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As4a 1965407
Aslmov on numbers
512.81
Aslmov
As4a 1965407
Asimov on numbers
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ESSAYS
ON SCIENCE
by Isaac Asimov
From The Magazine
of Fantasy
and Science Fiction
Fact and Fancy
View from Adding
a
a
Height
Dimension
Of Time and Space and Other Things
From Earth to Heaven Science Numbers and I The Solar System and Back The Stars in Their Courses The Left Hand of the Electron Asimov on Astronomy
The Tragedy
of the
Moon
Asimov on Chemistry
Of Matters Great and Small Asimov on Physics
The
Planet That Wasn't
Asimov on Numbers
From Other Only Is
Sources
a Trillion
Anyone There?
Today and Tomorrow and
.
.
Science Past— Science Future
.
ASIMOV ON NUMBERS
ASIMO V ON NUMBERS ISAAC ASIMOV
& Company, Inc. GARDEN CITY, NEW YORK Doubleday
1977
Library of Congress Cataloging in Publication Data Isaac, 1920Asimov on numbers.
Asimov,
Essays selected from the Magazine of fantasy and science fiction, Sept. 1959June 1966.
Includes index. 1.
Numbers, Theory
of
—Addresses,
I.
QA241.A76
essays, lectures.
Title.
51
2'. y'o8
isbn: 0385120745
Library of Congress Catalog Card
Copyright
Copyright©
©
Number 7623747
1977 by Isaac Asimov
1959, i960, 1961, 1962, 1963, 1964,
1965, 1966 by Mercury Press, Inc.
PRINTED
IN
ALL RIGHTS RESERVED THE UNITED STATES OF AMERICA
98765432
Dedicated to
CATHLEEN JORDAN and
a
new beginning
19S5407
volume
All essays in this
of Fantasy
and Science
originally appeared in
The Magazine
Fiction. Individual essays were in the
following issues.
Varieties of the Infinite
September 1959
A
May
Piece of Pi
The Imaginary That
Isn't
That's About the Size of Prefixing
It
Up
It
One, Ten, Buckle
i960
September i960
Tools of the Trade
My
Shoe
March 1961 October 1961
November 1962 December 1962
TFormation
August 1963
Forget
March 1964
It!
Nothing Counts
July 1964
The Days
August 1964
of
Our Years
Begin at the Beginning
January 1965
Exclamation Point!
July 1965
Water, Water, Everywhere—
December 1965
The ProtonReckoner
January 1966
Up
February 1966
and
The
Down
Isles of
the Earth
Earth
June 1966
.
CONTENTS
Introduction
Part
NUMBERS AND COUNTING
I
1.
Nothing Counts
2.
One, Ten, Buckle
3.
Exclamation Point!
30
4.
TFormation
44
5.
Varieties of the Infinite
58
3
My
Shoe
16
NUMBERS AND MATHEMATICS
Part II 6.
A Piece of Pi
73
7.
Tools of the Trade
85
8.
The Imaginary That
Part III 9.
10.
Forget
1 1
2.
The Days
97
111
It!
IV
Begin
Isn't
NUMBERS AND MEASUREMENT
Prefixing It
Part
1
xiii
Up
123
NUMBERS AND THE CALENDAR of
Our Years
1
at the
Beginning
153
39
CONTENTS
Xll
Part 1 3.
That's About the Size of
Part 14.
V NUMBERS AND BIOLOGY It
NUMBERS AND ASTRONOMY
VI
The ProtonReckoner
Part
VII
Water, Water, Everywhere—
16.
Up
17.
The
Down
Isles of
Index
185
NUMBERS AND THE EARTH
15.
and
169
the Earth
Earth
199
213
226
239
INTRODUCTION in 1959, I began writing a monthly science column for The Magazine of Fantasy and Science Fiction. I was given carte blanche as to subject matter, ap
Back
proach, style, and everything else, and
column that of
And
as
I do (and monthly essays.
the writing
all
pleasure as these
made
I
use of that.
full
have used the
I
through every science in an informal and very personal way so
to range
do
I
though that were not pleasure enough
complete seventeen
essays,
much
so
why, every time
in itself,
Doubleday & Company,
me
nothing gives
a great deal)
Inc., puts
them
my F & SF essays have essays. A thirteenth, of course,
I
book
into a
and publishes them. Twelve books of
been published
by now, containing
is
204 Few books, however, can be expected
enough
to be
a total of
on
its
way.
to sell indefinitely; at least, not well
worth the investment of keeping them forever in
The
print.
esti
at Doubleday have therefore (with some reluctance, for they me and know how my lower lip tends to tremble on these occa
mable gentlemen fond of
are
sions) allowed the
Out in is
my
of
first five
of hardback print,
I
books of essays to go out of print.
hasten to say. All
paperback, so that they are
still
about the hardback that
a cachet
that supplv the libraries;
and
of the books are flourishing
five
available to the public. Nevertheless, there I
am
for those
to their large personal collections of
reluctant to lose. It
who
really
want
a
the hardbacks
is
permanent addition
Asimov books* there
is
nothing
like a
hardback.
My
first
impulse, then, was to ask the kind people at Doubleday to put the
books back into print and gamble on a kind of second wind. This
my
odically in the case of
science fiction books, with success even
back editions are simultaneously available. But the case was different.
My
science fiction
is
could see that with
I
I
I
got to thinking.
.
.
done
when
my
interests
chance to
and
to give each
satisfy his or
her
own
is
inexorable.
member
of
my
particular taste
my own
heterogeneous audience a
now and
then.
The
result
on chemistry,
that each collection of essays has some on astronomy, some on physics, some on biology, some on mathematics, and so on.
*
Well,
start one!
essays
.
deliberately range widely over the various sciences both to satisfy
restless
peri
paper
ever fresh, but science essays do
tend to get out of date, for the advance of science
And then
is
is
some
INTRODUCTION
XIV
But what about the reader who
is
interested in science but
He has to read may be just up his
interested in one particular branch?
the book to find three or four that
Why
through
particularly
is
the articles in
all
alley.
not, then, go through the five outofprint books, cull the articles
and put them together
particular branch of science,
in a
more
on
a
specialized vol
ume.
Doubleday agreed and I made up a collection of astronomical articles which as Asimov on Astronomy. This introduction appeared at the start, explaining to the readers just what I had done and why I had done it, so that appeared
everything was open and on the table.
It
and might have done very poorly.
course,
It
was an experimental project, of did not;
it
did very well.
The
peo
and I got to work at once (grinning) and put together Asimov on Chemistry and Asimov on Physics, each with the same introDoubleday
ple at
rejoiced
duction, as in the
warned
of
what
is
first
one of
this series, so that the reader will
continue to be
going on.
These, too, did well, and
now
I
am
preparing a fourth and last** of these
volumes, Asimov on Numbers, which contains one article from
View from a Of Time Heaven. The
Height, seven articles from Adding a Dimension, four articles from
and Space and Other Things, and Aside from grouping the
from nine
articles into a
what more have
derly arrangement,
articles
five
from Earth to
not chronologically, but conceptually.
articles are arranged,
to sixteen years old
and
I
more homogeneous mass
done? Well, the
their age
in
shows here and there.
I
an
or
anvwhere
articles are
feel rather
pleased that the advance of science has not knocked out a single one of the
here included, or even seriously dented any, but minor changes must
articles
be made, and
I
have made them.
In doing this,
have tried not to revise the
I
would deprive you
of the fun of seeing
anyway, chew them
a little.
So
I
me
eat
articles
mv
themselves since that
words
now and
then, or,
have made changes by adding footnotes here
and there where something
make
Where doing
I said needed modification or where I was forced to change to avoid presenting misinformation in the course of the article.
a
it
it
was necessary to make a number of small changes in the
in footnotes
in that case
I
would be unbearablv clumsv,
warn the reader
In addition to that,
my
I
am
doing
good friends
I
statistics
and
did revise the article, but
so.
Doubleday have decided to prepare on the individual branches of science in a consistent and more elaborate format than they have use for my ordinary essay collections and have added illustrations to which I have written captions that give information above and beyond what is in the essays themselves. at
these books
* * There are, of course, seventeen essays in the five outofprint books that have appeared in none of the four collections, but they are a miscellaneous lot. Perhaps when more of my essay books go out of print, I can combine some of them with additional
articles
from the
later
books to add to the subjectcentered collections.
INTRODUCTION Finally, since the subject
matter
ordinary grabbag essay collections,
admit that
this will
be
is I
so
XV
much more homogeneous than
have prepared an index (though
less useful in this case
than in the
first
in I
my
must
three books of
the series). So, although the individual essays are old,
I
hope you
what
I
find the
And at least I have explained, have done and why. The rest is up to you.
enjoyable just the same.
Isaac
New
book new and
in all honesty, exactly
Asimov November
York,
10,75
ASIMOV ON NUMBERS
PART
I
Numbers and Counting
NOTHING COUNTS
1.
Roman
numerals seem, even after
five centuries of
obsolescence, to exert a
peculiar fascination over the inquiring mind.
my
It is
theory that the reason for this
peal to the ego.
WTien one
MCMXVIII,"
gives
it
eighteen" to one's
one
a sensation of
Whatever the
self.
is
that
passes a cornerstone
power
Roman which
numerals ap
savs:
"Ah,
to sav,
"Erected
ves,
nineteen
reason, thev are worth further dis
cussion.
The notion
of
number and
of counting, as well as the
names
of the
smaller and moreoftenused numbers, dates back to prehistoric times and
don't believe that there
is
a tribe of
ever primitive, that does not
With
human
I
beings on Earth today, how
have some notion of number.
the invention of writing (a step which marks the boundary line
between "prehistoric" and "historic"), the next step had to be taken—
numbers had bols for the
to
be written.
other word. In English "five"
One
can, of course, easily devise written sym
words that represent particular numbers,
we can
and the number of
digits
write the
on
all
number
as easily as for
of fingers on one
any
hand
as
four limbs as "twenty."
Early in the game, however, the kings' taxcollectors, chroniclers, and scribes
one
set
saw that numbers had the peculiarity of being ordered. There was way of counting numbers and any number could be defined by
counting up to
up
to the
Thus, as
'" ',
it.
Therefore
why not make marks which need be counted
proper number. if
we
let
"one" be represented
we can then work out the number
out trouble.
You can
see,
for
as
'
and "two"
as ",
and "three"
indicated by a given symbol with
instance,
that the 'symbol
stands for "twentythree." WTiat's more, such a symbol
is
"""""""'""""
universal.
What
NUMBERS AND COUNTING
4
number "twenty
ever language you count in, the symbol stands for the
three" in whatever sound your particular language uses to represent It gets
hard to read too
natural to break
it
many marks
an unbroken row, so
in
up into smaller groups.
If
we
it.
it is
only
are used to counting
on
it seems natural to break up the marks into groups "Twentythree" then becomes '"" '"" '"" '"" '". If we are more
the fingers of one hand, of
five.
and use both hands
sophisticated ,„
///////,„
jf
we
g Q b are f 00
t
anci use
we would write it """"" too, we might break numbers
in counting,
ur toes,
into twenties. All three
methods of breaking up number symbols into more
mark on the
dled groups have left their
kind, but the favorite was division into ten. are,
on the whole, too many
group produce too is
han
for easy grasping, while five symbols in
groups as numbers grow
one
larger. Division into ten
the happy compromise. It
seems a natural thought to go on to indicate groups of ten bv
arate mark. as
many
easily
number systems of manTwenty symbols in one group
various
"'"
"
There
no reason
is
when
every time,
to insist
on writing out
a separate mark, let us say
Once you've
started this way, the next steps are clear.
for instance
(a
group of ten
can be used for
,
the purpose. In that case "twenty three" could be written as
have ten groups of ten
a
a sep
'".
By the time you
hundred), vou can introduce another symbol,
+. Ten hundreds, or a thousand, can become = and so on. number "four thousand six hundred seventyfive" can be
In that case, the
written
===== — — — — — 
To make

j



'"".
such a set of symbols more easily graspable, we can take ad
vantage of the ability of the eve to form a pattern. (You can
tell
pattern
know how you
the numbers displaved by a pack of cards or a pair of dice by the itself.)
We
could therefore write "four thousand
six
hundred
seventyfive" as
And,
as a
matter of
fact,
the ancient Babylonians used just this system
of writing numbers, but they used cuneiform wedges to express
The
it.
Greeks, in the earlier stages of their development, used a system
similar to that of the Babylonians, but in later times an alternate
method
grew popular. They made use of another ordered system— that of the ters of
the alphabet.
let
.
}
NOTHING COUNTS natural to correlate the alphabet and the
It is
taught both about the same time
comes
." .
.
we
one
series ." .
.
the symbol
if
if
to
is
"ABCDEFG"
stands
then, since each symbol
for
"seven"

count the
different, only the last
is
G
is
like
/.en
'
need to be written. You
onecomponent symbol does
six
looks very

G
like
all
own
their
alphabet,
of
course,
but
let's
use
A=one. B = two,
our
C=
D=four, E = five, F=six. G=seven, H=eight, I=nine. and J=ten.
three,
We
could
alphabet
the letter
let
K
go on to equal "eleven," but at that rate our
only help us up through "twentysix."
will
better system.
The Babylonian
J=ten, than
The Greeks had
notion of groups of ten had
a
mark.
left its
equals not only ten objects but also one group of tens.
J
not. then, continue the next letters as
In other
numbering groups
words [=ten, K=twenty, L=thirtv. M=fortv.
of tens?
X=
fifty,
0=
P=seventv, Qmeighty, R=ninetv. Then we can go on to number
S=one hundred, T=two hundred, U=three hunW=five hundred, X=six hundred. Y=seven
groups of hundreds:
Y=four
dred,
hundred.
hundred. Z=eight hundred. dred, but
the
six
looks nothing at
alphabet here for the complete demonstration:
sixty,
see
i
The Greeks used
If
no
the seventh letter of the alphabet and
whereas F
;
and
letters
the work of a sevencomponent symbol. Furthermore, " "
Why
see, is
all
therefore stands for "seven." In this way, a
seven
bee,
"ay,
and there
all the commust be included without exmean "seven" and nothing else. On the other
can't confuse the fact that
(
are sys
"'"" use undifferentiated symbols such as for "seven,"
ception
much
We
svstem.
for the other.
ponents of the symbol are identical and
hand,
The
as glibly as "'one, two, three, four
difficulty in substituting If
number
childhood, and the two ordered
in
tems of objects naturally tend to match up. dee
z
we have run out
ampersand
we can
The
say that first

&

It
would be convenient
of letters.
However,
was sometimes placed
on
to nine
hun
oldfashioned alphabets
in
at the
to go
end of the alphabet,
so
&=nine hundred.
nine
letters,
in
other words, represent the units from one to
nine, the second nine letters represent the tens groups
from one
to nine,
the third nine letters represent the hundreds groups from one to nine.
(The Greek alphabet,
in classic times,
had only twentyfour
twentyseven are needed, so the Greeks to
till
out the
letters
where
use of three archaic letters
list.
advantages and disadvantages over the Babyloadvantage is that anv number under a thousand can be
This svstem possesses nian system.
made
One
its
given in three symbols. For instance, bv the system
I
have
just set
up with
NUMBERS AND COUNTING
6
our alphabet, is
six
hundred
seventyfive
is
XPE,
while eight hundred sixteen
ZJF.
One
disadvantage of the Greek system, however,
must
of twentyseven different symbols
numbers
of
to a thousand,
different symbols
be carefully
is
that the significance
memorized
for the use
whereas in the Babylonian system only three
must be memorized.
Furthermore, the Greek system comes to a natural end when the of the alphabet are used up.
number
est
Nine hundred ninetynine (&RI)
letters
the larg
is
that can be written without introducing special markings to
indicate that a particular symbol indicates groups of thousands, tens of
thousands, and so on.
A
will get
I
back to
this later.
rather subtle disadvantage of the
Greek system was that the same
symbols were used for numbers and words so that the mind could be distracted.
easilv
For instance, the Jews of GraecoRoman times adopted the
Greek system of representing numbers but, of course, used the Hebrew alphabet— and promptly ran into
difficulty.
naturally be written as "tenfive." In the
The number
Hebrew
five" represents a short version of the ineffable
would
"fifteen"
alphabet, however, "ten
name
of the Lord,
and the
Jews, uneasy at the sacrilege, allowed "fifteen" to be represented as "ninesix" instead.
Worse
yet,
words
instance, to use our
the alphabet system
symbols though,
as
in the
own
GreekHebrew system look
alphabet,
WRA
all,
wish.) Consequently, martial,
The the
"five
numbers. For
like
hundred ninetyone." In
doesn't usually matter in which order
it
we
shall see, this
came
numerals, which are also alphabetic, and ninetyone." (After
is
we can
it is
say "five
to
WAR also
in the
place the
means
"five
Roman hundred
hundred oneandninety"
easy to believe that there
and of ominous import
we
be untrue for the
number
"five
is
if
we
something warlike,
hundred ninetyone."
Jews, poring over every syllable of the Bible in their effort to copy
word of the Lord with the exactness that reverence required, saw numall the words, and in New Testament times a whole svstem of mys
bers in
ticism arose over the numerical interrelationships within the Bible. This
was the nearest the Jews came to mathematics, and thev called this numbering of words gematria, which is a distortion of the Greek geometria.
We now call Some poor
it
"numerologv."
souls,
even today, assign numbers to the different
letters
and
decide which names are lucky and which unlucky, and which bov should
marry which ences.
girl
and
so on. It
is
one of the more laughable pseudosci
NOTHING COUNTS
7
In one case, a piece of gematria had repercussions in later history. This bit of gematria
book
the last
be found
to
is
of the
in
"The Revelation
New Testament— a
The
fashion that defies literal understanding.
tical
John the Divine,"
of St.
book which
written in a mvs
is
reason for the lack of
seems quite clear to me. The author of Revelation was denouncing Roman government and was laying himself open to a charge of treason
clarity
the
and
subsequent crucifixion
to
quently, he made an
he made
if
effort to write in
his
words too
clear.
Conse
such a way as to be perfectly clear
"ingroup" audience, while remaining completely meaningless to the
to his
Roman
authorities.
In the thirteenth chapter he speaks of beasts of diabolical powers, in the eighteenth verse
derstanding count the
and
number
his
Clearly, this
Six
is
is
he
says,
number
"Here
is
and
wisdom. Let him that hath un
of the beast: for
hundred threescore and
it is
the
number
of a
man;
six."
designed not to give the pseudoscience of gematria holy
meant by the known, was writ
sanction, but merely to serve as a guide to the actual person
obscure imagery of the chapter. Revelation, as nearly as ten only a few decades after the
first
is
great persecution of Christians under
If Nero's name ("Neron Caesar") is written in Hebrew characters sum of the numbers represented by the individual letters does indeed come out to be six hundred sixtvsix, "the number of the beast." Of course, other interpretations are possible. In fact, if Revelation is
Nero. the
taken as having significance for
which
in
For
was written,
this reason,
show
to
it
it
may
all
time
time
as well as for the particular
also refer to
some
antiChrist of the future.
generation after generation, people have
made attempts name in an
that, by the appropriate jugglings of the spelling of a
appropriate language, and by the appropriate assignment of numbers to letters,
some
particular personal
enemy could be made
to possess the
num
ber of the beast. If ily
the Christians could apply
have applied
centuries later
it
it
it
to Nero, the Jews themselves
in the next century to
to
if
eas
they had wished. Five
Mohammed. At the time Martin Luther's name and found
could be (and was) applied to
of the Reformation, Catholics calculated it
Hadrian,
might
be the number of the beast, and Protestants returned the compliment
by making the same discover)
when
in the case of several popes.
were replaced by nationalistic ones, Napoleon Bonaparte and William II were appropriately worked out. What's more, a few minutes' work with my own system of alphabetnumLater
still,
religious rivalries
bers shows me that "Herr Adollf Hitler" has the number of the beast. need that extra "1" to make it work.)
(I
NUMBERS AND COUNTING
8
The Roman system
of
number symbols had
similarities
to
both the
Greek and Babylonian systems. Like the Greeks, the Romans used
them
of the alphabet. However, they did not use
few
which thev repeated
letters
as often as
system. Unlike the Babylonians, the for even tenfold increase of
Thus,
to begin with, the
did not invent a
just a
Babylonian
new symbol
well.
symbol
II, III,
Romans
but used
in the
number, but (more primitively) used new
symbols for fivefold increases as
"four," can be written
in order,
necessary— as
letters
and
"one"
for
is
I,
and "two," "three," and
IIII.
ROMAN NUMERALS This
is
a horoscope cast for Albrecht von Wallenstein, an Imperial gen
during the Thirty Years
eral
War, by the great astronomer Johann Kepler. make a living, just as a modern actor,
(Kepler cast horoscopes in order to
even a good one, might do commercials on the
side.)
Even though the numerals used are mostly Arabic, the twelve signs of the zodiac are numbered in Roman style for their greater effect. Roman numerals carried a cachet of stateliness for centuries after they were seen to be useless in computation.
Although our own familiar number system 10, the
for
Roman
1, 5, 10,
five fingers
based on 10 and powers of
numerals are based on both 5 and 10 with special symbols and 1,000. Obviously, this arises because we have
50, 100, 500,
on each hand and ten
fingers altogether.
doesnt take much of an intellectual jump to base a number system on 20. The Mayans of Central America
In a barefoot society decide to
is
it
counted by both tens and twenties and had special symbols for {20
2 ),
8,000 (20 3
),
Although there in
Western
160,000 (20 4 is
tradition,
),
so
common,
20, for
400
so on.
no formal vigesimal (by twenties) number system we still count by "scores" and say "four score and
seven" (four twenties and seven) is
and
in fact, that
when we mean
we speak
87.
Counting by twenties
and ask "What's
of keeping score
the score?" in connection with a ball game.
Duodecimal systems
are also used, in
because 12 can be divided evenly by therefore,
and
of grosses,
words anyway,
2, 3, 4,
where a gross
is
a
and
6.
if
We
not in symbols, speak of dozens,
dozen dozen, or
matter, the ancient Sumerians used a 60based system
60 seconds to the minute and 60 minutes to the hour.
144.
and we
For that still
have
NOTHING COUNTS The symbol
for five,
themselves no end trying
then, to
is
not
IIIII,
but V. People have amused
work out the reasons
for the particular letters
chosen as symbols, but there are no explanations that are universally cepted. However,
and that
V
For
pleasant to think that
might symbolize the hand
branch of the fingers.
it is
V
I
ac
represents the upheld finger
itself
with
all
five
fingers— one
would be the outheld thumb, the other, the remaining
"six," "seven," "eight,"
and "nine," we would then have VI,
VII, VIII, and Villi.
^orofcop mmgefidht tnsxci> loannem Kepplerum
1608.
The Granger
Collection
NUMBERS AND COUNTING
lO
For "ten" we would then have X, which (some people think) represents both hands held wrist to eight" would be
The symbol is
"Twentythree" would be XXIII, "forty
wrist.
XXXXVIII, and
for "fifty"
is
so on.
"one hundred"
L, for
M
is M. The C and are easy to understand, centum (meaning "one hundred") and is the
D, and for "one thousand"
for
C
the
is
first letter
first
letter of
C, for "five hundred"
is
M
of mille (one thousand).
For that very reason, however, those symbols are suspicious. As
may have come
they
initials
symbols for those
to oust the original lessmeaningful
numbers. For instance, an alternative symbol for "thousand" looks something like this (I). Half of a thousand or "five hundred" of the symbol, or I)
L which Now,
stands for "fifty," then,
we can
may have been converted don't know why it is used.
this I
advantage of writing numbers according to
write nineteen sixtyfour as
nineteen sixtyfour
If
Roman
write nineteen sixtyfour, in
doesn't matter in which order the
it is
the right half
is
As
into D.
numerals, as
not
if I
numbers
system
CDCLIIMXCICI,
would
it
add up the number values of each
anyone would ever scramble the
likely that
this
are written. If
time,
first
it
would then be much simpler
is
it
still
represent
letter.
However,
letters in this fashion.
did the
I
add the values of the
to
in fact, this order of decreasing value
that
is
decided to
I
the letters were written in strict order of decreasing value, as
And,
for the
MDCCCCLXIIII.
follows:
One
and
letters.
(except for special cases)
al
ways used.
Once
the order of writing the letters in
established convention, one can if it
will
make
is
if
"three," but
it is
the same way while
one of
we decide
larger value, the
that
when
two are added;
XC
is
Thus VI
is
"five" plus "one" or
LX
first
"six," while
minus "one" or "four." (One might even say that
"five"
is
made an
is
the symbol of smaller value precedes one of larger value, the
subtracted from the second.
IV
numerals
use of deviations from that set order
help simplify matters. For instance, suppose
a symbol of smaller value follows
while
Roman
II
V
is
conventional to subtract no more than one symbol.) In is
XL
"sixty" while
"ninety";
MC
is
is
"forty";
CX
is
"one hundred ten,"
CM
"one thousand one hundred," while
is
"nine hundred."
The
value of this "subtractive principle"
work of write
five.
Why
CM? The
write Villi
year
if
is
nineteen
sixtyfour,
do the
that two symbols can
you can write IX; or instead
DCCCC of
if
being
you can written
MDCCCCLXIIII (twelve symbols), can be written MCMLXIV symbols). On the other hand, once you make the order of writing
(seven letters
.
NOTHING COUNTS you can no longer scramble them even
significant,
instance,
MCMLXIV
if
thousand one hundred
The
11
you wanted
if
MMCLXVI
scrambled to
is
on and
off in
ancient times but was
One
interesting theory for
not regularly adopted until the Middle Ages.
IV
the delay involves the simplest use of the principle— that of first letters
IVPITER,
of
Romans may have had
the
For
sixtysix."
subtractive principle was used
These are the
to.
becomes "two
it
("four").
Roman
the chief of the
gods,
and
about writing even the beginning of
a delicacy
Roman
the name. Even today, on clockfaces bearing
represented as IIII and never as IV. This
is
numerals, "four"
is
not because the clockface does
not accept the subtractive principle, for "nine"
represented as
is
IX and
never as Villi.
With
we can go up
the symbols already given,
number "four
to the
thousand nine hundred ninetynine" in
Roman
MMMMDCCCCLXXXXVIIII
the subtractive principle
MMMMCMXCIX. Roman
than four times.
V;
was
letter
little
A new
symbol
used,
is
not quite
right. Strictly
always invented to prevent that: 11111=:
is
Well, then, what
for "five
in ordinary life for
had occasion
tax collectors
is
system never requires a symbol to be repeated more
was decided upon
need
numerals. This would be
that "five thousand" (the next
MMMMM, but this
XXXXX=L; and CCCCC=D. No
if
You might suppose
number) could be written speaking, the
or,
MMMMM?
numbers that high. And
for larger
down to the common man. One method of penetrating to
is
thousand/' In ancient times there if
scholars
and
numbers, their systems did not per
colate
'Jive
bar to represent thousands. Thus,
And LXVIICDLXXXII.
thousand" and beyond
would represent not
is
"five"
to use a
but
"five
sixtyseven thousand four hundred eightytwo would be
thousand."
'
But another method tive
V
symbol
(I) for
of writing large
numbers harks back
to the primi
"thousand." By adding to the curved lines we can
in
Thus "ten thousand" would be ((I) ), and "one hundred thousand" would be (((I))). Then just as "five hun
number by
crease the
dred" was
would be
I
ratios of ten.
or D, "five thousand"
I)
would be I)) and
) )
Just as the
Romans made
special
sands and for millions (or at least
Romans
also,
made
some
special
marks
Greek extreme is no
of the
didn't carry this to the logical
mans prided themselves on being it
thousand"
marks to indicate thousands, so did
the Greeks. What's more, the Greeks
the
"fifty
)
nonintellectual.
however, will never cease to astonish me.
for ten thou
writers did). surprise.
That
The Ro
That the Greeks missed
NUMBERS AND COUNTING
12
Suppose that instead of making special marks
one were If
we
to
make
stick to the
the one in which
only,
marks for every type of group from the units on.
system
I
introduced at the start of the chapter— that
stands for units,
'

for tens,
number with
could write every
groups =+'. Then
A
get by with only the letters from
thousand
five
set of
hundred
to
and write
it
for
We
the type of
off
BEHA. What's more
we could
fiftyfive"
is,
hundred eightyone" we could
five
I
=
nine symbols.
marking
a little heading,
for "two thousand
and
for hundreds,

thousands— then we could get by with but one
for "five
numbers
for large
special
EEEE. There E
write
would be no confusion with
all
would indicate that one was
a "five," another a "fifty," another a "five
the E's since the symbol above each
hundred," and another a "five thousand." By using additional symbols for
hundred thousands,
ten thousands,
ever large, could be written in this
Yet
it is
not surprising that
had thought of
same
this
and so on, any number, how
millions,
fashion.
would not be popular. Even
if
a
Greek
he would have been repelled bv the necessitv of writing
it
those tiny symbols. In an age of handcopying, additional symbols meant additional labor
and
scribes
would resent that
furiously.
Of course, one might easily decide that the symbols weren't The groups^ one could agree, could always be written right to creasing values. Trie units
thousand
five
hundred
hundred eightyone" and
fiftyfive"
even without the
no
EEEE
little
Here, though, a difficulty would creep of ten, or perhaps
left in in
at the right end, the tens next
the hundreds next, and so on. In that case,
left,
five
would be
necessary.
would be
on the
would be "two "five
thousand
symbols on top.
What
in.
units, in a particular
BEHA
if
there were no groups
number? Consider the number
"ten" or the number "one hundred and one." The former is made up of one group of ten and no units, while the latter is made up of one group of
hundreds, no groups of tens, and one unit. Using symbols over the
umns, the numbers could be written dare leave out the
little
symbols.
If
A
and
you did,
A
A, but
how
col
now you would not
could you differentiate
A
meaning "ten" from A meaning "one" or AA meaning "one hundred and one" from AA meaning "eleven" or AA meaning "one hundred and ten"? You might try to leave a gap so as to indicate "one hundred and one" by
A
A. But then, in an age of handcopying,
come AA, too,
or, for
that matter,
how would you
Greeks thought of
how
how
quickly might
quickly would that be
AA
become A A? Then, No, even if the
indicate a gap at the end of a symbol?
this system,
they must obviously have
come
to the con
NOTHING COUNTS numbers made
elusion that the existence of gaps in fication impractical.
SA
it
"one hundred and one" and
for
What no Greek that
They decided ever thought
13
was
to
this
Hades with
and
symbols.
little
of— not even Archimedes himself— was
wasn't absolutely necessary to work with gaps.
it
attempted simpli
safer to let J stand for 'ten"
One
could
fill
the
gap with a symbol by letting one stand for nothing— for "no groups."
Suppose we use $
made up
such a symbol. Then,
as
of one group of hundreds,
be written A$A.
If
don't need the
we do
"one hundred and one"
if
no groups of
that sort of thing,
tens,
and one
unit,
it
we
gaps are eliminated and
all
is
can
symbols on top. "One" becomes A, "ten" becomes
little
A$, "one hundred" becomes A$$, "one hundred and one" becomes A$A,
"one hundred and ten" becomes AA$, and so on. Any number, however large,
can be written with the use of exactly nine
symbol
letters plus a
for nothing.
Surely this
Yet
it
ning of
is
took
the simplest thing in the
men about
number symbols,
world— after you think
thousand
five
symbol
to think of a
of
it.
counting from the begin
years,
for nothing.
The man who
succeeded (one of the most creative and original thinkers in history)
unknown.
We know
only that he was some
Hindu who
lived
no
later
is
than
the ninth century.
The Hindus for
called the
symbol sunya, meaning "empty." This symbol
nothing was picked up by the Arabs,
who termed
it sifr,
language meant "empty." This has been distorted into our
which
in their
own words
"ci
pher" and, by way of zefirum, into "zero."
Very
slowly, the
new system
of numerals (called "Arabic numerals" be
them from the Arabs) reached the West
cause the Europeans learned of
and replaced the
Roman
system.
Because the Arabic numerals came from lands which did not use the Ro
man alphabet, the shape of the numerals was nothing like the letters of the Roman alphabet and this was good, too. It removed wordnumber confusion and reduced gematria from the even day occupation of anyone who could read, to a burdensome folly that only a few would wish to bother with.
The Arabic numerals 8,
9,
as
now
and the allimportant
used by us
Such
o.
is
are, of course, 1, 2, 3, 4,
5, 6, 7,
our reliance on these numerals
(which are internationallv accepted) that we are not even aware of the tent to
which we
rely
on them. For instance,
vaguely queer to you, perhaps
from using Arabic numerals
We
all
know
all
it
was because
if I
this
had
ex
chapter has seemed
deliberately refrained
through.
the great simplicity Arabic numerals have lent to arith
ARABIC NUMERALS Quite apart from the greater ease of computation with Arabic numerals, as
compared with any other system man has invented, there
pactness of
it.
Imagine
all
is
the com
the numerical information in the table given
Roman
It would bemake any sense of. For instance, it is obvious just in the number of digits of the number that 12,000 is greater than y8y. You can go down the final column of the illustration in one quick sweep of the eye and see at a glance that the greatest number of sales listed for any item in it is the 285,800 for Montgomery Ward. It happens to be the only sixdigit number in the column that starts with a numeral higher than 1. You dont even have to read the other digits to know it's highest. This cannot be done in any other number system. For instance, of the two numbers XVIII and XL, the twosymbol number is over twice as
here translated into
come
numerals (or any other kind).
a bulky mass that only an expert could
great as the fivesymbol number.
Of
course, there are disadvantages to the Arabic
There
is
no redundancy
every place has one
and you
a digit
in
it.
and only one
are lost.
numeral system,
too.
digit has
one and only one value, and
Drop
a single digit or interchange
value.
For instance, there
a letter out of "redundancy"
one
Every
is
redundancy in words. Leave 7
and you have "redundncy" and hardly anyword. Or invert two letters and you have the mistake and allow for it.
will fail to see the correct
"redundancy" and people see
On
the other hand, change 2835 to 235 by dropping the
by inverting two
made
and there
is
or to 2385
no sign that any mistake has been
or any chance of retrieving the correct value.
This 1929.
digits,
8,
illustration,
Midland
by the way, shows the famous stock market crash of
Steel
Products,
preferred,
dropped 60 points. Murray
Corp. was at 20 from a year's high of 100%. Oh, boy.
metical computation. all
The
unnecessary load they took
because of the presence of the zero,
fact
gone unnoticed
in the
is
off
the
human mind,
Nor has this The importance of the zero
simply incalculable.
English language.
is reflected in the fact that when we work out an arithmetical computation we are (to use a term now slightly oldfashioned) "ciphering." And when we work out some code, we are "deciphering" it.
So
if
you look once more at the
title
of this chapter, you will see that
I
M3gSS3g§§gS8§8H§£Sggg\?g§g*s^^ °s=~
a stag
«"'"•
—"• a's»


