Science Awakening I

335 Pages·1975·29.48 MB·English

by  B. L. Van Der Waerden (auth.)

SCIENCE AWAKENING
B.L.VAN DER WAERDEN
SCIENCE
AWAKENING
I
English translation
by
A rnold Dresden
with additions of the author
Fourth edition
KLUWER ACADEMIC PUBLISHERS, DORDRECHT, THE NETHERLANDS
SCHOLAR'S BOOKSHELF, PRINCETON JUNCTION, NEW JERSEY, U.S.A.
Hardcover edition published throughout the world, exclusive of
North America, by Kluwer Academic Publishers, Spuilboulevard 50,
P.O. Box 17, 3300 AA Dordrecht, The Netherlands
Paperback edition published throughout the world by The Scholar's
Bookshelf, 51 Everett Drive, P.O. Box 179, Princeton Junction,
New Jersey 08550, United States of America
Hardcover edition published in North America by
The Scholar's Bookshelf
Copyright © 1975 by Noordhoff International
Softcover reprint oft he hardcover 4th edition 1975
Publishing, a division of Kluwer Academic Publishers,
Dordrecht, The Netherlands
All rights reserved. No part of this publication may be
reproduced, stored in a retrieval system, or transmitted, in any
form by any means, electronic, mechanical, photocopying,
recording or otherwise without the prior permission of the
copyright owner.
First edition 1954
Second edition 1%1
Third edition 1969
Fourth edition 1975
Fifth edition 1988
Kluwer Academic Publishers
ISBN-13: 978-94-010-7115-4 ~ISBN-13: 978-94-009-1379-0
DOl: 10.1007/978-94-009-1379-0
First Scholar's Bookshelf hardcover printing, 1988
First Scholar's Bookshelf paperback printing, 1988
PREFACE TO THE ENGLISH EDITION
Soon after the publication of
my"Ontwakende W etenschap"the need for an English translation
was felt. We were very glad to find a translator fully familiar with
the English and Dutch languages and with mathematical terminol·
ogy. The publisher, Noordhoff, had the splendid idea to ask
H. G. Beyen, professor of archeology, for his help in choosing
a nice set of illustrations. It was a difficult task. The illustrations
had to be both instructive and attractive, and they had t~ illustrate
the history of science as well as the general background of ancient
civilization. The publisher encouraged us to find better and still
better illustrations, and he ordered photographs from all over
the world, with never failing energy and enthusiasm. Mr. Beyen's
highly instructive subscripts will help the reader to see the inter·
relation between way of living, art, and science of the ancient world.
Thanks are due to many correspondents, who have suggested
additions and pointed out errors. Sections on Astrolabes and
Stereographte Projection and on Archimedes' construction of
the heptagon have been added. The sections on Perspective and
on the Anaphorai of Hypsicles have been enlarged.
In the second English edition I have incorporated an important
discovery of P. Huber, which sheds new light upon the role of
geometry In Babylonian algebra (see p. 73). The section on Heron's
Metrics (see p. 277) was written anew, follOWing a suggestion of
E. M. Bruins.
Zurich. 1961 B. L. VAN DER WAERDEN
PREFACE
Why History of Mathematics?
Every one knows that we are living in a technological era. But it is not often
realized that our technology is based entirely on mathematics and physics.
When we ride home on the streetcar in the evening, when we turn on the
electric light and the radio, everything depends on cleverly constructed physical
mechanisms based on mathematical calculations. But more than that! We owe
to physics not only these pleasant articles of luxury, but, to a large extent, even
our daily bread. Apart from the fact that our grains come to us, chiefly by steamer
from overseas, our own agriculture would be far less productive without artificial
fertilizers. Such fertilizers are chemical products, and chemistry depends on
physics.l
But science has not brought us blessings only. The destructive armaments which
mankind uses at the present time to knock its own civilization to pieces are also
products to which the development of mathematics and physics have inevitably
led.
Our spiritual life is also influenced hy science and technology, in a measure but
rarely fully understood. The unprecedented growth of natural science in the 17th
century was followed ineluctably by the rationalism of the 18th, by the deification
of reason and the decline of religion; an analogous development had taken place
earlier, in Greek times. In a similar manner, the triumphs of technology in the
19th century were followed in the 20th by the deification of technology. Unfor
tunately, man seems to be overly inclined to deify whatever is powerful and
successful.
These considerations indicate that science has put its stamp on the whole of
our life, material and spiritual, in its beneficent and in its evil aspects. Science
is the most significant phenomenon of modern times, the principal ingredient of
our civilization - alas!
But if this be true, then the most important question for the history of culture
is: How did our modern natural science come about?
