>
\ |=o=| /
\__:|___/
(a)

_________________
/ o o \
/ @ ___*___ @ \
/ \/ \/ \
\={ \_______/ }=/
\ |=o=| /
\_______:|________/

(b)



Figure 9. Two essential types of men.


passage is open and the other closed. It is usually difficult to dis-
tinguish one of these cases from the other, especially in political
oratory; the resulting surface is in both of these cases topologically
the same. Technically, it is an identification space of a torus. (Only
one of the possible ways of including such a topological space in R<3>
is capable of being human; the other is the skin of a ring sausage
(Figure 10). Neither the ring sausage nor the orator is a manifold,
and the difficulties of this case force us to ignore it.)

179



,--.-.----.
/ :__)_ \
|___/ \ \
( \ \ _\
\_.'\ ____)/ )
\___\/ `. /
`------' ______________
(a) /------. _____\
/ \/_ \
\........'`.|..../
\_______`._|___/

(b)





Figure 10. (a) Ring sausage; (b) political orator.


The first essential type of man is the Quiet Man. His motto:
Silence is Golden. He is not given to flatulence or loquaciousness.
His sign is the Sun. The second essential type of man is the Loud
Man. His motto: Speech is Silver. He is flatulent, loquacious, and
given to overeating. He thinks and acts little, but speaks much. His
sign is Saturn, the planet with rings. Most men oscillate between
these two types.

Loud Men, abstractly, are capable of much more interesting com-
binations than Quiet Men. They may be linked together in chains. A
simple configuration of this type is shown in Figure 11.

Because of the theoretical possibility of this configuration, biolo-
gists have searched assiduously for an example. Their quest has been
fruitless, so far as authenticated records show. Loud Men, as such,
exist in abundance; linked Loud Men appear to be very rare. Why

180


###
/__/
&& < <
| \_____________> \___
|__|____ | |@|
| | |o| ###
### _|_|_________|_|_ |_|
\__\___/__@__o__*__o__@__\__| |
\______ | _____|
| :| |
/_________:|________\
___ / __________________ \
&__/\/_/ | | | | \ _\
__\/ / /|_|_________|_| | |__
&____|\_/ / < < |____&
\___/ \ \
\ \
(a) ###



(b) ________________
___/ ________ \
/ ______/ __#_ \ \
| / ___/ o/ \ \ \ |
/ / / | | \ \ | &| |
| \# \__| _|_/ #/ | | |
\ \___o___o___/ | &| /
\___ \ o\____/ / /
\__ \____#___/ /
\__________/



Figure 11. (a) Linked Loud
Men; (b) the same, simplified.

181

is this? Topological biology seems to have provided at least a partial
answer. We know from modern embryology that all men begin as
Quiet Men; that is, a man is at first a tiny sphere in the womb of his
mother. The potentiality of being Loud is developed somehow in
utero. If it can be shown that two fully developed Loud Men, not
already linked, can only become linked by a major surgical operation,
then the question, Why do we see no linked Loud Men? will have
been reduced to the two further questions: Why are twin Loud Men
never born linked; and, Why do no Loud Men choose to undergo a
surgical operation so as to become a pair of linked Loud Men?



QUESTION 30. Whether two men who customarily maintain the alimen-
tary tract in an open and ventilated state, by never shutting the mouth
or anus (hereafter called Loud Men), can become linked like the
links of a chain--so that each man passes down the oesophagus of the



____________
/ ______&_ \
/ / _______\ \_____
/ / / ______\ \_ & >
\ \/ / \ / /
\ / / \/ /
// / / /\
/# / \ / / _\
/ /\ \______/ / \ #\
\ \ \__________/ / /
\ \ / /
\ \___________/ /
\_@__o__*__o__@_/


Figure 12. Knotted Loud Men.



182

other, out through the vent, and back again--without a major
surgical operation?
(Note: It is helpful to refer to Figure 11.)


I ANSWER THAT two Loud Men, not yet linked, can only become so
through surgery.

The two Loud Men, not yet linked, will for simplicity be taken to
be geometrical tori. (This is a big assumption; in particular it assumes
that no Loud Man is tied in knots, as in Figure 12.) Similarly for the
linked Loud Men. The beginning and end of the hypothetical linking
thus appear as in Figure 13.

