A HISTORY OF 7T (PI)

Petr Beckmann
Electrical Engineering Department,
University of Colorado

ST. MARTIN'S PRESS

New York

TO [;MUDLA,

who never doubted the success of this book

Copyright © 1971 by THE GOLEM PRESS
All rights reserved. For information, write:
St. Martin's Press, Inc., 175 Fifth Ave., New York, N.Y. 10010.
Manufactured in the United States of America
Library of Congress Catalog Card Number: 74-32539

Preface

The history of 17 is a quaint little mirror of the history of man. It is the
story of men like Archimedes of Syracuse, whose method of calculating 17 defied substantial improvement for some 1900 years; and it is also
the story of a Cleveland businessman, who published a book in 1931
announcing the grand discovery that 17 was exactly equal to 256/81,
a value that the Egyptians had used some 4,000 years ago. It is the story
of human achievement at the University of Alexandria in the 3rd
century B.C.; and it is also the story of human folly which made
mediaeval bishops and crusaders set the torch to scientific libraries
because they condemned their contents as works of the devil.
Being neither an historian nor a mathematician, I felt eminently
qualified to write that story.
That remark is meant to be sarcastic, but there is a kernel of truth in
it. Not being an historian, I am not obliged to wear the mask of
dispassionate aloofness. History relates of certain men and institutions
that I admire, and others that I detest; and in neither case have I hesitated to give vent to my opinions. However, I believe that facts and
opinion are clearly separated in the following, so that the reader should
run no risk of being overly influenced by my tastes and prejudices.
Not being a mathematician, I am not obliged to complicate my
explanations by excessive mathematical rigor. It is my hope that this
little book might stimulate non-mathematical readers to become
interested in mathematics, just as it is my hope that students of physics
and engineering might become interested in the history of the tools they

4

PREFACE

are using in their work. There are, however, two sure and all too well
tried methods of how to make mathematics repugnant: One is to
brutalize the reader by assertions without proof; the other is to -hit him
over the head with epsilonics and proofs of existence and unicity.
I have tried to steer a middle course between the two.
A history of 17 containing only the bare facts and dates when who did
what to 17 tends to be rather dull, and I thought it more interesting to
mix in some of the background of the times in which 17 made progress.
Sometimes I have strayed rather far afield, as in the case of the Roman
Empire and the Middle Ages; but I thought it just as important to
explore the times when 17 did not make any progress, and why it did not
make any.
The mathematical level of the book is flexible. The reader who finds
the mathematics too difficult in some places is urged to do what the
mathematician will do when he finds it too trivial: Skip it.
This book, small as it is, would not have been possible without the
wholehearted cooperation of the staff of Golem Press, and I take this
opportunity to express my gratitude to every one of them. I am also
indebted to the Archives Division of the Indiana State Library for
making available photostats of Bill 246, Indiana House of Representatives, 1897, and to the Cambridge University Press, Dover Publications
and Litton Industries for granting permission to reproduce copyrighted
materials without charge. Their courtesy is acknowledged in the notes
accompanying the individual figures.
I much enjoyed writing this book, and it is my sincere hope that the
reader will enjoy reading it, too.
Boulder, Colorado
August 1970

Petr Beckmann

PREFACE

5

PREFACE TO THE SECOND EDITION
After all but calling Aristotle a dunce, spitting on the Roman Empire, and flipping my
nose at some other highly esteemed institutions, I had braced myself for the reviews that
would call this book the sick product of an insolent ignoramus. My surprise was therefore
all the more pleasant when the reviews were very favorable, and the first edition went out
of print in less than a year.
I am most grateful to the many readers who have written in to point out misprints and
errors, particularly to those who took me to task (quite rightly) for ignoring the recent
history of evaluating 17 by digital computers. I have attempted to remedy this
shortcoming by adding a chapter on 17 in the computer age.
Mr. D.S. Candelaber of Golem Press had the bright idea of imprinting the end sheets
of the book with the first 10,000 decimal places of 17, and the American Mathematical
Society kindly gave permission to reproduce the first two pages of the computer print-out
as published by Shanks and Wrench in 1962. A reprint ofthis work was very kindly made
available by one of the authors, Dr. John W. Wrench, Jr. To all of these, I would like to
express my sincere thanks. I am also most grateful to all readers who have given me the
benefit of their comments. I am particularly indebted to Mr. Craige Schensted of Ann
Arbor, Michigan, and M. Jean Meeus of Erps-Kwerps, Belgium, for their detailed lists of
misprints and errors in the first edition.
Boulder, Colorado
May 1971

P.B.