It will be conceded that most historical writings either do not consider this
question at all, or else deal with it in a very unsatisfactory manner. For example,
which are the histories of Greek culture that mention the names of Theaetetus
and of Eudoxus, two of the greatest mathematicians of all times? Who realizes
that, from the historical point of view, Newton is the most important figure of
the 17th century?
Every physicist will admit that the mechanics of Newton are the foundation of
modern physics. Every astronomer knows that modern astronomy begins with
Kepler and Newton. And every mathematician knows that the largest domain of
1 The ruder should bur in mind that this book was addressed originally to the public of the N,·thcrl.nds.
4 PREFACE
modern mathematics, the part most important for physics, is Analysis, which has
its roots in the Differential and Integral Calculus of Newton. Thus the work of
Newton constitutes the foundation for by far the greater part of modern exact
science.
It was Newton who discovered the fundamental laws of motion, to which
terrestrial as well as celestial objects are subject. He placed the crown on the task
of renovating antique astronomy, begun by Copernicus and Kepler. He discovered
a general method for solving all problems of differentiation and integration,
whereas Archimedes, the greatest genius of antiquity, had not progressed beyond
special methods for particular problems.
The work of Newton can not be understood without a knowledge of antique
science. Newton did not create in a void. Without the stupendous work of Ptole
my, which completed and closed antique astronomy, Kepler's Astronomia Nova,
and hence the mechanics of Newton, would have been impossible. Without the
conic sectIOns of Apollonius, which Newton knew thoroughly, his deVelopment
of the law of gravitation is equally unthinkable. And Newton's integral calculus
can be understood only as a continuation of Archimedes' determination of areas
and volumes. The history of mechanics as an exact science begins '.vith the laws
of the lever, the laws of hydrostatics and the determinatIOn of mass centers by
Archimedes.
In short, all the developments which converge in the work of Newton,"1hose of mathematics,
of mechanics and of astronomy, begin in Greece.
The History of Greek Mathematics,
from fhales to Apollonius, covers the four centuries from 600 B.C. to 200 B.C.
Until recently, the first three of these four centuries were enveloped in twilight,
because we possess only two original texts from this period: the fragment con
cerning the lunules of Hippocrates and that of Archytas on the duplication of the
cube. To this can be added two brief fragments of Archytas, a number of scattered
communications of Plato, Aristotle, Pappus, Proclus and Eutocius, and a self
contradictory set of Pythagorean legends. For this reason, the older works, such
a~ Cantor's Geschichte der Mathematik, contain little more about this period
than speculations concerning things of which we really do not know anything,
such as, for example, the "Theorem of Pythagoras"
In recent times however more light has penetrated into the darkness. In the
first place, as a result of the indefatigable industry of Otto Neugebauer and his
collaborators, we know now the mathematical cuneiform texts, which have thrown
an entirely new light not only on the Theorem of Pythagoras, but especially on
the earliest history of arithmetic and of algebra. Neugebauer, following in the
tracks of Zeuthen, succeeded in discovering the hidden algebraic element in Greek
mathematics and in demonstrating its connection with Babylonian algebra. No
PREFACE 5
longer does the history of algebra begin with Diophantus; it starts 2000 years
earlier in Mesopotamia. And, as to arithmetic, in 1937 Neugebauer wrote: "What
is called Pythagorean in the Greek tradition, had probably better be called
Babylonian"; and, a cuneiform text, concerning "Pythagorean numbers", dis
covered in 1943, showed that he was entirely right.
A second new impulse came from philosophically oriented philology. In 1927,
Stenzel and Toeplitz, with Neugebauer, established the penodical "~ellen und
Studien zur Geschichte der Mathematik, Astronomie und Physik". It was their
purpose to get to know more about the philosophy of Plato by an analysis of the
fundamental concepts of Greek mathematics, and, reciprocally, to learn more
about Greek mathematics by means of an analysis of Plato. This method has
enabled Becker, Reidemeister and others to obtain highly important results. At
an earlier date, Eva Sachs had rescued the excellent mathematician Theaetetus
from oblivion.
Another very fertile method was the analysis of the Elements of Euclid. This
work, written about 300 B.C., proves to be largely a compilation of mathematical
fragments, quite diverse in calibre and quite varied in age. By carefully taking
these fragments apart, by dusting them off and then replacing them in the mathe
matical historical environment from which they had originally come, it has become
possible to obtain a considerably clearer picture of Greek mathematICS of the
years 500---300 B.C.
These different things have not as yet been brought together in a book. We
have, it is true, an excellent book by O. Neugebauer on "Vorgriechische Mathe
matik", and there are excellent works on Euclid, Aristarchus, Archimedes and
Apollonius. As to these latter writers, we shall therefore be able to confine our
selves to the most interesting and the most important things; we shall pick tidbits
here and there from their works and we shall try to serve them in as tasteful a manner
as possible. But the principal purpose of the present book is to explain clearly
how Thales and Pythagoras took their start from Babylonian mathematics but gave it a
very different, a specifically Greek character;
how, in the Pythagorean school and outside, mathematics was brought to higher and ever
higher development and began gradually to satisfy the demands of stricter logic;
how, through the work of Plato's friends Theaetetus and Eudoxus, mathematics was
brought to the state of perfection, beauty and exactness, which we admire in the elements
of Euclid.