Having made this assumption, it is possible to define a belt of a
Loud Man. The man is generated as a surface of revolution by
revolving a circle, not intersecting the axis of revolution; the circle


___________
/ _______ \ ________
/ / \ \ / ____ \______
\ \_______/ / / / / \ o\ __ \
\___o___o___/ | | / / | | \ \
M[1] | | \ \_|__|__/ /
___________ | | \___o___o___/ M[2]
/ _______ \ \ \____/ o/
/ / \ \ M[1] \________/
\ \_______/ /
\___o___o___/ After
M[2]
Before



Figure 13. Hypothetical linking of Loud Man.




183

traced by any point of this circle as it is revolved about the axis is a
belt of the man. It will be assumed that, after linking, the belts are
in their right places again; i.e., that they are still belts. It can be shown
that this assumption is fair. It can also be shown that it can be assumed
without loss of generality that the whole procedure whereby the two
men are linked can be carried out without moving or changing man
M[2] at all. He can remain fixed while M[1] is twisted and tortured.

Suppose it were possible to link the two men without cutting,
tearing, or any other surgery. Take C[1] and C[2] to be belts which the
two men are wearing. Then the belts have also been linked without
surgery, and indeed without unbuckling them. The question has been
reduced to this: Can a belt be linked to a stationary belt without
breaking or unbuckling? By hypothesis, we have a homotopy

C[1] X [0, 1] -> R<3> ~ C[2]

in which the image of C[1] X {0} is a geometric circle not linked with
C[2], and the image of C[1] X {1} is a geometric circle linked with C[2].
If Q is the centre of C[1] X {0} in R<3> ~ C[2], then taking Q as origin of
coordinates and 1, say, as the radius of C[1], the map

(x, y, t) -> (tx, ty)

makes C[1] X {0} homotopic to {Q} in R<3> ~ C[2]. Hence there exists a
homotopy shrinking C[1] X {1} to a point. Taking cylindrical co-
ordinates in R<3> ~ C[2], with the axis of rotation about which C[2] may
be generated by rotating a point taken as the z-axis, we note that

(r, (theta), z) -> (r, z)

is a continuous function from R ~ C[2] to [0, (infinity)) X R ~ {(1, 0)}. Com-
posing this map with the homotopy that shrinks C[1] X {1} to a point
in R<3> -> C[2], we infer that a circle can be shrunk to a point in a plane
punctured at the centre of the circle; which is as much as to say that
the identity map

S<1> -> S<1>

is homotopic to a trivial map. But it is well known that the relevant
group is Z, and is generated by the homotopy class of the identity
map.

This conclusion, that Loud Men cannot be linked without surgery,
is one of theoretical interest, but it has for most of us little practical
importance. Two linked Loud Men would no doubt be as loud as

184

two unlinked ones, and no louder. If there is any practical inference to
be drawn from the theoretical fact that Loud Men cannot be linked,
it is one which brings a comforting thought to the mind, such as it is,
of every member of the Loud Fraternity. Since he cannot be linked,
the danger is averted that (relapsing momentarily into quiet and
closing his mouth) his link-mate might bite him through from mouth
to anus. Such a calamity is then impossible. If it could happen, it
would clearly change the toroidal Loud Man into a very quiet,
spherical Quiet Man (Figure 14).

What looks like a halo is the toroidal soul ascending whence it came.
His grave must have a marker: let us put, 'May he ever rest in peace.'


__________
/__________\
((__________))
\__________/

_________
/ -- -- \
/#-___*___-#\
/ / |:::| \ \
\ \ !|:::|! / /
\ \!|:::|!/ /
\&!_____!&/
! !


/---\
|:::| Endoderm
\---/

/---\
| ! | Mesoderm
\---/


Figure 14. Transformation of toroidal Loud Man.



185


3. Geometry


Civilised man lives close by the brink of the water. To the great
civilisations that developed along the banks of the great rivers--the
Indus, the Euphrates, the Cam--can be traced much of that artistic
and scientific heritage which adorns modern life. The earliest
recorded discovery of a useful art occurred in Eden, near the head
of the Euphrates. This was the discovery of both nakedness and its
complementary opposite, clothes; both of which arts remain im-
portant industries in the great cities of today. The art of falling was
discovered by a river-valley man, Isaac Newton, although the actual
discovery took place at Grantham, Lincolnshire, and not at Cam-
bridge (the centre of the great Cam civilisation). This event, known
as the Fall of the Apple, occurred much later than the other, the
Fall of Man. Before either of them occurred the Fall of Lucifer,
which was not an art, though it became the subject-matter of several
great works of art. From the Fall of the Apple was developed the
great art of Potential Theory.

