Asimov on Numbers

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512.81

Aslmov

As4a 1965407

Aslmov on numbers

512.81

Aslmov

As4a 1965407

Asimov on numbers

SEa.

trcr

3 PUBLIC LIBRARY Fort Wayne and Allen County, Indiana

D

ALLEN COUNTY PUBLIC LIBRARY

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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. 1959-June 1966.

Includes index. 1.

Numbers, Theory

of

—Addresses,

I.

QA241.A76

essays, lectures.

Title.

51

2'. y'o8

isbn: 0-385-12074-5

Library of Congress Catalog Card

Copyright

Copyright©

©

Number 76-23747

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

T-Formation

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 Proton-Reckoner

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.

T-Formation

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

Pre-fixing 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 Proton-Reckoner

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 out-of-print 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 out-of-print 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 subject-centered collections.

INTRODUCTION Finally, since the subject

matter

ordinary grab-bag 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 more-often-used 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' tax-collectors, 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 "twenty-three." 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 "Twenty-three" 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 seventy-five" 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

seventy-five" 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

one-component 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 "twenty-six."

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 seven-component 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-

old-fashioned 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 twenty-four

twenty-seven 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

seventy-five

is

XPE,

while eight hundred sixteen

ZJF.

One

disadvantage of the Greek system, however,

must

of twenty-seven 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 ninety-nine (&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 Graeco-Roman times adopted the

Greek system of representing numbers but, of course, used the Hebrew alphabet— and promptly ran into

difficulty.

naturally be written as "ten-five." 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

Greek-Hebrew system look

alphabet,

WRA

all,

wish.) Consequently, martial,

The the

"five

numbers. For

like

hundred ninety-one." In

doesn't usually matter in which order

it

we

shall see, this

came

numerals, which are also alphabetic, and ninety-one." (After

is

we can

it is

say "five

to

WAR also

in the

place the

means

"five

Roman hundred

hundred one-and-ninety"

easy to believe that there

and of ominous import

we

be untrue for the

number

"five

is

if

we

something warlike,

hundred ninety-one."

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 inter-relationships 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 pseudo-sci-

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

"in-group" 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 three-score and

it is

the

number

of a

man;

six."

designed not to give the pseudo-science 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 sixtv-six, "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

anti-Christ 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 alphabet-numLater

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 60-based 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

"Twenty-three" 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 less-meaningful

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 sixty-four as

nineteen sixty-four

If

Roman

write nineteen sixty-four, 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

sixty-four,

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

sixty-six."

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 ninety-nine" 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

sixty-seven thousand four hundred eighty-two 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

)

non-intellectual.

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

fifty-five"

is,

hundred eighty-one" 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 hand-copying, 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 eighty-one" and

fifty-five"

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 hand-copying,

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 word-number 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 all-important

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 six-digit 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 two-symbol 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 five-symbol 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 old-fashioned) "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

'-'

•*«'•*

aa-if**

v x"-8s'*

3-2'"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 to-one 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

fully-equipped 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 double-headed 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

one-to-one 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

one-to-one 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 one-to-one 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 "trillion-integers," 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

"trillion-integers,"

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 one-to-one 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

one-to-one 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 integer-points 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 one-to-one

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 one-to-one 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 one-to-one 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 intellect-shaking 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 one-for-one

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 "aleph-one," "aleph-two," and

This could be svmbolized

as: Xn»

X,>

X

would be "aleph-null," 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 aleph-null. 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 aleph-one 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 aleph-one, 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 one-to-one 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 aleph-two, but that hasn't been proved yet, either.

And

that

is

all.

Assuming that the endlessness of

and the endlessness of points aleph-two,

we have come

variety of endlessness

to

integers

is

aleph-null,

aleph-one, and the endlessness of curves

is

the end.

Nobodv

which could correspond

is

has ever suggested any

to aleph-three (let alone to

aleph-thirty or aleph-three-million).

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 aleph-null?

"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: Prentice-Hall, 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

top-flight

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 hen-coop 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

foot-rule,

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 pyramid-builders 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 foot-rule. (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 well-known 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 twenty-four, 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 sixty-two 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 unit-radius 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

well-known 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 marked-off 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 man-hours 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 -f-Bx 2 -f-

Cx-f-D=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: Ax-|-B=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

-B-x/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

one-to-one 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 one-half 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 one-half 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 one-half 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 4-3=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 5x4-6=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 x-j~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—104-6=0, which

is

2 a solution of the equation.

we

if

— 5x4-6=0,

substitute

for x, then x 2

3

9—154-6=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

4-nx— 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 two-dollar 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

second-degree 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

set-to

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

V-4

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 -f-i, — 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 (-f-1)

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

-f-i.