"s'"'*
s"" m
''
•*«'•*
aaif**
v x"8s'*
32'"g""S"' s'a
a"
•*
a'**
•
'8
"a a
1
a^^^?^, *± :^K^a «.
!
:
*.
J
"^^if: JP*..JPtf,
11,
behind an
left
x—x=x If
you started with
series of
odd
series, so
that
five;
or a
oc+n=oo.
integers to the
even integers, you would simply have the unending
integers, or:
great.
the infinite series of integers, you would have elimi
series of
you would
In fact,
endless.
Suppose we consider only
the even integers only and added one odd integer, or two, or trillion,
25 or the
still
infinite series and could therefore be written: same way, the odd integers would form an infinite
out every even integer you came yi
integer,
startling than that.
We would have a series
and could be written:
Now,
is
would be an
endlessly. It .
the series of integers
where n represents any
we can be more
the even integers.
2, 4, 6,
off the first 3 integers or the first left of
with 1000000000001, 1000000000002, and go on
start, say,
x — n=x,
So
endlessly.
is
unending
series of all
x(x = x.
this point,
however,
of pulling a fast one.
it is
just possible that
some
of you
may
suspect
VARIETIES OF THE INFINITE After
63
in the first 10 integers, there are 5 even integers
all,
and
5
odd
1000 integers, there are 500 even integers and 500 odd integers; and so on. Xo matter how many consecutive integers we take,
ones; in the
first
and half are odd.
half are always even
Therefore, although the series
2,
6,
4,
.
.
endless, the total can
is
.
onlv be half as great as the total of the also endless .
.
And
.
the same
is
true for the series
onlv half as great as the series of
less, is
And
so (you
the set of
might think)
in subtracting
certain satisfaction,
To answer
x— x= 1
as:
that objection,
tired of
let's
lecture hall?"
He are
You
thinks a
no empty
5, 6.
the set of even integers from
odd
integers,
what we
are doing
2
a
who
has been doing
"How manv
asks,
unknown
all
audi
our counting,
seats are there in the
answer, "640."
and
little
seats
,
go back to counting the
and
turns to vou
it,
3, 4,
"makes sense."
ence at the lecture. Our bright bov, is
2,
x. That, you might think with
1
2
5,
3,
1,
which, though end
all integers.
integers to obtain the set of
all
can be represented
and
1,
series
...
"Well,
says,
and there
is
see that every seat
I
You, having equally good eyesight,
say,
taken.
There
"That's right."
"Well, then," says the bov, "why count them
now that there And he's correct.
is
no one standing." as they leave.
We
know
are exactly 640 spectators."
right
If two series of objects (A series and B series) just match up so that there is one and only one A for even B and one and onlv one B for even A, then we know that the total number of A objects is
just equal to the total
In fact, this
many
is
number
of
B
objects.
teeth there are in the fullv equipped
each tooth one and onlv one
number
ber to one and onlv one tooth. (This toone correspondence.") that the series
this, so
with the
series
If we want to know how human mouth, we assign to
what we do when we count.
1,
We 2,
3,
(in order) is
find that
...
,
and we apply each num
called placing
we need
30,
31,
two
series into
onlv 32 numbers to do
32 can be exactly
one tooth, next tooth, next tooth,
"one
.
.
.
,
matched
next tooth, next
tooth, last tooth.
And
therefore,
mouth to
put
is
it
we
the same as the
terselv
Now we
and
number of teeth in the number of integers from
sav, the
fullyequipped 1
to 32 inclusive. Or,
succinctlv: there are 32 teeth.
can do the same for the set of even integers.
down
the even integers and give each one a number.
write
down
all
human
the even integers, but
we can
write
We
can write
Of course, we can't down some and get
NUMBERS AND COUNTING
64
We
started anyway.
above
directly
it,
can write the number assigned to each even integer
with a doubleheaded arrow, so:
1234 We
6
4
2
8
5
67
8
9
10...
10
12
16
18
20
14
.
.
.
can already see a system here. Every even integer
particular
number
number and no
and you can
other,
by dividing the even integer by
is
number 19 number 12309
has the
assigned to
the
assigned to
the series of
all
assigned one particular
Thus, the even integer 38 and no other. The even integer 24618 has
it
In the
it.
2.
same way, any given number
integers can be assigned to
The number 538 number 29999999
is
what the
tell
one and onlv one even
in
integer.
The
is
applied to even integer 1076 and to no other.
is
applied to even integer 59999998 and no other; and
so on.
number
Since every
and only one number series are in
in
the series of even integers can be applied to one
in the series of all integers
and
The number
onetoone correspondence and are equal.
integers then
is
equal to the
number
of
all integers.
vice versa, the
Bv
a similiar
two
of even
argument,
number of odd integers is equal to the number of all integers. You may object bv saving that when all the even integers
the
integers) are used up, there will left over.
Maybe
so,
of even integers (or
Therefore,
"odd
odd
when we
integers," this
thrown
but
is
this
still
be
(or
odd
fully half the series of all integers
argument has no meaning since the
series
integers) will never be used up.
sav that "all integers" like saving
oc
minus "even integers" equals
— x = oc,
and terms
% °°
like
can De
out.
In fact, in subtracting even integers from
out every other number and thus, the series
is
still
in a
all
integers,
way, dividing the
we
are crossing
series
by
2.
Since
unending, 00/2=00 anyway, so what price half of
in
finity? if we crossed out everv other integer in the series of even we would have an unending series of integers divisible bv 4; and if we crossed out every other integer in that series, we would have an unend
Better yet,
integers,
ing series of integers divisible bv
8,
and
so
on
endlessly.
Each one
"smaller" series could be matched up with the series of
onetoone correspondence.
bv
2 endlesslv, If
and
still
If
an unending
series of integers
of these
integers in
all
can be divided
remain endless, then we are saving that 00/00
you doubt that endless
series that
have been
drastically
can be put into onetoone correspondence with the just consider those integers that are multiples of
= 00.
thinned out
series of all integers,
one
trillion.
You
have:
.
VARIETIES OF THE INFINITE 2,000,000,000,000;
1,000,000,000,000;
matched up with
are
2,
3,
.
.
all
many
.
.
by
These
00.
;
in the set of
one and only one
is
numone and only one number
is
is
4856. For any
342,000,000,000,000. There
this case,
integers divisible
.
For any number
00.
integers, say 342, there
in the set of "trillionintegers," in fore, there are as
3,000,000,000,000; ,
which, in this case,
in the set of all integers,
ber in the set of
.
say 4,856,000,000,000,000, there
"trillionintegers,"
number
1,
6c
a trillion as there are integers
altogether.
It
works the other way around, too.
midway
the
=
In fact,
00.
the fourths, then
keep the resulting integers so that
This
up
If
1%,
1,
you place between each number 1
2%,
2,
3 /2 ,
3,
series
.
.
.
you
00,
,
and yet
are,
new
this
can be put into onetoone correspondence with the set of integers,
so that 200 all
,
doubling the number of items in the
in effect, series
y2
fraction, thus:
00
you keep on doing
if
the eighths, then
all
00
=
00
2
it
indefinitely, putting in
the sixteenths, you can
still
onetoone correspondence with the set of
series in
•
all
=
all
00
may seem too much to swallow. How can all the fractions be lined we can be sure that each one is getting one and only one num
so that
ber. It
is
easy to line up integers,
1,
2,
3,
or even integers,
or even
2, 4, 6,
But how
can you line up fractions and 2, 3, 5, 7, 11. be sure that all are included, even fancy ones like 1489 %725523 an ^
prime numbers
.
.
.
689444473,/
There
are,
Suppose we
however, several wavs to make up an inclusive first list all
nator add up to
the fractions in
There
2.
is
onlv one of these:
%. Then
where the numerator and denominator add up these:
%
to
3.
list
see,
of fractions.
those fractions
There are two of
have %, %, and y3 where the numerator and to 4. Then we have %, %, %, and Y4 In each group, we place the fractions in the order of decreasing numerator and
and
y2 Then we .
,
denominator add up you
list
which the numerator and denomi
.
increasing denominator. If
we make such
%,%,
and
a
list:
Ylt
V2
,
%, %, %,
ft,
fraction 1489
% 72 552
be included
will
3 will
be
in that
of the group. in
Similarly,
if
we proceed
68944447%
wm
it
will
will thus
have
its
fraction,
far in
enough.
which the
be the 2725523rd
be the second fraction
which the numerator and the denominator add up
Every possible fraction series.
,
group of fractions
numerator and denominator add up to 2740422, and group
%, %, %, %, 4/2 %,
so on endlessly we can be assured that any particular
no matter how complicated,
The
%,
to
in
the
689444475.
particular assigned place in the
NUMBERS AND COUNTING
66 It follows,
own number and that no number has its own fraction and
then, that every fraction has
fraction will be left out. Moreover, every
no number
is
The
left out.
correspondence with the fractions
(In the
Y2
Thus,
equal to the
is
list
an d
number
%
number
%, %, and
of fractions
1.
%
is
All this
repeated
in a line.
A
endlessly,
all
number
It
shows that the
total
of integers even though
reluctantly decided that
that "infinity"
line
you imagine the
if
same
all
times; in fact, endlessly.)
is
can be marked are
no mat
"infinity"
equal inter
off at
numbered
marked %, 1%,
unend
all
1,
2 a/2,
2,
3,
and
The mid
line continuing endlessly.
points between the integerpoints can be
when
are equal in value.
each particular fraction, and
and the marks can represent points which
on
some
all right.
is
many
By now you may have more or less ingness is the same unendingness and ter what you do to it. Not so! Consider the points
all
not only have the same value but that
equal to the total
is
integral values as well,
so
of
of all integers.
in the series of fractions, the value of
vals,
put into a onetoone
is
and thus the number
are listed as different fractions, but both have the
that of an integer,
is
series of all fractions
series of all integers,
of fractions above, you will see that
value. Fractions like
value
its
•
•
•
,
and
the thirds can be marked and the fourths and the fifths and indeed
the unending
number
of fractions can be assigned to
some
particular
point. It
would seem then that even point
or other assigned to after
it.
in
the line would have
some
fraction
Surely there would be no point in the line left out
an unending number of fractions had been assigned to
it?
Oh, wouldn't there? There
is
on the
a point
line,
you
see, that
would be represented by
a
value equal to the square root of two (V2). This can be shown as follows. If you construct a square on the line with each side exactlv equal to
the interval of one integer already marked off on the onal of the square would be just equal to \/i.
on the
line, starting
cides with the point
Now
the catch
fraction;
by any
line
which can be
for
it
then the diagis
laid
down
set equal to \/i.
that the value of \fi cannot be represented by a
fraction;
bv any conceivable
the ancient Greeks and the proof
word
line,
that diagonal
from the zero point, the end of that diagonal coin
on the
is
If
is
here to save room. Well,
fraction.
simple but if
all
I'll
This was proved by ask you to take
my
the fractions are assigned to
VARIETIES OF THE INFINITE various points in the line, at least
67
one point, that which corresponds
to
y/2, will be left out.
numbers which can be represented
All
num
as fractions are "rational
two numbers, the numerator and the denominator. Numbers which cannot be represented as fractions
bers" because a fraction
numbers" and \fi
are "irrational
although
it
really the ratio of
is
was the
is
bv no means the only one of those,
such to be discovered. Most square roots, cube
first
roots, fourth roots, etc., are irrationals, so are
are
etc., so
numbers involving
most
sines, cosines, tangents,
pi (tt), so are logarithms.
numbers
In fact, the set of irrational
unending.
is
can be shown that
It
between any two points represented bv rational numbers on a ever close those two points are, there
is
how
line,
always at least one point repre
sented by an irrational number.
Together, the rational numbers and irrational numbers are spoken of
numbers."
as "real
made
It
can be shown that any given
to correspond to
any point
in the line
one and only one point
number. In other words, tion,
made
can be
to correspond to
a point in a line
can always be assigned an irrational.
real
number can be
in a given line;
and that
one and only one
which can't be assigned
No
real
a frac
point can be missed by both
categories.
The
numbers and the series of points and are equal.
series of real
in a line are there
fore in onetoone correspondence
Now
the next question
is:
numbers, or of
all real
all
The answer is, No! how you arrange your
series of integers.
can be shown that no matter
It
the series of
being equivalent), be set into a onetoone cor
points in a line (the two
respondence with the
Can
numbers or
real
your points, no matter what conceivable system you use, an endless number of either real numbers or points will always be left out.
we
that
are in the
audience in which
same
situation as that in
all seats
are taken
which we
and there
The
result
is
are faced with an
are people standing.
We
more people than seats. And so, in conclude that there are more real numbers,
are forced to conclude that there are
same way, we are forced
the
to
or points in a line, than there are integers.
we want
If
want
to use the
symbol
up with
integers
tied
all
bol
C
is
by symbols, we don't "and so on endlessly," since this has been
to express the endless series of points oo
for
and
rational
numbers
generally. Instead, the sym
usually used, standing for continuum, since
line represent a
We can
continuous
all
the points in a
line.
therefore write the series: Point
1,
Point
2,
Point
3,
.
.
.
,
C.
GEORG CANTOR To
designate Cantor by nationality
Leningrad, in fact
was called
(it
St.
is difficult.
He was
born in Russia, in
Petersburg then), on
March
3,
1845.
His father had emigrated to Russia from Denmark, however, and then Russia for
left
Germany when young Georg was
only eleven. In addition,
the family was of Jewish descent, though his mother was born a
Roman
Catholic and his father was converted to Protestantism.
Even
as a schoolboy
Cantor showed talent for mathematics, and evenhe made mathematics his profession.
tually (over his father's objections)
In i86y he obtained Berlin.
He
his
Ph.D. magna cum laude from the University of
obtained an academic position at the University of Halle,
advancing to a professorial appointment in 1872.
was
It
in 1874 that
Cantor began to introduce
his intellectshaking con
had caught glimpses of it, but Cantor was a complete logical structure in which a whole series of
cepts of infinity. Earlier, Galileo
the
first
to erect
transfinite ity,
numbers was postulated, representing
different orders of infin
so to speak.
Not much can be done with can be described. The
numbers
is
the different orders in terms of sets that
set of integers
equal to the
is
higher, the set of functions
is
higher
still,
first,
the set of real
and there we must
stop.
Cantors views were not accepted by
all
his colleagues.
In particular
Leopold Kronecker, who had been one of Cantor's teachers, attacked Cantor's work with great vigor. Inspired by professional jealousy, Kronecker prevented Cantor's advancement, keeping him from a post at the University of Berlin. Cantor's mental health broke in 1884 under the strains of the controversy,
ony, on January
Now we
6,
and he died
in a
mental hospital in Halle, Sax
1918.
have a variety of endlessness that
is
different
and more
in
tensely endless than the endlessness represented bv "ordinary infinity."
This new and more intense endlessness also has
For instance, the points with the points in a long a solid. In fact, let's
are as
many
in a short line line, or
its
peculiar arithmetic.
can be matched up oneforone
the points in a plane, or the points in
not prolong the agony, and say at once that there
points in a line a millionth of an inch long as there are points
in all of space.
About 1895 tne German mathematician Georg Cantor worked out the
The Granger
Collection
arithmetic of infinity and also set up a whole series of different varieties of endlessnesses,
He the
which he called
first letter
The
of the
numbers."
Hebrew alphabet and which
looks like this:
is
X
various transfinities can be listed in increasing size or, rather, in
increasing intensity
ning with zero. there
"transfinite
represented these transfinite numbers by the letter aleph, which
of endlessness bv giving each one a subscript, begin
The
very lowest transfinite
would be "alephone," "alephtwo," and
This could be svmbolized
as: Xn»
X,>
X
would be "alephnull," then so on, endlessly.
X
NUMBERS AND COUNTING
70
number
Generally, whatever you do to a particular transfinite
way
A
of adding, subtracting, multiplying, or dividing, leaves
change comes only when you
equal to
raise a transfinite to a transfinite
(not to a transfinite power
itself
in the
unchanged.
it
less
than
power
Then
itself).
it
is
increased to the next higher transfinite. Thus:
Hq
What we
=X,; XT
=X
and
?;
so on.
usually consider as infinity, the endlessness of the integers, has
been shown to be equal to alephnull. In other words:
°°=X
And