PREFACE TO THE THIRD EDITION
Some more errors have been corrected and the type has been re-set for the third
edition.
A Japanese translation of this book was published in 1973.
Meanwhile, a disturbing trend away from science and toward the irrational has set in.
The aerospace industry has been all but dismantled. CoIIege enroIIment in the hard
sciences and engineering has significantly dropped. The disoriented and the guIlible flock
in droves to the various Maharajas of Mumbo Jumbo. Ecology, once a respected scientific
discipline, has become the buzzword of frustrated housewives on messianic ego-trips.
Technology has wounded affluent intellectuals with the ultimate insult: They cannot
understand it any more.
Ignorance, anti-scientific and anti-technology sentiment have always provided the
breeding ground for tyrannies in the past. The power of the ancient emperors, the
mediaeval Church, the Sun Kings, the State with a capital S, was always rooted in the
ignorance of the oppressed. Anti-scientific and anti-technology sentiment is providing a
breeding ground for encroaching on the individual's freedoms now. A new tyranny is on
the horizon. It masquerades under the meaningless name of "Society."
Those who have not learned the lessons of history are destined to relive it.
Must the rest of us relive it, too?

Boulder, Colorado
Christmas 1974

P.B.

Contents

1. DAWN
9
2. THE BELT
20
3. THE EARLY GREEKS
36
4. EUCLID
45
5. THE ROMAN PEST
55
6. ARCHIMEDES OF SyRACUSE
62
7. DUSK
73
8. NIGHT
78
9. AWAKENING
87
10. THE DIGIT HUNTERS
99
11. THE LAST ARCHIMEDEANS
110
12. PRELUDE TO BREAKTHROUGH
121
13. NEWTON
134
14. EULER. "
147
15. THE MONTE CARLO METHOD
158
16. THE TRANSCENDENCE OF 17 • • • • • • • • • • • • • • • • • • • • • • • 166
17. THE MODERN CIRCLE SQUARERS
173
18. THE COMPUTER AGE
183
Notes
190
Bibliography
193
Chronological Table
196
Index
198

DAWN
History records the names of royal
bastards. but cannot tell us the origin of
wheat.
JEAN HENRI FABRE
(1823-1915)

million years or so have passed since the tool-wielding animal
called man made its appearance on this planet. During this
time it learned to recognize shapes and directions; to grasp the
concepts of magnitude and number; to measure; and to realize that
there exist relationships between certain magnitudes.
The details of this process are unknown. The first dim flash in the
darkness goes back to the stone age - the bone of a wolf with incisions
to form a tally stick (see .figure on next page). The flashes become
brighter and more numerous as time goes on, but not until about 2,000
B.C. do the hard facts start to emerge by direct documentation rather
than by circumstantial evidence. And one of these facts is this: By
2,000 B.C., men had grasped the significance of the constant that is
today denoted by TT, and that they had found a rough approximation
of its value.
How had they arrived at this point? To answer this question, we
must return into the stone age and beyond, and into the realm of
speculation.
Long before the invention of the wheel, man must have learned to
identify the peculiarly regular shape of the circle. He saw it in the pupils
of his fellow men and fellow animals; he saw it bounding the disks of
the Moon and Sun; he saw it, or something near it, in some flowers; and
perhaps he was pleased by its infinite symmetry as he drew its shape in
the sand with a stick.
Then, one might speculate, men began to grasp the concept of
magnitude - there were large circles and small circles, tall trees and
small trees, heavy stones, heavier stones, very heavy stones. The
transition from these qualitative statements to quantitative measure-