In the valley of the Indus a great civilisation developed, to which
we owe Sanskrit and the Indo-European language. The most impor-
tant word in this language is lox, or, if you prefer, Lakshmi. She is
the goddess of smoked salmon and bagels. A bagel is a torus, and
has been encountered already in the chapter on topology. It is eaten
with lox, and is a topological group R<2>/Z<2>. Along with lox, bagels,
and language came the great Aryan race; to them we certainly owe
the discovery of the square. It has long been unknown whether the
Aryans developed the torus, or bagel, from the square, or vice versa.
Which came first? Square enthusiasts point to the shape of the
Aryan head as an indication that they must have been forced at an
early date to contemplate this figure. At first they would have found
it natural to bake their bread in this shape; later, because the torus
is obtained by an identification of opposite edges of a square, it
naturally occurred to their rather simple, straightforward minds to
make bagels, or toroid bread as a kind of elegant variation. That is
how one segment of opinion reconstructs the story, which must
nevertheless in its more intimate details remain dark for us. It is

186

noteworthy that one branch of the Aryan race, the famous Aryan
heretics, persevered in making and eating exclusively square bread,
and eschewed bagels. (Bagels are hard to chew, but it is foolish to
eschew them). The Aryan race were also interested in the wheel,
which they called a wheel-wheel, whirr-whirr, a GLGL, a quirquel, or
a circle. It is possible that the wheel was developed from the bagel;
that when the early Aryans began to think about the bagel, they
realised that it was a compact topological group and began searching
for other compact topological groups, and that this search brought
them on the trail of the wheel, or circle. It cannot be overlooked that
others derive the bagel not from the square but from the circle.
When the Aryans heard of Tychonoff's theorem (the product of
compact spaces is compact) they applied it to the wheel, and got a
new compact group:

BAGEL = WHEEL X WHEEL.

So much for the Indus Valley and the immense heritage we possess
as its successors.

The Yellow River civilisation gave us the I Ching, or index of
permutations, from which developed the symmetric group on n letters
and the non-conservation of parity, and was responsible for the
Chinese Remainder Theorem, space travel, dragons, Pekingese dogs,
and fortune cookies.

The Nile Valley has a civilisation so ancient that the name given
to the process of suspended animation discovered on the banks of
this river is mummification, and the subjects of it, who were provided
with picture-books of instructions and light entertainment against
their future reanimation, were designated as mummies. These names
make it absolutely certain that the Nile civilisation dates from before
the Fall of Man, when the distinction between mummies and daddies
was discovered. Pyramids, another Egyptian invention, were prin-
cipally a tourist attraction, like the Eiffel Tower. They were copied
in America, where they were used for heart transplant operations,
flower shows, and football games. Pyramids were also influential in
the history of Greek geometry. Among all the ideas of the Nilesiders
--mummies, men with bird's faces, cats with women's faces, the
feeding of non-mummies to crocodiles--only the pyramid is of
mathematical significance, and this only through the Greeks.

187

The Greeks were a Hellenic people; hence, although they were
good at mathematics, poetry, politics, architecture, navigation, and
many useful arts, they were overly fond of beautiful women. Fond of
measurement, they measured feminine pulchritude in Hellens. Rather
oddly for a civilised people, they based their unit of measurement on
the curious idea that the more beautiful was a woman, the larger
was the number of ships her face could launch, and that the largest
number of ships that could be so launched was 1000, or 10^3. This
quantity of beauty was called one Hellen. The practical result of
this idea was that the Greek women spent much time on the seaside
toughening their facial features by pushing boats into the water,
and in the end the Hellenes lost interest in women and invented
Platonic love. At the same time, the huge fleet of ships thus launched
made the Hellenes a nautical nation, so that they did much sailing.
The sailors would often bring home little trinkets, among which were
small replicas of the great pyramids, something like little Eiffel
Towers cast in lead that visitors bring home from Paris as souvenirs.
These knicknacks had far-reaching effects on later Greek thought, as
we shall see. Most important of all, doubtless, was the simple habit
of spending time at the beach. At first one wandered down to the
beach for the idle amusement of watching the ladies at their competi-
tive launching. But the men, to pass the hours, would chat lying on
the sand, and scratch crude drawings in its smooth surface. Later,
when Platonism had replaced the mythical Hellen at the centre of the
Greek imagination, and feminine beauty no longer played the leading
ro^le, the beach remained a gathering point. A preoccupation with
form, with the shape of things, remained; but now it was abstract
forms that were contemplated, and geometric shapes were drawn in
the sand. Geometry dawned. Her day is not yet spent.