so
the tremendous vastness of ordinary infinity turns out to be the very smallest of
That
the transfinites.
all
which we have symbolized
variety of endlessness
represented by alephone so that
No
C=X
,
but
this
be
1
mathematician has yet been able to prove that there
series
C may
as
has not been proved. is
any
infinite
which has an endlessness more intense than the endlessness of the
integers but less intense than the endlessness of the points in a line.
How
any mathematician been able# to prove that such an
inter
ever, neither has
mediate endlessness does not exist.** If
the continuum
is
equal to alephone, then
we can
finally write
equation for our friend "ordinary infinity" which will change Finally,
it
has been shown that the endlessness of
be drawn on a plane in a line. In
is
all
the curves that can
even more intense than the endlessness of points
other words, there
is
no way of
up the curves so that
lining
thev can be matched onetoone with the points in a
out an unending
an
it:
series of the curves.
line,
without leaving
This endlessness of curves
may be
equal to alephtwo, but that hasn't been proved yet, either.
And
that
is
all.
Assuming that the endlessness of
and the endlessness of points alephtwo,
we have come
variety of endlessness
to
integers
is
alephnull,
alephone, and the endlessness of curves
is
the end.
Nobodv
which could correspond
is
has ever suggested any
to alephthree (let alone to
alephthirty or alephthreemillion).
As John E. Freund says in his book A Modern Introduction to Mathematics^ (a book I recommend to all who found this article in the least interesting), "It seems that our imagination does not permit us to count
beyond three when dealing with Still,
we
if
we now
infinite sets."
return to the
title
Invaders from the Infinite,
are entitled to ask, with an air of phlegmatic calm,
Just alephnull?
"Which
New
think
Nothing more?"
[** Since this article first appeared, it has been shown that the statement one can be neither proved nor disproved by any method.] f
I
infinite?
York: PrenticeHall, 1956.
C— Aleph
PART
II
Numbers and Mathematics
_,...
A
6.
my
In
"Those Crazy
essay
OF
PIECE
Ideas,"
which appeared
PI
my book
in
Fancy (Doubleday, 1962), I casually threw in a footnote to the 1^= 1. Behold, a good proportion of the comment which
—
thereafter dealt not with
more
reader,
the essay
sorrow than
in
itself
Fact and effect that I
received
but with that footnote (one
proved the equality, which
in anger,
I
had
neglected to do).
My Since
conclusion
am, too
I
the impulse in this
is
is
that
(albeit
I
some
am
irresistible to
readers are interested in these
not
really a
first
place,
what
else),
pick up one of them, say v, and talk about
chapter and the next. In Chapter
In the
odd symbols.
mathematician, or anything
is tt?
Well,
8, I will
it is
discuss
the Greek letter pi and
it
repre
sents the ratio of the length of the perimeter of a circle to the length of
diameter. Perimeter
is
it
i.
its
from the Greek perimetron, meaning "the measure
ment around," and diameter from the Greek diametron, meaning "the measurement through." For some obscure reason, while it is customary to use perimeter in the case of polygons,
Latin circumference in speaking of
it is
circles.
also
customary to switch to the
That
is
all right,
I
suppose
(I
am no
purist), but it obscures the reason for the symbol ?r. Back about 1600 the English mathematician William Oughtred, in cussing the ratio of a circle's perimeter to its diameter, used the Greek ter
tt
to symbolize the perimeter
ize the
diameter.
They were the
and the Greek
first letters,
letter 8 (delta) to
dislet
symbol
respectively, of perimetron
and
diametron.
Now
mathematicians often simplify matters by setting values equal to
unih whenever they can. For instance, they might
talk of a circle of unit
diameter. In such a circle, the length of the perimeter to the ratio of perimeter to diameter.
(This
is
is
numerically equal
obvious to some of you,
I
)
NUMBERS AND MATHEMATICS
74
suppose, and the rest of you can take
my word
for
Since in a circle of
it. )
unit diameter the perimeter equals the ratio, the ratio can be symbolized
by
7T,
the symbol of the perimeter.
And
since circles of unit diameter are
very frequently dealt with, the habit becomes quickly ingrained.
The
first
topflight
man
to use
symbol
as the
k
of a circle's perimeter to the length of
for the ratio of the length
diameter was the Swiss mathe
its
matician Leonhard Euler, in 1737, and what was good enough for Euler
was good enough
Now
for everyone else.
can go back to calling the distance around a circle the circumfer
I
ence.
But what
is
the ratio of the circumference of a circle to
its
diameter
in
actual numbers?
This apparently
is
concerned the ancients even
a question that always
long before pure mathematics was invented. In any kind of construction past the hencoop stage you
ments,
if
must calculate
nut, these
beams
are
all
advance
all sorts
half a foot too short." In order to
urements, the universe being what value of
in
of measure
you are not perpetually to be calling out to some underling, "You
7T
in multiplications.
Presumably, the
first
the meas
Even when you're not dealing with
but only with angles (and you can't avoid angles) you empirical calculators
was important determined the
make
you are forever having to use the
it is,
ratio
who
by drawing
will
is
a circle
handled by the usual wooden
which
footrule,
a tricky is
circles,
into
it.
realized that the ratio
and actually meas
uring the length of the diameter and the circumference. uring the length of the circumference
bump
Of
course, meas
problem that can't be
far too inflexible for the
purpose.
What
the pyramidbuilders and their predecessors probably did was to
lay a linen cord along the circumference very carefully,
at the point line
make
a little
and measure
it
wooden footrule. (Modem and make haughty remarks such
with the equivalent of
theoretical mathematicians frown at this
a
"But you are making the unwarranted assumption that the
as
same length when est
workman
mark
where the circumference was completed, then straighten the
it is
straight as
when
it
was curved."
I
line
is
the
imagine the hon
organizing the construction of the local temple, faced with
such an objection, would have solved matters by throwing the objector into the river Nile.
Anyway, by drawing urements,
it
in the
game, that the
words,
if
one
circles of different size
and making enough meas
undoubtedly dawned upon architects and
circle
had
ratio
was always the same
a diameter twice as long or
artisans, very early
in all circles. In other
1%
as
long as the diam
PIECE OF
A eter of a second,
would
it
The problem
as long.
also
have
75
1%
circumference twice as long or
a
boiled down, then, to finding not the ratio of the
you were interested
particular circle
PI
in using,
would hold
for all circles for all time.
in his head,
he would never have
but
a universal ratio that
Once someone had
the value of *
determine the ratio again for any
to
cir
cle.
As
determined by measurement, that
to the actual value of the ratio, as
depended, in ancient times, on the care taken by the person making the
measurement and on the value he placed on accuracy
much
ancient Hebrews, for instance, were not
and when the time came
engineers,
building
(
would use round fractions,
figures only, seeing
and refusing
when the House
of
way
no point
The
of construction
one important
to build their
Phoenician architect.
call in a
be expected, then, that the Hebrews
to
It is
them
for
Solomon's temple), thev had to
in the abstract.
in the
in describing the
in stupid
temple
and troublesome
be bothered with such petty and niggling matters
to
God was
in question.
Thus, in Chapter 4 of 2 Chronicles, they describe a "molten sea" which was included in the temple and which was, presumably, some sort of container in circular form. verse of that chapter
The beginning
and
from brim to brim, round
and
circle
thirtv cubits
and
equal to exactly
There literal
of
7T
is
They
as well.
in so
compass
it
is
in the
second
molten sea of ten cubits
a
the height thereof;
five cubits
round about."
did not realize that in giving the diameter of a
see,
(as ten cubits or as
cumference
compass, and
in
a line of thirty cubits did
The Hebrews, you
of the description
he made
reads: "Also
anything else) they automatically gave the felt it
cir
necessary to specify the circumference as
doing revealed the fact that they considered  to be
3.
some
always the danger that
may
words of the Bible,
in consequence.
I
wonder
consider if
Fortunately, the
bill
?r
wedded
to the
be the divinely ordained value
mav not have been the motive of who some years back, introduced a
this
simple soul in some state legislature
which would have made
individuals, too
3 to
legallv equal to 3 inside the
did not pass or
bounds of the
the bill
state.
the wheels in that state (which
all
would, of course, have respected the laws of the
august legislators)
state's
would have turned hexagonal. In anv case, those ancients
who were
architecturally sophisticated
well,
from their measurements, that the value of
than
3.
The
really isn't
best value they
bad and
Decimally,
 2 7
is still is
had was
22 7
(or 3 1
was
orld.
One
of them, Ptolemy, established a dynasty
that was to rule over Egypt for three centuries.
He
converted his capital
at Alexandria into the greatest intellectual center of ancient times,
and
one of the first luminaries to work there was the mathematician Euclid. Very little is known about Euclid's personal life. He was born about 325 b.c, we dont know where, and the time and place unknown.
His name
of his death are
indissolubly linked to geometry, for he wrote a textbook
is
(Elements) on the subject that has been standard, with some modifica
went through more than
tions, of course, ever since. It
tions after the invention of printing,
and he
a thousand edi
undoubtedly the most suc
is
cessful textbook writer of all time.
And search. did,
yet, as a
Few
mathematician, Euclid's fame
and what made him
mathematics to
in
is
his
was to take
great,
time and codify
he evolved, as a starting point, a
all
own
not due to his
of the theorems in his textbook are his own.
What
re
Euclid
the knowledge accumulated
it
into a single work. In doing so,
series of
axioms and postulates that were
admirable for their brevity and elegance.
In addition to geometry, his text took up ratio and proportion and what is
now known
etry, too,
One
as the theory of
by dealing with
light rays as
He made
optics a part of
though they were straight
geom
lines.
about him involves King Ptolemy who was studying
story told
geometry and who asked easier to follow.
tle
numbers.
if
Euclid couldnt make his demonstrations a
Euclid
said,
uncompromisingly, "There
is
lit
no royal
road to geometry."
on mud, on blackboard, or on paper, using pen. (Beauty must be swathed in drapery
gainlv approximations in wax, a in
pointed
stick, chalk, pencil, or
mathematics,
alas, as in life.)
Furthermore, in order to prove some of the ineffably beautiful properties of various geometrical figures, lines
line
it
was usually necessary to make use of more
than existed in the figure alone.
through a point and make
second
line. It
double the
it
might be necessary
size of
an angle.
It
might be necessary
to
draw
a
new
parallel or, perhaps, perpendicular to a
to divide a line into equal parts, or to
The Granger
To make
all
this
ments must be used.
drawing
as neat
and
It follows naturally,
Collection
as accurate as possible, instru
think, once
I
you get into the
Greek way of thinking, that the fewer and simpler the instruments used for the purpose, the closer the
approach to the
ideal.
Eventually, the tools were reduced to an elegant is
a straightedge for the
drawing
you, with inches or centimeters of
wood
of straight lines.
marked
off
on
(or metal or plastic, for that matter)
it.
minimum
This It is
is
not
of two. a ruler,
One mind
an unmarked piece
which can do no more than
guide the marking instrument into the form of a straight
line.
NUMBERS AND MATHEMATICS
88
The second
tool
the compass, which, while most simply used to draw
is
circles, will also serve to
secting arcs that
and
mark
mark
off
equal segments of lines, will draw inter
a point that
is
equidistant from two other points,
so on.
presume most of you have taken plane geometrv and have
I
utilized
these tools to construct one line perpendicular to another, to bisect an angle, to
an
circumscribe a circle about a triangle, and so on. All these tasks and
infinite
number
and compass
By
of others can
be performed by using the straightedge
in a finite series of manipulations.
Plato's time, of course,
it
was known that bv using more complex
could be simplified; and,
tools, certain constructions
some con
in fact, that
structions could be performed which, until then, could not be performed
by straightedge and compass alone. That, to the Greek geometers, was
something
shooting a fox or a sitting duck, or catching
like
worms, or looking but
it
answers in the back of the book.
at the
just wasn't the
gentlemanlv thing to do.
The
It
with
fish
got results
straightedge and com
pass were the only "proper" tools of the geometrical trade.
Nor was
that this restriction to the compass and straightedge un
it felt
duly limited the geometer. tools of the trade;
it
might be tedious at times
It
might be
to stick to the
by using other de
easier to take a short cut
but surely the straightedge and compass alone could do
vices;
it all, if
you
were only persistent enough and ingenious enough.
For instance, to represent the
if
vou are given
numeral
and straightedge alone, exactly twice that length
pass
another line to represent 2 7 1 %3 In fact,
ber
a line of a fixed length
(i.e.,
allowed
5
or 500 or
H
or
V3
or
com
to represent 2, or
y5
or
%
or
2%
or
any integer or fraction) could be duplicated geometrically. You
make make it
to
or
is
bv using compass and straightedge onlv, anv rational num
could even alas)
3
which
possible to construct another line, by
1, it is
use of a simple convention (which the Greeks never did, possible to represent both positive
and negative
rational
which no
definite
numbers.
Once
irrational
numbers were discovered, numbers
fraction could be written,
would
fail,
it
for
might seem that compass and straightedge
but even then thev did not.
For instance, the square root of
2
has the value 1.414214
and on without end. How, then, can you construct one 1.414214
know
.
.
.
exactly
Actually,
times as long as another
how manv
it's
easy.
when you cannot
times as long vou want
Imagine
a given line
it
.
.
line
.
and on
which
is
possibly ever
to be.
from point
A
to point B. (I can
TOOLS OF THE TRADE do
without
this
a
diagram,
the lines as you read.
think, but
I
you
if
won't be hard.) Let
It
feel
89 the need you can sketch
this line,
AB, represent
Now
Next, construct a line at B, perpendicular to AB. lines
forming a right angle. Use the compass to draw
where the two
ter at B,
lines
of the wellknown properties of the circle, line
AB, and
is
Finally,
That
also
we can
A and C
as
can be proven by geometry,
AB
or
BC, and
to obtain
represented bv
line.
exactly
is
it is
now
AC
that
infinite
\/T times
only necessary to measure
in the
hands of imperfect
men and
is
\/T,
and not
AC
itself in
It
AC
number
is
all
is
only a
and compass
and compass alone
constructions boiled
ing certain numbers, tool could tails
to
of other irrational quantities.
num
could be represented by a line that could be constructed by use
of straightedge all
in
the ideal line
In fact, the Greeks had no reason to doubt that any conceivable
ber at
as
actual reality.
possible, in similar fashion, to use the straightedge
represent an
the
C. Because
exactly equal to line
crude approximation of the ideal figures they represent.
is
call
an exact value of y/T. The construction was drawn
by imperfect instruments
It
cen
its
It will cut
therefore represents the irrational quantity
Don't, of course, think, that
AB
is
with a third straight
AC,
long as either
terms of
BC
with
1.
connect points
line,
a circle
meet, and passing through A.
perpendicular line you have just drawn at a point
L.
you have two
it
down was
in a finite
number
of steps.
And
since
to the construction of certain lines represent
that anything that could be done with any
felt
be done bv straightedge and compass alone. Sometimes the de
of the straightedge
and compass construction might be elusive and felt, given enough ingenu
remain undiscovered, but eventually, the Greeks ity,
insight,
worked
intelligence, intuition,
and
luck,
the construction could be
out.
For instance, the Greeks never learned teen equal parts bv straightedge
The method was not
how
and compass
to divide a circle into seven
alone.
Yet
it
could be done.
discovered until 1801, but in that year, the
German
mathematician Karl Friedrich Gauss, then only twentyfour, managed it. Once he divided the circle into seventeen parts, he could connect the points of division by a straightedge to form a regular polygon of seventeen sides (a "septendecagon"). The same system could be used to construct a regular polygon of 257 sides,
with
still
more
formula which
sides, the I
and an
number
won't give here.
infinite
number
of other polygons
of sides possible being calculated by a
KARL FRIEDRICH GAUSS Gauss, the son of a gardener, was born in Braunschweig, Germany, on
He was an infant prodigy life. He was capable of
April 30, iyyy.
a prodigy
all
his
in
mathematics who remained
great feats of
memory and
of
mental calculation. At the age of three, he was already correcting his father s sums. His unusual mind was recognized and he was educated at the expense of
Duke Ferdinand
of Brunswick. In ijg^
Gauss entered the
University of Gbttingen.
While
still
including the
in his teens he
"method
made
a
number
of remarkable discoveries,
of least squares," which could determine the best
curve fitting a group of as few as three observations. university,
While
still
in the
he demonstrated a method for constructing an equilateral
polygon of seventeen sides and, more important, showed which polygons
would not be
so constructed— the
first
demonstration of a mathematical
impossibility.
In 1799 Gauss proved the fundamental theorem of algebra, that every algebraic equation has a root in the form of a complex number,
and
in
1801 he went on to prove the fundamental theorem of arithmetic, that
number can be represented and only one way. every natural
as the product of primes in
All this required intense concentration.
was told
in
There
is
a story that 1
one
when he
i8oj that his wife was dying, he looked up from the problem
that was engaging
him and muttered, "Tell her
to wait a
moment
till
Ym
throughr His agile mind never seemed to cease. At the age of sixtytwo he taught himself Russian. Personal tragedy dogged him, though. Each of his two wives died young, and only one of his in
six
children survived him.
He
died
Gbttingen on February 23, 1855.
If
the construction of a simple thing like a regular septendecagon could
elude the great Greek geometers and yet be a perfectly soluble problem in the end, it
why
could not any conceivable construction, however puzzling
might seem, yet prove soluble
in
the end.
As an example, one construction that fascinated the Greeks was Given This
a circle, construct a square of the is
called "squaring the circle."
same
area.
this:
The Granger
There are
several
this. Here's one method. Measure the most accurate measuring device you have
ways of doing
radius of the circle with the say, just for fun, that
the radius proves to be one inch long precisely. (This
method
will
plicity.)
Square that radius, leaving the value
work
Collection
for a radius of
anv length, so why not luxuriate still
1,
since
1X1
is
1,
goodness, and multiply that by the best value of w you can find.
you wondering when of
tt,
I'd get
back to
tt?)
If
you use 3.1415926
as
the area of the circle proves to be 3.141 5926 square inches.
in sim
thank
(Were
your value
NUMBERS AND MATHEMATICS
92
Now,
take the square root of that, which
1.7724539 inches, and draw
is
a straight line exactly 1.7724539 inches long, using vour measuring device to
make
sure of the length. Construct a perpendicular at each end of the
line,
mark
two
points.
Voila!
off
1.7724539 inches on each perpendicular, and connect those
You have
may
feel uneasy.
ther
is
Your measuring device
isn't infinitely
the value of n which you used. Does not this
ing of the circle
Yes, but
it is
is
course,
you
accurate and nei
mean
that the squar
only approximate and not exact?
We
not the details that count but the principle.
sume the measuring
device to be perfect, and the value of
used to be accurate to an infinite number of places. After as justifiable as
Of
a square equal in area to the given circle.
assuming our actual drawn
n
all,
can
as
which was this
is
just
lines to represent ideal lines,
considering our straightedge perfectlv straight and our compass to end in
two perfect points. In
we have indeed
principle,
perfectly squared the
circle.
Ah, but we have made use of
a
measuring device, which
is
the only two tools of the trade allowed a gentleman geometer.
you
as a
not one of
That marks
cad and bounder and vou are hereby voted out of the club.
Here's another
method
of squaring the circle.
suming the radius of your presenting \/^.
A
What
1, is
you
really need, as
another straight line
re
square built on such a line would have just the area of
a unitradius circle. line equal to
circle to represent
How
to get such a line? Well,
you could construct
if
a
n times the length of the radius, there are known methods,
using straightedge and compass alone, to construct a line equal in length to the square root of that line,
hence representing the V?r, which we are
after.
But
it is
simple to get a line that
is n
times the radius. According to a
wellknown formula, the circumference of the twice the radius times line
the
and line.
let's
Now
make
tt.
So
a little
let us
mark
circle
imagine the
where the
at the point
slowly turn the circle so that
is
it
equal in length to
circle resting
on a straight
circle just
touches
moves along the line (with
out slipping) until the point you have marked makes a complete circuit
and once again touches the
Make
line.
another mark where
it
again
touches. Thus, you have marked off the circumference of the circle on a straight line
and the distance between the two marks
Bisect that markedoff line by the usual
compass geometry and you have
a
line
is
twice
methods of straightedge and
representing
tt.
square root of that line and vou have \Ar. Voila!
By
n.
that act, you have, in effect, squared the circle.
Construct the
TOOLS OF THE TRADE But no. I'm
afraid you're
still
mark on
out of the club.
93
You have made
use of a
and that comes under the heading of an instrument other than the straightedge and compass. rolling circle with a
The
point
it
that there are any
is
number
of ways of squaring the circle,
but the Greeks were unable to find any way of doing
and compass alone
in a finite
number
of steps.
how many manhours of time searching on
it
it,
might
search, they
seem an
all
came
and new theorems, which were circle
new more
across all sorts of far
with straightedge I
don't
know
method, and looking back
for a
exercise in futility
it
(They spent
now, but
it
wasn't. In their
curves, such as the conic sections,
valuable than the squaring of the
would have been.)
Although the Greeks
failed to find a
method, the search continued and
continued. People kept on trying and trying and trying and trying—
And now
change the subject
let's
for a while.
Consider a simple equation such as
x=Y2 zero.
2X— 1=0. You can for 2(Y2 — 1 is
make a true statement out of it, No other number can be substituted will
)
see that setting
indeed equal to
for x in this equation
and
yield
a true statement.
By changing
the integers in the equation (the "coefficients" as they are
called) x can
be made to equal other
3X— 4=0,
equal to
x
is
%; and
in
specific
numbers. For instance,
yx+2=o, x= — %.
in
In fact, by choosing
the coefficients appropriately, you can have as a value of x any positive or negative integer or fraction whatever.
But
in
such an "equation of the
first
degree," you can only obtain
tional values for x. You can't possibly have an B=o), where A and B are rational, such that x
V2,
equation of the form will turn
ra
Ax+
out to be equal to
for instance.
The
thing to do
pose you try x cause
it
2
is
to try a
— 2=0,
which
more complicated is
variety of equation. Sup
an "equation of the second degree" be
involves a square. If you solve for x you'll find the answer,
V2,
substituted for x will yield a true statement. In fact, there are two for x will also yield a true possible answers, for the substitution of
when
— V2"
statement.
You can
build
up equations
of the third degree, such as
Ax 3 fBx 2 f
CxfD=o, or of the fourth degree (I don't have to give any more examples, do I?), or higher. Solving for x in each case becomes more and more difficult, but will give solutions involving cube roots, fourth roots, and so on.
:
NUMBERS AND MATHEMATICS
94
In any equation of this type (a "polynomial equation") the value of x
To
can be worked out by manipulating the coefficients. case, in the general
— B/A.
is
equation of the
first
take the simplest
degree: AxB=o, the value of x
In the general equation of the second degree:
Ax 2 +Bx+C=o,
there are two solutions.
One
!
is
,
—
and the
Bx/B — — 4AC 2
.
.
other
is
2A
Solutions get progressivelv tions of the fifth degree
though
specific solutions can
however, that
by use of a
in all
finite
more complicated and
eventuallv, for equa
and higher, no general solution can be be worked out.
still
The
given,
al
principle remains,
polynomial equations, the value of x can be expressed
number
of integers involved in a finite
number
of opera
tions, these operations consisting of addition, subtraction, multiplication,
power ("involution"), and extracting
division, raising to a
roots ("evolu
tion").
These operations are the onlv ones used therefore called "algebraic operations."
from the integers bv bination
Xow
it
number
called an "algebraic
is
number
braic
a finite
is
in ordinary algebra
Any number which it
in reverse,
any
alge
some polynomial equation.
happens that the geometric equivalent of
so
are
com
of algebraic operations in any
number." To put
a possible solution for
and
can be derived
all
the algebraic
operations, except the extraction of roots higher than the square root, can
be performed bv straightedge and compass alone. 1,
therefore,
it
straightedge and compass in a finite k
sible that
be
bers
number
number
that
of manipulations.
does not seem to contain any cube roots (or worse), it
is
it
pos
can be constructed bv straightedge and compass? That might
algebraic
if
given line represents
no root higher than the square root can be constructed by
involves
Since
If a
follows that a line representing any algebraic
numbers included
which cannot be solutions
all
to
numbers. But do they? Are there num
anv polynomial equation, and are there
fore not algebraic?
To
begin with,
tions of the
first
all
possible rational
degree, so
Then, certainly some
all
irrational
numbers
easv to write equations for which y/z or
But can there be a single
each of
one of the all
irrational
numbers can be solutions
rational
numbers
to equa
are algebraic numbers.
are algebraic numbers, for
^15—3
numbers which
will
not serve
number of different polynomial equations number of degrees possible?
In 1844 the French mathematician Joseph Liouville finally found a of
showing that such nonalgebraic numbers did
is
as a solution to
infinite
the infinite
it
are solutions.
exist.
(No,
I
don't
in
way
know
)
TOOLS OF THE TRADE how he
did
it,
but
if
must warn him not
any reader thinks
to overestimate
95
can understand the method, and
I
me, he
welcome
is
to send
However, having proved that nonalgebraic numbers could
still
that a for
The
not find a specific example.
number
nearest he
existed, Liouville
came was
to
show
represented by the symbol e could not serve as the root
any conceivable equation of the second degree.
(At
point
this
e because, as
equation
I
am tempted
I
^=—1.
But
1873, the French mathematician Charles
showed that
about e
of analysis that
Chapter
in
ber,"
number.
was, in fact,
It
one which transcends (that
is
(That
yT
is,
called a "transcendental
num
goes beyond) the algebraic operations
irrational
is
Hermite worked out
and hence was actually not an
and cannot be produced from the integers by any operations.
had
not be the root of any con
e could
what
is,
I
3.
ceivable equation of any conceivable degree
algebraic
the famous
is
temptation because, for one thing,
resist
I'll
method
to say
number
to launch into a discussion of the
said at the start of the previous chapter, there
some things Then, in a
I
in.)
it
finite
number
of those
but can be produced by a single
algebraic operation, taking the square root of
The
2.
value of
e,
on the
other hand, can only be calculated by the use of infinite series involving
an infinite number of additions, divisions, subtractions, and so on.)
Using the methods developed bv Hermite, the German mathematician Ferdinand Lindemann
in
1882 proved that
ir f
too,
was
a transcendental
number. This
is
crucial for the purposes of this chapter, for
segment equivalent
to
it
meant
that a line
n cannot be built up by the use of the straightedge
number of manipulations. The circle cannot be squared by straightedge and compass alone. It is as impossible to do this as to find an exact value for \/i, or to find an odd number that is an
and compass alone
in a finite
exact multiple of 4.
One odd
point about transcendental
They were
difficult to find,
numbers—
but now that they have been, they prove to
be present in overwhelming numbers. Practically any expression that
in
not
ar
volves either e or
tt
transcendental, provided the expression
is
ranged so that the e or
tt
cancel out. Practically
(which involve
logarithms
e)
and
trigonometric functions (which involve involving
numbers
raised to
all
practically all tt)
is
expressions involving expressions involving
are transcendental. Expressions
an irrational power, such
as
x^,
are tran
scendental. In fact,
if
you
refer
back to Chapter
5,
you
will
understand
me when
I
NUMBERS AND MATHEMATICS
96 say that
it
has been proved that the algebraic numbers can be put into
onetoone correspondence with the integers, but the transcendental
num
bers can not.
This means that the algebraic numbers, although lowest of the transfinite numbers,
>$
,
infinite,
while the transcendental numbers
^ r There
belong, at the least, to the next higher transfinite, infinitely
To be lished
more transcendental numbers than
squarers,
who
if
alone,
And
I
it's
?r
is
now
well estab
for nearly a century doesn't stop the ardent circle
continue to work away desperately with straightedge and
compass and continue So
are thus
there are algebraic numbers.
sure, the fact that the transcendentality of
and has been
belong to the
you know
a
to report solutions regularly.
way
to square the circle
bv straightedge and compass
congratulate you, but you have a fallacy in your proof somewhere.
no use sending
it
to
me, because I'm a rotten mathematician and
couldn't possibly find the fallacy, but
I tell
you anyway,
it's
there.
THE IMAGINARY THAT
8.
ISN'T
WTien
I
was
whom
I
ate lunch every day. His 11 a.m. class
a
mere
slip of a lad
and
absolutely refused to take, steadfastly refused to
and attended
my
college,
was
11 a.m. class
take— so we had
had
I
a friend with
in sociology,
which
I
was calculus, which he
as
and meet
at
to separate at eleven
twelve.
As
it
happened,
his sociologv professor
was
a scholar
who did things in The more eager
the grand manner, holding court after class was over.
students gathered close and listened to fifteen
him
pontificate for an additional
minutes, while they threw in an occasional log in the form of a
question to feed the flame of oracle.
Consequently, when
mv
calculus lecture was over,
I
had
to enter the
room and wait patientlv for court to conclude. Once I walked in when the professor was listing on the board his classification of mankind into the two groups of mystics and realists, and under
sociology
mystics he
had included the mathematicians along with the poets and
theologians.
One
student wanted to
know why.
"Mathematicians," said the professor, "are mystics because they believe in
numbers that have no
Now
reality."
nonmember boredom, but now
ordinarily, as a
suffered in silent
of the class, I
I
sat in the corner
rose convulsively,
and
said,
and
"What
numbers?"
The
professor looked in
minus one. believe
it
It
my
direction
said,
"The square
root of
imaginary. But they
call it
has some kind of existence in a mystical way."
"There's nothing mystical about
minus one
The
and
has no existence. Mathematicians
is
just as real as
it," I said, angrily.
"The square
root of
any other number."
professor smiled, feeling he
had
a live
one on
whom
he could now
— NUMBERS AND MATHEMATICS
98
proceed to display his superiority of intellect
my own and
know
I
(I
how he felt). He who wants to prove
exactly
young mathematician here
have since had said, silkily,
have
a
the reality of the square
Come, young man, hand me
root of minus one.
classes of
"We
the square root of minus
one pieces of chalk!" I
reddened, "Well, now, wait—"
"That's
he
all,"
plished, both neatly
But of
raised
I
my
said,
waving
and
sweetly.
voice. "I'll
minus one pieces of chalk,
The
do if
hand. Mission, he imagined, accom
his
it.
I'll
do
you hand
I'll
it.
me
hand you the square root
a onehalf piece of chalk."
professor smiled again, and said, "Very well," broke a fresh piece
of chalk in half,
and handed
me one
"Now
of the halves.
end of
for your
the bargain."
"Ah, but wait," of chalk you've
others to see.
"you haven't
said,
"Wouldn't you
piece of regulation length.
if
"Now
said,
I
accept
it,
You have one
one piece
up
for the It cer
piece of chalk
are
is
a
But even
vou're springing an arbitrary definition on me.
you willing to maintain that
this
And
a onehalf piece of
is
can you really consider
minus one, when you're
a
hazy on the meaning of one half?"
But by now the professor had final
One
it.
yourself qualified to discuss the square root of
left
is
it
that's half the regulation length."
chalk and not a 0.48 piece or a 0.52 piece?
little
held
I
was one piece of chalk?
say this
all
the professor wasn't smiling. "Hold
I
vour end. This
fulfilled
a onehalf piece."
two or three."
tainly isn't
Now
I
handed me, not
argument was unanswerable.
He
said,
(laughing) and thereafter waited for
Twenty
equanimity altogether and
lost his
years have passed since then
my
and
"Get the
hell
his
out of here!"
I
friend in the corridor. I
suppose
I
ought to
finish the
argument
The
Let's start with a simple algebraic equation such as x 43=5.
pression,
x,
represents
some number which, when
substituted for
x,
the expression a true equality. In this particular case x must equal
2+3=5, an d so vve h ave The interesting thing There
is
This
no number but
is
2,
since
''solved for x."
about 2
this solution
which
will give 5
it
polynomial equation of the
first
is
that
when
true of any question of this sort,
tion" (because in geometry
first
ex
makes
it is
which
the only solution.
added
3 is is
to
it.
called a "linear equa
can be represented as a straight line) or "a degree."
No
polynomial equation of the
degree can ever have more than one solution for
x.
THE IMAGINARY THAT
I
S
N T
99
There are other equations, however, which can have more than one an example: x 2 5x46=0, where x 2 ("x square" or "x
—
solution. Here's
squared") represents x times
x.
This
Latin word for "square," because
a
called a "quadratic equation,"
is
it
involves x square. It
polvnomial equation of the second degree" because of the for x itself, that could
be written x
and taken
and that
for granted,
is
1 ,
except that the
why xj~3=5
is
1
from
also called "a
is
2
little
in x 2
As
.
alwavs omitted
is
an equation of the
first
degree.
is
take the equation x 2
If
we
4,
while 5X
However,
10, so that
is
making
correct,
and substitute
the equation becomes
2 for x, then x 2
4—1046=0, which
is
2 a solution of the equation.
we
if
— 5x46=0,
substitute
for x, then x 2
3
9—1546=0, which
equation becomes
is
is
9 and 5X
also correct,
is
15, so that
making
3 a
the
second
solution of the equation.
Now
no equation
more than two third degree?
of the second degree has ever been
solutions,
These are equations containing x 3 ("x cube" or "x cubed"),
which are therefore
also called "cubic equations."
resents x times x times
The equation substitute in
1, 2,
each case.
found which has
but what about polynomial equations of the
x3
or
No
The
expression x 3 rep
x.
— 6x
3 for
2
4nx— 6=0
has three solutions, since you can
come up with
x in this equation and
a true equality
cubic equation has ever been found with more than three
solutions, however.
In the
of the fourth degree can be
same way polynomial equations
constructed which have four solutions but no more; polynomial equations of the fifth degree,
might as
which have
sav, then, that a
manv
as
five solutions
but no more; and so on. You
polvnomial equation of the nth degree can have
n solutions, but never more than
Mathematicians craved something even
n.
prettier than that
and by about
1800 found it. At that time, the German mathematician Karl Friedrich Gauss showed that even equation of the nth degree had exactly n solutions,
not onlv no more, but also no
However, of
in order to
what constitutes
make
less.
the fundamental theorem true, our notion
a solution to
an algebraic equation must be
drastically
enlarged.
To so on.
begin with,
This
is
units generally. 2
1
,
children,
men
accept the "natural numbers" only:
1,
2,
3,
and
adequate for counting objects that are only considered as You can have 2 children, 5 cows, or 8 pots; while to have
5%
cows, or
%%
pots does not
make much
sense.
NUMBERS AND MATHEMATICS
lOO
In measuring continuous quantities such as lengths or weights, however, fractions
The
essential.
them sneered like
$y
2,
Such
wood
is
Egyptians and Babylonians managed
though these were not very
fractions,
by our own standards; and no doubt conservative scholars among
efficient
is
became
work out methods of handling
to
at the mystical mathematicians
which was neither
5
nor
believed in a
whole numbers. To say
fractions are really ratios of
2%
who
number
6.
yards long, for instance,
plank of
a
to say that the length of the plank
is
to the length of a standard yardstick as 21
is
to
The
8.
Greeks, however,
discovered that there were definite quantities which could not be expressed as ratios of
of
2,
whole numbers. The
commonly
multiplied by
expressed as
itself,
gives
pressed as a ratio; hence,
Only thus
2.
it is
far did the
to
first
\f?.,
There
be discovered was the square root
which is
is
that
number which, when
such a number but
less
than nothing?
How
cannot be
ex
an "irrational number."
notion of number extend before modern times.
How can there equation x+5=3 had
Thus, the Greeks accepted no number smaller than
be
it
To them,
consequently, the
zero.
any number and have 3 as a result? number (that is, to zero), you would have 5 as the sum, and if you added 5 to any other number (which would have to be larger than zero), you would have a sum greater than 5. The first mathematician to break this taboo and make systematic use
no
solution.
Even
of
if
can you add
you added
numbers
less
than zero was the
less
you own
than nothing.
dollars of your
to
5 to the smallest
there can be less than nothing. If all
5
in the If
A
world
Italian,
debt
is
own (assuming you
you end with three
are an honorable
x+5=3,
number
less
all,
you have two dollars
given five dollars,
debts). Consequently, in the equation a
Girolamo Cardano. After than nothing.
a twodollar debt,
is
you are then
where the minus sign indicates
less
man who
pays his
x can be set equal to
—2,
than zero.
Such numbers are called "negative numbers," from a Latin word meaning "to deny," so that the very
name carries the traces of the Greek denial Numbers greater than zero are "positive
of the existence of such numbers.
numbers" and these can be written +1, +2, +3, and
From
a practical standpoint, extending the
negative numbers simplifies
all
sorts
so on.
number system by
including
of computations; as, for example,
those in bookkeeping.
From
a theoretical standpoint, the use of negative
every equation of the less.
first
numbers means that
degree has exactly one solution.
No
more; no
3
THE IMAGINARY THAT ISNT If
we
pass on to equations of the second degree,
would agree with and
us that the equation x
They would
3.
one solution,
1.
for x
1
1—4—5 = 3. No
and xa
— qx— 5=3
while 4*
1,
is
number
other
has two solutions,
x2
is
C
Greeks
find that the
— 5X— 6=c
however, that the equation
say,
Substitute
equation becomes
we
1
1
2
has onlv
so that the
4,
will serve as a solution,
long as you restrict yourself to positive numbers.
as
However, the number are
worked out
—5
is
a solution,
if
we
consider a few rules that
connection with the multiplication of negative numbers.
in
In order to achieve consistent results, mathematicians have decided that
the multiplication of a negative
number by
number yields number by
positive
a
negative product, while the multiplication of a negative
number
negative
—
comes
— 4X— 5=0, — — or — 25, while
times
5
that there are two solutions to this equation,
Sometimes,
only
number
the
if
—
3
solution of the equation
happen
— 6x— o,=c,
show that there
(x— 31=0 and each (x— 3) equation
are, therefore,
Allowing
which
substituted for
is
be identical. Thus, x
to
—1
times
We
true.
is
x.
then x 2 be
—
or
5,
^3
2
and —5. be
will
a single
a true equalitv
if
and
However, the mechanics of
x.
are actually
— 6x— 9=0
—20.
would saw then,
equation does indeed seem to have but
a quadratic
example, x 2
—4
becomes
4.x
5,
substituted for
is
5
The equation becomes 25—20—5=0, which
root, as, for
a
yields a positive product.
the equation x 2
in
If,
a
two
solutions,
can be converted to
yields a solution.
The two
which
(x—
solutions of this
and +3.
for occasional duplicate solutions, are
we readv
to say
then that
seconddegree equations can be shown to have exactly two solutions if negative numbers are included in the number system!' 2 Alas, no! For what about the equation x 2 — i=c. To begin with, x must be —1, since substituting —1 for x 2 makes the equation — 1 — i=c, which all
is
correct enough.
But ~i
myself.
tive
The
To
must be the famous square root
x
First,
it.
give
—1. But
quantities,
—1
times
allow anv solution at it is
its
is
all
—1;
that
is
is
is
new kind
of
it
—1
times
equation x
is
2
—1
— = i
If
c.
is
—1.
let
alone two
no positive number
absolutely essential to define a
number; an imaginary number,
square equal to —1.
multi
in the set of posi
the reason the sociology professor
secondlv,
for the
either,
minus one
number which when
no such number
necessary to get past this roadblock.
do and no negative one
completely
there
and that
+1
of
between the sociologv professor and
setto
square root of minus one
itself will
solutions,
with
—1, then
and negative
scorned
will
is
which occasioned the
),
bv
plied
x2
if
(\ /=
if
vou
like;
one
NUMBERS AND MATHEMATICS
102
We The
could,
we
if
and the minus
new number and
could use an asterisk for the
—
times *i was equal to
However,
new kind
wished, give the
plus sign does for positives
of
number
say that *i
(for "imaginary")
i
was introduced by the Swiss mathematician Leonhard Euler was thereafter generally adopted. So we can write i=\/
Having defined
i
in this fashion,
negative number. For instance, or
we
—
1
or
V4
we can
In this way,
series of
we would have
ii, 31, 41.
V —n=\/n
.
.
1.
V — n,