All

x4

(-f-i)

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 so-called

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 -f-i, +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

"real-number 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 imaginary-number

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 too-ardent 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 thirty-five 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'Uk-es

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 right-angle

tated through 180 turns.

which

-j-i,

(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, one-dimensionally.

we used

onlv the imaginary-number

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 imaginary-number 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 -f-i-foz

real

— i-f-c*.

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 "keeper-upper" 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 above-quoted 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

Model-T 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 man-sized 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

two-digit

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+10-1-9=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 English-speaking 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 long-used 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

half-crowns 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,

anti-British,

pro-French 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

well-being 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 Man-land 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 Man-land 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

three-month 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 hair-splitting 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, anti-British 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 ounce-pieces. Joachim's German. Joachimsthal. and the coins were called "Joachims-

north-uvstern Czechoslovakia)

Valley

is.

in

thalers" or. for short, "thalers." or. in English, "dollars."

In colonial times. Spanish coins of about the value of the weU-k

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 old-fallen,

and

it

when

it is

falling

so on.

using adjective-noun 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, new-fallen

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 "pre-fixing

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 sub-units 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 1-95 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 sub-unit a thousand times larger

be the largest useful one, while

a sub-unit 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 electron-volts), the British call a

"Gev"

call a

"Bev"

(bil-

(giga-electron-volts).

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 non-Greek roots

1962, but

I

will

PRE-FIXING

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 light-year

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

light-year

the

is

dis-

about 5,880,000,000,000 miles

the distance at which a star would appear

one second of

arc (parallax-second, get it),

and

equal to 3.26 light-years, 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 thirty-three 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

full-size

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 light-years. 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 sky-gazer, 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

thirty-first

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 present-day 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

PRE-FIXING 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 tobacco-mosaic 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 man-like 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 continuous-creation

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 one-celled animal

is

in

usually considered the

most

primitive of the type. It has no fixed shape as other one-celled 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-

("amoeba-within: cell-dissohing"

cell,

smallest free-lixing 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 light-propagation "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 omega-meson 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

PRE-FIXING The

entire range of time, then,

IT

UP

135

from the lifetime of an omega-meson to

that of a red-dwarf star covers a range of 40 orders of magnitude. In other

words, during the normal

life

of a red dwarf,

some 10 40 omega-mesons

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 well-known entertainer to attend the session.

made one

entertainer

He was

condition, however.

a

fol-

coup by

The well-known

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 even-one

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 ready-made 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 time-telling

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 day-unit,

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, ready-made— 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 pre-agricultural 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 29.5-day :

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 30-day months, the third new Moon, and the sixth year will start two days before the new Moon. To properly religious people, this would alternate 29-day

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] —::::: :::

29.5-day

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 calendar-making

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 30-year cycle there are nineteen 354-day years years,

it

Moon, naked-eye 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 355-day

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

"solar-year."

month behind This

is

about years

11 days shorter fall

in roughly 365 days, than the season-cycle, 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 season-cycle.

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 dav-light hours. But each vear,

Ramadan

falls

a bit earlier in the cvcle of the seasons,

and

at 33-vear 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

19-vear 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 12-month years and seven 13-month

cycle

to

new

added that 7-month discrepancy

through the 19-year 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

lunar-solar 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 'lunar-solar 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 constant-length 365-day 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 12-month 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 365-dav 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 ultra-conservative

Egyptians would have none of such a radical innovation.

Roman

Meanwhile, the

Republic had a lunar-solar 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 30-dav months and

Romans

considered February an unlucky

we ended with a silly arrangement months, and one 28-day 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 31-day months, four 30-day

lished even- fourth year with a length of 366 days.

calary

we should

31-dav 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

365-day 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

366-day 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 366-day 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 13-day 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 soul-searching 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 twenty-four-hour 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

four-hour 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 twenty-four-hour 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

non-Jewish 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 24-hour

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 24-hour 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. 20:1-16), 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

t-o, t-0-0,

and

t-w-o,

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 not-to-be-taken-seriouslv 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 best-known 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 present-dav 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 American-Soviet 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 negative-numbered 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 four-year 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 vear-dating many of them could be accurately synchronized as a

systems, so that result.

The most important

vear-dating 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 so-called 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 over-all 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

David-and-Goliath 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 sulfur-bottom.

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 Sheraton-Boston 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 Mid-Atlantic

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 one-quarter 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

seventy-five 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 long-sustained 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

4-3

1

6,948

24,442

4.64

7> 2 5 2

30,184

5-7 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 low-high 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 cross-sectional outline

would be an

would be from

axis (or shortest possible Earth-radius)

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 well-known "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

*3-3

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

5-5°

8,850

55°

23,200

4.40

-,050

6o°

17,700

3-35

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 21,300-meter-high 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 good-humored 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 one-half 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 Anglo-Saxon "eglond," and

"water-land"; that

is

this

may

"land surrounded by water."