One of the most difficult of the geometric concepts invented by the
geometric Greeks was the geometric line. A line was said to have no
thickness or breadth, and to be generated by a moving point. A
point was said to have no length, breadth, or thickness, and to be
indivisible. Thus, we are forced to interpret a point as a singleton
{x}; i.e., a universally attracting object in the category of sets. That
is simple enough. But how are we to interpret the line?



188

QUESTION 31. Whether, if three distinct points lie on a line, one of them
must lie between the other two?

Objection 1. A point has no extent; i.e., no length, breadth, or
thickness. That which has no length obviously cannot lie, sit, or
stand. Hence, a point cannot lie on a line.

Reply to Objection 1. It is quite clear that what is meant by the
statement, 'a point lies on a line', is not that the point reclines there,
but that there exists a monomorphism from the singleton to the line,
and that this monomorphism is among those considered to be inclu-
sions. Alternatively, there exists a map f from the line to itself such
that for any map g from the line to itself

fg = f.

It is merely another way of saying that the relation of incidence
holds between the point and the line.

Objection 2. This would seem obvious. By the Krein-Milman
theorem, the closed convex hull of {P, Q, R} being clearly compact
and convex, it is the closed convex hull of its extreme points. But
since only two of P, Q, R can be extreme, the other is in the closure
of their convex hull--which is as much as to say, in their convex
hull itself.

Reply to Objection 2. It is true that by the Krein-Milman theorem,
each of P, Q, R lies in the closed convex hull of the extreme points
of the closed convex hull of {P, Q, R}; but the objection assumes
without proof that this set can have only two extreme points. If it
has three, the argument fails.

Objection 3. You could try moving the points around. Keep two of
the points fixed firmly in their right places, but let the third point
move freely on the line. If the third point is between the other two,
it cannot go on for ever, either in one direction or the other, without
bumping into one of the other points; if it is not between the other
two, then the moving point can go on for ever in at least one direc-
tion, and never encounter either of the other points. Let one of the
points, which we may call P, be supposed (as a hypothesis) not to
lie between the other two. Move P toward Q with a large velocity,
so that when P strikes Q, the particle Q will continue in the same
direction with the same velocity. If Q never strikes the third point R,

189

then the whole line has been swept out without encountering R,
and hence R does not exist. It follows that Q is between P and R.

Reply to Objection 3. The objection is stated in terms with a
pseudo-particle-dynamical flavour, but we need not insist on look-
ing at the objection in this light; if we did, it would of course have
to be rejected as non-mathematical. The idea of a moving point is a
highly sophisticated one by the time it is given a mathematical
expression. The billiard-ball behaviour of Q when struck by P in-
volves differential equations, and the points are even treated as
impermeable to one another. Worst of all, the objection assumes
without proof that a line has only two directions. Just what is the
justification for this bold assumption?

I ANSWER THAT it is the case that if three points on a line are distinct,
then one lies between the two others. The general idea of the proof
is very simple. Since P, Q, R (the three points) lie on a line, we may
draw arrows from P to Q, from Q to R, and from R to P. It is
necessary to show that either two of these arrows go to the right,
or two of them go to the left. Since the line is an affine space, there
exists a way of associating to each pair of points a vector or arrow:
to (A, B) we associate the vector AB. Since the line is by Euclid's
definition a one-dimensional affine space, the vectors in question
lie in a one-dimensional space--i.e., module--over a field of charac-
teristic 0--i.e., characteristic (infinity)--which possesses an archimedean
order. (It was the great Eureka who discovered that lines have got
to have an archimedean order. This was on one of the many occa-
sions when Eureka was singing in the bath. Since he had very bad
pitch combined with good volume, Eureka's singing was always in-
distinguishable from his shouting.) We cannot be sure, and need not
worry about, just which archimedean ordered field F is being used,
but since it contains (root)2, it cannot be the universally repelling archi-
medean ordered field Q, though it certainly contains Q. Since
PQ + QR + RP = PP = 0 by the axioms of affine spaces, and since
the vector space, being a one-dimensional free module over F, is
isomorphic as an abelian group to the abelian group structure of F,
the fact that none of PQ, QR, RP is 0 enables us to conclude that by
the use of a suitable permutation (in the symmetric group on
{P, Q, R}) it can be arranged that two of PQ, QR, RP are positive

190

without loss of generality. These can be PQ, QR, so that Q is
between P and R.