*>
can be
i.
ordinarv or "real numbers." For .
=—
number times the
picture a whole series of imaginarv
analogous to the i,
is,
1777 and
in 2
can De written \/4 times x/
written as the square root of the equivalent positive
square root of minus one; that
i
can express the square root of any
In general, any square root of a negative number,
ii.
we
one")
("star
1.
was not done. Instead, the symbol
this
a special sign.
sign for negatives; so
numbers exactly 1, 2,
3, 4,
%
This would include fractions, for
...
,
would
15i/ %\ 15/17 by 17 and so on. It would also include irrationals, for y/2 would be matched by \fii and even a number like w (pi) 2
be matched by
,
would be matched bv These are
What
about negative numbers? Well, why not negative imaginaries, too?
—
For —1, —2, So
iri.
comparisons of positive numbers with imaginary numbers.
all
3,
—4,
now we have
.
.
.
,
there would be
—
i,
—ii, —31, —41.
.
.
.
four classes of numbers: 1) positive real numbers, 2)
negative real numbers, 3) positive imaginary numbers, 4) negative imaginary numbers. (When a negative imaginarv is multiplied by a negative
imaginary, the product
Using
is
negative.)
further extension of the
this
number
system,
necessary two solutions for the equation x 2 +i=o. First
\i
times
\i
equals
—
and secondly —i times
1,
we can find the are +i and — i. — equals —1, so
They
i
—1+1=0,
that in either case, the equation becomes
which
is
a true
equality.
number system to find all four solutions for an equation such as x The solutions are fi, — 1, +£, and — To show this, we must remember that any number raised to the fourth power is equal to the square of that number multiplied by itself. That is, n 4 equals n2 times n 2 In fact, you can use the
same extension
of the
4
— 1=0.
i.
.
Now
let's
so that x 4
substitute each of the suggested solutions into the equations
becomes successively
First (f1)
that becomes
4
equals
+1
times
(
+ 1)
2
(
+ 1)
times
+1, which
is
4 ,
(
(
— 1)
+ 1)
+1.
2 ,
4 ,
(fi)
and
4 ,
and
since
(
(— i) 4
+ 1)
2
.
equals
+1,
HE
I
M AGINARY THAT
equals
(
— 1)
T
Second,
+1,
(
— 1)
4
the expression
is
as
—1,
(— z) 4
times
suggested
four
becomes times
solutions,
— 1=0, give the expression It
might seem
all
,
and since
(
— 1)
2
also equals
+1.
we have denned {+i) 2
—1 times — 1, (— z) 2 which ,
or
+1.
is
also
rules of
f 1
when
substituted
— 1=0, which
—1
times
—
1,
equation
the
into
correct.
is
numbers— for
very well to talk about imaginary
a
some defined quantity can be made subject to manipulation that do not contradict anything else in the mathe
mathematician. As long
as
matical system, the mathematician it
or
2
times (+i) 2 and
(— i) 2
equals
— 1)
+1,
times
2
(
10 3
fi.
All
x4
(fi)
so that the expression
Fourth, or
+1
again
Third, (+*) 4 equals
2
ISN T
is
happy.
He
doesn't really care what
"means."
my
Ordinary people do, though, and that's where
sociologist's charge
of mysticism against mathematicians arises.
And
yet
it
the easiest thing in the world to supply the socalled
is
"imaginary" numbers with a perfectly
real
and concrete
imagine a horizontal line crossed by a vertical line and
Now
intersection zero.
you have four
angles from that zero point.
You
significance. Just call
the point of
lines radiating out at
mutual right
can equate those lines with the four kinds
of numbers. If
is marked off at equal intervals, and so on for as numbered fi, +2, +3, +4, if we only make the line long enough. Between the
the line radiating out to the right
the marks can be
long as
we
wish,
markings are
shown that one positive
all
.
on such
number, and
and only one point on the
The
.
,
the fractions and irrational numbers. In fact,
to every point real
.
a line there corresponds
for every positive real
it
can be
one and only
number
there
is
one
line.
line radiating out to the left can
be similarly marked
with the
off
negative real numbers, so that the horizontal line can be considered the
"realnumber axis," including both positives and negatives. Similarly, the line radiating
upward can be marked
off
with the positive
imaginary numbers, and the one radiating downward with the negative imaginary numbers.
Suppose we bols,
The
vertical line
label the different
but by the directions
positive real
in
numbers can be
tion of extension
on
is
then the imaginarynumber
numbers not by the usual
which the
lines point.
The
called East because that
a conventional map.
The
signs
axis.
and sym
rightward line of
would be
its
direc
leftward line of negative
LEONHARD EULER Euler, the son of a Calvinist minister, was born in Basel, Switzerland,
April 15, 1707.
He
received his master
s
on
degree at the age of sixteen from
the University of Basel.
Euler went to
St.
Petersburg, Russia, in 1727, for there Catherine I
widow of Peter the Great) had recently founded the Petersburg Academy and Euler spent much of his life there. In 17^ he lost the sight of his right eye through tooardent observations of the Sun in an attempt (the
to
work out a system of time determination. In 1741 Euler went to Berlin to head and revivify the decaying Acad
emy
new
of Sciences but didn't get along with the
erick II.
tember
He
returned to
St.
Prussian King, Fred
Petersburg in 1766 and there he died on Sep
18, 1783.
Euler was the most prolific mathematician of
time, writing on every
all
branch of the subject and being always careful to describe his reasoning and to list the false paths he had followed. He lost the sight of his remaining eye in 1766 but that scarcely seemed to stop
down,
for
which would
some
of
him
or even slow
he had a phenomenal memory and could keep
them
fill
He
several blackboards.
quite long,
and
mind
him that
published eight hundred papers,
enough papers
at the time of his death left
behind to keep the printing presses busy
in
for thirtyfive years.
Euler published a tremendously successful popularization of science 1768, one that remained in print for ninety years.
in
working out certain mathematical problems
after
in
He
died shortly
connection with
ballooning, inspired by the successful flight of the Montgolfier brothers.
He
introduced the symbol "e" for the base of natural logarithms, "i" for
the square root of minus one,
real
and
"/(
" for functions.
)
numbers would be West; the upward
would be North; and the downward
line
of positive imaginaries
would
line of negative imaginaries
be South.
Now
if
we
agree that
on the compass
signs as
+1 I
times
+1
times East equals East. Again since
times
—
i
West
times
equals East. Then, since
—
times South.
i,
equals
+1, and
if
we
concentrate
have defined them, we are saying that East
—1 \i
—1
times
times
\i
then North times North equals
also equals f 1 *
equals
—
West and
1,
and
West
so does
so does South
LEOKAM© Zenim PubtUhtd
a* the
.let Obrmeor.
E [7UER. tWJ& * 18i> I ,b\
J.
it'Ukes
The Granger
Collection
NUMBERS AND MATHEMATICS
106
We
—
equals
when If we
times
1
West
times North equals South.
the possible combinations as compass points, abbreviating
all
initial letters,
we can
set
up the following system:
ExE=E
ExS=S
EXW=W
SXE=S
SxS=W
SXW=N
SxN=E
WxE=W
WxS=N
WxW=E
WxN=S
NxE=N
NxS=E
NXW=S
NxN=W
There of o°.
a very orderlv pattern here.
is
unchanged, so that East
left
is
On
ExN=N
Anv compass
point multiplied bv
as a multiplier represents a rotation
West
the other hand, any compass point multiplied by
is
ro
("about face"). North and South represent rightangle
tated through 180 turns.
which
ji,
(since positive times negative vields a negative product even
i
list
—
other combinations such as
imaginaries are involved), so that
those points by
East
make
can also
Multiplication bv South results in a go° clockwise turn
North
face"); while multiplication bv
results in a
("right
counterclockwise
90
turn ("left face").
Now
so
it
happens that an unchanging direction
ment, so East (the positive
real
numbers)
is
is
the simplest arrange
handle and more
easier to
West
comforting to the soul than any of the others.
(the negative real
numbers), which produces an about face but leaves one on the same at least,
is
less
line
comforting, but not too bad. North and South (the imagi
nary numbers), which send you off in a
new
direction altogether, are least
comfortable.
But viewed
compass points, vou can see that no
as
more "imaginary"
Now
consider
we
long as
number be true
that matter,
"real" than
numbers
only,
numbers
is
any other.
two number axes can be. As
we can move along the realThe same would
backward and forward, onedimensionally.
we used
onlv the imaginarynumber
Using both, we can define
number
more
useful the existence of
deal with the real
axis, if
or, for
how
set of
axis.
point as so far right or
a
left
on the
real
up or down on the imaginarynumber axis. This will place the point somewhere in one of the quadrants formed by the two axes. This is precisely the manner in which points are located on the axis
and
earth's surface
We
so far
by means of latitude and longitude.
can speak of a number such
point reached
when you marked
Or you can have —7+6/
or
+5+5*, which would East followed by
+0.5432— 9.1151 or
Such numbers, combining numbers."
as
off 5 units
real
+V2+V3
and imaginary
represent the 5
units North.
*•
units, are called
"complex
THE IMAGINARY THAT ISN
T
10"*
axes, any point in a plane (and not merely on a line) can correspond to one and only one complex number. Again every to
Using both
made
be
number can be made
conceiyable complex
one point on
to correspond to
one and only
a plane.
numbers themselves
In fact, the real
are only special cases of the
plex numbers, and so, for that matter, are the imaginary numbers.
numbers
represent complex
numbers
real
are
equal to zero.
only, has
than merely
if
the complex numbers in
polynomial equation
in a
complex numbers are considered
is
equal
as solutions, rather
— 1=0
The two
x?+i=o
solutions of
four solutions of x4
The
z.
all
numbers and imaginary numbers. For instance the two are +1 and — 1, which can be written as fifoz
real
— ifc*.
and
you
been of inestimable use to the mathematician.
degree only
solutions of x 2
o—
If
then the
\a\bu
to zero.
For instance, the number of solutions its
form
use of the plane of complex numbers, instead of the lines of real
numbers to
of the
imaginary numbers are
which a happens to be equal
The
numbers
those complex numbers in which b happens to be
all
And
as all
com
— 1=0
are
are
\i
and
—
four complex
all
z,
o+z and
or
numbers
just
listed.
In boil less, is
all
these yen simple cases, the complex
down is
to either real
not always
+i+oz
tions are
so.
numbers or
to
In the equation x3
(which can be written simply
—
J 1/ 2 r 2\^i and
1/
numbers contain
zeros
and
imaginary numbers. This, neverthe
— 1=0 as
be
sure,
+1), but the other two
solu
one solution,
to
—%—%y/$i
The Gentle Reader with ambition can take the cube of either of these expressions (if he remembers how to multiply polynomials algebraically) and satisfy himself that it will come out +1. Complex numbers are of practical importance too. Many familiar measurements involve "scalar quantities" which
One yolume
is
real
is
only in magnitude.
greater or less than another; one weight
than another; one density
one debt
differ
is
is
greater or less
greater or less than another. For that matter,
greater or less than another. For
all
such measurements, the
numbers, either positiye or negatiye, suffice.
Howe\er, there are tude and direction.
being greater or
A
less,
also 'Vector quantities"
velocity
may
but in being
for forces, accelerations,
and
differ in
which possess both magni
from another velocity not only
in
another direction. This holds true
so on.
For such yector quantities, complex numbers are necessary to the mathematical treatment, since complex numbers include both magnitude and
.
1
NUMBERS AND MATHEMATICS
08
direction (which was
types of
my
reason for making the analogy between the four
numbers and the compass points)
Now, when my
demanded "the square root of minus a scalar phenomenon for which
sociology professor
one pieces of chalk," he was speaking of the real numbers were sufficient.
On had
me how
the other hand, had he asked
"Go two hundred
said,
to get
from
his
room
on the campus, he would probably have been angered
certain spot
yards."
He would
have asked, with
to a if
I
asperity,
"In which direction?"
Now, you which the
see,
real
he would have been dealing with
numbers
are insufficient.
two hundred yards northeast," which plus
100V2
Surely
i
it is
of
is
a vector quantity for
could satisfy
him by
equivalent to saying
saying
"Go
"Go
ioo\/2
yards." as ridiculous to consider the square root of
nary" because you can't use
number 200
I
as
it
to
count pieces of chalk
"imaginary" because by
one point with reference
to another.
itself it
minus one "imagias to consider the
cannot express the location
PART
III
Numbers and Measurement
:
9.
The
other day
I
An
cal Science:
FORGET
was looking through
new textbook on biology (Biologinumber of contributing Brace & World, Inc., in 1963). I
a
Inquiry into Life, written by a
authors and published by Harcourt,
found
IT!
fascinating.
it
Unfortunately, though,
I
read the Foreword
first
I'm one of that
(yes,
kind) and was instantly plunged into the deepest gloom. Let
from the
first
me
quote
two paragraphs
"With each new generation our fund of scientific knowledge increases ... At the current rate of scientific advance, there is about four times as much significant biological knowledge today as in 1930, and about sixteen times as much as in 1900. Bv the year 2000, at this rate of increase, there will be a hundred times as much biology to 'cover' in the fivefold.
introductorv course as at the beginning of the century."
Imagine
think
Then falls
about it.
my
Still
Homo
me.
am
I
a professional "keeperupper" with
manic, ebullient, and carefree moments,
to
ears.
I
dont keep up with
science.
when I'm
all
Worse,
even
cant keep up
I
through sorrowing for myself,
worrying about the world generally.
sapiens?
We're going
to
What
is
I
devote a few
going to become
smarten ourselves to death. After our brain
we will all die of pernicious education, with crammed to indigestion with facts and concepts, and with
while,
formation exploding out of our
a
cells
blasts of in
ears.
would have it, the very day after I read the Foreword Biological Science I came across an old, old book entitled Pikes Arith
But then, to
I
fairly well.
worse, I'm falling farther behind every day.
finally,
moments
this affects
my more
read something like the abovequoted passage and the world
I
with
of
in
succeed
I
And
how
and
science
as luck
NU
112
At
metic.
least that
M BERS AND MEASUREMENT the
is
Complete System
name on
of Arithmetic
titles
were
Composed
the United States, by Nicolas Pike, A.M."
but the copy
I
have
a large
book
On
the spine.
those days
itself a bit better, for in
the
for the
was
It
title
page
goes
titles. It
it
spreads
"A New and
Use of the Citizens of first
published in 1785,
only the "Second Edition, Enlarged," published in
is
1 797
It
is
with no
relief
of over 500 pages,
whatever
way
in the
crammed
of small print
full
of illustrations or diagrams. It
slab of arithmetic except for small sections at the verv
is
and
a solid
end that introduce
algebra and geometry. I
was amazed.
I
have two children
They
days. fifth
are
(and once
in grade school*
I
was
know what arithmetic books are like these as large. They can't possibly have even one
and
in grade school myself),
I
nowhere near
the wordage of Pike.
Can So
it
And
be that we are leaving anything out?
went through Pike and, you know, we
I
there's
enough
On
nothing wrong with that.
page
five
Now
19, for instance,
is
we're not leaving
Pike devotes half a page to a listing of
Roman
numerals, extending the
to
list
num
numbers
as
hundred thousand.
Arabic numerals reached Europe
once thev came on the scene the
moded
are leaving something out.
trouble
out.
bers as expressed in
high as
The
(see
Chapter
They
1).
Roman
in the
High Middle Ages, and
numerals were completely out
lost all possible use, so infinitely superior
was the new Arabic notation. Until then who knows
how many reams of Roman nu
paper were required to explain methods for calculating with merals. Afterward the
same
dredth of the explanation.
And
yet five
merals, Pike translate
hundred
still
them
no instructions
calculations could be performed with a hun
No
knowledge was lost— only
is
Roman
nu
included them and expected his readers to be able to
into Arabic numerals for
learning
and
vice versa even
how to manipulate them. In Roman numerals are still
years after Pike, the
daughter
inefficient rules.
years after the deserved death of the
fact,
though he gave
nearly two hundred
being taught.
My
little
them now.
But why? Where's the need? To be sure, vou will find Roman numerals on cornerstones and gravestones, on clockfaces and on some public buildings
[* is
and documents, but
This
article first
it
isn't
used for any need at
all.
appeared in March 1964, and time marches on.
nou well along in college.]
It is
My
used for
younger child
FORGET show, for status, for antique
H^
IT!
some kind
flavor, for a craving for
phony
of
classicism.
dare say there are some sentimental fellows
I
Roman
of the
is
them would be
scrapping
but
numerals
like
that everyone
who
the wheel of a
ModelT Ford
Roman
knowledge
feel that
gateway to history and culture; that knocking over what is left of the Parthenon,
We
have no patience with such mawkishness.
I
who
a kind of
so
numerals? Forget
he could get the
as well suggest
cardom.
flavor of early
—And make room
it!
might
be required to spend some time at
learns to drive a car
instead for
new and
valuable material.
Why
But do we dare forget things?
not?
We've
much; more
forgotten
than you imagine. Our troubles stem not from the fact that we've gotten, but that
A
we remember
great deal of Pike's
too well;
book
we don't
for
forget enough.
we have
consists of material
imperfectly
is why the modern arithmetic book is shorter than Pike. we could but perfectly forget, the modern arithmetic book could
That
forgotten.
And
if
grow
still
shorter.
For instance, Pike devotes many pages to tables— presumably important
he thought the reader ought
tables that
be familiar with. His
fifth table
labeled "cloth measure."
is
Did you know that nails
make
2%
No, wait
Those
a while.
to
make
Scotch
Now
French
a
ell.
make make an
inches
a yard; while 12 nails
takes 20 nails (45 inches) to
a
to
make
And
16
an English
ell,
make
a
and 24
nails plus 1^5 inches
Flemish
inches)
(37H
ell.
It
(54 inches)
nails
make
ell.
you're going to be in the business world and import and export
if
cloth, you're going to
some way
have to know
of getting the
ell
all
those ells— unless you can figure
out of business.
Furthermore, almost every piece of goods
You speak
of a firkin of butter, a
and
of butcher's meat,
number
Well, they do.
ell.
(27 inches)
12 nails
Then, 16
a "nail"?
of
so on.
punch
is
measured
so on),
its
own
units.
of these quantities weighs a certain
Each
pounds (avoirdupois pounds, but there
and apothecary pounds and
in
of prunes, a fother of lead, a stone
and Pike
are also troy
pounds
carefully gives all the equiv
alents.
Do
you want to measure distances? Well,
make
inches
make
1
1
link; 25 links
furlong;
and
Or do vou want in
Colonial times.
make 1 make
8 furlongs to
measure
You have
to
how about
pole; 4 poles 1
make
1
9
7 %oo chain; 10 chains this:
mile. line of
work
know the language, of course. Here
it is:
ale or
beer— a
very
common
— NUMBERS AND MEASUREMENT
114
make
2 pints
that
and 4 quarts make
a quart
a gallon. Well,
we
still
know
much anyway.
In Colonial times, however, a mere gallon of beer or ale was but a
You had to know how to speak of mansized quantities. Well, 8 gallons make a firkin— that is, it makes "a firkin of ale in London." It takes, however, 9 gallons to make "a firkin of beer in London." The intermediate quantity, 8% gallons, is marked down as "a That was
starter.
beer"— presumably outside
of ale or
firkin
for infants.
where the provincial
were
citizens
less
of the environs of
London
finicky in distinguishing
between
the two.
But we go on:
make
sure)
2 firkins
a kilderkin
(I
and
suppose the intermediate kind, but I'm not
make 1 hogshead; 2 barrels make Have you got all that straight? But
still
Don't
3 barrels
better.
Here, 2 pints pottle.
Then 1% barrels make a butt.
a barrel.
a puncheon; and
dry measure in case your appetite has been sharpened for
let's try
something
make
2 kilderkins
make
tell
a quart
me
and
2 quarts
make
a pottle.
(No, not bottle,
you've never heard of a pottle!) But
proceed.
let's
make a peck, and 4 pecks make a bushel. (Long breath now.) Then 2 bushels make a strike, 2 strikes make a coom, 2 cooms make a quarter, 4 quarters make a chaldron (though in the demanding citv of London, it takes 4% quarters to make a chaldron). Finally, 5 quarters make a wey and 2 weys make a last. Next, 2 pottles
make
I'm not making
this up.
Were orize
people
all this?
pound
a gallon, 2 gallons
I'm copying
who were
right out of Pike,
Apparently, yes, because Pike spends
addition. That's right,
You see, Compound
it
is
48.
a lot of
mem
time on com
compound addition.
the addition you consider addition addition
page
studying arithmetic in 1797 expected to
something stronger and
I
is
just
will
"simple addition."
now
explain
it
to you.
Suppose you have 15 apples, your friend has 17 apples, and a passing and you decide to make a pile of them. Having done
stranger has 19 apples so,
you wonder how many you have altogether. Preferring not
you draw upon your college education and prepare begin with the units column and find that divide 21
by 10 and
find the quotient
put down the remainder, I
seem
1,
to hear loud yells
the fevered demand.
and
2 plus a
remainder of
therefore 1,
so
you
carry the quotient 2 into the tens col
"What
does this 'divide by 10' jazz
this
to count,
add 15+17+19. You
5+7+9=21. You
from the audience.
"Where
Ah, Gentle Readers, but
is
to
is
exactlv
is all
this?"
comes
come from?"
what vou do whenever you add.
THE YARDSTICK The
yardstick
one of those accompaniments of
is
we tend
that
life
to take
Few people have any idea how difficult it was to produce one and how many subtle concepts had to be embraced before the yardstick for granted.
became
The
possible.
natural
manner
of measuring lengths in early times was to use var
body
ious portions of the
We
for the purpose.
measuring the heights of horses, and of a "span by the outstretched
bow"
A
"hands" in
for the length represented
"cubit" from a Latin word meaning
"el
the distance from fingertips to elbow, and a "yard" (related to
is
"girth") of a
fingers.
talk of
still
7
is
the distance from fingertips to nose, or the waist measurement
man.
The
trouble with using portions of the body as a measuring device
that the lengths to person.
yard, but
The
my
is
and measurements of those portions varies from person length from
waist
my
measurement
fingertip to
is
my
nose
distinctly greater
is
quite close to a
than a yard.
occurred to people to establish a "standard yard" and never
It finally
mind what your own measurements
are.
According to tradition, a standard
yard was originally adjusted to the length from the fingertips of King
Henry
I
of
England
to his nose.
(And
the standard foot
is
supposed to be
based on the foot of Charlemagne.) Naturally,
the King of England cant travel from village to village
measuring out lengths of cloth from his nose to his stick is held up against him and marks are made at gertips.
The
distance between the marks
can be measured
off
is
fingertips.
his
nose and his
a standard yard.
against that standard
Instead a
Other
fin
sticks
and can become secondary
standards to be sent to every village for use in checking the activities of the local merchants.
It is
who
only that the kindly souls
on the number 10
tion based
it
number
divided by 10, the
is
in
first
devised our Arabic system of numera
such a way that
when any
twodigit
digit represents the quotient
and the
second the remainder.
For that reason, haying the quotient and remainder out dividing,
we can add
automatically.
we put down 1 and cam' down 7 and carried 5, and
2; if it
If
in
our hands with
the units column adds up to 21,
had added up
to 57,
we would have put
so on.
The Granger
Collection
NUMBERS AND MEASUREMENT
Il6
The
mind
only reason this works,
each column of
vou,
is
column
resents a value ten times as great as the
column and
one
units, the
is
to
this
combination of
from column
ratio
to
for this reason that
Now and 10
as
it is,
one
to
its left is
hundreds,
and
calls
dozen and 8 apples, your friend has
1
1
dozen
8 units
dozen
10 units
1
dozen
9 units
we put down
12,
8+10+9=27, we must
tient of 2 plus a
And
since the
we can no
carry 2?
Not
number svstem we
We
sum by
The
12, since
are using
the ratio of the value of
find that 27 divided
by 12
we put down 3 and 3, 1 + 1 + 1+2=5. Our total therefore so
gives a quo
carry is
5
2.
In the
dozen and
apples.
Whenever
a ratio of other
compound
addition
than 10
if
vou
trv to
add
8 ounces, for there are 16 ounces to a
yards 2 feet 6 inches to
to a foot,
and
You do
3 feet to a
the former
if
used so that vou have to
is
1
5
pound. You are stuck again
do the
to; 111
latter. First,
and
3+1 + 1 = 5. Your
Now why
1.
remainder of
As 2,
for the feet,
put down
answer, then,
is
on Earth should our unit
number system
is
so firmly based
5
2
and
a
carry
6 inches and 8
remainder of
2+2+1 = 5.
and carry
2
a
you
yard.
vou care
you put down
and
if
vard 2 feet 8 inches, for there are 12 inches
1
1
ac
pounds 12 ounces and 6 pounds
inches are 14 inches. Divide 14 bv 12, getting
and get
make
vou have "compound addition." You must indulge
tual divisions in adding,
have
at all?
not 10 but
is
longer let the digits do our think
divide that
remainder of
dozens column we get
3
and
7
"dozens" column to the "units" column
the columns; in this case, 12.
add
a
We have to go long way round.
ing for us.
3
dozen
as follows:
there are 12 units to a dozen.
If
1
dozen and 9 apples. Make
1
1
based on 10 and not on
It is
"simple addition."
it,
a passing stranger has
and add them
ten and a value
makes addition very simple.
of ten that
Pike
Since 8+1019=27, do ratio of the
in
rightmost
number system based on
a
column
suppose you have apples,
pile of those
and
tens, the
its left is
rep
The
before.
so on.
It is
is
that in adding a set of figures,
from the right and working leftward)
digits (starting
1.
Divide
5
2,
by
so 3
In the yards, you
yards 2 feet 2 inches.
ratios varv all over the lot,
on 10? There are many reasons
when our (valid in
)
)
FORGET their time)
we
surely
odd
for the use of
forget
compound
3,
sophisticated
exclusive (or nearly exclusive) ratio. If
pleasure
II7
!
ratios like 2,
now advanced and
are
IT
8,
4,
we could do
so,
and
12, 16,
enough
20,
but
to use 10 as the
we could with much
about compound addition— and compound subtraction,
compound
multiplication,
division, too.
(They
also exist, of
course.
To be sible.
us
when nature makes
sure, there are times
the universal ten impos
In measuring time, the day and the year have their lengths fixed for
by astronomical conditions and neither unit of time can be abandoned.
Compound
addition and the rest will have to be retained for such special
cases, alas.
But who
remember
ells?
and
pottles
are purelv
that measures
in this world. It
things in firkins
manmade measurements, and we must were made for man and not man for measures.
These
happens that there
It so
on ten
we must measure
in blazes savs
and Flemish
is
a svstem of
is
measurement based
called the metric svstem
and
exclusively
used
it is
all
over
the civilized world except for certain Englishspeaking nations such as the
United States and Great Britain. for we gain nothown measurements. The loss in time
By not adopting the metric system, we waste our time ing,
not one thing, by learning our
(which
(To be tools it
is
expensive indeed)
sure,
but
it
it
is
balanced bv not one thing
would be expensive
to convert existing instruments
would have been nowhere nearlv
a century ago, as
we should
there
is
They have
as
expensive
and
we had done
if
have.
There are those, of course, who object ished measures.
can imagine.
I
given up
to violating our longused cher
cooms and chaldrons but imagine
something about inches and feet and pints and quarts and pecks
and bushels that
is
"simpler" or "more natural" than meters and
liters.
There may even be people who find something dangerously foreign and radical (oh, for that vanished ric
system— yet In
it
word of opprobrium, "Jacobin")
in the
met
was the United States that led the way.
1786, thirteen years before the wicked French revolutionaries de
signed the metric system,
Thomas
Jefferson (a notorious "Jacobin" accord
ing to the Federalists, at least) saw a suggestion of his adopted by the in
fant United States. \\ Tiat
The
we had been
nation established a decimal currency.
using was British currency, and that
is
a
fearsome
and wonderful thing. Just to point out how preposterous it is, let me say that the British people who, over the centuries, have, with monumental patience, taught themselves to endure anything at
all
provided
it
was
"tra
NUMBERS AND MEASUREMENT
Il8 ditional"
verting
—are
now
sick
and
tired of their currency
(They
to the decimal system.
it
can't agree
and
are debating con
on the exact
details of
the change.)**
But consider the
make
things
British currency as
it
penny; 12 pennies make
1
To
has been. shilling,
1
begin with, 4
and 20
shillings
far
make
1
pound. In addition, there
is
coins, such as ha'pennies
and thruppences and sixpences and crowns and
a virtual farrago of terms,
not always actual
if
knows what other
halfcrowns and florins and guineas and heaven
devices
with which to cripple the mental development of the British schoolchild
and
line the pockets of British
and attempt
Needless to
pounds,
say,
shillings,
dividing
and pence— and very shillings, 7
10 mills
dollar; 10 dollars
make make
culations, stick to dollars
The
result?
be treated
as
to call
1
1
3.
to
manipulate
Try
Quick now!
system, as originally established,
is
as
make 1 dime; 10 dimes make 1 modern Americans, in their cal
cent; 10 cents eagle. Actually,
and cents
how
special instructions they are.
pence by
money
In the United States, the follows:
come
tourists
Pike gives careful instruction on
pounds, 13
5
tradesmen whenever
to cope with the currency.
onlv.
American monev can be expressed can any other decimals.
in
An American
decimal form and can child
who
has learned
decimals need only be taught to recognize the dollar sign and he
is all set.
In the time that he does, a British child has barely mastered the fact that
thruppence ha'penny equals 14 farthings.
What came
a pity that
when, thirteen vears
into being, our original
the metric system
later, in 1799,
antiBritish,
proFrench feelings had not
enough to allow us to adopt it. Had we done so, we would happy to forget our foolish pecks and ounces, as we are now have forgotten our pence and shillings. (After all, would you
lasted just long
have been
happy like to
to
as
go back to British currency
What
I
would
Everywhere. I
to
like to see
a
a
locality all
one form of money do
for all the world.
I
and
may be accused because I
I
just don't
want
I
insist
to
on them
for
I
visited
Great Britain in 1974,
I
am
constitute a
keep provincialisms that were
[** Since this article was written, the British have carried through the change. I
I
have no objection to local customs and
local dietaries. In fact,
by myself.
of this of wanting
mold, and of being a conformist. Of course,
conformist (heavens!).
local dialects
our own?)
Why not?
appreciate the fact that
pour humanity into
not
is
in preference to
was greatly disappointed at not being able
When to deal
with threepences and half crowns. They are also adopting the metric system, leading the intransigent United States virtually alone in opposition.]
FORGET enough
well
world which
me
human
wellbeing in a
9c minutes in circumference.
you think provincialism
If
let
now
lig
time but that interfere with
in their is
IT!
is
cute and gives humanity color and charm,
quote to you once more from Pike.
Money"
"Federal
1
dollars
and cents
1
had been introduced eleven
years
before Pike's second edition, and he gives the exact wording of the law that established
and discusses
it
it
in
detail— under the decimal svstem
and not under compound addition. Naturally, since other systems than the Federal were
had
to
to another.
Here
the
is
list.
I
won't give you the actual
of reductions that were necessary, exactly as he
To
I.
cut,
still
in use, rules
be formulated and given for converting (or "reducing") one svstem
New
reduce
lists
rules, just
the
list
them:
Hampshire, Massachusetts, Rhode Island, Connecti
and Virginia currency; 1.
2. 3.
4. 5.
6. 7. 8.
To Federal Money To New York and North Carolina currency To Pennsylvania, New Jersey, Delaware, and Manland To South Carolina and Georgia currency To English money To Irish money To Canada and Nova Scotia currency To Livres Tournois (French money To Spanish milled dollars
currency