13
~

10

CHAPTER ONE

ment was the dawn of mathematics. It
must have been a long and arduous road,
but it is a safe guess that it was first taken
for quantities that assume only integral
values - people, animals, trees, stones,
sticks. For counting is a quantitative
measurement: The measurement of the
amount of a multitude of items.
Man first learned to count to two, and
a long time elapsed before he learned to
count to higher numbers. There is a fair
amount of evidence for this, l perhaps
none of it more fascinating than that
preserved in man's languages: In Czech,
until the Middle Ages, there used to be
two kinds of plural - one for two items,
another for many (more than two) items,
and apparently in Finnish this is so to
this day. There is evidently no connection
between the (Germanic) words two and
half; there is none in the Romance languages (French: deux and moitie> nor in
the Slavic languages (Russian: dva and
po!), and in Hungarian, which is not an
Indo-European language, the words are
kettlJ and f~l. Yet in all European languages, the words for 3 and 113,
4 and 114, etc., are related. This suggests
that men grasped the concept of a ratio,
and the idea of a relation between a
number and its reciprocal, only after they A stone age tally stick. The tibia
(shin) of a wolf with two long
had learned to count beyond two.
The next step was to discover relations incisions in the center. and two
between various magnitudes. Again, it series of 25 and 30 marks.
Found in Vestonice. Moravia
seems certain that such relations were (Czechoslovakia) in 1937. 2
first expressed qualitatively. It must have
been noticed that bigger stones are heavier, or to put it into more
complicated words, that there is a relation between the volume and the
weight of a stone. It must have been observed that an older tree is taller,
that a faster runner covers a longer distance, that more prey gives more
food, that larger fields yield bigger crops. Among all these kinds of

.-..

DAWN

11

relationships, there was one which could hardly have escaped notice,
and which, moreover, had no exceptions:
The wider a circle is "across." the longer it is "around."
And again, this line of qualitative reasoning must have been followed
by quantitative considerations. If the volume of a stone is doubled, the
weight is doubled; if you run twice as fast, you cover double the
distance; if you treble the fields, you treble the crop; if you double the
diameter of a circle, you double its circumference. Of course, the rule
does not always work: A tree twice as old is not twice as tall. The reason
is that "the more ... the more" does not always imply proportionality;
or in more snobbish words, not every monotonic function is linear.
Neolithic man was hardly concerned with monotonic functions; but it
is certain that men learned to recognize, consciously or unconsciously,
by experience, instinct, reasoning, or all of these, the concept of
proportionality; that is, they learned to recognize pairs of magnitude
such that if the one was doubled, trebled, quadrupled, halved or left
alone, then the other would also double, treble, quadruple, halve or
show no change.
And then carne the great discovery. By recognizing certain specific
properties, and by defining them, little is accomplished. (That is why
the old type of descriptive biology was so barren.) But a great scientific
discovery has been made when the observations are generalized in such
a way that a generally valid rule can be stated. The greater its range of
validity, the greater its significance. To say that one field will feed half
the tribe, two fields will field the whole tribe, three fields will feed one
and a half tribes, all this applies only to certain fields and tribes. To say
that one bee has six legs, three bees have eighteen legs, etc., is a
statement that applies, at best, to the class of insects. But somewhere
along the line some inquisitive and smart individuals must have seen
something in common in the behavior of the magnitudes in these and
similar statements:
No matter how the two proportional quantities are varied. their ratio
remains constant.
For the fields, this constant is 1 : 112 = 2 : 1 = 3 : Ph = 2. For the
bees, this constant is 1 : 6 = 3 : 18 = 1/6. And thus, man had discovered a general, not a specific, truth.
This constant ratio was not obtained by numerical division (and
certainly not by the use of Arabic numerals, as above); more likely, the
ratio was expressed geometrically, for geometry was the first mathematical discipline to make substantial progress. But the actual tech-

12

CHAPTER ONE

nique of arriving at the constancy of the ratio of two proportional
quantities makes little difference to the argument.
There were of course many intermediate steps, such as the discovery
of sums, differences, products and ratios; and the step of abstraction,
exemplified by the transition from the statement "two birds and two
birds make four birds" to the statement "two and two is four." But
the decisive and great step on the road to 17 was the discovery that
proportional quantities have a constant ratio.
From here it was but a dwarfs step to the constant 17: If the
"around" (circumference) and the "across" (diameter) of a circle
were recognized as proportional quantities, as they easily must have
been, then it immediately follows that the ratio
circumference : diameter = constant for all circles.
This constant circle ratio was not denoted by the symbol 17 until
the 18th century (A.D.), nor, for that matter, did the equal sign ( = )
come into general use before the 16th century A.D. (The twin lines
as an equal sign were used by the English physician and mathematician Robert Recorde in 1557 with the charming explanation that
"noe .2. thynges, can be moare equalle.") However, we shall use
modern notation from the outset, so that the definition of the number 17 reads
17

=

C
D

where C is the circumference, and D the diameter of any circle.
And with this, our speculative road has reached, about 2,000 B.C.,
the dawn of the documented history of mathematics. From the
documents of that time it is evident that by then the Babylonians
and the Egyptians (at least) were aware of the existence and significance of the constant TT as given by (1).