(Tri)gonometry

One of the most important elementary subjects under the general
heading of geometry is gonometry. The origins of trigonometry,
which is the most elementary form of gonometry and is followed by
tetragonometry, pentagonometry, hexagonometry, etc., are to be
found in the great Ur-city, Ur. The Ur-civilisation was a great river-
civilisation, and invented trigonometry, which is therefore a useful
art. Its citizens, as we know from plastic art, wore beards in the
shape of rectangular parallelepipeds, and since Ur was situated on
the Euphrates, we know that they wore clothes and divided their
children into girls and boys. The boys had beards, and the girls had
none. The great interest in triangles that was such a great part of the
cultural life of Ur is evidenced by their writing, in which all letters
were formed of triangles. This is called cuneiform. Even little bearded
boys of the lowest social orders ran about the streets of the great
Ur-city studying triangles. Because of their characteristic beards,
these children were called Ur-chins. Nowadays, the name remains
despite the fact that few Urchins any longer possess the beard to
which it refers.

Clearly, if one is to do gonometry, which is the measurement of
angles, one requires some means of measuring angles. Hence the
question whether the Ur-menschen could have constructed a pro-
tractor was long a thorn in the side of the historian (see Figure 15).



QUESTION 32. Whether the Ur-menschen (as the inhabitants of Ur
were called) could have constructed a protractor?

Objection 1. It is well known that the Ur-menschen counted in
sixties. Since it is easy to construct an angle of 60 degrees, they
would naturally have done so, and this angle would then have been
divided in sixtieths, or angles of one degree. But an angle of one
degree is the basis of the protractor. Hence the Ur-menschen con-
structed a protractor.

191

Reply to Objection 1. This merely shows that it would come
naturally to the Ur-mensch to construct a protractor. But an Ur-
mensch is far from being a Natur-mensch. The latter was not invented
until all the Ur-menschen had disappeared. Hence the intersection
of Ur-menschen and Natur-menschen is empty, and no Ur-mensch
always acted naturally. Hence it is possible that no Ur-mensch
constructed a protractor.



\|______
| _/.|
| |@ > _ _
|_/ (__ \\//
\ __| |##/
_|_|____/ /
| || __/
|/_/|||
/ /_|||
\ \/,-|
|\##:_|
| |\ \
/ / \ \_/\
|___> |___/



Figure 15. Ur-chin, studying angle.



I ANSWER THAT no man has at any time constructed a protractor.
The cosine of 40 degrees obviously satisfies

8 cos^3(40 degrees) - 6 cos(40 degrees) + 1 = 0,

which clearly makes the construction of a protractor an impossibility.

192

How then did the men of Ur measure their angles? Possibly they
knew that one can integrate

x dt
(integral) ---------------
1 (root)(1 - t^2)

for 1 >= x >= -1, and that when this (obviously continuous mono-
tonic) function is inverted, it is capable of providing material for a
homomorphism from the topological group of real numbers to the
(multiplicative) topological group of non-zero complex numbers
whose image is the circle. If so, the measurement of angles would
become a triviality for them. But there is no record of a theory of
divergent Riemann integrals among the Urmen, and perhaps they
did it via Taylor's series.

The line, the circle, the angle--to these basic ideas of geometry
must be added a fourth.


Area of a rectangle

Almost everything is made in the shape of a rectangle. Two of the
most obvious examples are a book and a farmer's field. Why this
should be is difficult to say, but it is worthy of attention that the
farmer ploughs his field in furrows, and that the writer writes and the
reader reads the book in lines, which are very like furrows. The
author has never understood why books are not written even more
plough-fashion than they are. Why not write like this?

Oh, what are the zeros of zeta of s?
!ssefnoc ylurt I, uoy llet nac ydoboN

It would be so much easier to read--the eye would not travel back
to the beginning of the line emptyhanded, as it were, but would get
some reading done on the way back. If farmers ploughed their fields
as inefficiently as we read, we should all likely starve. However that
may be, the consideration of furrows only partly helps to explain
why everything is a rectangle. All it explains is why everything is a
product of something times a unit interval. Perhaps, in some dark
half-conscious way, the farmer is trying to prove that the fence on
one side of his field is homotopic to the fence on the other. Who
knows? In any case, we have not explained why his field is not an
infinite cylinder, or why this book is not a punctured disc.

193

Why is it that when a man has a rectangle, the first thing he does
with it is to calculate its area? Is it because he knows how to get
the right answer? That hardly seems likely.