9.
II.
To reduce Federal Money to New England and Virginia currency. To reduce New Jersey, Pennsylvania, Delaware, and Manland cur
III.
rency: 1.
To New Hampshire,
Massachusetts,
Rhode
Island,
Connecticut,
and Virginia currency 2.
To New York and
Oh, the heck with
Can anvone
it.
.
You
.
.
get the idea.
possibly be sorrv that
all
that cute provincial flavor has van
ished? Are vou sorrv that even time you travel out of state you don't have 
to
throw yourself into
to
make
a
purchase?
state invades yours
forgotten
Then
fits
Or
and
whenever you want someone from another
of arithmetical. discomfort
into similar tries to
fits
every time
dicker with vou 9 \\ "hat a pleasure to have
all that.
tell
me
divorce laws?
what's so wonderful about having
fifty sets
of marriage
and
1
NUMBERS AND MEASUREMENT
20
In 1752, Great Britain and her colonies (some two centuries later than
Catholic Europe) abandoned the Julian calendar and adopted the astro
nomically more correct Gregorian calendar (see Chapter 11). Nearly half a
century
Pike was
later,
giving rules for solving complex calendar
still
based problems for the Julian calendar it
Wouldn't
it
by adopting to the
be nice
we could
if
forget
Gregorian. Isn't
day of the week and have itself
most of calendrical complications
would
a rational calendar that
perpetual one, repeating a
as well as for the
nice to have forgotten the Julian calendar?
tie
would enable
I
would
do
us to
be nice
if
everyone— whatever
English fluently.
would help
It
That would
Why
come
major language.
start.
room
save a lot of
would
communications and perhaps, eventu
for other things.
Secondly, far more science
be
English,
it
be even more
will
To
it
communication
critical
How
up.
word
a
in
mad
why
we
spell colour, color,
and
funnv
is
almost
as possible for
become more and
we'll
and "will"
is
critical
today
people to speak
and grammar.
No
Chinese ideograms.
a set of
it
is
at the letters that
so terribly necessary to spell
centre, center,
a lot of
Who or
if
lost
all
hic
those
throo, thoh, cawf, hic
ruff,
already write hiccup and
intelligent,
not have
And grammar? "shall"
spelling
pronounced by looking
is
to a Britisher
and save
a
combination "ough"?
letter
cup, and lokh; but
rest, too,
science that
its
looks funny, perhaps, to spell the words
result looks
we have
do you pronounce: rough, through, though, cough,
cough, and lough; and
sounds with the
so
tomorrow.
spelled today,
it is
on Earth,
reported in English than in any
we ought to make it as easv which means we should rationalize
one can be sure how
make
is
is
sure,
English, as
ing,
It
also speak
English? Well, for one thing more people speak English as either
other language and
the
in
own language was— could
his
Not
into worldwide use.
or second language than anv other language
first
head
It
is
everyone would just choose to speak English.
ally,
and
as a
just this.
necessarilv as the onlv language or even as the just
firmly
a lot of useful forgetting.
English language
like to see the
month
over and over everv three months? There
world calendar proposed which would do It
the day of the
threemonth calendar serve
a single
it
doesn't look funnv.
and shew, show and
but we are used to
it.
We
grey, gray.
can get used to
wear and tear on the brain.
intelligence
is
We The
We
would
measured bv proficiency
all
at spell
one thing.
needs the eternal hairsplitting arguments about
"which" and "that"? The uselessness of
it
can be
AMERiCAY It is rather heartbreaking that the
BILLS
United
States, tuning started resolutely
in the right direction, did not continue.
Immediately after the Revolutionary War, antiBritish feeling j strong to cause many Americans to want to do away with anything triv
enough ial
would remind them of the hated
that
foe.
The
"rights of English
men" were not trivial, so those were congealed into the Bill of Rights. The f) zemof coinage, however familiar, was trivial. The key person in this respect was a Pennsvlvanian. Gouverneur had been a over the quarreling
Federalist,
an advocate of a strong central government states who made up what was wrongly
and disunited
called the "United" States immediately after the Revolution.
He
was a
more than anyone d :. responsible for the actual wording of the Constitution and the casting : :: into a clear and simple phraseology, devoid of fustian and melo
member
It
of the Constitutional Congress and.
was
he, also,
who
suggested that the United States adopt a
coinage based on a decimal system.
form of which
is
so familiar
to
is it
lyOO, silver
The
presented in the illustration, thougfi all
of us) , gets
from the
silver
its
name
mines in
new
basic unit, the "dollar" (the paper it
scarcely needs
it,
way round. Back about Joachims Valley (in what is now the long
was used to coin ouncepieces. Joachim's German. Joachimsthal. and the coins were called "Joachims
northuvstern Czechoslovakia)
Valley
is.
in
thalers" or. for short, "thalers." or. in English, "dollars."
In colonial times. Spanish coins of about the value of the weUk
The Spaniards called them "pesos" the English and the Americans adopted the name and began coining them in
'dollars"
dollars existed.
\:
_ :
:_
NUMBERS AND MEASUREMENT
122
demonstrated by the
fact that virtually
no one
gets
it
straight anyway.
Aside from losing valuable time, blunting a child's reasoning instilling
him
faculties,
or her with a ravening dislike for the English language,
and
what
do vou gain?
some who think that such blurring of fine distinctions will I would like to point out that English, before the grammarians got hold of it, had managed to lose its gender and its declensions almost everywhere except among the pronouns. The fact that we have only one definite article (the) for all genders and cases and times instead of If
there be
ruin the language,
three, as in
French
den, des) in no
(le, la, les)
way blunts
rablv flexible instrument. to
or
six, as in
We
We
must make room
Forget
it, I
But why
for
the
new and
sav, forget it
am
I
dem,
it is
as
we
are used
really follies.
expanding knowledge, or at
as possible. Surelv
less as it is to learn
(der, die, das,
cherish our follies onlv because
them, and not because they are not
much room
German
the English language, which remains an admi
least
make
important.
more and more. Forget
getting so excited?
No
as
important to forget the old and use
one
is
it!
listening to a
word
I
say.
PREFIXING IT UP
10.
I
go through
do
supported and bolstered by
life
One
of us.
all
my own
of
many comforting
myths, as
particularly cherished articles of faith
that
is
no arguments against the metric system and that the common make up an indefensible farrago of nonsense that we keep only out
there are units
of stubborn folly.
Imagine the sobering
by
artificial,
(and
I
bitterly
and not geared
sterile,
don't quote exactly),
A
the thing.
liter of
having recently come across a letter
effect, then, of
gentleman who
a British
beer
is
if
too
denounced the metric system
human
to
one wants
as
being
needs. For instance, he said
to drink beer, a pint of beer
much and
half a liter
too
is
little,
is
but a
pint, ah, that's just right.*
As
far as
I
can
tell,
the gentleman was serious in his provincialism, and
in considering that that to
ural law.
against
It
me
reminds
which he
accustomed has the force of a nat
is
of the pious
foreign languages by holding
all
woman who
set her face firmly
up her Bible and
saying, "If the
English language was good enough for the prophet Isaiah, and the apostle Paul,
it is
good enough
But mainly
it
for
reminds
me."
me
that
I
want
to write
an essay on the metric
system.
In order to do so,
system does not that
it is
want
for
to begin
bv explaining that the value of the
the actual size of the basic units.
a logical system.
The
Its
worth
is
this:
units are sensibly interrelated.
measurements with which
All other sets of
names
I
lie in
I
am
acquainted use separate
we we volume,
each unit involving a particular type of quantity. In distance,
ourselves have miles, feet, inches, rods, furlongs,
and
so on. In
have pecks, bushels, pints, drams. In weight, we have ounces, pounds, tons, grains. It
*
is
like the
Before you write to
though
it is
Eskimos,
tell
larger than
me
who
are supposed to
that half a liter
an American
pint,
it is
is
larger
have
I
than a pint,
don't
know how
me
explain that
let
smaller than a British pint.
NUMBERS AND MEASUREMENT
12 4
many dozens of words for snow, or when it is lying there, when it or oldfallen,
and
it
when
it is
falling
so on.
using adjectivenoun combinations.
term
for all kinds of
snow, and so on. \\ Tiat's the advantage? did not see before. Second,
we can
snow and the
wet snow, drv snow, hard snow,
adjective describing the specific variety:
we
for
loose or packed, wet or dry, newfallen
is
We ourselves see the advantage in We then have the noun as a general soft
word
a different
we
see a generalization
same
adjectives for other
First,
use the
nouns, so that we can have hard rock, hard bread, hard heart, and consequently see a
The
new
generalization, that of hardness.
metric system
is
the only system of measurement which, to
knowledge, has advanced to Begin with
metrum
.an arbitrary
measure of length, the meter (from the Latin
or the Greek metron, both
meaning "to measure"). Leave that
the generic term for length, so that entiate
mv
cident,
I
prefixes.
it
up
as
units of length are meters. Differadjective. That, in
right.
sure, the adjectives in the metric
system
(lest
they get lost by ac
suppose) are firmly jointed to the generic word and thus become (Yes, Gentle Reader, in doing this to the
they were "prefixing
The
all
one unit of length from another bv means of an
opinion, would be fixing
To be
my
this stage.
prefixes
the following
it
measurement system,
up.")
were obtained out of Greek and Latin
in
accordance with
little table:
ENGLISH
Now, if we save we have:
LATIN
GREEK
thousand
chilioi
mille
hundred
hecaton
centum
ten
deka
decern
the Greek for the large units and the Latin for the small
ones,
meters
1
kilometer**
equals
1COO
1
hectometer
equals
lOO
meters
i
dekameter
equals
10
meters
i
meter
equals
1
meter
i
decimeter
equals
O.l
meter
i
centimeter
equals
0.01
meter
i
millimeter
equals
o.ooi
meter
**The Greek ch has the guttural German ch sound. The French, who invented the metric system, hare no such sound in their language and used k instead as the nearest approach. That is why chilioi becomes kilo. Since we don't hare the guttural ch either, this suits us fine.
f
PREFIXING how
doesn't matter
It
as defined. If
long a meter
IT UP
is;
125
the other units of length are
all
you happen to know the length of the meter
in
terms of yards
or of wavelengths of light or of two marks on a stick, you automatically
know
the lengths of
the other units. Furthermore, by having
all
all
the
becomes very easv (given our decimal number system to convert one into another. For instance, I can tell vou right off that there are exactly one million millimeters in a kilometer. Now subunits varv bv powers of ten,
it
)
vou
tell
And
me
how manv
right off
again, once
type of measurement. ity,
sorts of units or
at
about
all
it,
a
large a unit
know
still
is
a
"kilobuck"
In one respect and, to
who
memorized, they
it
a "poise"
mv
is.
that a centipoise
hundred is
or
is
even what, exactly, viscosity
vou
a poise, that a hectare
and even that
prefixes
vou are told that
If
how
doesn't matter
it
inches there are in a mile.
vou have the
how
is
is
Perhaps thev
mark
felt
some measurable
for
any
viscos
related to other
Without knowing anything is
equal to a hundreth of a
ares, that a decibel
is
a tenth of a bel;
equal to a thousand dollars.
mind,
in
only one were the French scientists
established the metric system in 195 shortsighted.
past the thousand
do
measure of
a
it
will
They
did not go
in their prefix system.
that once a convenient basic unit was selected for
quantity, then a subunit a thousand times larger
be the largest useful one, while
a subunit a
thousandth
as large
would
would be
Or perhaps thev were influenced by the fact that there is no word in Latin for anv number higher than a thousand. (Words like
the smallest. single
million and billion were invented in the late middle ages and in early
em
mod
times.)
The
later Greeks, to
sible to sav
be
sure,
used myrias for ten thousand, so
"mvriameter" for ten thousand meters, but
this
is
it is
pos
hardly ever
used. People say "ten kilometers" instead.
The
net
result, then,
offers prefixes that
"kilo,"
is
that the metric system as organized originally
cover only
six orders of
one million (io*) times
the exponent,
it is
is
6,
mav do
The
largest unit,
as great as the smallest unit "milli."
and
that marks the orders of magnitude.
Scientists could not, however, stand
tude
magnitude.
still
for this. Six orders of
magni
advance of instrumentation carried science into the very large and very small in almost even field of measurement, the svstem simply had to stretch. for everyday life,
but
as the
came into use for units above the kilo and below the and of course that meant the danger of nonconformity (which is a
Unofficial prefixes milli
f If anyone wants to write that a millipede is a thousandth of a pede and that one centipede equals ten millipedes, by all means, do but I wont listen.
—
NUMBERS AND MEASUREMENT
26
bad thing
in scientific language).
For instance, what we
lion electronvolts), the British call a
"Gev"
call a
"Bev"
(bil
(gigaelectronvolts).
In 1958, then, an extended set of prefixes, at intervals of three orders of
magnitude, was agreed upon by the International Committee on Weights
and Measures thrown
Here they
at Paris.
with a couple of the older ones
are,
in for continuity:
trillion
(10
GREEK ROOT
PREFIX
SIZE 12 )
9
tera
teras
("monster"
giga
gigas ("giant")
million (10 6 )
mega
megas ("great")
thousand (10 3 )
kilo
billion (10
)
one (io°) thousandth (io 3 ) 6 millionth (io
milli
micro
mikros ("small")
(io~ 9 )
nano
nanos ("dwarf")
(io~ 12 )
pico
)
billionth trillionth
The
prefix pico does not
have
a
Greek
Well, then, we have a "picometer"
gram"
as a billionth of a
"teradyne" as a
trillion
root.
as a trillionth of a meter, a
dynes. Since the largest unit, the tera,
now
times the smallest unit, the pico, the metric svstem
merely over
6,
but over a
"nano
gram, a "gigasecond" as a billion seconds, and a
full
is
10 24
stretches not
24 orders of magnitude.
15 and atto for a In 1962 femto was added for a quadrillionth (io ) 18 quintillionth (io Neither prefix Greek root.* This extends the
has a
).
metric svstem over 30 orders of magnitude. too
Is this
The it
was
units
A
much? Have we overdone
metric unit of length fixed at
is
its
is
perhaps? Well,
it,
the meter.
I
let's see.
won't go into the story of
1.093611 yards or 39.37 inches.
kilometer, naturally,
comes out
is
to 0.62137 miles.
A
of a mile.
mile
is
a
thousand times
We
won't be
that, or 1093.6 yards,
far off
if
sometimes said to equal "twenty
kilometer would represent
halfway between 66th and 67th
12%
streets to
which
kilometer
%
citv blocks"; that
is,
we
call a
the distance between, let us sav, 59th Street and 79th Street in If so, a
how
precise length, but that precise length in terms of familiar
Manhattan.
city blocks, or the distance
from
79th Street.
when this article first appeared in November now. Pico is from the Spanish word for "small." Femto and atto are from the Danish words for "fifteen" and "eighteen" respectively.] [* 7
did not give the nonGreek roots
1962, but
I
will
PREFIXING
IT
UP
12 7
For a megameter we increase matters three orders of magnitude and equal to 621.37 miles. This
The
ments.
California,
air
is
convenient unit for planetary measure
a
is
distance from Boston, Massachusetts, to San Francisco,
just
4%
about
megameters.
The diameter
megameters and the circumference of the earth
And
the
finally,
it is
moon
is
of the earth
12%
is
about 40 megameters.
380 megameters from the earth.
is
we have
Passing on to the gigameter,
a unit 621,370 miles long,
and
Venus
this
comes
in
handy
closest
is
42 gigameters awav and Mars can approach us as closely as 58
The sun
is
newly extended metric system,
to the limit of the
the terameter, equal to 621,370,000 miles. This will allow us to
embrace the entire instance,
The
its
its
is
by stretching
Finally,
we have
at
145 gigameters from the earth and Jupiter, at 640 gigameters distant; at its farthest, 930 gigameters away.
gigameters. closest,
for the nearer portions of the solar system.
The extreme width
solar system.
of Pluto's orbit, for
not quite 12 terameters.
is
solar system, however,
distances to the stars, the
just a
is
two most
speck in the Galaxy. For measuring
common
units are the lightyear
and
the parsec, and both are outside the metric system. What's more, even the
new
extension of the system can't reach them.
tance that light travels in one year. This or 9450 terameters. to us to
that
is
have
The
parsec
a parallax of
is
is
The
lightyear
the
is
dis
about 5,880,000,000,000 miles
the distance at which a star would appear
one second of
arc (parallaxsecond, get it),
and
equal to 3.26 lightyears, or about 30,000 terameters.
Even these nonmetric
units err
on the small
side. If
one were
to
draw
a
sphere about the solar system with a radius of one parsec, not a single
known
star
would be found within that sphere. The nearest
stars,
those of
the Alpha Centauri system, are about 1.3 parsecs away. There are only thirtythree stars, out of a
hundred
billion or so in the Galaxy, closer to
our sun than four parsecs, and of these only seven are visible to the naked eye.
There are many whole has
a
beyond this— far beyond
stars
diameter which
is,
at
its
this.
The Galaxy as a Of course, we
longest, 30,000 parsecs.
might use the metric prefixes and say that the diameter of the Galaxy
is
30 kiloparsecs.
But then the Galaxy
is
only a speck in the entire universe.
The
nearest
which are 50 kiloparsecs our own is Andromeda, which
extragalactic structures are the Magellanic Clouds,
away, while the nearest is
yond
at a distance of
fullsize
galaxy to
And there are hundreds many megaparsecs.
700 kiloparsecs away.
of billions of galaxies be
ANDROMEDA GALAXY The Andromeda distinction. It
eye— so
if
is
Galaxy, mentioned briefly in this
anyone asks you how
far
you can see (with
him 2,300,000 lightyears. The Andromeda looks like a faint, fuzzy
nearsighted),
magnitude.
article,
has one unusual
the farthest object that can be seen with the unaided glasses on,
if
you
re
tell
It is
object of about the fourth
not likely to be noticed by a casual skygazer, but
it
was
star maps of some of the Arab astronomers of the Middle Ages. The first to describe it among our Western astronomers was the
noted in the
German
observer
Simon Marius,
in 1612.
In the next century, a French observer, Charles Messier, was interested
permanently fuzzy objects in the sky so that they not
in recording all the
be mistaken for comets. (Messier was interested in cornets.) The Andromeda was
thirtyfirst
on
his
list,
and
its
alternate
name,
still
often used,
is
M31.
Andromeda looked like a French astronomer Pierre Simon de Laplace
In the simple telescopes of the 1700s, the whirling cloud of gas, and the
thought that was indeed what
it
was. In a popular book
on astronomy he
wrote in the early 1800s, he made the suggestion in an appendix. Stars our Sun and the planets that accompany them originated out of a
like
condensing cloud of gas like that of the Andromeda. The Andromeda was then called the Andromeda Nebula (from the Latin word whirling, u
for
cloud
,y
and Laplace's suggestion has always been
),
called the ''nebu
lar hypothesis."
In recent years a vastly more sophisticated form of the nebular hypothhas come to be accepted as the origin of the solar system, but the
esis
Andromeda
is
no cloud of own Milky
larger than, our
gas. It
Way
is
a collection of stars as large as, or
Galaxy, and farther beyond are billions
of other galaxies.
The at
farthest galaxies that
about two
verse, as of
billion parsecs,
now, has
Suppose, now,
we
have been made out have distances estimated
which would mean that the
entire visible uni
diameter of about 4 gigaparsecs.** consider the units of length in the other direction—
a
toward the very small. [** Since this article was written, quasars have been detected at distances of 4 gigaparsecs so the visible universe has a diameter of 8 gigaparsecs.]
The Granger
Collection
NUMBERS AND MEASUREMENT
I30
A
micrometer
is
good unit of length
a
dinary optical microscope.
micrometers
in diameter.
Drop down
The body
(A micometer
is
nanometer (often
to the
for objects visible
shortest violet light
is
is
or
about 4
often called a "micron.")
called a "millimicron")
be conveniently used to measure the wavelengths of wavelength of the longest red light
under the
for instance, average
cells,
and
it
visible light.
can
The
760 nanometers, while that of the
380 nanometers. Ultraviolet light has a range of
wavelengths from 380 nanometers Shrinking the metric system
down
still
to
nanometer.
1
further,
we have
the picometer, or a
atoms have diameters of from 100 to 600
trillionth of a meter. Individual
And soft gamma rays have wavelengths of about 1 picometer. The diameter of subatomic particles and the wavelengths of the hard
picometers.
gamma The
go well below the picometer
rays
thing like
level,
however, reaching some
femtometer.
1
range of lengths encountered by presentday science, from the
full
known
diameter of the
universe at one extreme, to the diameter of a sub
atomic particle at the other, covers a range of 41 orders of magnitude. In other words, the
known
What
it
would take 10 41 protons
laid side
by
side to stretch across
universe.
about mass?
The fundamental
unit of mass in the metric system
derived from the Greek
gramma, meaning
a small unit of weight, equivalent to
sand grams,
is
%
8 .35
equal to 2.205 pounds, and a
is
the gram, a word
a letter of the alphabet.f It
A
ounces.
megagram
is
kilogram, or a thouis
therefore equal to
2205 pounds.
The megagram is almost equal to the long ton (2240 pounds) in our own units, so it is sometimes called the "metric ton" or the "tonne." The latter gives it the French spelling, but doesn't do much in the way of differentiating the pronunciation, so
A and
gigagram this
is
And
1000 metric tons and a teragram
is
1,000,000 metric tons
enough by commercial standards. These don't even begin, scratch the surface astronomically. Even a comparatively small
like the
81 times
prefer metric ton.
large
however, to
body
is
I
moon
has a mass equal to 73
more massive and has
the sun, a merely average
a
trillion
teragrams.
The
earth
is
mass of nearly 6 quadrillion teragrams.
star,
has a mass 330,000, times that of the
earth.
f The Greeks marked small weights with letters of the weight, for they used letters to represent numbers, too.
alphabet to indicate their
PREFIXING Of
we might
course,
use the sun
IT
UP
itself as a
1^1
unit of weight. For instance
the Galaxy has a total mass equal to 150,000,000,000 times that of the sun,
and we could therefore sav that the mass of the Galaxy
150 gigasuns. Since
it
is
also estimated that in the
mean
minimum
a
equal to
is
universe there
assuming ours to be of average
are at least 100,000,000,000 galaxies, then,
mass, that would
known
mass of the universe equal to
total
15,000,000,000 terasuns or 100 gigagalaxies.
Suppose, now,
A
we work
in the other direction.
milligram, or a thousandth of a gram, represents a quantitv of matter
easilv visible to the
naked
eve.
A
drop of water would weigh about 50
milligrams.
Drop
to a
microgram, or
gram, and we are
a millionth of a
An amoeba would weigh
scopic range.
in the
in the micro
neighborhood of
five
micro
grams.
The cells of our bodv are considerably down to the nanogram, or a billionth of a a
them we drop
smaller and for
gram.
The
average liver
cell
has
weight of about two nanograms.
Below the
cells are
trillionth of a
for instance,
Xor
is
the viruses, but even
if
we drop to the picogram, a The tobaccomosaic virus,
gram, we do not reach that realm.
weighs onlv 66 attograms.
that particularly near the
bottom
cules far smaller than the smallest virus,
of the scale.
There
are mole
and the atoms that make up the
molecules and the particles that make up the atom. Consider the following table:
WEIGHT hemoglobin molecule
ATTOGRAMS
0.1
uranium atom
0.0004
0.00000166
proton
0.0000000009
electron All told, the range in
IX
mass from the electron
to the
minimum
total
mass of the known universe covers 83 orders of magnitude. In other words, it
would take
10* 3 electrons to
make
a
heap
as
massive as the total
known
universe.
In
some ways, time (the
sidering) possesses the
third of the types of
most familiar
units,
measurement
because that
where the metric svstem introduced no modification
at
is
all.
the second, the minute, the hour, the dav, the vear, and so on.
I
am
con
the one place
We
still
have
NUMBERS AND MEASUREMENT
332
This means, too, that the units of time are the only ones used bv scientists tell,
The
result
week
or the
that lack a systematic prefix system.
number of seconds in number of days in fifteen
offhand, the
in a year or the
The fundamental
unit of time
a
years.
that you cannot
Neither can
we
the second and
is
is
number
of minutes
scientists.
could,
if
we
wished,
build the metric prefixes on those as follows: 1
equals
kilosecond
equals
16
megasecond
equals
n
2
1
gigasecond
equals
32
years
1
terasecond
equals
32,000
years
1
sobering to think that
is
second
1 2
1
It
second
I
3
3
have lived only
minutes days
a little oyer
i
1
^ gigasec
onds*; that civilization has existed for at most about 250 gigaseconds; and
may not have existed for more than 18 teraseconds doesn't make much of an inroad into geologic time
that manlike creatures altogether.
and even
The
that
Still,
less of
an inroad into astronomic time.
system has been in existence for about 150,000 teraseconds
solar
and may well remain tional teraseconds. fuel supply
and
The
a red
without major change for 500,000 addi
in existence
smaller the
dwarf
may
star,
the
more
carefully
it
without undue change for
last
hoards as
its
long as
3,000,000 teraseconds. As for the total age of the universe, past and future, I
say nothing.
boys consider
There its
is
no way
of estimating,
lifetime to be eternal.**
have one suggestion to make
I
tion
which
I
and the continuouscreation
don't think
is
for
astronomic time, however (a sugges
me). The sun, accord
particularly original with
ing to reasonable estimates, revolves about the galactic center once every
200,000,000 vears. This year."
we could
seconds.
On
year"
call a "galactic
(An ugh" word, but never mind!) One
galvear
the other hand, a "picogalvear"
is
is
or, better, a "gal
equal to 6250 tera
equal to
1
hour and 45
minutes. If
we
stick to galyears then, the entire fossil record covers at
the total
3 galyears;
the total
But now
life
I've got to trv
for small units of time.
[*
Since this article
never mind,
of the solar system thus far
oka red dwarf
life
it's
first
as a red
dwarf
is
is
most only
onlv 25 galvears; and
perhaps 500 galvears.
the other direction, too, and see what happens
Here
appeared,
at least there are
my
no
age has increased to
common 1%
units to con
gigaseconds, alas, but
better than the alternative.
[** Since this article was written, the continuous creation theory has about been wiped out,
and
it
isn't likely that
the Universe, in
its
present form, at
least, is eternal.)
AMOEBA The amoeba
and
a onecelled animal
is
in
usually considered the
most
primitive of the type. It has no fixed shape as other onecelled animals l
"protozoa"
(
Greek
is
.
It
moxes bx means of these pseudopods and that
considered the most primitixe form of animal locomotion.
The
shape
fact that its
name, which of
hare but can bulge at anx point to form a "pseudopod'
i
for "false foot"
is
not
but
fixed,
from the Greek word
is
is
changeable,
for "change."
amoeba we commonlx mean when the name
tion
is
"Amoeba
proteus" which
his
shape at
is
is
the basis of
is
used without qualifica
name
the
its
particular species
found on decaying organic matter
The word "proteus"
streams and ponds.
who could change
is
The
of a
in
Greek demigod
will.
There are numerous other species of amoeba, some of which are parasitic,
and
tolytica
which can parasitize man. One of them. Entamoeba ), causes ameobic dysentery.
six of
Although the amoeba
is
mentioned
organism,
it is
within
its
single cell, include all the
of
A human
life.
an amoeba has
not
(
as
cell, far
2. jog
in the article as the type of small
also indicated
is
more

a small
cell.
The amoeba must,
machinery for the essential functions
specialized, can afford to be smaller. Thus,
times the xolume of a typical body
human
25,000 times the xolume of the smallest
The
his
("amoebawithin: celldissohing"
cell,
smallest freelixing cells are the bacteria,
cell
and about
the spermatozoon.
and the amoeba has
210,
000,000 times the xolume of the smallest bacteria.
The
smallest objects that can be considered alixe
tion onlx within cells they parasitize 2 rpo, 000, 000, 000
1
we
are to the
,w3>'
.&h

K^£. The Granger Collection
The amoeba has The amoeba is as large
are the viruses.
the yolume of the smallest virus.
to that smallest \irus as
(though they func
amoeba.
NUMBERS AND MEASUREMENT
34
fuse us. Scientists have therefore been able to use millisecond
second
freely,
and now they can
join to that
and micro
nanosecond, picosecond, fem
tosecond, and attosecond.
These small units of time
When
Glenn
a Gagarin or a
aren't very useful in the macroscopic world. circles
the earth at
miles a second, he trav
5
than 9 yards in a millisecond and less than a third of an inch in a microsecond. The earth itself, moving at a velocity of i^y2 miles a second els less
in
about the sun, moves only a
travels
its
little
over an inch in a microsec
ond. In other words, at the microsecond level, ordinary motion
However, the motion of
light
is
frozen out.
more rapid than any ordinary motion,
is
while the motion of some speeding subatomic particles
is
nearly as rapid
as that of light. Therefore, let's consider the small units of
time in terms
of light.
DISTANCE COVERED BY LIGHT
second
186,200
millisecond
186
miles
microsecond
327
yards
nanosecond
1
1 1
1 1
1
Now, you may
when
it
foot
Vso inch
picosecond
think that at picosecond levels subatomic motion and
even lightpropagation "frozen"
miles
is
"frozen." After
moved an
inch.
all, I
How much
dismissed earth's motion as
more
so,
when
then,
thou
sandths of an inch are in question.
However, there ^500,000,000 of
its
is
a difference.
own
diameter.
The
A
its
own
years.
its
own
diameter.
To
moving an
inch,
moves
speeding subatomic particle moving at
almost the speed of light for a distance
000 times
earth, in
travel a
moves
120,000,000,
hundred and twenty
billion times
y80
of an inch
diameter, the earth would have to keep on going for 1,500,000
For Gagarin or Glenn to have traveled
billion times their
own
hundred and twenty
for a
diameter, thev would have had to stay in orbit a
full year.
A
subatomic particle traveling
"frozen," and has time to
make
%
is
therefore anything but
number
of collisions with other
of an inch
a fabulous
subatomic particles or to undergo internal changes. As an example, neutral pions break
down
in a
matter of
0.1
femtoseconds after formation.
What's more, the omegameson breaks down attoseconds
or,
roughly, the time
of an atomic nucleus
and back.
it
would take
in
something
like 0.0001
light to cross the diameter
PREFIXING The
entire range of time, then,
IT
UP
135
from the lifetime of an omegameson to
that of a reddwarf star covers a range of 40 orders of magnitude. In other
words, during the normal
life
of a red dwarf,
some 10 40 omegamesons
have time to come into existence and break down, one after the other.
To summarize,
the measurable lengths cover a range of 41 order of mag
nitude, the measurable masses 83 orders of magnitude,
times 40 orders of magnitude. Clearly,
we
and the measurable
are not overdoing
ing the metric system from 6 to 30 orders of magnitude.
it
in
expand
PART
IV
Numbers and Calendar
the
THE DAYS OF OUR YEARS
11.
A
group of us meet for an occasional evening of talk and nonsense,
lowed by coffee and doughnuts and one of the group scored persuading a wellknown entertainer to attend the session.
made one
entertainer
He was
condition, however.
a
fol
coup by
The wellknown
not to entertain, or even
be asked to entertain. This was agreed to.*
Now
there arose a problem.
someone was sure
If
the meeting were left to
own
its
entertainment had to be supplied, so one of the boys turned to said, "Say,
I
objected at once.
said,
I
''How can
I
stand up there
audience and
talk with everyone staring at this other fellow in the
wishing he were up there instead? You'd be throwing
But they I
all
smiled very toothily and told
(Somehow evenone
give.
me and
you know what?"
knew what and
I
and
devices,
to begin badgering the entertainer. Consequently, other
putty as soon as the flattery
me
me
to the wolves!"
about the wonderful
quickly discovers the fact that is
turned on.
thrown to the wolves. Surprisingly,
it
)
In
no time
at
I
all, I
talks
soften into
agreed to be
worked, which speaks highly for the
audience's intellect— or perhaps their magnanimity.
As
it
happened, the meeting was held on "leap day" and so
conversation was readymade and the gist of
It forces itself
upon the awareness
humanoids. However, the dav
[* I I
went
suppose there's no question but that the
I
was the day. of
it
didn't
name
and asked
known
for his
entertainer."]
topic of
as follows:
earliest unit of timetelling
of even the
most primitive
not convenient for long intenals of
this article first appeared in August 1964, because Mas wrong, because when I met him again months autograph, he wrote "To Isaac, with best wishes, from a well
the entertainer
thought he wouldn't want
later
is
my
me
when
to. I
NUMBERS AND THE CALENDAR
I4O
Even allowing a primitive lifespan of thirtv years, a man would some 11,000 days and it is very easy to lose track among all those davs. time.
Since the Sun governs the dayunit,
it
seems natural to turn to the next
most prominent heavenlv bodv, the Moon, self at
One offers itThe Moon waxes from
for another unit.
once, readymade— the period of the phases.
nothing to a
Moon and
full
This period of time
word "moon")
more
or
then to nothing in
called the
is
"month"
in
a definite period of time.
English (clearly from the
we have
the "lunar month," since
specificallv,
live
other months, representing periods of time slightlv shorter or slightlv longer than the one that
The
month
lunar
is
to 29 days, 12 hours,
is
strictlv tied to
roughly equal to
davs.
More
it
mav
well have been that
man was
more convenient
it is
equal
no
special signifi
the month, which remained only a convenient de
itself to
vice for measuring moderately long periods of time.
primitive
exactlv,
44 minutes, 2.8 seconds, or 29.5306 days.
In preagricultural times,
cance attached
the phases of the moon.
29%
probablv something
like
The
life
expectancy of
350 months, which
is
a
much
figure than that of 11,000 davs.
In fact, there has been speculation that the extended lifetimes of the
may have
patriarchs reported in the fifth chapter of the
Book
arisen out of a confusion of years with lunar
months. For instance, sup
of Genesis
pose Methuselah had lived 969 lunar months. This would be just about 79 years, a very reasonable figure. However, once that got twisted to 969 years by later tradition we gained the "old as Methuselah" bit.
However, seriously
I
mention
by any
this
only in passing, for this idea
biblical scholars. It
is
much more
Flood. It is
.
.
my
.
But
am
I
off
really taken
the subject.
feeling that the
month gained
with the introduction of agriculture.
and precariouslv
closely
Nomads
ing society was. ers
not
hangover from Babylonian tradition about the times before the
are a
more
is
likely that these lifetimes
had
to
stay
a
An
new and enhanced importance was much
agricultural society
tied to the seasons
than a hunting or herd
could wander in search of grain or grass but farm
where thev were and hope
for rain.
To
increase their
chances, farmers had to be certain to sow at a proper time to take advantage of seasonal rains and seasonal warmth; and a mistake in the sowing
period might easily spell disaster. What's more, the development of agriculture
made
possible a denser population,
and that
intensified the scope
of the possible disaster.
Man was
had
still
came
in
to
pay attention, then, to the cvcle of seasons, and while he
the prehistoric stage he must have noted that those seasons
full cycle in
roughly twelve months. In other words,
if
crops were
THE DAYS OF OUR TEARS

planted at a particular time of the year and
all went well, then if twelve months were counted from the first planting and crops were planted again, all would again go well. Counting the months can be tricky in a primitive societv, especiallv
when
miscount can be ruinous, so
a
it isn't
surprising that the count
usually left in the hands of a specialized caste, the priesthood.
The
was
priests
could not only devote their time to accurate countings but could also ose
and
their experience
seasons was by
to propitiate the gods. After
skill
no means
as rigid
and unvarying
as
and night or the cvcle of the phases of the moon. of rain could blast that seasons crops,
bound lieved It
to follow
mistake in
little
ritual
frost or a failure
flaws in weather
(at least so
men
were
often be
the priestly functions were of importance indeed.
,
is
any
was the cvcle of dav
A late
and since such
the cycle of the
all,
not surprising then, that the lunar month grew to have enormous
religious significance.
There were new
Moon
and
festivals
called the "svnodic
to be
and twelve lunar months
there
month."
The cycle of seasons make up a "lunar
fore
is
called the "year"
year."
referred to as the use of a
people in
modem
The
use of lunar years
"lunar calendar."
The
ir.
measuring time
Mohammedan
made up
of 29
vears
is
made up
and 30 days
:
;
onlv impc::r.: r:?up of
times, using a strict lunar calendar, are the
medans. Each of the are. in turn, usuallv
special priestly
month came
proclamations of each one of them, so that the lunar
of iz
Moham
months which
in altematicr.
Such months average 29.5 davs, but the length of the true lunar month [Ve pointed out 29.5336 davs. The lunar vear built up out of twelve 29.5day :
months
is
354 days long, whereas twelve lunar months are actually
days long.
54 57
say "So what?" but don't A true lunar year should always start on the dav of the new Moon. If, however, vou start one lunar year on the
You may
new Moon and then simplv
and 30day months, the third new Moon, and the sixth year will start two days before the new Moon. To properly religious people, this would alternate 29day
year will start the dav before the
be unthinkable.
N actly
:
•
it
so
happens that 30 true lunar vears come out to be almost
an even number of da] —::::: :::
29.5day
months come
with the
Moon. For
to 10.62c
days— just
that reason, the
TTiirtv years built
11 days short of keeping
Mohammedans
ex
up out of
scatter 11
time days
through the 32 years in some fixed pattern which prevents any individual years from starting as much as a full dav ahead or behind the New Moon.
THE CRESCENT MOON The
crescent
Moon, which marked
month in ancient Moon, was responsible
the beginning of the
times, together with the remaining phases of the
for the birth of astronomy, for surely the regularly changing shape of the
Moon
was the
The
object in the sky that roused man's curiosity.
first
and value
necessities
of calendarmaking
must have urged man on
to
develop mathematics and religion out of the lunar cycle.
There was something
The
else, too.
.
.
.
ancient Greek philosophers found
it
vide the Universe into two parts: the Earth
do
aesthetically satisfying to di
and the heavenly
bodies.
they sought for fundamental differences in properties. Thus:
so,
heavenly bodies were
all
luminous, while the earth was nonluminous.
The Moon, however, had
to be
relationship of the phases of the
Moon and Sun made
it
To The
clear
an exception to
Moon
this general rule.
even in ancient times that the
only by reflected sunlight. That meant that, of
and nonluminous as the Earth. What's more, when the Moon
The
to the relative positions of the
its
own, the
Moon shone Moon was as
dull
is
phase and
in its crescent
thin sliver of curling light, as in the illustration, the rest of the
sometimes seen shining with a dim ruddy out that from the the
Moon
was
as
as the
in Earthlight. Earth, too, reflected light
and
it
could be seen to be a world of some
in diameter to appear to be as large as
that distance. In short, thanks to the
might many other heavenly bodies
An
seemed from
if
the
sufficed
Moon
was
be.
In each 30year cycle there are nineteen 354day years years,
it
Moon, nakedeye astronomy
to establish the doctrine of "plurality of worlds," for a world so
and
Greeks had already determined the distance of
quite accurately,
two thousand miles
is
own. Galileo pointed
Moon.
too, the ancient
Moon
just a
Moon
the Earth was seen in the full phase and that
was shining dimly
luminous
Then, the
Moon,
light of its
is
and eleven 355day
and the calendar remains even with the Moon. extra day, inserted in this
movements
of a heavenly body,
way is
to
keep the calendar even with the
called
an "intercalary day"; a day
in
serted "between the calendar," so to speak.
The ever,
lunar year, whether
match the
it is
354 or 355 days in length, does not, howBy the dawn of historic times the
cycle of the seasons.
)
NUMBERS AND THE CALENDAR
144
Babylonian astronomers had noted that the Sun moved against the back
ground of
stars.
This passage was followed with absorption because
grew apparent that
complete
a
complete cycle of the seasons
circle of the skv
it
by the Sun matched the
(This apparent influence of the stars
closely.
on the seasons probably started the Babylonian fad of astrology— which is still
with us today.
The Sun makes
complete cycle about the zodiac
its
so that the lunar year
Three lunar
"solaryear."
month behind This
is
about years
11 days shorter fall
in roughly 365 days, than the seasoncycle, or
more than
33 days, or a little
important. is
If
vou use
and
a lunar calendar
start it so that the first
planting time, then three years later you are planting a
too soon, and by the time a decade has passed you are planting in
midwinter. After 33 years the first day of the year is back where posed to be, haying traveled through the entire solar year.
This
month
a full
the seasoncycle.
day of the year
month
is
is
exactly
of the
holy because
what happens
Mohammedan
it
was the month
revelation of the Koran. In
in
year in
is
Mohammedan
the
Mohammed
which
Ramadan,
therefore,
The
year.
named Ramadan, and
it is
sup
ninth
especially
it is
began
to receive the
Moslems
abstain from
food and water during the davlight hours. But each vear,
Ramadan
falls
a bit earlier in the cvcle of the seasons,
and
at 33vear intervals
found
this
time abstaining from drink
in the
hot season of the vear; at
particularly wearing,
and Moslem tempers grow
The Mohammedan vears when Mohammed
the date
are fled
it is
to
be is
particularly short.
numbered from the Hegira; that is, from from Mecca to Medina. That event took
place in a.d. 622. Ordinarily, vou might suppose, therefore, that to find the
number of the Mohammedan year, one need only subtract 622 from the number of the Christian vear. This is not quite so, since the Mohammedan year is shorter than ours. I write this chapter in a.d. 1964 and it is
now
1342 solar years since the Hegira. However,
the Hegira, so that, as
I
write, the
I've calculated that the
tian vear in
Mohammedan
vear
will
it is
1384 lunar years since
a.h. 1384.
is
year will catch up to the Chris
about nineteen millennia. The vear
20,874, and the Moslems
minimum
Moslem
a.d.
20,874
"^
a ^ s0
^e
AH