We shall see moreover that the mathematical method of proof served as a
prototype for Plato's dialectiCS and for Aristotle's logic.
The history of mathematics should not be detached from the general history of
culture. Mathematics is a domain of intellectual activity, intimately related not
only to astronomy and mechanics, but also to architecture and technology, to
philosophy, and even to religion (Pythagoras!).
6 PREFACE
Political and social conditions are of very great importance for the flowering
of science and for its character. This will become very clear in Chapter VII when
the mathematics of Alexandria is compared with that of the classical period, and
still more in Chapter VIII in the discussion of the causes for the decay of Greek
mathematics.
The plan of this book.
It is the intention to make this book scientific, but at the same time accessible
to anyone who has learned some mathematics in school and in college, and who
is interested in the history of mathematics. It is to be scientific in the sense that it
is to be based on a study of the sources and that its conclusions are to be supported
by arguments, so as to enable the reader to judge the conclusions for himself.
The naive reader may take the use of such a method for granted. But - how
often has it been sinned against! How frequently it happens that books on the
history of mathematics copy their assertions uncritically from other books, without
consulting the sources! How many fairy tales circulate as "universally known
truths" !
%
Let us quote an example. In 90 of all the books, one finds the statement that
the Egyptians knew the right triangle of sides 3, 4 and 5, and that they used it for
laying out right angles. How much value has this statement? None! What is it
based on! On two facts and an argument of Cantor. The facts are the following:
"rope-stretchers" took part in laying out an Egyptian temple, and the angles at
the base of temples and pyramids are nearly always, very accurately, right angles.
Now Cantor reasons as follows: these right angles must have been constructed by
the rope-stretchers, and I (Cantor) can not think of any other way of constructing
a right angle by means of stretched ropes than by using three ropes of lengths
3, 4 and 5, forming a right triangle. Therefore the Egyptians must have known
this triangle.
Is this not incredible? Not that Cantor at one time formulated this hypothesis,
but that repeated copying made it a "universally known fact". This is nevertheless
the fact.
To avoid such errors, I have checked all the conclusions which I found in mo
dern writers. This is not as difficult as might appear, even if, as is my case, one
cannot read either the Egyptian characters or the cuneiform symbols, and one
is not a classical philologist. For reliable translations are obtainable of nearly all
texts. For example, Neugebauer has translated and published all mathematical
cuneiform texts. The Egyptian mathematical texts have all been translated into
English or German. Plato, Euclid, Archimedes, ... of all these, good translations
exist in French, German and English. Only in a few doubtful cases it became
necessary to consult the Greek text.
Not only is it more instructive to read the classical authors themselves (in
translation if necessary), rather than modern digests, it also gives much greater
PREFACE 7
enjoyment. If my book should lead the reader to do this, it will fully have accom
plished its purpose. For this reason, I advise the reader emphatically not to accept
anything on my say-so, but to verify everything.
I have tried to consider the great mathematicians as human beings living in
their own environment and to reproduce the impression which they made on
their contemporaries. In some cases, the scarcity of source material made this
impossible. but striking personalities such as Pythagoras, Archytas, Theaetetus
ana Archimedes can be made to stand out clearly. It is also possible to get an
impression of the character of Thales, Eudoxus and Eratosthenes. Of the Egyptian
and Babylonian mathematicians not even the names are known.
What is new in this book,
In Chapter II.
A hypothesis of Freudenthal on Indian number symbols.
In Chapter III.
Freudenthal's interpretation of a Babylonian textbook.
In Chapter IV.
A new way of looking at the mathematics of Thales.
In Chapter V.
Reconstruction of the Pythagorean theory of numbers from the arithmetical
books of the Elements.
Connections between the Babylonian and the Greek mathematics, particularly
the Pythagorean mathematics.
The irrationality proofs of Theodorus of Cyrene.
In Chapter VI.
The feeble logic of Archytas of Taras.
Mathematics and the theory of harmony in the Epinomis.
An analysis of Book X of the Elements and a reconstruction of the mathematical
work of Theaetetus.
In Chapter VII.
The history of the Delian problem, actually and according to the dialogue
Platonicus.
In Chapter VIII.
The cause of the decay of Greek mathematics.
Acknowledgements.
I thank Dr. Brinkman, Professor Freudenthal and especially Dr. Dijksterhuis,
who have read the manuscript critically and have made numerous useful remarks.