BUT

the Babylonians and the Egyptians knew more about 17
than its mere existence. They had also found its approximate value.
By about 2,000 B.C., the Babylonians had arrived at the value

(2)
and the Egyptians at the value
17

(3)

13

DAWN

c
Ea
A~...L-----'--~'-'------'--I---"4----J

B

How to measure 7T in the sands
oftheNile

E

How did these ancient people arrive at these values? Nobody
knows for certain, but this time the guessing is fairly easy.
Obviously, the easiest way is to take a circle, to measure its
circumference and diameter, and to find 7T as the ratio of the two.
Let us try to do just that, imagining that we are in Egypt in 3,000
B.C. There is no National Bureau of Standards; no calibrated measuring tapes. Weare not allowed to use the decimal system or
numerical division of any kind. No compasses, no pencil, no paper;
all we have is stakes, ropes and sand.
So we find a fairly flat patch of wet sand along the Nile, drive in a
stake, attach a piece of rope to it by loop and knot, tie the other end
to another stake with a sharp point, and keeping the rope taut, we
draw a circle in the sand. We pull out the central stake, leaving a
hole 0 (see figure above). Now we take a longer piece of rope, choose
any point A on the circle and stretch the rope from A across the hole
o until it intersects the circle at B. We mark the length AB on the
rope (with charcoal); this is the diameter of the circle and our unit of
length. Now we take the rope and lay it into the circular groove in
the sand, starting at A. The charcoal mark is at C; we have laid off
the diameter along the circumference once. Then we lay it off a
second time from C to D, and a third time from D to A, so that the
diameter goes into the circumference three (plus a little bit) times.
If, to start with, we neglect the little bit, we have, to the nearest
integer,

17=3

(3)

To improve our approximation, we next measure the little left-over
bit EA as a fraction of our unit distance AB. We measure the curved