This Anglo-Saxon 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 man-made 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 well-traveled 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 Greek-Egyptian 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 Europe-Asia-Africa

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 mid-1960s' 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 three-quarters 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 man-made Panama Canal,

can travel from Alaska to Patagonia without

crossing salt water.

[** Mid-1970s' 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

old-fashioned)

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 one-sixth 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 continental-islands.

Table 18

The

Continental-Islands

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 thirtv-one island nations; thirtv-one 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 continent-nation by ordi-

is

are listed in

Table

The

thirtv-one

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 twenty-one 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

self-governing 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 fifth-most-pop-

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 low-lying

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, 13-14, 48,

Aepyornis, 177-78

112

Africa, 229

Aral Sea, 207

African elephant, 170, 174

Archimedes,

Albatross, 178-79

death

Aleph numbers, 69-70 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, 147-48, 159

death

of, 148,

162

4 Bacon, Roger, 150

Calendar month, 146

Bacteria, 133

Campbell, John, 58

Badwater, 216

Cantor, Georg, 68

Baffin Island, 232-33

Cardano, Girolamo, 100

Baikal, Lake, 21c

Cards, playing, 40

permutations

Baluchitherium, 174 Beast,

number

Bells,

33-34

of,

Carstenz,

Mount, 217

Bering, Vitus, 230

Caspian Sea, 206-7

Bering

Catherine

Strait,

Bible, large

202

numbers

in,

38-39, 4-

Carolingian Renaissance, 164

of the, 7

60

I

of Russia, 104

Celestial Mechanics, 160

pi and, 75

Cells, 133

week, and, 155-56

Ceulen, Ludolf von, 82

year numbering and,

Challenger, 220

Challenger Deep, 220

159-60

Change,

Big bang, 194 Binary system,

i6ff.

ringing, 33-34

Charlemagne,

115, 163, 164

Biological Science, 111

Charles River, 226

Blue whale, 170-71

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

153-54

Sea, 207-8

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, 106-7

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, 176-77

Currency, 117, 121

e,

31-32, 95, 104

Earth, 222 Earthlight, 142 Easter, 145-46

East Indies, 232

D

Eggs, bird, 178

Egyptian calendar, 146 Eight-based 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

Eleven-based systems, 27-28

41

INDEX

242

English language, 120

Fish, large, 177

ENIAC,

Fractions, 65-66, 100

84

Equations, 93-95

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, 44-45

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, 80-81

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

Godwin-Austen, Henry

Fibonacci, Leonardo, 47, 50

Fibonacci numbers, 48-50

fish,

180

Haversham, 214 Godwin-Austen, Mount, 214

^

:

INDEX

---

Goliath beetle, 181

Hillary,

Googol, 44

Hilo. 22c

Googolplex, 56

Himalaya Mountains, 214

Gosainthan, Mount. 215

Hindu number symbols,

Great Britain, 233-34

Hispaniola. 233-34

Great Lakes, 209-11

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, 67-68,

Irrational

Kilauea, 218, 220

Kilimanjaro,

82-83, 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

79-80

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, 52-53

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, 117-18, 1238.

Lunar-solar 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, 232-33

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,

140-42

Gouvemeur,

121 2i4ff.

Era, 163

Mark Antony, 164 Martinique, 220

Mass, measure

of,

130-31

Mathematics and the Imagination, 44

N

Matter-waves, 190-91

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, 200-1,

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

One-to-one 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

Nine-based systems, 25

Nine

Tailors, 33, 34

Noon, 155 Norgay, Tenzing, 215 Nuclei, atomic, 191

Numbers, 88-89 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, 53-55

Peter the Great, 20, 104 Pi, ?3ff-

o

algebraic

numbers and,

94-95 irrationality of, 83

Observable Universe,

18!

192-94

of, 204,

gases in, 202

of, 79ff.

Pico Alto, 218

Ocean, 202 depth

value

Piccard, Jacques, 220

219-21

Pike, Nicolas, 112

Pike

s

Arithmetic, 111-12

59

INDEX Pisa,

2

Rational numbers, 67

47

Plato, 88

Real numbers, 67, 103

Points, 66-67

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, 51-53

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.,

112-13

Roman Roman Roman

calendar, 147 Era, 162

numerals,

8ff.,

112-13

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

Sand-Reckoner, 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

Ten-based systems, 25-26

Sharp, Abraham, 82

Ternary system, 24

Shrews, 172, 175, 178

Time, measure

Sirius,

146

T-n umbers, 46

Sitter,

Willem

Skewes,

S.,

de, 188

of,

131-32

Tokyo, 236 Transcendental numbers, 95

57

Skewes' number, 57

Transfinite numbers, 69

Sodom and Gomorrah, 208

Tropical year, 146

Solar calendar, 146

Twelve-based systems, 27-28

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, 228-29

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

229-30 201

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, 155-56 Weekend, 156

Zama,

Whale

Zerah, 60

shark, 177

WTiite shark, 177

battle of, 170

Zero, 13