then be able to switch to our year with a
of trouble.
But what can we do about the lunar year with the seasons and the solar year? for then the next year
would not
cient Babylonians, for instance, a
in order to
We can't
start
just
with the
new Moon
add
make
it
new Moon and
start
was
keep even
11 days at the end,
essential.
to the an
THE DAYS OF OUR YEARS However,
we
if
new Moon and
with the
start a solar year
find that the twentieth solar year thereafter starts
the
new Moon. You
M5 wait,
we
will
once again on the day of
19 solar years contain just about 235 lunar
see,
months.
Concentrate on those 235 lunar months. That
(made up
years
We
could, then,
hammedans
equivalent to 19 lunar
is
months each) plus 7 lunar months left over. we wanted to,' let the lunar years progress as the Mo
of 12 lunar if
had passed. At
do, until 19 such years
time the calendar
this
would be exactly 7 months behind the seasons, and by adding 7 months the 19th year (a 19th year of 19
months— very
19vear cycle, exactly even with both the
The Babylonians were months behind the
neat)
Moon and
seasons. Instead, they
Each
The
month" was added
"intercalary
themselves
let
and 19th year of each
fall
7
as nearly evenly as pos
had twelve 12month years and seven 13month
cycle
to
new
added that 7month discrepancy
through the 19year cycle, one month at a time and sible.
start a
the seasons.
however, to
unwilling,
we could
nth,
in the 3rd, 6th, 8th,
cycle, so that the vear
years.
14th, 17th,
was never more than about 20
days behind or ahead of the Sun.
Such
based on the lunar months, but gimmicked so
a calendar,
keep up with the Sun,
The Babylonian it
is
lunarsolar calendar
was popular
in
ancient times since
adjusted the seasons while preserving the sanctity of the
Hebrews and Greeks both adopted
this calendar
basis for the Jewish calendar today.
calendar are allowed to
month is why the
added,
is
slightly
Yom
Moon. The
fact, it
is still
the
individual dates in the Jewish
slightly
until the intercalary
ahead of the Sun. That
Kippur occur on different days of
calendar (kept strictlv even with the Sun) each year. These holi
same dav
early Christians
centuries,
of the vear each year in the Jewish calendar.
continued to use the Jewish calendar for three
and established the day
passed, matters
solar calendar
of Easter
on that
grew somewhat complicated,
becoming Christian
in swelling
and were puzzled
Some formula had
to
basis.
for the
As the centuries
Romans (who were
numbers) were no longer used at the erratic
to a lunar
jumping about of Easter.
be found by which the correct date for Easter could
be calculated in advance, using the It
The
and, in
behind the Sun
when they suddenly shoot
days occur on the
The
fall
holidavs like Passover and
civil
as to
a 'lunarsolar calendar."
Roman
calendar.
in a.d. 325 (by which time Easter was to fall on the that Christian), officially
was decided at the Council of Nicaea,
Rome had become Sunday
after the first full
the vernal
Moon
after the vernal equinox, the date of
equinox being established as March
21.
However, the
full
1
NUMBERS AND THE CALENDAR
46
Moon
referred to
is
is
not the actual
is
Moon"
the "Paschal Full
Hebrew word
the
Dominical
The
Letters,
result
and can
fall
for Passover).
according
calculated
is
to
which
a
Moon, but
one called
a fictitious
The
Moon
date of the Paschal Full
formula involving Golden Numbers and
won't go
I
that Easter
into.
jumps about the days of the
still
March
as early as
full
("Paschal" being derived from Pesach, which
Many
22 and as late as April 25.
year
civil
other
church holidays are tied to Easter and likewise move about from year
to
year.
Moreover,
all
Christians have not always agreed on the exact formula
bv which the date of Easter was to be calculated. Disagreement on
this
was one of the reasons for the schism between the Catholic Church
detail
of the
West and
Ages there was
Church of the East. In the early Middle Church which had its own formula.
the Orthodox
a strong Celtic
Our own calendar is inherited from Egypt, where seasons were unimThe one great event of the year was the Nile flood, and this took place (on the average) every 365 davs. From a very early date, certainly portant.
as early as
2781
B.C.,
Moon
the
was abandoned and
a "solar calendar,"
adapted to a constantlength 365day year, was adopted.
The
solar calendar kept to the tradition of 12
year was of constant length, the 30 days each. This
meant
a "calendar
Of
of constant length,
that the
month, but the Egyptians didn't is
months, however. As the
too— new Moon could fall on any day of the care. (A month not based on the Moon months were
month.")
course 12
months
of 30 davs each
the end of each 12month cycle,
5
add up only
to 360 davs, so at
additional days were added and treated
as holidays.
The
solar year, however,
is
not exactly 365 davs long. There are several
kinds of solar years, differing slightly in length, but the one upon which the seasons depend
the "tropical year," and this
is
is
about 365% days
long.
This means that each year, the Egyptian 365dav year
hind the Sun. As time went on the Nile flood occurred the vear, until finally
it
had made
a
complete
%
dav be
and
later in
falls
later
circuit of the year. In
1460
would be 1461 Eg\ptian years. This period of 1461 Egyptian years was called the "Sothic cycle," from Sothis, the Egyptian name for the star Sirius. If, at the beginning of one tropical years, in other words, there
Sothic cvcle, Sirius rose with the Sun on the vear,
it
would
rise later
and
later
first
day of the Egyptian
during each succeeding vear until
finally,
THE DAYS OF OUR YEARS new cycle would Day once more.
1461 Egyptian years later, a
Sun on
New
Year's
The Greeks had learned about when Eudoxus of Cnidus made getes, the
Macedonian king that quarter
take
dar to
H7
begin as Sothis rose with the
that extra quarter day as early as 380 B.C.,
the discovery. In 239 b.c. Ptolemv Euer
of Egypt, tried to adjust the Egyptian calen
day into account, but the ultraconservative
Egyptians would have none of such a radical innovation.
Roman
Meanwhile, the
Republic had a lunarsolar calendar, one in
which an intercalary month was added every once officials in
in a while.
The
priestly
charge were elected politicians, however, and were by no means
as conscientious as those in the East.
or not according to
annually elected
The Roman
whether thev wanted
officials in
power were of
(when they were not). By 46 b.c, the
their
Roman
added a month (when the other
priests
a long year
own
party) or a short one
calendar was 80 days be
hind the Sun.
Caesar was
Julius
sense.
He had
just
in
power then and decided
to
put an end to
this
non
returned from Egypt where he had observed the con
venience and simplicity of a solar year, and he imported an Egyptian
tronomer, Sosigenes, to help him'. Together, they
let
46
b.c.
as
continue for
445 days so that it was later known as "The Year of Confusion." However, this brought the calendar even with the Sun so that 46 b.c. was the last year of confusion.
With 45 which the
b.c.
the
five extra
Romans adopted
a
modified Egyptian calendar in
days at the end of the year were distributed through
out the vear, giving us our months of uneven length. Ideally,
have seven 30dav months and
Romans
considered February an unlucky
we ended with a silly arrangement months, and one 28day month.
month and shortened
In order to take care of that extra
%
as
February
so that
and Sosigenes estab(Under the numbering
day, Caesar
of the years of the Christian era, every year divisible
day— set
it,
of seven 31day months, four 30day
lished even fourth year with a length of 366 days.
calary
we should
31dav months. Unfortunately, the
five
by 4 has the
29. Since 1964 divided by 4
is
inter
491, without a
remainder, there
This
is
is a February 29 in 1964.) the "Julian year," after Julius Caesar. At the Council of Nicaea,
the Christian
Church adopted the
Julian calendar. Christmas
was
finally
accepted as a Church holiday after the Council of Nicaea and was therefore given a date in the Julian year. It does not, therefore,
from vear to vear
as Easter does.
bounce about
JULIUS CAESAR
whom
Julius Caesar, for
known among
ter
He was born man of ancient who,
wastrel,
the Julian calendar
in 102 B.C.,
He
times.
in
middle
and he was
man
was a
life,
named,
is
many
the general public for
is,
of course, far bet
other reasons.
about the most remarkable
just
enormous courage,
of
a playboy
and
turned to leading armies and proved himself
who never lost a battle. He was a great orator, second among the Romans, and a great writer. And he was a suc
to be a great general
only to Cicero
cessful politician.
His charm was legendary. In j6 tured by pirates
he set
B.C.
sail for
On
under the best Greek teachers.
in order to study
the island of
Rhodes
the way, he was cap
held him for ransom of about Sioo,ooo in modern
who
money. While the money was being scraped up by friends and
relatives,
Caesar charmed his captors and had a great time with them. While they
were engaged in friendly conversation, Caesar told them that once he was fleet and hang every one of them. The and when Caesar was paid for and freed, he fleet and hang them all.
he would return with a
set free,
pirates laughed at the joke,
did indeed return with a
With ficult
the
Roman
to rule the
[in the course of
Republic slowly decaying as
empire
proved increasingly in civil
which he entered Egypt and had a famous love
with Cleopatra) and
man
it
had gathered, Caesar engaged
it
finally
emerged
as sole ruler
dif
war
affair
and
dictator of the Ro
He
firmly believed that
realm.
Here was where
his
one great
failing
showed
an enemy forgiven was an enemy destroyed. fought on the other conspired against
up.
He
forgave
many who had
and gave them high positions in the him, and on March 15, 44 b.c. (the Ides side
state.
of
They
March),
they assassinated him.
The
365day year
February
is
weeks and
just 52
6, for instance,
is
on
a
Sunday
next year, on a Tuesday the year
day in
years,
is
day long. This means that
one
and
year,
it is
on
a
Monday
if
the
so on. If there were only 365
then any given date would move through the days of the week
steady progression.
52 weeks
after,
1
in
and
If a
2 days long,
on Thursday the year
366day year
and
after.
if
is
involved, however, that year
is
on Tuesday that
it
February 6
The day
is
year,
has leaped over Wednesday.
It
NUMBERSANDTHECALENDAR
150 is
29
for that reason that the 366day year
and February
called "leap year"
is
"leap day."
is
would have been well
All
days long; but
it isn't.
The
the tropical year were really exactly 365.25
if
tropical year
is
365 days,
46 seconds, or 365.24220 days long. The Julian vear minutes 14 seconds, or 0.0078 days, too long.
may not seem much, but
This
day on the tropical year
it
means that the
in 128 years.
hours, 48 minutes,
5
on the average,
is,
11
Julian year gains a full
As the Julian year
gains, the vernal
equinox, falling behind, comes earlier and earlier in the year. At the
Council of Nicaea a.d.
453
in a.d. 325, the vernal
was on March
it
20,
by
a.d. 581
equinox was on March
on March
and
19,
1263, in the lifetime of Roger Bacon, the Julian vear
days on the Sun and the vernal equinox was on Still
not
fatal,
and Easter was
March
so on.
By
21.
Bv
a.d.
had gained eight
1 3.
but the Church looked forward to an indefinite future
tied to a vernal
equinox at March
were allowed
21. If this
go on, Easter would come to be celebrated in midsummer, while
to
Christmas would edge into the spring. In 1263, therefore, Roger Bacon wrote a
letter to
Pope Urban IV explaining the
situation.
The Church,
however, took over three centuries to consider the matter.
By 1582 the equinox was First,
Julian calendar
falling
on March
had gained two more days and the vernal 11.
Pope Gregory XIII
he dropped ten days, changing October
5,
finally
took action.
1582 to October
15, 1582.
That brought the calendar even with the Sun and the vernal equinox 1583 fell on March 21 as the Council of Nicaea had decided it should.
The
next step was to prevent the calendar from getting out of step
again. Since the Julian year gains a full day every 128 years, full
in
days in 384 years
centuries.
That means
or, to
approximate
slightly, three full
it
gains three
days in four
that every 400 days, three leap years (according to
the Julian system) ought to be omitted.
Consider the century years— 1500, 1600, 1700, and so on. In the Julian by 4 and are therefore leap years. Every
year, all century years are divisible
400 years there are 4 such century years, so why not keep 3 of them ordinary years, and allow only one of them (the one that is divisible by 400) to be a leap year? This arrangement will match the year more closely to the Sun and give us the "Gregorian calendar."
To summarize:
Every 400
years,
the Julian calendar allows 100 leap
years for a total of 146,100 days. In that
same 400
years, the
calendar allows only 97 leap years for a total of 146,097 days.
Gregorian
Compare
these lengths with that of 400 tropical years, which comes to 146,096.88.
)
^
THE DAYS OF OUR YEARS Whereas,
in that stretch of time, the Julian year
had gained
3.12 days
on
the Sun, the Gregorian year had gained only 0.12 days. 0.12 days
Still,
is
nearly
Gregorian calendar
5000 we
will
and
hours,
3
this
have gained a
will
full
means that in 3400 years the day on the Sun. Around a.d.
have to consider dropping out one extra leap year.
But the Church had waited the job a century
earlier, all
without trouble. By
much
1582, however,
a.d.
Had
a little too long to take action.
it
done
western Europe would have changed calendars
Europe had
of northern
turned Protestant. These nations would far sooner remain out of step with the Sun in accordance with the dictates of the pagan Caesar, than consent to be corrected by the Pope. Therefore they kept the Julian year.
The
year 1600 introduced no
crisis.
was divisible by 400. Therefore,
it
was
was a century year but one that
It
by both the Julian and
a leap year
Gregorian calendars. But 1700 was a different matter. dar had
it
and the Gregorian did
as a leap year
Julian calendar
was going
days altogether).
to
not.
The
Julian calen
By March
1700, the
1,
be an additional day ahead of the Sun (eleven
Denmark, the Netherlands, and Protestant Germany
gave in and adopted the Gregorian calendar.
Great Britain and the American colonies held out until 1752 before in. Because of the additional day gained in 1700, they had to drop
giving
eleven days
and changed September
2,
1752 to September
13, 1752.
There
many people came quickly to were riots all over England the conclusion that they had suddenly been made eleven days older by as a result, for
legislation.
"Give us back our eleven days!" they cried
(A more
rational objection
was the
in despair.
fact that
although the third quarter
of 1752 was short eleven days, landlords calmly charged a full quarter's rent.
As
a
result of this,
turns out that
it
"Washington's birthday."
He was
Washington was not born on 22, 1732 on the
born on February
Gregorian calendar, to be sure, but the date recorded in the family Bible
had
to
place,
be the Julian date, February
Washington— a remarkably
When the changeover took man— changed the date of his
11, 1732.
sensible
birthday and thus preserved the actual day.
The
Eastern Orthodox nations of Europe were more stubborn than the
Protestant nations.
The
years
1800 and 1900 went by. Both were leap
by the Julian calendar, but not by the Gregorian calendar. By 1900, then, the Julian vernal equinox was on March 8 and the Julian calendar
years
was 13 days ahead of the Sun.
It
was not
until after
World War
I
that the
NUMBERS AND THE CALENDAR
152
Soviet Union, for instance, adopted the Gregorian calendar.
the Soviets
so,
made
made
Some which It
a slight modification of the leap vear pattern
matters even more accurate.
day on the Sun until
The
is
why
fullv 35,000 vears pass.)
the Orthodox Christmas
December 25 by
in
falls
cling to the Julian vear,
on Januarv 6 on our calendar.
me—
was myself born at a time when the Julian calendar was
the— ahem— old country.** Unlike George Washington,
the birthdate and, as a result, each year earlier
than
And it
still
their calendar.
In fact, a horrible thought occurs to I
this
I
celebrate
my
making myself 13 days older than 13day older me is in all the records and
I
which
Soviet calendar will not gain a
of the Orthodox churches, however,
is still
(In doing
should,
I
still
in force
never changed
birthday 13 days
I
have to be.
I
can't ever
change
back.
Give
me
back
my
13 days! Give
Well, the Soviet Union,
if
me
you must know.
back
I
my
13 days! Give
came here
at the age of
me
3,
back
.
.
.
Each da}
AT THE BEGINNING
BEGIN.
12.
year,
New
another
upon
follows hard
Year's
New
Day
upon
falls
and because
us;
always a doubled occasion for great and somber soulsearching on
Perhaps
I
can
make my
consciousness of passing time
who
thinking more objectively. For instance, Year's
What is there day? What makes
Day?
any other In fact,
my
birth
Year's Day, the beginning of the year
New
about
January
when we chop up time
1
less
poignant by
on
New
different
from
says the year starts
Day
Year's
that
is
is
my part.
so special?
into any kind of units,
how do we
decide
with which unit to start?
For instance,
begin at the beginning (as
let's
I
dearly love to do)
and
consider the day.itself.
The day
is
composed
two
of
parts, the
daytime* and the night. Each,
separately, has a natural astronomic beginning. sunrise; the night begins
the night but that
is
a
with sunset.
mere
daytime plus night
is,
is
live,
however, both daytime
(one growing longer
as the
therefore, a certain convenience in using
start at sunrise or at sunset?
since in a primitive society that
very annoying that "day"
upon
The com
of nearly constant duration.
Well, then, should the day
* It is
twilight encroach
as a single twentyfourhour unit of time.
bination of the two, the day,
first,
humanity
in length during the year
other grows shorter) and there
argue for the
(Dawn and
begins with
detail.)
In the latitudes in which most of
and night change
The daytime
means both the
is
when
sunlit portion of time
fourhour period of daytime and night together. This
is
You might the workday
and the twenty
a completely unnecessary short
coming of the admirable English language. I understand that the Greek language contains separate words for the two entities. I shall use "daytime' for the sunlit period and "day" for the twentyfourhour period.
NUMBERS AND THE CALENDAR
154
On
begins.
is when the worknew beginning. and some the other. The Egyptians, for
the other hand, in that same society sunset
day ends, and surely an ending means
Some
groups
began the day
instance,
The
made one
decision
a
at sunrise, while the
latter state of affairs
is
Hebrews began
reflected in the very
first
it
at sunset.
chapter of Genesis
which the davs of creation are described. In Genesis 1:5 it is written: the evening and the morning were the first day." Evening (that is,
in
"And
night) comes ahead of morning (that
is,
daytime) because the dav
starts
at sunset.
This arrangement holidays
still
of Judaism
maintained
is
in
Judaism to
and Jewish
day,
this
begin "the evening before." Christianity began as an offshoot
and remnants
of this sunset beginning cling even
now
some
to
nonJewish holidays.
The
expression Christmas Eve,
if
taken
literally, is
the evening of De
know it really means the evening of December 25, which it would 24— naturally mean if Christmas began "the evening before" as a Jewish holiday would. The same goes for New Year's Eve. cember
but
as
we
all
Another familiar example
All Hallows' Eve, the evening of the day
is
before All Hallows' Dav, which
is
Hallows' Eve
is
Dav
on the evening of
therefore
is
that All Hallows' Eve
commemoration of all 1, and All October 31. Need I tell you
given over to the
the "hallows" (or "saints"). All Hallows'
known by
better
its
is
on November
familiar contracted
form of
"Halloween."
As
matter of
fact,
though, neither sunset nor sunrise
ning of the day.
The
period from sunrise to sunrise
a
is
is
now
slightly
the begin
more than
24 hours for half the vear as the daytime periods grow shorter, and slightly less
than 24 hours for the remaining half of the year
grow
longer. This
is
also true for the period
as the
from sunset
daytime periods
to sunset.
Sunrise and sunset change in opposite directions, either approaching
each other or receding from each other, so that the middle of daytime
(midday) and the middle of night (midnight) remain intervals
fixed at 24hour
throughout the year. (Actually, there are minor deviations but
these can be ignored.)
One
can begin the day at midday and count on a steady 24hour cycle,
but then the working period
is
split
better to start the day at midnight
and
that, in fact,
Astronomers,
is
what we
who
are
all
dates.
Far
decent people are asleep;
do.
among
at midnight, long insisted
between two different
when
the indecent minority not in bed alseep
on starting
their
day at midday so as not to
break up a night's observation into two separate dates. However, the
spirit
BEGIN AT THE BEGINNING was not to be withstood, and
of conformity
1
55
1925, they accepted the
in
inconyenience of a beginning at midnight in order to get into step with the rest of the world.
All the units of time that are shorter than a
no problem. You
offer
you
day;
start
day depend on the day and
counting the hours from the beginning of the
start
counting the minutes from the beginning of the hours, and
so on.
Of
when
course,
the start of the day changed
its
position, that affected
and the night were
the counting of the hours. Originally, the daytime
each di\ided into twelve hours, beginning
The hours changed
sunset.
and night
and
sunrise
length with the change in length of daytime
that in June
so
respectively,
at,
(in
the northern hemisphere)
the daytime
was made up of twelve long hours and the night of twelve short hours,
December the
while in
situation was reversed.
This manner of counting the hours
"Tierce" ("three") is
3
is
the dav
is
is
is
the term for 6 a.m.
is
12 a.m.,
and "none" ("nine")
located in the middle of the afternoon
The warmest
warmest.
Church
survives in the Catholic
("one")
9 a.m., "sext" ("six")
p.m. Notice that "none"
when to
still
"canonical hours." Thus, "prime"
as
part of the day might well be
felt
be the middle of the day, and the word was somehow switched to the
astronomic midday so that
we
call 12
a.m. "noon."
This older method of counting the hours also plays a part
in
one of the
parables of Jesus (Matt. 20:116), in which laborers are hired at various
times of the dav, up to and including "the eleventh hour."
hour referred to
in the parable
is
day ends. For that reason, "the eleventh hour" has come to
moment lost
on
which something can be done. The force
in
us,
however, for
we
and
too late— we ought to be asleep by then.
the last
of the expression
11 a.m.
is
rest.
(The
rationale was that
it
was an unlucky day.)
Jews, captive in Babvlon in the sixth century B.C., picked
notion and established rather than of
ill
it
on
fortune.
a religious basis,
They explained
making
its
it
a
up the
dav of happiness
beginnings in Genesis 2:2
where, after the work of the six days of creation— "on the seventh day
ended
To
his
is
originated in the Babylonian calendar where one day out of
seven was devoted to
The
mean
too early in the day to begin to feel panicky,
a.m. or 11 p.m.,
The week
eleventh
think of the eleventh hour as being either 11
while 11 p.m.
is
The
one hour before sunset when the working
work which he had made; and he
rested
God
on the seventh day."
those societies which accept the Bible as a book of special
signifi
.
NUMBERS AND THE CALENDAR
156
Hebrew word
cance, the Jewish "sabbath" (from the
defined as the seventh, and
marked Saturday on our of a
new week.
Sunday
The
first
and Sunday,
calendars,
therefore,
thus
day
first
columns with
and Saturday seventh.
early Christians
began to attach special significance to the
had taken place on
a
became important
to
Sunday. Then, too,
them
first
was the "Lord's day" since the Resur
it
as
Christians began to think of themselves as something sect, it
is
the one
is
the
is
All our calendars arrange the days in seven
day of the week. For one thing, rection
for "rest")
day of the week. This day
last,
to
have distinct
time went on and more than a Jewish
own. In
rituals of their
became the day Saturday and Sunday are
Christian societies, therefore, Sunday, and not Saturday,
(Of
of rest.
course, in our
both days of
and
rest,
are
modern effete times, lumped together as the "weekend,"
a period
celebrated by automobile accidents.)
The
work week begins on Monday causes
fact that the
people to think of that
lowing children's puzzle (which neatly the
You
first
time
heard
I
a great
many
dav of the week, and leads to the
as the first
mention only because
I
trapped
it
fol
me
it)
ask your victim to pronounce
to, t00,
and
two,
one
at a time,
thinking deeply between questions. In each case he says (wondering what's
up) "tooooo."
Then you
say,
"Now
pronounce the second day of the week" and
face clears up, for he thinks he sees the trap. will say
"toooosday"
like a
With
lowbrow.
He
is
sure you are hoping
his
he
exaggerated precision, therefore,
he says "tyoosday."
At which you look gently puzzled and pronounce it Monday."
The month, being
say, "Isn't that strange?
I
always
Moon, began, in ancient times, at a fixed The month can start at each full .Moon, or each first quarter, and so on. Actually, the most logical way is to begin each month with the new Moon— that is, on that evening when tied to the
phase. In theory, anv phase will do.
the
first
sliver of
after sunset. at that
the growing crescent makes
To any
logical primitive, a
time and the month should
Nowadays, however, the month year,
which
is
years, the first
in
is
visible
immediately
clearly being created
start then. is
freed of the
Moon and
is
tied to the
turn based on the Sun. In our calendar, in ordinary
month
begins on the
on the 32nd day of the the fourth
itself
new Moon
first
vear, the third
month on the
day of the
month on
91st day of the year,
year, the
second month
the 60th day of the year,
and
so
on— quite
regard
)
BEGIN AT THE BEGINNING less of
third
the phases of the
onward
start a
But that brings
day
Moon.
late
(In a leap year,
all
*57 the
months from the
because of the existence of February
29.
When
us to the year.
does that begin and why? must have been first aware of the year as a succession of seasons. Spring, summer, autumn, and winter were the morning, midday, evening, and night of the year and, as in the case of the Primitive agricultural societies
day, there
seemed two equally
qualified candidates for the post of begin
ning.
The beginning
of the
returns to the earth
work year
beginning of the year in general?
end of the work in
the time of spring,
is
when warmth
and planting can begin. Should that not
year,
On
the other hand,
with the harvest
to
(it is
also
be the
autumn marks
be devoutly hoped)
the
safely
hand.
With
With
the development of astronomy, the beginning of the spring season
the work year ended, ought not the
new
year begin?
was associated with the vernal equinox which, on our calendar,
March
20,
while the beginning of
equinox which
Some
Among
falls,
autumn
on
associated with the autumnal 23.
chose one equinox as the beginning and some the other.
societies
came
the Hebrews, both equinoxes
One
Year's Day.
is
on September
half a year later,
falls
of these
fell
on the
to
be associated with a
New
day of the month of Nisan
first
(which comes at about the vernal equinox). In the middle of that month
comes the
feast of Passover,
Since, according
to
which
is
thus tied to the vernal equinox.
the Gospels, Jesus' Crucifixion and Resurrection (the Last Supper was a Passover
occurred during the Passover season seder),
Good
Friday and Easter are also tied to the vernal equinox (see
Chapter 11).
The Hebrews
also celebrated a
Tishri (which falls at about the
New
Day on
the
first
two days of
autumnal equinox), and
this
became the
Year's
more important of the two occasions. It is celebrated by Jews today as "Rosh Hashonah" ("head of the year"), the familiarly known "Jewish
New Year."
A much
later
example of
autumnal equinox came September
22,
in
a
New
Year's
Day
in
connection with the
connection with the French Revolution.
1792, the French monarchy was abolished and
On
a republic
new epoch in human history had begun, a new calendar was needed. They made September 22 the New Year's Day and established a new list of months. The first proclaimed.
The
Revolutionary
idealists
felt
that since a
month was Vendemiare, so that September 22 became Vendemiare 1. For thirteen years, Vendemiare 1 continued to be the official New Year's
NUMBERS AND THE CALENDAR
1^8
Day
French Government, but the calendar never caught on outside
of the
France or even
among
up the struggle and
the people inside France. In 1806 Napoleon gave
officially reinstated
There are two important
the old calendar.
solar events in addition to the equinoxes. After
the vernal equinox, the noonday Sun continues to until
maximum
reaches a
it
and
solstice,
rise
higher and higher
height on June 21, which
summer
the
is
consequence, has the longest daytime period of
this day, in
the year.
The
height of the noonday Sun declines thereafter until
autumnal equinox.
position of the
and farther solstice
it
till
minimum
reaches a
solstice
at about the
24). This
is
not of
is
summer
A Midsummer
Night's
December
21, the winter
of the vear.
much
significance.
"Midsummer Day"
solstice (the traditional English day
time for gaietv and carefree
a
reaches the
it
then continues to decline farther height on
and the shortest davtime period
The summer falls
It
Dream
is
joy,
even
an example of
folly.
a plav
June
is
Shakespeare's
devoted to the
kind of nottobetakenseriouslv fun of the season, and the phrase "mid
summer madness" mav have arisen similarly. The winter solstice is a much more serious from day
to
dav, and
astronomical laws, tinue
and
its
it
affair.
The Sun
declining
is
to a primitive society, not sure of the invariability of
might well appear that
this time, the
Sun
decline and disappear forever so that spring will never
will con
come
again
all life will die.
Therefore, as the Sun's decline slowed from dav to day and
on December
halt
and began
and
joy which, in the end,
to turn
marked by gaiety and
became
21, there
came
must have been great
to a relief
ritualized into a great religious festival,
licentiousness.
The bestknown examples of this are the several davs of holiday among the Romans at this season of the year. The holiday was in honor of Saturn (an ancient Italian god of agriculture) and was therefore called the "Saturnalia." It to
was
men, even
a
to
time of feasting and of giving of presents; of good
will
the point where slaves were given temporary freedom
while their masters waited upon them. There was also a lot of drinking at Saturnalia parties.
In fact, the terized
word "saturnalian" has come
to
mean
dissolute, or charac
by unrestrained merriment.
There
is
logic, then, in
beginning the year at the winter solstice which
marks, so to speak, the birth of a
new Sun, as the first appearance of a new Moon. Something like this
crescent after sunset marks the birth of a
BEGIN AT THE BEGINNING
*59
may have been in Julius Caesar's mind when he reorganized the Roman made it solar rather than lunar see Chapter 11). The Romans had, traditionally, begun their vear on March 15 (the
calendar and

"Ides of March"), which was intended to
but which, thanks
originallv
upon the vernal equinox in which the Romans
fall
way
the sloppv
to
maintained their calendar, eventuallv moved
the vear to Januarv
1
instead, placing
out of synchronization
far
moved
with the equinox. Caesar adjusted matters and
the beginning of
nearlv at the winter solstice.
it
This habit of beginning the vear on or about the winter
solstice did
not
become universal, however. In England and the American colonies March 25. intended to represent the vernal equinox, remained the official i

beginning of the year until 1752.
was onlv then that the January
It
1
beginning was adopted.
The beginning
of a
new Sun
way, too. In the davs of the tianity
found
its
Empire, the
most dangerous competitor
Mithraism,
in
a cult that
was
The ritual centered who represented the Sun, on December 25— about the time of the
and was devoted
Persian in origin
modern times in another rising power of Chris
reflects itself in
Roman to
sun worship.
about the mythological character of Mithras,
and whose birth was celebrated winter solstice. This was a
Romans were used
good time
for
a
anyway, for the
holiday,
to celebrating the Saturnalia at that time of year.
Eventual!}", though, Christianity stole Mithraic
the birth of Jesus on
December
25
(
there
is
no
so that the period of the winter solstice has
thunder by establishing
biblical authority for this),
come
mark the
to
birth of
whom Roman
both the Son and the Sun. There are some presentdav moralists (of I
am
one
who