CHAPTER ONE

14

23

~n i'~~ ~:0M~ ~J~
~~~ TD~'" ~~~~ I t,~~ 'M~~

'M~~~ ~~;

t~~;~ '~N ~~: ~~~ Q~~~

28.

mm 'M~'i?

Kat (TrOCT/af TTfV {)at..aaaav OfKa fV 1TTfXH aTTO TOU XHt..OU"; aUT7Jt;

((,It; TOV t/iHt..OVt; aUTTfS, aTpOyyUt..OV KUft..W TO aUTO.

VI/10F, aVTTfS' Kat aVvTfYllfVTf TpHS

"at TpWKOvra

TTfVTf fV TTTfXH TO

fV TTtXH.

21 Hizo asimismo un mar de fundicion. de diez codos del uno al otro
lado. redondo. y de cinco codos de
alto. y cefiialo en derredor un cordon de treinta codos.

2:l. II fit uussi une mer de fonte, de
dix coudeeH d'un bord jusqu'a I'autre,
qui ctuit toutc ronde: eUe avalt cinq
l'OUd«(PH de haut, et elle etait environndc tout h I'cntour d'un cordon de
tn'ntl' coudees.

23. IId15lal tet mafe slite, deslti laket ad jednaha kraje k druhemu,
o!uollhlJ l'ukal. a pet laket byla uysakast jeha, a okolek jeha tFicet
I "J.>e I 1'17kal.

23. Unb er mad)te ein 9Jtecr, gegoffen,
non einem 9\anb ~um an bern 5d)tt (f({en
roeit, runbum~er, unb fiinf ~Uen ~od),
nnb eine 6d)nur breiBig ~Uen lang war
bas 9JtaB ringsum.

23. And he made a molten sea, ten cubits from the one brim
to the other; it was round all about, and his height was five
cubits: and a line of thirty cubits did compass it round about.

DAWN

15

length EA and mark it on a piece of rope. Then we straighten the
rope and lay it off along AB as many times as it will go. It will go
into our unit distance AB between 7 and 8 times. (Actually, if we
swindle a little and check by 20th century arithmetic, we find that 7
is much nearer the right value than 8, i.e., that E 7 in the figure on
p. 13 is nearer to B than E s , for 1/7 = 0.142857 , 118 = 0.125,
and the former value is nearer 17 - 3 = 0.141592
However, that
would be difficult to ascertain by our measurement using thick,
elastic ropes with coarse charcoal marks for the roughly circular
curve in the sand whose surface was judged "flat" by arbitrary
opinion.)
We have thus measured the length of the arc EA to be between
1/7 and 118 of the unit distance AB; and our second approximation
is therefore

(4)
for this, to the nearest simple fractions, is how often the unit rope
length AB goes into the circumference ABeD.
And indeed, the values
7T

are the values most often met in antiquity.
For example, in the Old Testament (I Kings vii.23, and 2 Chronicles iv.2), we find the following verse:
"Also, he made a molten sea of ten cubits from
brim to brim. round in compass, and five cubits the
height thereof; and a line of thirty cubits did compass
it round about."

The molten sea, we are told, is round; it measures 30 cubits round
about (in circumference) and 10 cubits from brim to brim (in diameter); thus the biblical value of 17 is 30/10 = 3.
The Book of Kings was edited by the ancient Jews as a religious
work about 5SO B.C., but its sources date back several centuries. At
that time, 7T was already known to a considerably better accuracy,
but evidently not to the editors of the Bible. The Jewish Talmud,
which is essentially a commentary on the Old Testament, was published about SOO A.D. Even at this late date it also states "that
which in circumference is three hands broad is one hand broad."

16

CHAPTER ONE

The molten sea as reconstructed by Gressman
from the description in 2 Kings vii.'

In early antiquity, in Egypt and other places, the priests were
often closely connected with mathematics (as custodians of the calendar, and for other reasons to be discussed later). But as the process
of specialization in society continued, science and religion drifted
apart. By the time the Old Testament was edited, the two were
already separated. The inaccuracy of the biblical value of 17 is, of
course, no more than an amusing curiosity. Nevertheless, with the
hindsight of what happened afterwards, it is interesting to note this
little pebble on the road to confrontation between science and religion, which on several occasions broke out into open conflict, and
about which we shall have more to say later.
Returning to the determination of 17 by direct measurement using
primitive equipment, it can probably safely be said that it led to
values no better than (4).
From now on, man had to rely on his wits rather than on ropes
and stakes in the sand. And it was by his wits, rather than by
experimental measurement, that he found the circle's area.

THE

ancient peoples had rules for calculating the area of a
circle. Again, we do not know how they derived them (except for one
method used in Egypt, to be described in the next chapter), and once

DAWN

17

Calculation of the area of a circle by
integral calculus. The area of an elementary ring is ciA =211pdp; hence the area of
the circle is
r

A = 211

.r pdp

11r

2

•

(J

more we have to play the game "How do you do it with their
knowledge" to make a guess. The area of a circle, we know, is
A =

11r

(5)

where r is the radius of the circle. Most of us first learned this
formula in school with the justification that teacher said so, take it
or leave it, but you better take it and learn it by heart; the formula
is, in fact, an example of the brutality with which mathematics is
often taught to the innocent. Those who later take a course in the
integral calculus learn that the derivation of (5) is quite easy (see
figure above). But how did people calculate the area of a circle
almost five millenia before the integral calculus was invented?
They probably did it by a method of rearrangement. They calculated the area of a rectangle as length times width. To calculate
the area of a parallelogram, they could construct a rectangle of equal
area by rearrangement as in the figure below, and thus they found
that the area of a parallelogram is given by base times height. The
age of rigor that came with the later Greeks was still far away; they

The paraIIelogram and the rectangle have equal areas,
as seen by cutting off the shaded triangle and reinserting it as indicated.

18

CHAPTER ONE
2rrr
2Trr

(hi
(a)

2 Tr r

-_
(
...

2 Trr
_------------------

_
Determination of the area of a circle by rearrangement.
The areas of the figures (b). (c). (d) equal exactly double
the area of circle (a).

did not have to know about congruent triangles to be convinced by
the "obvious" validity of the rearrangement.
So now let us try to use the general idea of rearrangement as in
the figure above to convert a circle to a parallelogram of equal area.