QUESTION 33. Whether it be true that the area of a rectangle whose
sides have lengths B and H respectively, is the product of those
numbers, BH?

This is one question of which every intelligent person ought to be
able to guess the right answer, at least after he has read the correct
definition of 'rectangle'. A rectangle is a geometric figure composed
of two pairs of parallel line segments, perpendicular to each other,
such that each line segment meets each of the perpendicular line
segments in its endpoints. Basically then, a rectangle is made up of
four lines. And yet people say that the area of a rectangle is the
product of base times height!

I ANSWER THAT the area of a rectangle is never the product of base
times height. This is because the base of a rectangle is the length of
a line segment, and so cannot be 0; neither can the height be 0. But
the area of a rectangle is always 0, since it is obvious that a line seg-
ment has area 0, and hence so has the union of four line segments.
If we are to speak of areas, we ought to speak of the area of the
convex hull of a rectangle, or the inside of a rectangle. The fence
round the farmer's field is a rectangle, perhaps; the field is its convex
hull.



QUESTION 34. Whether the area of the convex hull of a rectangle,
whose sides have lengths B and H, is BH?

Objection 1. The correct answer is not Yes; neither is it No. The
correct answer is Maybe. For example, the rectangle that has base
1 and height 1 might be said to have area 1; but it would make just
as much sense to say that it had area 197 X 54. This would, of
course, necessitate giving a rectangle of base 2 and height 2 exactly
the area 790 X 16.

194

Reply to Objection 1. The objection is perfectly correct, except
that it has already been mentioned (tacitly) in the question, which
asked what was 'the area of the convex hull of a rectangle, normalised
in the obvious way, needless to say'. Since it was needless to say that
the area was normalised in the obvious way, the words 'normalised
in the obvious way, needless to say' were omitted from the question.
It is very important that the question be phrased as it is here, because
if normalisation is not mentioned then one may indeed give the
unit square the area 197 X 54.

Objection 2. The answer Yes would seem to be obvious, since the
Jacobean determinant of the linear transformation sending ortho-
gonal basis elements e[1], e[2] to Be[1], He[2], respectively is merely the
determinant of the linear transformation itself, namely BH. It is
well known that under a transformation obeying suitable conditions
areas are multiplied by the Jacobean determinant, so that the rect-
angle of sides B and H which is the image of the unit square under
this transformation has area BH.

Reply to Objection 1. This fact is only well known after the area
of a rectangle is well known. Hence this objection is illogical.


I ANSWER THAT the area of a rectangle is base times height, at least
if the rectangle has sides parallel to the coordinate axes. (It is
difficult to see what are the base and height of a rectangle, if the
rcctangle is at an angle to the coordinate axes, anyway.)

The area of a one-by-one rectangle is taken as 1. This is purely for
simplicity's sake; if anyone disagrees, it will be easy to alter the areas
of all rectangles so that the one-by-one rectangle has area equal to
whatever suits one's fancy. It is not assumed that the area of any
one-by-one rectangle whatever is 1; that would be rather bold.
However, it will be necessary to assume the existence of at least one
such rectangle with that area.

If a rectangle is moved about, its area remains the same. It is
desirable to move the rectangles gently, as they are not rigid figures.
Preferably, they are to be moved by an affine transformation of the
form

c + I,

where of course I is the identity endomorphism of the vector space

195

R<3>. It is of course far from obvious that the area of a rectangle does
remain the same when it is moved, since not every measure on a
locally compact topological group is Haar measure. Clearly, one of
the tacit words in the question was Haar: the question asked about
the Haar area of a rectangle with sides B and H. (The words 'convex
hull', the reader will notice, have become tacit by now.)

If a rectangle is (the boundary of the union of the convex hulls of)
two other rectangles joined together, and if the two other rectangles
do not overlap (so that their convex hulls have disjoint interiors)
then its area is the sum of theirs. This is because the words 'finitely
additive' appear tacitly in the question.

It is obvious that if axes have been chosen through two adjacent
sides of the basic unit rectangle, the relation given holds so long as
the vertices of the rectangles have rational coordinates. The rather
mild assumption that countable additivity or continuity from above
at (psi) holds for rectangles suffices to finish the job. It has not been
shown that rectangles really have areas; what has been shown is that
if they have areas which are as described in the question (mostly by
tacit expressions hidden therein) then the areas are as stated. The
question still remains: are areas of rectangles all nonsense?

















196

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207