Saturnalia in the
find
something unpleasantly reminiscent of the
modern
secular celebration of Christmas.
But where do the years begin? years,
but where do we
start the
It is
numbers? In ancient times,
was not highly developed,
of history
it
was
sufficient to begin
the years with the accession of the local king or ruler.
would begin over again with each new king. \\ Tiere
years
by
its
\\ Tien
stance,
by the name of the magistrate
m (
all,
an annually
but merely
Athens named
its
manner. For
in
archons.
the Bible dates things at II
Kings 16:1,
the son of Remaliah, reign."
for that year.
numbering
The numbering
a city has
chosen magistrate, the year might not be numbered at identified
number the when the sense
certainly convenient to
it
is
all,
it
does
it
in this
written: "In the seventeenth year of
Ahaz the son
of
Jotham king
Pekah was the contemporary king of
Israel.)
Pekah
of Judah began to
NAPOLEON BONAPARTE who went from
Napoleon,
peror, to exile,
is
mentioned
modern experiment
the only
Corsican rebel, to French general, to briefly in this article as having
in novel calendars.
He was
Em
put an end to
distantly involved
in science in other respects.
In iSoj,
when
his
no statue
prise that
conquests brought him to Poland, he expressed
sur
Copernicus had ever been erected, and one was
to
put up in consequence.
When
was, no Catholic priest
it
would agree
to
on the occasion.
officiate
Napoleon patronized scientists such as Lagrange and Laplace, promoted them and honored them. Once, when he was holding British prisoners of war, he released them only after Edward Jenner (the discoverer of vaccination against smallpox) added his name to those petitioning for the release.
When tists
Napoleon invaded Egypt in 1798, he brought a number of scienwith him to investigate its ancient civilization. The Rosetta stone,
inscribed in both Greek
and Egyptian, was discovered on that
and Egyptian was eventually deciphered
occasion,
knowledge of ancient history was greatly expanded. Once Emperor, Napoleon supported French science vigorously in an attempt to
with British science.
and a
It
was similar
so that our
make
it
compete more
successfully
to the AmericanSoviet rivalry a century
half later.
The most famous Napoleonic
tale
nection with the astronomer Laplace,
with respect to science was in con
who was
putting out the
first
few
his Celestial Mechanics which completed the work of Newton and described the machinery of the Solar system. Napoleon leafed through the book and remarked there was no mention of God. "I had no need
volumes of
of that hypothesis," said Laplace.
And is
in
Luke
dated only
2:2, the
as follows:
was governor of
time of the taxing, during which Jesus was born,
"And
was
first
made when Cyrenius
Syria."
Unless you have accurate
how many
this taxing
years each
was
of kings and magistrates and know just power and how to relate the list of one
lists
in
region with that of another, vou are in trouble, and that so
many
a date as
ancient dates are uncertain— even (as
important
as that of the birth of Jesus.
it is I
shall
for that reason
soon explain)
'
mwW"
\A ;
The Granger
Collection
NUMBERS AND THE CALENDAR
l62
A much
better system
would be
to pick
some important date
in the
past (preferably one far enough in the past so that vou don't have to deal
with negativenumbered years before that time) and number the years in progression thereafter,
The Greeks made
without ever starting over.
use of the
Olympian Games
was celebrated every four years so that
The Olympiads were numbered
piad."
was the This
ist, is
for that purpose.
a fouryear cycle
This
was an "Olym
progressively and the year
itself
2nd, 3rd or 4th year of a particular Olympiad.
needlessly complicated, however,
and
the time following
in
Alexander the Great something better was introduced into the Greek world.
The
and one
ancient East was being fought over bv Alexander's generals,
of them, Seleucus, defeated another at Gaza.
By
mined
to
number
That year became Year
Era," and later years continued in succession as
more elaborate than
The
He
deter
the years from that battle, which took place in the 1st
year of the 117th Olympiad.
ing
victory
this
Seleucus was confirmed in his rule over a vast section of Asia.
of the "Seleucid
1
and
2, 3, 4, 5,
so on.
Noth
that.
Seleucid Era was of unusual importance because Seleucus and his
descendants ruled over Judea, which therefore adopted the system. Even
under the leadership of the
after the Jews broke free of the Seleucids
Maccabees, they continued to use the Seleucid Era
dating their com
in
mercial transactions over the length and breadth of the ancient world.
Those commercial records can be tied in with various local veardating many of them could be accurately synchronized as a
systems, so that result.
The most important
veardating system of the ancient world, however,
was that of the "Roman Era." This began with the year was founded. According to
Olympiad, which came
tradition, this
be considered
to
as
"Anno Urbis Conditae";
"a.u.c." stands for
in
which
Rome
was the 4th Year of the 6th 1
a.u.c.
that
is,
(The abbreviation "The Year of the
Founding of the City.") Using the
Roman
finally defeated,
Era, the Battle of
was fought
Zama,
in
which Hannibal was
in 553 a.u.c, while Julius
Caesar was
assassi
nated in 710 a.u.c, and so on. This system gradually spread over the ancient world, as
Rome waxed
supreme, and lasted well into early medieval
times.
The
early Christians, anxious to
those of Greece and
Rome,
that of either the founding of
Games.
A
Church
show
that biblical records antedated
strove to begin counting at a date earlier than
Rome
or the beginning of the
historian, Eusebius of Caesarea,
who
lived
Olympian
about 1050
BEGIN AT THE BEGINNING
163
Abraham, had been born 1263 years Rome. Therefore he adopted that year as his Year became 2313, Era of Abraham.
a.u.c, calculated that the Patriarch,
before the founding of 1,
so that 1050 a.u.c.
Once world,
the Bible was thoroughly established as the book of the western
was possible
it
to carry matters to their logical
extreme and date
The medieval
the years from the creation of the world.
had taken place 3007
that the creation of the world
Jews calculated years before the
founding of Rome, while various Christian calculators chose years varying
from 3251 to 4755 years before the founding of Rome. These are the ious
"Mundane
The
Eras" ("Eras of the World").
Jewish
var
Mundane Era
used today in the Jewish calendar, so that in September 1964, the Jewish year 5725 began. is
The Mundane Eras have one important early
enough
so that there are verv few,
factor in their favor.
if
that have to be given negative numbers. This Era, for instance.
The founding
of the
the reign of David, the building of
founding of
Rome and
They
start
any, dates in recorded history is
not true of the
Roman
Olvmpian Games, the Trojan War, the Pyramids, all came before the
have to be given negative year numbers.
The Romans wouldn't have
cared, of course, for
none of the ancients
were very chronologv conscious, but modern historians would. In
modern
Roman
historians are even worse off than they
fact, if
the
Era had been retained.
About 1288 from
would have been
a.u.c., a
biblical data
been born
in
754
monk named
Syrian
and secular
a.u.c.
Dionysius Exiguus, working
records, calculated that Jesus
must have
This seemed a good time to use as a beginning for
counting the years, and in the time of Charlemagne (two and a half centuries after
The
Dionysius ) this notion
vear 754 a.u.c.
became
ing "the vear of the Lord"). of
Rome
first
took place in 753
Olympiad was
312 b.c,
This
in
won
a.d.
By
b.c.
out.
Anno Domini, mean
(standing for
1
this
new
"Christian Era," the founding
("before Christ").
776 b.c, the
first
The
first
year of the
year of the Seleucid Era was in
and so on.
is
the system used today, and means that
all
of ancient history
from
Sumer to Augustus must be dated in negative numbers, and we must forever remember that Caesar was assassinated in 44 b.c. and that the next is number 43 and not 45. Worse still, Dionysius was wrong
year
clearly states that "Jesus
Herod the king." This Herod
is
in
his
calculations.
Matthew
2:1
Bethlehem of Judea in the days of the socalled Herod the Great, who was
was born
in
CHARLEMAGNE Charlemagne
mentioned here
is
as the
adoption of the modern Christian
numbering
versally uses in
era,
moving
spirit
behind the
official
which the world today almost
uni
its years.
Under Charlemagne, born in Aachen, Germany, about 742, the FrankEmpire reached its apogee. He ruled over what is now France, Belgium, the Netherlands, Switzerland, most of Germany, most of Italy, and ish
The Roman Empire in the West was revived (after 800 he was made Emperor, thus beginning a tradition
even some of Spain. a fashion),
and
in
that was to last just over a thousand years
Napoleon
s
and end
in
1806 as a result of
conquests in Germany.
Charlemagne
importance in the history of science was
s
midst of the period candles once more.
known
He
as the
Dark Age, he did
himself was
illiterate as
that, in the
his best to light the
was almost everyone but
churchmen. In adulthood, however, he managed
to learn
to read but
could not persuade his fingers to learn to make the marks necessary for writing.
He
recognized the value of learning in general, too, and in 789 began
to establish schools in
which the elements of mathematics, grammar, and
ecclesiastical subjects
could be taught under the overall guidance of an
named
English scholar
The
result of
Alcuin.
Charlemagne's
efforts
is
sometimes termed the
gian renaissance." It was a noble effort but a feeble one, and last
the great
Emperor
himself.
he was succeeded by his to as "the Pious
7
much
it
u
Carolin
did not out
He
died in Aachen, January 28, 814, and
less
talented son, Ludwig, usually referred
because he was entirely in the hands of the priesthood
and could not control
his family or the nobility.
The coming
of the Vik
ing terror completed the disintegration of the abortive renaissance.
born about 681 714
a.u.c.
He
a.u.c.,
and was made king of fudea bv Mark Antony in is known as certainly as any ancient date is
died (and this
known) later
in 750 a.u.c, and therefore Jesus could not have been born any than 750 a.u.c.
But 750 a.u.c, according
to the system of Dionysius Exiguus,
and therefore you constantly 4 b.c; that
is,
find in
lists
is
4
B.C.,
of dates that Jesus was born in
four years before the birth of Jesus.
;>.
Inc.
NUMBERS AND THE CALENDAR
l66 In fact, there
Herod
year that
attempt to be
no reason
died. In
kill Jesus,
to
be sure that Jesus was born
Matthew
ordered
all
2:16,
least
fore have
been born
Which
forces I
is
me
as earlv as 6 b.c. Indeed,
to
written that Herod, in an
as indicating that Jesus
two vears old while Herod was
the birth of Jesus as early as 17
beginning,
it
in the very
male children of two years and under to
This verse can be interpreted
slain.
been at
is
still
alive,
may have
and might
there
some estimates have placed
b.c.
admit sadly that although
can't alwavs be sure
I
where the beginning
love to begin at the is.
PART V Numbers and Biology
ABOUT THE
THAT'S
13.
OF
No
matter
SIZE
IT
how much we tell ourselves that quality is what counts, sheer The two most popular types of animals in any
remains impressive.
size
zoo are the monkeys and the elephants, the former because they are embarrassingly like ourselves, the latter simply because the} are huge.
We
laugh at the monkeys but stand in silent awe before the elephant.
And
even
among
the monkeys,
one were
if
would outdraw every other primate
Gargantua
to place
in the place. In fact,
This emphasis on the huge naturally makes the even puny.
The
mankind has
fact that
unparalleled domination of the planet as a
he
human
he
in a cage,
did.
being
feel small,
nevertheless reached a position of
is
consequently presented very often
DavidandGoliath saga, with ourselves as David.
And
yet this picture of ourselves
we view
is
not quite accurate,
as
we can
if
the statistics properly.
First, let's
consider the upper portion of the scale. I've just mentioned
the elephant as an example of great
"Big as an elephant"
is
a
common
size,
and
this
hallowed by
is
animal can be expected
Even
diluted.
the ground to size. If
ing out
its
if
it
to.
On
land, an animal
were not a question of
and moving
it
more or
motionlessly as an oyster,
reasons, but it
to death.
one
is
that
its
it
No
land
fight gravity, un
bulk several feet
off
that fight sets sharp limits
as lying flat
masses of tissue upward with every breath.
strangles
must
lifting its
less rapidly,
an animal were envisaged life as
cliche.
phrase.
But, of course, the elephant does not set an unqualified record.
several
see
on the ground, and
would
A
still
beached whale
own weight upon
liv
have to heave
its
dies
for
lungs slowly
ELEPHANTS The most glamorous interconnection of elephant and man came in ancient times when the elephant was used in warfare as the living equivalent of the modern tank. It could carry a number of men, together with im1
portant assault weapons.
and
tusks,
Most
legs.
of
the opposing forces,
to
animals.
The
It all,
could do damage on it
who
its
own, with
its
trunk,
represented a fearful psychological hazard
could only with
difficulty
the giant
face
greatest defects of the use of elephants lay in the fact that
elephants were intelligent enough to run from overwhelming odds, and in their panic [especially
wounded) could prove more damaging
if
own side than to the enemy. The West came across elephants
for the
first
time in 326
to their
b.c.
Alexander the Great defeated the Punjabi King, Porus, despite the
when latter s
use of two hundred elephants. For a century afterward, the monarchs
who
succeeded Alexander used elephants. Usually only one side or the other had elephants, but at the Battle of Ipsus in 301
b.c.
between
were elephants on both
rival generals of
sides,
the late Alexander
nearly three
hundred
in
s
army, there African
all.
ele
phants were sometimes used, though Asian elephants were more common.
The
African elephants, however, were native to north Africa and were
The north African
smaller than the Asian elephants.
and when we speak
tinct,
of African elephants today,
variety
eastern Africa,
which are the giants of the species and the
mammal
It is
alive.
the giant African elephant that
is
we mean is
now
ex
those of
largest land
shown
in the
il
lustration.
The Greek 280
b.c.
beasts,
general Pyrrhus brought elephants into southern Italy in
to fight the
Romans; but the Romans, though
The
fought resolutely anyway.
last
terrified of the
elephant battle was that of
Zama, where Hannibal's elephants did not help him defeat the Romans.
In the water, however,
that
buovancv
largely negates gravity,
would mean crushing death on land
is
and
a
mass
supported under water with
out trouble.
For that reason, the be found record
is
among
largest creatures
the whales.
the blue whale or, as
And it is
on
earth, present or past, are to
the species of whale that holds the
alternatively
known, the sulfurbottom.
1^;v
7 8 4
2.6l
4,200
13,680
2.59
V7
Kerintji
12,484
2.36
3,807
New
Cook
12 >349
2.34
3,662
Zealand
Carstensz
a continent.
portion of
METERS
Sumatra