We are still using sticks to draw pictures in the sand, but this time
we do this only to help our imagination, not to perform an actual
measurement.
We first cut up a circle into four quadrants as in (a) above, and
arrange them as shown in figure (b). Then we fill in the spaces
between the segments by four equally large quadrants. The outline of
the resulting weird figure is vaguely reminiscent of a parallelogram.
The length of the figure, measured along the circular arcs, is equal
to the circumference of the original circle, 2TTr. What we can say with
certainty is that the area of this figure is exactly double the area of
the original circle.
If we now divide the circle not into four, but into very many
segments, our quasi-parallelogram (c) will resemble a parallelogram

DAWN

19

The rearrangement method
used in a 17th century Japanese document. 4

much more closely; and the area of the circle is still exactlly one half
of the quasi-parallelogram (c).
On continuing this process by cutting up the original circle into a
larger and larger number of segments, the side formed by the little
arcs of the segments will become indistinguishable from a straight
line, and the quasi-parallelogram will turn into a true parallelogram
(a rectangle) with sides 2 TTr and r. Hence the area of the circle is
half of this rectangle, or 1Tr 2•
The same construction can be seen in the Japanese document
above (1698). Leonardo da Vinci also used this method in the 16th
century. He did not have much of a mathematical education, and in
any case, he could use little else, for Europe in his day, debilitated
by more than a millenium of Roman Empire and Roman Church,
was on a mathematical level close to that achieved in ancient Mesopotamia. It seems probable, then, that this was the way in which
ancient peoples found the area of the circle.
And that should be our last speculation. From now on, we can rely
on recorded history.

THE BELT
Accurate reckoning - the entrance into
the knowledge ofall existing things and all
obscure secrets.
AHMES THE SCRIBE
17th century B.C.

AN is not the only animal that uses tools on his environment; so do chimpanzees and other apes (also, some birds).
As long as man was a hunter, the differences between the
naked ape and the hairy apes was not very radical. But roughly
about 10,000 B.C., the naked ape learned to raise crops and to tame
other animals, and thereby he achieved something truly revolutionary: Human communities could, on an average, produce so much
more food above the subsistence minimum that they could free a
part of their number for activities not directly related to the provision of food and shelter.
This Great Agricultural Revolution first took place where the
geographical conditions were favorable: Not in the north, where the
winters were long and severe, and the conditions for farming generally adverse; nor in the tropics, where food was plentiful, clothing
unnecessary, shelter easily available, and therefore no drastic need
for improvement; but in the intermediate belt, where conditions were
sufficiently adverse to create pressures for change, yet not so adverse
as to foil the attempts of farming and livestock raising.
This intermediate belt stretched from the Mediterranean to the
Pacific. The Great Revolution first took place in the big river valleys
of Mesopotamia; later the Belt stretched from Egypt through Persia
and India to China. States developed. Specialists came into being.
Soldiers. Priests. Administrators. Traders. Craftsmen. Educators.
And Mathematicians.

II)

THE BELT

21

The hunters had neither time nor need for ratios, proportionalities
or conic sections. The new society needed surveyors and builders,
navigators and timekeepers (astronomers), accountants and stockkeepers, planners and tax collectors, and, yes, mumbo-jumbo men to
impress and bamboozle the uneducated and oppressed. This was the
fertile ground in which mathematics flourished; and it is therefore
not surprising that the cradle of mathematics stood in this Belt.

SINCE Mesopotamia was the first region of the Belt where the
agricultural revolution occurred and a new society took hold, one
would expect Babylonian mathematics to be the first and most
advanced. This was indeed the case; the older literature on the
history of mathematics often saw the Egyptians as the founders of
mathematics, but this was due to the fact that more and earlier
Egyptian documents than any others used to be available. The research of the last few decades has changed this, and as a small
sidelight, we find a better approximation for 17 in Mesopotamia than
in Egypt.
One of the activities for which the new society freed some of its
members was, unfortunately, organized warfare, and the various
peoples inhabiting the region at different times, Sumerians, Babylonians, Assyrians, Chaldeans and others, warred against each other
as well as against outsiders such as Hittites, Scythians, Medes and
Persians. The city of Babylon was not at all times the center of this
culture, but the mathematics coming from this region is simply
lumped together as "Babylonian."
In 1936, a tablet was excavated some 200 miles from Babylon.
Here one should interject that the Sumerians were first to make one
of man's greatest inventions, namely, writing; through written communication, knowledge could be passed from one person to others,
and from one generation to the next and future ones. They impress~d their cuneiform (wedge-shaped) script on soft clay tablets
with a stylus, and the tablets were then hardened in the sun. The
mentioned tablet, whose translation was partially published only in
1950,5 is devoted to various geometrical figures, and states that the
ratio of the perimeter of a regular hexagon to the circumference of
the circumscribed circle equals a number which in modern notation
is given by 57/60 + 36/(60)2 (the Babylonians used the sexagesimal
system, i.e., their base was 60 rather than 10).

22

CHAPTER TWO

r

r

c

r

The Babylonian value of

TT.

The Babylonians knew, of course, that the perimeter of a hexagon
is exactly equal to six times the radius of the circumscribed circle, in
fact that was evidently the reason why they chose to divide the circle
into 360 degrees (and we are still burdened with that figure to this
day). The tablet, therefore, gives the ratio 6r/C, where r is the
ra...