Mount
MILES
Carstensz
Whom
New
is
the highest mountain in the world that
it is
named
Guinea and
for the
Dutch
is
for
I
do not know, but
part of the Nassau
royal family.
I
I
don't
know what
or,
it is
is
not on
in the western
Mountain Range, which
suspect that by
renamed both the range and the mountain the original names, but
1
more
now
Indonesia has
likely,
has restored
these might be.*
Mount Cook is just a little west of center in New Zealand's southern land. It is named for the famous explorer Captain Cook, of course, and Maori name is Aorangi.
All the heights
I
have given
for the
mountains, so
far,
is
its
are 'above sea
level."
[* I
man
have found out since. Range.]
Mount
Carstensz
is
now
officially
Mount
Djaja of the Sudir
NUMBERS AND THE EARTH
2l8
However, remembering the manager of the SheratonBoston Hotel,
let's
improve the fun by qualifying matters. After of
all,
base.
its
the height of a mountain depends a good deal upon the height
The Himalayan mountain
in the world; there
they
is
no disputing
upon the Tibetan
sit
plateau,
peaks are bv far the most majestic
that. Nevertheless,
which
is
also true that
it is
The
the highest in the world.
Tibetan "lowlands" are nowhere lower than some 12,000 feet above sea level. If its
we
This
Mount
Everest's height,
we can
only 17,000 feet above the land mass upon which
it rests.
subtract 12,000 feet from
peak
is is
not exactly contemptible, but bv
this
new standard
say that
(base to top,
instead of sea level to top) are there any mountains that are higher than
Mount
Everest? Yes, indeed, there
is
new champion
one, and the
not
is
in
the Himalayas, or in Asia, or on anv continent.
This stands to reason after
That
tively small island.
all.
island
wouldn't look impressive because
depth and with the ocean lapping This
is
Suppose you had
may be it
was standing with
The
an area of 4,021 square miles (about twice the
of
That
actually the case for a particular island.
huge mountain
rising
its
who knows how many
the largest single unit of the Hawaiian Islands.
tually a
mountain on
a
a rela
the mountain, and the mountain
out of the
base in the ocean
up
feet
island
its
is
slopes.
Hawaii—
island of Hawaii, with
size of
Pacific. It
Delaware)
comes
is
ac
to four peaks,
which the two highest are Mauna Kea and Mauna Loa (See Table 13). The mountain that makes up Hawaii is a volcano actually, but most of
it is
Mauna Loa
extinct.
est single
mountain
how
can imagine
must
alone remains active.
in the
world
large the
in
It, all
by
the larg
whole mountain above and below sea
level
be.
The
central crater pit of
Mauna Loa
is
sometimes active but has not
comes from open
actually erupted in historic times. Instead, the lava flow ings
itself, is
terms of cubic content of rock, so you
on the
The
sides.
largest of these
is
Kilauea, which
is
on the eastern
Mauna
sea level.
Loa, some 4,088 feet (0.77 miles, or 1,246 meters) above Kilauea is the largest active crater in the world and is more than
two miles
in diameter.
side of
As though these distinctions peaked mount we whole.
If
are
not enough,
Hawaii becomes
totally
one plumbs the ocean depths, one
land base that If
call
is
this
finds that
if
viewed
fouras a
Hawaii stands on
a
over 18,000 feet below sea level.
the oceans were removed from Earth's surface
please), then
tremendous
astounding
no
single
(only temporarily,
mountain on Earth could possibly compare with
UP AND
DOWN THE EARTH
the breathtaking towering majesty of Hawaii.
mountain on Earth, counting from base
It
21 9
would be by
to peak. Its height
would be 32,036 feet (6.08 miles or 9,767 meters). It tain on earth that extends more than six miles from base
The peak
on that
the only
is
basis
moun
to tip.
vanishing of the ocean would reveal a similar, though smaller,
Ocean; one that
in the Atlantic
Range. For the most
tain
far the tallest
because
it is
part,
we
is
part of the MidAtlantic
Moun
unaware of the mountain range
are
drowned by the ocean, but
it
is
larger
and longer and more
spectacular than any of the mountain ranges on dry land, even than the
Himalayas.
Some
It is
7,000 miles long and 500 miles wide and that's not bad.
above the surface of the Atlantic.
The
their heads
and have
a total land area of
888
than that of Rhode Island.
less
Pico Island in the Azores stands the highest point of land on the
island group. This
Pico Alto ("High Mountain"), which reaches 7,460
is
However,
(1.42 miles or 2,274 meters) above sea level.
down sea
poke
Azores, a group of nine islands and
are about 800 miles west of Portugal
square miles, rather
feet
to
(belonging to Portugal), are formed in this manner. Thev
several islets
On
do manage
of the highest peaks of the range
the slopes of the mountain and proceed
the way
all
if
you
down
slide
to the
bottom, you find that only onequarter of the mountain shows above
water.
The
total height of Pico Alto
from underwater base
27,500 feet (5.22 miles or 8,384 meters), which makes
it
peak
to a
is
about
peak of Hima
layan dimensions.
While we have the oceans temporarily gone from the Earth, we might
how deep
as well see
About sea level,
are a
1 .2
the ocean goes.
per cent of the sea bottom
and where
number
this
of these,
lies
more than 6,000 meters
below
happens we have the various "Trenches." There
most of them
in the Pacific
Ocean. All are near
island chains and presumably the same process that burrows out the deeps also heaves
The
up the
island chain.
greatest depth so far recorded in
to the material available to
The
figures for the
me)
is
some
of these deeps (according
given in Table 14.
depth of the deeps are by no means
those for the heights of mountains, of course,
and
I
can't
as reliable as
tell
when some
oceanographic ship will plumb a deeper depth in one or more of these deeps. The greatest recorded depth in the Mindanao Trench— and in the
world— was plumbed only
as
recently as
oceanographic vessel the Vityaz.
March, 1959, by the Russian
,
KILAUEA Xlauna Loa
is
Mauna Kea
virtually
the largest mountain mass in the world and together with
makes up the island of Hawaii. Xlauna Loa means "long mountain," which is a good name since from end to end it is
seventyfive miles wide.
usually snowcapped.
volcano, but
which
is
u
Mauna Kea means
The snowcap
Mauna Loa
active,
is
white mountain" because
indicates that
and near
Mauna Kea
is
a
it is
dormant
Kilauea {here illustrated)
it is
the largest crater in the world, with an area of four square miles.
Kilauea
is
known
not
for spectacular eruptions. Rather,
mering, and within the crater
and overflows
casionally rises
is
its
it is
always sim
a boiling lake of molten rock that oc
Occasionally such a lava flow
sides.
is
copious and longsustained and in 1935 eyen threatened the city of Hilo,
on the
largest
island.
The native Hawaiians considered the most active vent of Kilauea to be the home of Tele, the fire goddess— which makes sense if we assume the existence of divine beings.
The Hawaiian
Here, however, rests a peculiar coincidence.
and on the island two names have nothing is
tague Telee" tain, if
is
of Martinique
Tele,
is
in
you want
to preserve the
two minor eruptions, one lava ital
May
8,
It
The "Mon"pealed" moun
sound), because the top was denuded.
mountain
was not taken
in 1792
and one
is
a volcano
and that
as a very serious volcano, for
in 1851, impressed
nobody. But
An
avalanche of
mountain suddenly exploded.
1902, the
goddess
fire
called Telee.
despite appearances, since
fateful, for the
rolling
collection of 30,000 corpses.
The
mountain
down the side right down onto St. Tierre, then the capMartinique. One moment it was a city of 30,000 people; the next a
went of
a
merely "Bald Mountain" in Trench (or
The name, however, was why the top is denuded.
then, on
common
is
greatest depth in
Jacques Piccard and
Only two people
survived.
the Mariana Trench was actually reached by
Don Walsh
in
person on January 23, i960, in the
bathyscaphe Trieste. This has been named the "Challenger Deep"
honor of the oceanographic ducted a lished
scientific cruise
vessel
from 1872
the H.M.S. Challenger, which conto
1876
all
over the oceans and estab
modern oceanography.
In any case, the oceans are deeper than anv is
in
the point
I
want
to
make, and
in several places.
mountain
is
high, which
The Bettmann Archive
Consider the greatest depth of the Mindanao Trench.
If
Mount
Everest
it and made to nestle all the way in, the mountain would sink below the waves and waters would roll for 7,000 feet (1%
could be placed
miles I
over
its
in
peak.
If
the island of Hawaii were
and sunk
location, 4,500 miles westward,
moved from its present Mindanao Trench, it
into the
too would disappear entirely and 4,162 feet of water
flow above
4 (
:>
of a mile)
its tip.
Table 14
Some Ocean Trenches DEPTH
GENERAL
TRENCH
LOCATION"
Cuba
Bartlett
S.
of
Java
S.
of Java
FEET
MILES
METERS
22,788
43
1
6,948
24,442
4.64
7> 2 5 2
30,184
57 1
9>39 2
9,800
Puerto Rico
N
Japan
S.
32,153
6.O9
Kurile
E. of
Kamchatka
34>5 8°
6.56
i°o43
Tonga
E
35>597 35,800
10,853
E. of
Xew Zealand Guam
6.75
Mariana
6.9
10,915
Mindanao
E. of Philippines
36,198
6.86
11,036
of Puerto Rico
of Japan
of
would
.
NUMBERS AND THE EARTH
222 If sea level is
on Earth,
the standard, then the lowest bit of the solid land surface
just off the Philippines,
bit at the top of
Mount
Everest.
about 3,200 miles east of the highest
is
The
total difference in height
is
65,339
feet (12.3 miles or 19,921 meters).
This sounds like a lot but the diameter of the Earth
about 7,900
is
miles so that this difference in lowhigh makes up onlv 0.15 percent of
the Earth's total thickness.
mv
the Earth were shrunk to the size of
If
diameter), the peak of
Mount
librarv globe (16 inches in
Everest would project only 0.011 inches
above the surface and the Mindanao Trench would sink onlv 0.014 inches
below the
surface.
You can
see then that despite
been talking about, the surface of the Earth, viewed size of the Earth,
in
I
have
proportion to the
would be smooth, even
very smooth. It
is
downs
the extreme ups and
all
if
the oceans
were gone and the unevennesses of the ocean bottom were exposed. With the oceans
filling
up most of the Earth's hollows (and concealing the
worst of the unevennesses), what remains
But
let's
think about sea level again.
versal ocean,
it
nothing.
is
the Earth consisted of a uni
If
would take on the shape of an
thanks to the fact that the planet
rotating.
is
ellipsoid of revolution,
It
wouldn't be a perfect
ellipsoid because, for various reasons, there are deviations of a
few
feet
here and there. Such deviations are, however, of only the most academic interest
and
for our purposes
This means that
if
we can be
satisfied
with the ellipsoid.
Earth were bisected by a plane cutting through the
center,
and through both
ellipse.
The minor
poles, the crosssectional outline
would be an
would be from
axis (or shortest possible Earthradius)
the center to either pole, and that would be 6,356,912 meters. radius, or is
major
axis,
is
from the center
6,378,388 meters (on the average,
the equator
The is
itself
very slightly elliptical
farther
The
longest
anv point on the equator. This
we wish
to allow for the fact that
)
equatorial sea level surface, then,
13.3 miles)
This
is
if
to
is
21,476 meters (70,000 feet or
from the center than the polar sea
level surface
is.
the wellknown "equatorial bulge."
However, the bulge does not from center to sea
exist at the
level surface increases
poles to the equator. Unfortunately, extra length of the radius
I
equator alone.
smoothly
as
The
distance
one goes from the
have never seen any data on the
(over and above
its
minimum
length at the
poles) for different latitudes. I
have therefore had
to calculate
it
for myself,
making use
of the
manner
UP AND in
which the gravitational
find figures
DOWN THE EARTH from latitude
field varies
on the gravitational
field.
)
The
15
The
"0,000
*33
21,400
69,500
13.2
21,200
10°
68,000
12.9
20,800
65,500
12.4
20,000
20°
62,300
11.8
19,000
*5°
58,000
11.0
17,700
30°
52,800
10.0
16,100
35°
4~o 00
9.0
14,500
40°
41,100
7.8
45°
35,100
6.65
12,550 10,00
50°
29,000
55°
8,850
55°
23,200
4.40
,050
6o°
17,700
335
5>4 00
65°
12,500
2 3"
3,800
70°
8,250
1.56
2,500
75°
4,800
0.91
1,460
8o°
2,160
0.41
660
53°
0.10
160
85
0.00
(poles)
we measure
heights of mountains not from just any old
but from the polar sea
level.
This would serve to compare
from the center of the Earth, and mate way of comparing mountain heights. tances
we
If
did
this,
we would
could
are ap
Earth's Bulge
5°
Suppose, now,
(I
hope
15.
o° (equator)
90
I
EXTRA LENGTH OT EARTH'S RADIUS FEET MILES METERS
LATITUDE
sea level,
to latitude.
which
results,
proximately right anyway, are included in Table
Table
223
certainly that
instantly get a completely
is
new
another
dis
legiti
perspective on
matters.
For instance, the Mindanao Trench dips down sea level; but that c
means below the
sea level at
its
to 11,036 meters
own
latitude,
below
which
is
level is 20,800 meters above the polar sea level so that the greatest depth of the Mindanao Trench is still some 9,800 meters
io.o
X. That sea
(6.1 miles
)
above the polar sea
In other words,
was
six
level.
when Peary stood on
the sea ice at the North Pole, he
miles closer to the center of the Earth than
if
he had been
in a
bathyscaphe probing the bottom of the Mindanao Trench. Of course, the Arctic Ocean has a depth of its own. Depths of 4,500
.
NUMBERS AND THE EARTH
224 meters
(2.8
have,
miles)
means that the bottom
we have
point of view,
this
a
out of the running in
it is
And the mountains? Mount Everest is at Mount
Everest
mountain
But
is
only 2.2
it is
is
just
above
level.
own
its
the Arctic. This
Mindanao Trench, and mark of "deepest
for the
up bv the continent of Ant
this respect.)
a latitude of
meters higher than polar sea
filled
in
nearly nine miles closer
is
of the
new candidate
deep." (The south polar regions are arctica, so
Ocean
bottom
to the center of the Earth than the
from
been recorded
believe,
I
of the Arctic
about 30.0
Add
.
Sea level there
sea level
mark and vou
find that the
about 25,000 meters (15.5 miles) above polar sea miles above equatorial sea level.
In other words,
when
a ship
is
crossing the equator,
its
Mount
level.
passengers are
onlv 2.2 miles closer to the center of the Earth than Hillarv was stood on
16,100
is
that to the 8,886 meters that
when he
Everest's peak.
Are there mountains that can do better than Mount Everest by
new
standard?
The
other towers of Asia are in approximately
Mount Aconcagua and some
Everest's latitude. So are
this
Mount
of the other high
peaks of the Andes (though on the other side of the equator)
Mount McKinley some level
of
a
is
little
over 60.0
5,000 meters above polar sea level. is
N
so that
Its total
only 11,200 meters (7.0 miles), which
Mount
is
its
sea level
is
only
height above polar sea
less
than half the height
Everest.
No, what we need are some good high mountains near the equator, where thev can take midriff.
A
manjaro.
full
advantage of the
good candidate
It is
about
3.0
S and
the 21,300meterhigh bulge
above polar sea
level
maximum
bulge of the Earth's
the tallest mountain of Africa,
is
it
is
5,890 meters high.
stands on, so that
(16.9 miles), or nearlv a
To
this
Mount
Kili
one can add
some 27,200 meters mile and a half higher
it is
Mount Everest, counting from the center of the Earth. And that is not the best, either. My candidate for highest peak by these standards is Mount Chimborazo in Ecuador. It is part of the Andes Mountain Range, in which there are at least thirty peaks higher than Mount Chimborazo. Mount Chimborazo, is, however, at 2.0 S. Its height above its own sea level is 6,300 meters. If we add the equatorial bulge, we have
than
a total height of 27,600 meters
above polar sea
level (17.2 miles) .**
first appeared, Chimborazo has been announced as the highest world by others and the news has been printed with considerable excitement by Scientific American and by Newsweek. Neither took note of this article's
[** Since this article
mountain
in the
precedence.]
a
UP AND If
we go by
DOWN THE EARTH
225
the distance from the center of the Earth, then,
from the bottom of the Arctic Ocean
and increase that distance by 32,100 meters,
we can
pass
Mount Chimborazo
to the top of
or just about 20.0
miles—
nice even number.
By changing the point of view then, we have three different candidates mountain on Earth: Mount Everest, Mauna Kea, and Mount
for tallest
Chimborazo.
We
also
have two different candidates
the bottom of the Arctic
But
let's
heights single
it.
What
for the deepest deep:
of the
Mindanao Trench.
counts in penetrating extreme depths or extreme
not mere distance, but
measure of
pressure; is
is
face
Ocean and the bottom
difficultv in
and the greatest
attainment.
difficultv of
plumbing depths
single
measure of
is
The
greatest
the increase of water
difficulty in
climbing heights
the decrease of air pressure.
Bv
that token, water pressure
Trench, and
air
pressure
is
is
highest at the bottom of the
lowest at the top of
fore those are the extremes in practice.
Mount
Everest,
Mindanao and
there
THE
17.
One
OF EARTH
ISLES
of the nicest things about the science essays
brings
me— almost invariably goodhumored and
Consider, for instance, the previous chapter
which
in
I
maintain that Boston's Prudential
on continental North America
(as
opposed
write
I
the mail
is
it
interesting.
"Up and Down is
the Earth,"
the tallest office building
to the higher ones
on the
Manhattan). The moment that essay first appeared I received a card from a resident of Greater Boston, advising me to follow the Charles
island of
and Neponset Rivers back
to their source
and
see
if
Boston could not be
considered an island. I
The
followed his advice and, in a way, he was right.
flows north of Boston
and the Neponset
Charles River
River flows south of
In south
it.
western Boston thev approach within two and onehalf miles of each other. Across that gap there
meanders
I
live
in)* are surrounded on
Tower, and to
mv
a
stream from one to the other so
some western suburbs (including the one
that most of Boston and parts of
all
sides
by surface water. The Prudential
house, too, might therefore be considered, by a purist,
be located on an
island.
Well!
But before
I
grow panicky,
me
let
stop and consider.
What
is
an island,
anvway?
The word mean,
"island"
literally,
comes from the AngloSaxon "eglond," and
"waterland"; that
is
this
may
"land surrounded by water."
This AngloSaxon word, undergoing natural changes with time, ought to
have come down to us
as
"eyland" or "iland."
inserted, however, through the influence of the
[*
Not any more. As
I said before, I've
returned to
New
An
word
York.)
"s" was mistakenly "isle,"
which
is
syn
THE ISLES OF EARTH onymous with
''island" yet,
oddly enough,
is
22 7
not etymologically related to
it.
For "isle" we have to go back to classical times. The ancient Greeks in their period of greatness were a seafaring people who inhabited many islands in the Mediterranean Sea as well as sections of the mainland. They, and the Romans who followed, were well aware of the apparently fundamental difference
between the two types of land.
An island was to them a relatively small bit of land surrounded bv the sea. The mainland (of which Greece and Italy were part) was, on the other hand, continuous land with no
To be finite
known boundary.
Greek geographers assumed that the land surface was and that the mainland was surrounded on all sides by a rim of ocean sure, the
beyond the
but, except for the west, that was pure theory. In the west, Strait of Gibraltar, the
broad ocean.
No
Mediterranean Sea did indeed open up into the
Greek or Roman, however, succeeded
in traveling over
land to Lapland, South Africa, or China, as to stand on the edge of land and, with his
own
eyes, gaze at the sea.
In Latin, then, the
holds together."
The
mainland was
notion was that
terra continens; that
when you
is,
"land that
traveled on the mainland,
there was always another piece of land holding on to the part you were traversing.
There was no end. The phrase has come down
to us as the
word "continent."
On
the other hand, a small bit of land, which did not hold together
with the mainland, but which was separate and surrounded by the sea was terra in salo or
"land in the sea." This shortened to insula in Latin; and,
by successive steps, to
The
isola in Italian, "isle" in English,
and
ile in
French.
(and by extension the
meaning word word "island"), is that of land surrounded by salt water. Of course, this is undoubtedly too strict. It would render Manhattan's status rather dubious since it is bounded on the west by the Hudson River. Then, too, there are of the
strict
"isle,"
then
certainly bodies of land, usually called islands, nestled within lakes or rivers,
which are certainly surrounded by
fresh water.
However, even such
must be surrounded by a thickness of water that is fairly large compared to the island's diameter. No one would dream of calling a large tract of land an island just because a creek marked it off. So Boston is islands
not an island, practically speaking, and Manhattan
However, for the purposes of the remainder of stick to the strict definition of the
are surrounded
bv
salt
water—
term and
is.
this article
I
am
going to
discuss only those islands that
NUMBERS AND THE EARTH
228
we do
If
this,
however, then, again
the land surface
strictly speaking,
no continents
of the Earth consists of nothing but islands. There are
the
literal
meaning
The mainland
of the word.
in
The
never endless.
is
Venetian traveler Marco Polo reached the eastern edge of the anciently
known mainland
in
1275; the Portuguese navigator
Bartholomew Diaz
reached the southern edge in 1488; and Russian explorers marked
off the
northern edge in the seventeenth and earlv eighteenth centuries.
The mainland
refer to here
I
where there
is
making up the
usually considered as
is
three continents of Asia, Europe,
and
Africa.
But why three continents,
only a single continuous sheet of land,
one ignores
if
rivers
and the manmade Suez Canal?
The multiplicity of continents dates back to Greek days. The Greeks of Homeric times were concentrated on the mainland of Greece and faced a hostile second mainland to the east of the Aegean. The earliest Greeks had no reason
to suspect that there
was any land connection between the two
mainlands and they gave them two different names; their own was Europe, the other Asia.
These terms are of uncertain
origin but the theory
like best suggests
I
they stem from the Semitic words assu and erev, meaning "east" and
"west" respectively. (The Greeks the Phoenicians, by alphabet.) literally,
Of
the
way
mav have picked up
of Crete, just as they picked
The Trojan War of 1200 b.c. begins West and the East; a confrontation that
course,
Greek explorers must have learned,
these words from
up the Phoenician
the confrontation is still
with
early in the
game, that
there was indeed a land connection between the two mainlands.
myth
of Jason
probably
and the Argonauts, and
reflects
their pursuit of the
Golden
trading expeditions antedating the Trojan
of,
us.
The
Fleece,
War. The
Argonauts reached Colchis (usuallv placed at the eastern extremity of the Black Sea) and there the two mainlands merged.
Indeed, as
we now know,
there
is
some
fifteen
hundred miles of land
north of the Black Sea, and a traveler can pass from one side of the Aegean Sea to the other— from Europe to Asia and back— by way of this fifteen hundred mile connection. Consequently, Europe and Asia are separate continents only by geographic convention and there is no real boundary
between them at
all.
The combined
land mass
is
frequently spoken of as
"Eurasia."
The
Ural Mountains are arbitrarily set as the boundary between Europe
and Asia
in the
geography books. Partly,
a mild break in a
Germany
huge plain stretching
to the Pacific Ocean,
this
is
because the Urals represent
for over six
thousand miles from
and partly because there
is
political sense
THE ISLES OF EARTH in considering Russia (which, until
229
about 1580, was confined to the region
west of the Urals) part of Europe. Nevertheless, the Asian portion of Eurasia
so
is
much
larger than the
European portion that Europe
is
often
looked upon as a mere peninsula of Eurasia. Africa
is
much more
nearly a separate continent than Europe
only land connection to Eurasia
about a hundred miles wide, and Still
lized
in
Its is
ancient times was narrower.
the connection was there and
men (and
is.
the Suez Isthmus, which nowadays
is
it
was
armies too) crisscrossing
rarely crossed the land north of the Black
a welltraveled one, with civi
now and again— whereas thev Sea. The Greeks were aware of
it
the link between the nations thev called Svria and Egvpt, and thev therefore considered
Egypt and the land west of
The matter was
different to the
it
to
be a portion of Asia.
Romans. They were
farther
from the
Suez Isthmus and throughout their earlv historv that connection was merely of academic interest to them. Their connection with Africa was entirely
by way of the
Furthermore, as the Greeks had once faced
sea.
sea between, so— a thousand years Romans faced the Carthaginians on opposing mainlands with the sea between. The struggle with Hannibal was every bit as momentous to the Romans as the struggle with Hector had been to the Greeks. The Carthaginians called the region about their citv by a word which, in Latin, became Africa. The word spread, in Roman consciousness, from the immediate neighborhood of Carthage (what is now northern Tunisia) to the entire mainland the Romans felt themselves to be facing. The
Troy on opposing mainlands with the later— the
geographers of
Roman
times, therefore— notably the GreekEgyptian Ptol
emy—granted Africa the dignity of being a third continent. But let's face facts and ignore the accidents of history. If one ignores the Suez Canal, one can travel from the Cape of Good Hope to the Bering Strait, or to Portugal or Lapland, without crossing salt water, so that
the whole body of land forms a single continent. This single continent has no generally accepted
sometimes
We can it
is
finite
felt
the urge to do,
think of
and
name and
it is
it
this
is
to call
a
I
have
ridiculous.
bounded on
name for it; one It is "the World Island." The name seems to imply that is
"Eurafrasia," as
way, though. This tract of land all
sides
that
is
is
enormous but
by ocean. Therefore
land; a vast one, to be sure, but an island. If
then there
it
we
it is
an
is
take that into account
sometimes used by geopoliticians.
the triple continent of EuropeAsiaAfrica
makes up the whole world and, you know, it nearly does. Consider Table 16. (And let me point out that in this and the succeeding tables in this
NUMBERS AND THE EARTH
2 30
the figures for area are good but those for population are often
article,
quite shaky.
my
library,
when such
have
I
but
choke mid1960s' population figures** out of
tried to
haven't always been able to succeed. Furthermore, even
I
marked "estimate"
figures are given they are all too frequently
and may be quite
far off the truth.
.
.
.
But
let's
The World
Table 16
do our
best.)
Islandf
area
population
(square miles) Asia
16,
Africa
11,500,000
336,000,000
3,800,000
653,000,000
31,800,000
2,939,000,000
Europe
The World
The World
Island contains a
Even more
of the globe.
Earth's population.
The
It
and population
fair
it
half the total land area
contains threequarters of the
claim to the name.
the American
is
manv thousands
primitive Asians
a.d.,
Giovanni Caboto (John Cabot
to
services)
in
...
1497.
mainland,
and
finally
World
Island
discovered bv
bv the Icelandic
by the
Italian navigator
the English nation, which employed
don't mention
I
covered only islands prior to 1497.
first
of vears ago, again
navigator Leif Ericsson in 1000
his
more than
little
significantlv,
has a
1,950,000,000
only tract of land that even faintlv compares to the
area
in
Island
$00,000
He
Columbus because he
dis
did not touch the American main
land until 1498.
Columbus thought deed
that the
might have been.
it
not demonstrated
till
sailed
)
new mainland was
now
is
and
so in
the Danish navigator \ ltus Bering (em
explored what
through what
part of Asia,
complete phvsical independence of Asia was
when
1728,
ployed by the Russians
and
Its
now known
as the
Bering Sea
called the Bering Strait, to
show that
is
and Alaska were not connected. There is, therefore, a second immense
Siberia
traditionally,
divided
into
America. These, however, are connected,
and
a
man
island
two continents: if
on Earth and
this
one
is,
North America and South
one ignores the manmade Panama Canal,
can travel from Alaska to Patagonia without
crossing salt water.
[** Mid1970s' population figures in this printing.' [j
This
article
first
appeared in June 1966. In the years since, populations hare, of changed all the tables in this article, therefore, to reflect that
course, increased. I hare increase.}
THE ISLES OF EARTH There
no convenient name
is
called "the
would
and
reject
I
like to suggest a
it
name
(if
The 17.
vital statistics
As you
World
oldfashioned)
the
see,
Island,
what
Island that there
New
England, or between York and
little
Table 17
a
New
sort of relationship is
be
York.
on the New World Island are presented in New World Island has about half the area
but only a
is
Island."
phrase "the
same
New World
between the World Island and the
New
can be
It
for
my own— "the New World
of
for the Americas. It also indicates the
tween England and
term
a plural
for that reason.
common
This capitalizes on the
World"
combined continents.
Americas" but that makes use of
single tract of land I
for the
2 3*
Table of the
over onesixth the population.
The New World
Island
area (square miles)
North America South America
The New World There are two other tinents,
and one
tract
Island
9,385,000
321,000,000
7,035,000
190,000,000
16,420,000
511,000,000
enough
tracts of land large
which
is
population
borderline and
is
be considered con
to
usually considered too
small to be a continent. These are, in order of decreasing area; Antarctica
(counting
its
ice cap), Australia,
Since Greenland ity)
to
lump
it
in
islands" just to get
is
and Greenland.
almost uninhabited,
I
would
like (as a
with the group of what we might it
We
out of the way.
call
pure formal"continental
can then turn to the bodies ot
land smaller than Greenland and concentrate on those as a group.
Table 18
lists
the data on the continentalislands.
Table 18
The
ContinentalIslands
AREA
The World Island The New World Island
(square miles)
POPULATION
31,800,000
2,939,000,000
16,420,000
511,000,000
Antarctica
5,100,000
Australia
2,970,000
12,550,000
840,000
47,000
Greenland
The bodies of land that remain— all smaller than Greenland— are what we usually refer to when we speak of "islands." From here on in, then, when I speak of "islands" in this essay, I mean bodies of land smaller than Greenland and entirely surrounded bv the
sea.
NUMBERS AND THE EARTH
232
There are many thousands of such of the land surface of the globe that (as nearly as
I
can estimate
and they represent a portion no by means negligible. Altogether
islands is
the islands have a total area of about
it)
2,500,000 square miles— so that in combination they are continental in size,
The
having almost the area of Australia.
total
400,000,000, which is even more clearly continental above the total population of North America. Let's put
this
it
human
way, one
population in
is
about
being well
size,
being out of every ten
on an
lives
island smaller than Greenland.
There are some useful
we can
statistics
the islands. First and most obvious islands in terms of area are listed in
Table
19
is
Table
The
bring up in connection with
the matter of area.
The
five largest
19.
Largest Islands*
(square miles)
New
Guinea
12 >3 2 9 290,285
3
Borneo Madagascar
The
largest island,
of 1,600 miles.
from It
and the
Sumatra
l6 3>M5
New
Guinea, spreads out over an extreme length
to Denver. In area,
has the largest and
Island in
230,035 201,600
superimposed on the United
If
New York
Baffin
tallest
New World
it is
States,
it
would
fifteen per cent larger
stretch
than Texas.
mountain range outside those on the World
Island,
and some of the most primitive people
the world.
Two other New Guinea.
islands of the It,
first
five are
members
Borneo, and Sumatra are
called the "East Indies"
all
grouping
one million square
in the world.
miles,
same group
what used
to
as
be
an archipelago stretching across four thousand
miles of ocean between Asia and Australia, and largest island
of the
part of
The
making up by
and thus contains about
the island area in the world.
The
far the
archipelago has an area of nearly forty per cent of all
archipelago bears a population of perhaps
121,000,000 or about thirty per cent that of
all
the island population in
the world.
[*
These days it is becoming more customary to use the geographic names used by the Borneo is really Kalimantan, for instance. I will, however, use the more
inhabitants. familiar
name
in every case.]
)
THE ISLES OF EARTH In a way, Madagascar
like
is
2 S3
an East Indian island displaced four thou
sand miles westward to the other end of the Indian Ocean. the shape of
Even
Sumatra and
native population
its
is
has roughly
It
midway between Sumatra and Borneo. more closely akin to those of 'Southeast
in size
is
Asia than to those of nearby Africa.
Only
Baffin Island of the five giants falls outside this pattern. It
member
of the archipelago lying in the north of Canada.
It
is
a
located
is
between the mouth of Hudson Bay and the coast of Greenland.
Oddly enough, not one
of the five largest islands
is
a giant in respect to
population. There are three islands, indeed (not one of which
the
among
is
largest), which, among them, contain well over half of
five
all
the
The most populous is probably not known by many Americans. It is Honshu and, before you register a
island people in the world.
name
to very
blank, let
me
explain that
on which Tokyo
The
is
it is
the largest of the Japanese islands, the one
located.
three islands are given in Table 20.
Table 20
The Most Populous
Islands
POPULATION
AREA square miles
Java
Honshu
91,278
6
83,000,000
Java
48,504
12
78,000,000
Great Britain
88,133
7
56,000,000
easily the
is
RANK
most densely populated of the large islands. (I say Manhattan.) It has a
"large islands" in order to exclude islands such as
density of 1,600 people per square mile which makes
denselv populated as Europe.
It
is
1%
it
just
Netherlands, Europe's most thickly people nation. This
remarkable since the Netherlands
is
all
highly industrialized and Java
the more is
largely
all,
Netherlands' standard of living
Lagging
far
is
much
higher than Java's.)
behind the big three are four other
than ten million population. These are given the way,
as the
one usually expects an industrialized area to support population than an agricultural one would. (And, to be sure, the
agricultural. After a larger
is
nine times as
times as densely populated
is
in
islands,
Table
each with more 21.
(Kyushu, by
another of the Japanese islands.
Notice that the seven most populous islands are all in the Eastern Hemisphere, all lying off the World Island or between the World Island and Australia.
The most populous
island of the
Western Hemisphere
one which most Americans probably can't name.
It
is
is
again
Hispaniola, the
is
NUMBERS AND THE EARTH
34
Table
The Moderately Populous
21
Islands**
AREA
RANK
(square miles)
Sumatra
l6 3>M5
POPULATION 20,000,000
5
Formosa
13>855
34
1
5,000,000
Ceylon
2 5>33 2
2
4
1
5,000,000
Kyushu
M'79 1
3
1
12,000,000
land on which Haiti and the Dominican Republic are located. lation
is
One
It's
popu
9,000,000.
generally thinks of great powers as located on the continents. All
but one in history among the continental great powers were located on the
World The
Island.
(The one exception
great exception
powers
is,
to
is
the United States.)
the rule of continentalism
of course, Great Britain. f In
more recent
among
the great
times, Japan proved
another. In fact, Great Britain and Japan are the only island nations that
have been completely independent throughout medieval and modern
his
tory.
Nowadays, however (unless are
I
have miscounted and
I
am
sure that
if I
I
will
be quickly enlightened by a number
no
less
than thirtvone island nations; thirtvone independent nations,
have
that
is,
whose
territory
to
is
of Gentle Readers), there
be found on an island or group of
who lack any significant base on World Island. One of these nations, Australia,
either the
nary convention, but
here to be complete.
I'll
include
land nations (including Australia
it )
World
and
islands,
New
Island or the
actuallv a continentnation by ordi
is
are listed in
Table
The
thirtvone
22, in order of
is
popu
lation.*
Some
little
explanatory points should accompany this table. First the
discrepancy between Great Britain's area as an island and as a nation
caused by the fact that as a nation
home
Virtually
[**
Formosa
all
is
I
includes certain regions outside
previously referred to as the East Indies.
the island people are
more properly known
as
now
part of independent island na
Taiwan; Ceylon as
I'm not going to distinguish between England, Great and the British Isles. I can if I want to, though, never you i
[*
When
this article first
nations. Since then, ten I
is
its
Northern Ireland. Indonesia includes most but not
island, notably
the archipelago
all
it
Sri
Lanka.]
Britain, the
United Kingdom,
fear!
appeared nine years ago, there were only twentyone island islands or island groups hare become independent, and
more
have prepared a revised table, with updated population
figures, to reflect that fact.]
THE ISLES OF EARTH Table 22
The
2 35
Island Nations
AREA (square miles) 735,268
121,000,000
Japan
142,726
10 5,000,000
94,220
56,000,000
11 5»7°7
37,000,000
Great Britain Philippines
Formosa
13,885
Australia
2,971,021
12,500,000
2 5>33 2
12,500,000
Ceylon
Cuba Madagascar
44,218
8,500,000
230,035
7,500,000
10,714
5,000,000
Dominican Republic
18,816
4,000,000
Ireland
2 7> J
Zealand
35
3,000,000
103,376
2,900,000
Singapore
224
2,200,000
Jamaica
4> 2 3 2
2,000,000
Trinidad
1,980
950,000
720
900,000
Cyprus
3>57 2
630,000
Fiji
7>°55 122
330,000
864 166
275,000
2 5.6
230,000
5.382
215,000
Mauritius
Malta
Comoro Barbados Bahrain
Bahamas Iceland
Maldive Islands
J1 5
1 1
Grenada
*33
110,000
Tonga
269
100,000
Sao Tome'
37 2 8
69,000
largest island area that
frankly don't
know how
a considerable extent
may
210,000
^°97
is
still
New
Guinea.
but
qualify as the
island remaining.
is
counted
as
5,OO0
8,000
It
is
Papua
New
Guinea,
has an area of 182,700
administered by Australia.
to classify Puerto Rico. It
if it is
50,000
1
colonial
square miles, a population of 2,300,000, and
it
300,000
Western Samoa
the eastern half of the island of
I
575,000
39,768
Nauru
The
5,000,000
1
Haiti
New
tions.
POPULATION
Indonesia
is
selfgoverning to
an Ameiican colony,
I
think
most populous (pop. 2,700,000) nonindependent
NUMBERS AND THE EARTH
25 6
As you
see
from Table
most populous
22, the
Japan nor Great Britain, but Indonesia. ulous nation in the world.
United States
The
(all
giants in
It
is,
island nation
than a single island are Haiti
less
and the Dominican Republic (which share Hispaniola) and the six northeastern counties are
land nation which has part of
is
is
The
where
only
islands belonging to nations based
is
on
Indonesia. Part of the island of Borneo (most of which
new nation
New
on nearby Asia. The eastern half of is
Ireland,
part of Great Britain.
still
its
Indonesian) makes up a portion of the
which
neither
Onlv China, India, the Soviet Union, and the area) are more populous than Indonesia.
only island nations that occupy
some continent
is
in fact, the fifthmostpop
of Malaysia, based
Guinea (the western half of
Indonesian) belongs to Australia.
There are eighteen
cities
within these island nations which contain one
more people. These are listed, Table 23— and I warn you that some
million or
in order of decreasing population,
in
of the figures are not particularly
trustworthy.
Table 23 CITY
Of
these,
I
Island Cities
POPULATION
NATION
Tokyo London
Great Britain
8,200,000
Djakarta
Indonesia
4,500,000
11,400,000
Japan
Osaka
Japan
3,000,000
Sydney
Australia
2,800,000
Melbourne
Australia
2,400,000
Yokohama
Japan
2,300,000
Nagoya
Japan
2,000,000
Taipei
Formosa
1,750,000
Havana Kyoto Kobe
Cuba
1,700,000
Japan
1,400,000
Japan
1,300,000
Manila
Philippines
1,300,000
Surabaja
Indonesia
1,300,000
Bandung
Indonesia
1,100,000
Kitakyushu
Japan
1,050,000
Sapporo
Japan
1,000,000
Birmingham
Great Britain
1,000,000
Tokyo
in the world.
The
say
is
certainly remarkable since
"may be" because
post— Shanghai. Population
statistics
there
is
for the
a
it
may be
the largest city
second candidate for the
Chinese People's Republic
THE ISLES OF EARTH
2
(Communist China) are shaky indeed but there is population of Shanghai— a continental city— may be though
figures as
New
York
low
a possibility that the as
high as 11,000,000,
as 7,000,000 are also given.
City, the largest city
on the
New World
Island,
no better
is
than a good fourth, behind Tokyo, Shanghai, and Greater London.
York
located mostly on islands, of course. Only one of
is
the Bronx,
is
indisputably on the mainland.
nation in the same sense that Tokyo or If
we exclude New York
in the
as a
London
have a population of over leaves only
Still
it
is
its
New
boroughs,
not on an island
is.
doubtful case, then the largest island
Western Hemisphere, and the only one
That
37
a million,
one item. In
is
in that half of the
Havana.
citv
world to
,
restricting the discussion of islands to
those which are surrounded bv salt water, have
we been
forced to neglect
any important fresh water islands? In terms of size (rather than population) there
mentioning.
It
can be aware
is
a river island that very
of.
It
is
is
mouth
of the
one hundred miles across and has an area of
miles. It
is
larger than
islands of the sea
which
in the
is
it
Formosa and
if
it
is
worth
world (outside Brazil)
bad
Amazon fifteen
were counted
would be among the top
certainly not
only one that
the island of Marajo, which nestles like a huge
basketball in the recess formed by the It
few
is
for a river island.
River.
thousand square
among
the true
thirty islands of the world,
However,
it
is
a lowlying
piece of land, swampy, often flooded, and right on the equator. Hardly
anyone Its is.
lives there.
mere
existence, though,
shows what
a
monster of
a river the
Amazon
INDEX
Allosaurs, 176
Amazon
River, 237
Amoeba,
131, 133
Abacus, 50
Andromeda Galaxy, 128
Abraham,
Animals, size
60, 163
of, 1696?.
Aconcagua, Mount, 216
Antarctic Ocean, 201
Adelard of Bath, 50
Arabic numerals, 1314, 48,
Aepyornis, 17778
112
Africa, 229
Aral Sea, 207
African elephant, 170, 174
Archimedes,
Albatross, 17879
death
Aleph numbers, 6970 86, 162,
of,
76
pi and, 78
Arctic Ocean, 201
Arithmetic, fundamental
170
theorem
Alexandria, 86 Algebra, fundamental
theorem
46n, 60, 76,
185
Alcuin, 164
Alexander the Great,
13,
of,
90
Algebraic numbers, 94
of,
Asa, 60 Asia, 228
Asimov
series, 31
90
INDEX
240 Atlantic Ocean, 201
Brachiosaurus, 176
Autumnal equinox, 157
Buoyancy, principle
of,
76
Azores, 219
B Cabot, John, 230
Babylonian number symbols,
Caesar, Julius, 14748, 159
death
of, 148,
162
4 Bacon, Roger, 150
Calendar month, 146
Bacteria, 133
Campbell, John, 58
Badwater, 216
Cantor, Georg, 68
Baffin Island, 23233
Cardano, Girolamo, 100
Baikal, Lake, 21c
Cards, playing, 40
permutations
Baluchitherium, 174 Beast,
number
Bells,
3334
of,
Carstenz,
Mount, 217
Bering, Vitus, 230
Caspian Sea, 2067
Bering
Catherine
Strait,
Bible, large
202
numbers
in,
3839, 4
Carolingian Renaissance, 164
of the, 7
60
I
of Russia, 104
Celestial Mechanics, 160
pi and, 75
Cells, 133
week, and, 15556
Ceulen, Ludolf von, 82
year numbering and,
Challenger, 220
Challenger Deep, 220
15960
Change,
Big bang, 194 Binary system,
i6ff.
ringing, 3334
Charlemagne,
115, 163, 164
Biological Science, 111
Charles River, 226
Blue whale, 17071
Chicago, 2i3n
Borneo, 232, 236
Chimborazo, Mount, 224
Boston, 213
Chomolungma, 214
INDEX
2
Christian Era, 163
beginning
of,
Christmas, 152, 159
Day, Donald, 58
Christmas Eve, 154
Dead
15354
Sea, 2078
Cicero, 148
Death Valley, 216
Cipher, 13
Descartes, Rene, 54
Circle, squaring the,
c)off.
Diaz, Bartholomew, 228
Cleopatra, 148
Dinosaurs, 176
Coefficients, 93
Dionysius Exiguus, 163
Columbus, Christopher, 230
Djaja,
Mount, 2i7n
Compass, 88
Dollar, 121
Complex numbers, 1067
Dominican Republic, 236
Compound
Drake Passage, 204
Computer,
addition, 114 20, 26,
29
Duodecimal system,
8
Continent, 227
Continuous creation, 194
Continuum, 67 Cook, Mount, 217 Copernicus, Nicolas, 160
Council of Nicaea, 150 Crocodiles, 17677
Currency, 117, 121
e,
3132, 95, 104
Earth, 222 Earthlight, 142 Easter, 14546
East Indies, 232
D
Eggs, bird, 178
Egyptian calendar, 146 Eightbased system, 29
Mount, 216
Dapsang, 214
Elbrus,
Dark Age, 164
Electrons, 190
Dase, Zacharias, 84
Elements, 86
Davis, Philip
Elephants, 170
Day, 139
J.,
57
Elevenbased systems, 2728
41
INDEX
242
English language, 120
Fish, large, 177
ENIAC,
Fractions, 6566, 100
84
Equations, 9395
Frederick
I
Equatorial bulge, 222
Frederick
II of Prussia,
Era of Abraham, 163
French Revolutionary
Ericsson, Leif, 230
of Prussia, 20
104
calendar, 157
Euclid, 51, 85, 86
Fresh water, 205
Eudoxus, 147
Freund, John E., 70
Euler, Leonhard, 74, 102, 104 Eurafrasia, 229
Eurasia, 228
Europe, 228 Eusebius, 162 Everest, George, 214
Everest,
Mount,
214, 218, 222,
224 Exhaustion, method
of,
78
Exponent, 4445
Galactic year,
1
32
Galaxies, 188 Galileo, 54, 68
Gamow,
George, 59
Gasherbrun, Mount, 21 5n Gauss, Karl Friedrich, 89, 90,
99 Gelon, 185 Gematria, 6
Geometric progressions, 8081
H
George
),io4
I
of Great Britain, 20
Fact and Fancy, 73
Giant clam, 180
Factorial, 37
Giant squid, 180
Fairy
flies,
181
Gillies,
Donald
B., 53
Ferdinand of Brunswick, 90
Goby
Fermat's Last Theorem, 30
GodwinAusten, Henry
Fibonacci, Leonardo, 47, 50
Fibonacci numbers, 4850
fish,
180
Haversham, 214 GodwinAusten, Mount, 214
^
:
INDEX

Goliath beetle, 181
Hillary,
Googol, 44
Hilo. 22c
Googolplex, 56
Himalaya Mountains, 214
Gosainthan, Mount. 215
Hindu number symbols,
Great Britain, 23334
Hispaniola. 23334
Great Lakes, 20911
Honshu,
Great Rift Valley, 208
Hours. 155
Great Salt Lake, 207
Hubble, Edwin Powell.
Greeks, geometry and, S5
Hubble Radius, 194
number symbols
of,
Greenland, 231
Gregory XIII, Pope. 15c
Percival, 215
Hubble's Law. 1
13
23
Hubble's constant.
5
Gregorian calendar, 120,
Edmund
;
:

:
i

1
Hummingbirds,
i~ ;
; .
:
:
Huygens. Christian, 54
H
Hadrian, i:2. lGLf
l.
Haiti, 236
Halloween, 154


205
Hannibal, 162, i~:
Imaginary numbers,
Havana, 237
Index to the Science Fiction
Hawaii, 218
Magazines. 5S
Hegira, 144
Indian Ocean.
Heinlein, Robert, 34
Indonesia, 234. 236
Henry
I
Infinity, 58ft
Henry
II
of England, 115 of England, 50
Inland Seas,
:
::
salt in.
Hermite, Charles, 95
Intercalary day, 142
Herod, 163
Intercalan
Hieron
II,
6
loiff.
207
month, 145
Ipsus, battle of. 17a
INDEX
244
numbers, 6768,
Irrational
Kilauea, 218, 220
Kilimanjaro,
8283, 100
Mount, 216
Kiwi, 178
Island, 226
Kodiak bear, 175
Komodo
monitor, 177
Kosciusko, Mount, 217
Kosciusko, Thaddeus, 217
Kronecker, Leopold, 68
Japan, 234 Jason, 228 Java, 233 Jefferson,
Thomas, 117
Jenner, Edward, 160
Jewish calendar, 145 Jewish
Mundane
Era, 163
Lambert, Johann Heinrich, 83 Laplace, Pierre
Julian calendar, 120, 150
de, 128,
160
Jewish numbering system, 6
Jordan River, 208
Simon
Leap day,
Leap
50
1
year,
1
Leatherback
50 turtle,
177
Leibniz, Gottfried Wilhelm, 19, 20,
Leibniz
7980
series,
80
Length, measure
of,
124
Liber Abaci, 48, 50
K
Light, motion of,
1
34
Lindemann, Ferdinand, 95 Linear equations, 98
Kalimantan, 232n
Liouville, Joseph,
Karakorum Mountains, 215
Lobster, 180
Kasner, Edward, 44
Logarithms, 173
Kepler, Johann, 8
Lord's Day, 156
94
INDEX Lore of Large Numbers, The,
2
Mersenne, Marin,
52,
54
Mersenne primes, 5253
57 Ludolf's number, 82
Messier, Charles, 128
Ludwig the
Method
Pious, 164
of least squares, 90
Lunar calendar, 141
Methusaleh, 140
Lunar month, 140
Metric system, 11718, 1238.
Lunarsolar calendar, 145
Lunar
year, 141
Luther, Martin, 7
prefixes of, 124
Midsummer
day, 158
Midsummer
Night's Dream,
A, 158 Millay,
Edna
Vincent, 85
St.
Mindanao Trench,
219, 221,
22 3
Mithraism, 159
M
Moa, 178
Modern
Introduction to
Mathematics, A, 70
McKinley, William, 216
Mohammed,
McKinley, Mount, 216
Mohammedan
Madagascar, 23233
Month, 140
Magellan, Ferdinand, 201
beginning
7,
of,
144 calendar, 141
156
Manhattan, 213
Moon, phases
Marajo, 237
Morris,
Mariana Trench, 220
Mountains, highest,
Marius, Simon, 128
Mundane
of,
14042
Gouvemeur,
121 2i4ff.
Era, 163
Mark Antony, 164 Martinique, 220
Mass, measure
of,
13031
Mathematics and the Imagination, 44
N
Matterwaves, 19091
Mauna
Kea, 218, 220
Mauna
Loa, 218, 220 158, 160
Mayans, 8
Napoleon,
Measures, 1135.
Nassau Mountains, 217
7,
45
1
INDEX
246 Natural numbers, 99
size of, 2001,
Nebular hypothesis, 128
solids in,
Negative numbers, 100
volume
204
202
of,
205
Of Matters Great and Small,
Neponset River, 226 Nero, 7
57 n
Neutron, 190
Olympiads, 162
New
Onetoone correspondence,
Guinea, 217, 232
Newman, Newton,
James, 44
Isaac, 20,
One, Two, Three,
76
New World Island, 331 New Year's Eve, 1 54 New York City, 237
Infinity,
Oughtred, William, 73
Ninebased systems, 25
Nine
Tailors, 33, 34
Noon, 155 Norgay, Tenzing, 215 Nuclei, atomic, 191
Numbers, 8889 large,
60
symbols
of, 3fL, 1
Numerology, 6
Pacific
Ocean, 201
Papua
New
Guinea, 235
Passover, 157 Pele, 220
Pelee,
Mount, 220
Pendulum
clock, 54
Perfect numbers, 5355
Peter the Great, 20, 104 Pi, ?3ff
o
algebraic
numbers and,
9495 irrationality of, 83
Observable Universe,
18!
19294
of, 204,
gases in, 202
of, 79ff.
Pico Alto, 218
Ocean, 202 depth
value
Piccard, Jacques, 220
21921
Pike, Nicolas, 112
Pike
s
Arithmetic, 11112
59
INDEX Pisa,
2
Rational numbers, 67
47
Plato, 88
Real numbers, 67, 103
Points, 6667
Recher, 84
Polo, Marco, 40, 228
Recreational Mathematics,
Polynomial equations, 93,
Magazine, 49,
Red
98^9
5111,
Theory
188
Porus, 170
Relativity,
Positive numbers, 100
Repeating decimals, 81
Prime numbers, 5153
Reticulated python, 177
Proton, 190
Revelation,
radius of, 192
Pseudopod, 133 Pteranodon, 180 Ptolemy, 86
Ptolemy Euergetes, 147 Puerto Rico, 235
Book
of,
of,
7
Rio de Janeiro, 60
Roman (s),
Protozoa, 133
53
Sea, 208
76
number symbols
of, 8ff.,
11213
Roman Roman Roman
calendar, 147 Era, 162
numerals,
8ff.,
11213
Pvrrhus, 170
Rosetta stone, 160
Pythagoras, 30
Rosh Hashonah, 157 Rotifers, 182
Quadratic equation, 101
R
Sabbath, 156 St. Pierre,
220
Rainfall, 206
Salamanders, 180
Ramadan, 144
SandReckoner, The, 185
47
:
INDEX
48 Saturn, 158
Superior, Lake, 209
Saturnalia, 158
Synodic month, 141
Sayers, Dorothy, 33
Scalar quantities, 107
Schmidt, Maarten, 189 Seasons, 141
Seleucid Era, 162 Seleucus, 162
Seven Seas, 201 Shakespeare, William, 158
Shanghai, 236 Shanks, William, 84
Tenbased systems, 2526
Sharp, Abraham, 82
Ternary system, 24
Shrews, 172, 175, 178
Time, measure
Sirius,
146
Tn umbers, 46
Sitter,
Willem
Skewes,
S.,
de, 188
of,
13132
Tokyo, 236 Transcendental numbers, 95
57
Skewes' number, 57
Transfinite numbers, 69
Sodom and Gomorrah, 208
Tropical year, 146
Solar calendar, 146
Twelvebased systems, 2728
Solar year, 144
Solomon, 60 Sosigenes, 147
Sothic cycle, 146
Sperm whale, 175 Standard measures, 115 Straightedge, 87
u
Sudirman Range, 217 Suez Isthmus, 229 Sumerians, 8
Summer
solstice,
Universe, i86ff.
158
particle
numbers
in,
192
Sunrise, 153
Ural Mountains, 22829
Sunset, 153
Urban IV, Pope, 150
INDEX
249
Whitney, Josiah Dwight, 216 Whitney, Mount, 216 William
Germany, 7 Wimsey, Lord Peter, 34
V
II
of
Winter
Solstice,
World
Island,
World Ocean,
158
22930 201
Vector quantities, 107 Vega, George, 84 Vernal equinox, 150, 157 Vieta, Francois, 79, 82
Vigesimal system, 8 Viruses, 133
Vityaz, 219 Yardstick, 115
Year, 141
beginning
of, 157ft.
numbering
of, i59ff.
W Wallenstein, Albrecht von,
Walsh, Don, 220 Washington, George, 151
Week, 15556 Weekend, 156
Zama,
Whale
Zerah, 60
shark, 177
WTiite shark, 177
battle of, 170
Zero, 13