Table Of ContentAn Introduction to the
History of
Structural Mechanics
Part II
Edoardo Benvenuto
An Introduction to the
History of
Structural Mechanics
Part II: Vaulted Structures and Elastic
Systems
With 115 Illustrations
Springer-Verlag
New York Berlin Heidelberg London
Paris Tokyo Hong Kong Barcelona
Edoardo Benvenuto
Universita di Genova
Ordinario di Scienza delle Costruzioni
Facolta di Architettura di Genova
Genova, Italy
Mathematics Subject Classification: Ol-xx, 73xx, 82xx
Library of Congress Cataloging-in-Publication Data
Benvenuto, Edoardo.
An introduction to the history of structural mechanics I Edoardo
Benvenuto.
p. cm.
Contents: v. 1 Statics and resistance of solids-v. 2. Vaulted
structures and elastic systems.
ISBN-13:978-1-4612-7751-4 e-ISBN-13:978-1-4612-2994-0
001: 10.1007/978-1-4612-2994-0
I. Structural analysis (Engineering)-History. I. Title.
TA646.B46 1990
624.1 '71 '09-dc20 89-26230
CIP
Printed on acid-free paper.
This work was originally published in Italian by G.C. Sansoni, 1981.
© 1991 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover 1st edition 1991
All rights reserved. This work may not be translated or copied in whole or in part without the written
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The use of general oescriptive names, trade names, trademarks, etc. In this publication, even if the
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Text photocomposed using the LATEX system.
9 8 7 6 5 4 3 2 1
ISBN-13:978-1-4612-7751-4
To my mother, Giovanna
Acknowledgments
This book draws its origin from my textbook on the science of structures
and its historical development which I published with the Sansoni Publish
ers in 1981 (La Seienza delle Costruzioni e il suo Sviluppo Storieo, Florence,
1981). I am indebted to Professor Clifford Truesdell, who kindly appreciated
my attempt to outline a history of the relation between rational mechan
ics and structural engineering and presented my work to the late Mr. W.
Kaufmann-Buhler of Springer-Verlag for an English edition. In fact, this
book is not a translation of the original. Mr. Kaufmann-Buhler suggested
that I transform the text into a general introduction to the history of struc
tural mechanics concentrating on some specific topics and expanding their
historical references. I wrote the new text in Italian. I am very grateful
to my colleague and friend Prof. Aurelia V. von Germela for having care
fully interpreted my complex academic style and transformed it into fluent
English. I thank very much also Mrs. Molly Wolf for her clever and pre
cious final copy-editing, which made my manuscript more dry and light for
American preferences. Very useful to me was Dr. Peter Barrington Jones'
kind collaboration, and I am particularly grateful to my assistant, Arch.
Massimo Corradi, for his beautiful drawings, hand-made in "old style".
Finally, I am glad to thank my colleagues Prof. Gianpietro Del Piero and
Prof. Paolo Podio Guidugli for their useful suggestions regarding the topics
in the first volume.
Foreword
This book is one of the finest I have ever read. To write a foreword for· it
is an honor, difficult to accept.
Everyone knows that architects and master masons, long before there
were mathematical theories, erected structures of astonishing originality,
strength, and beauty. Many of these still stand. Were it not for our now
acid atmosphere, we could expect them to stand for centuries more. We
admire early architects' visible success in the distribution and balance of
thrusts, and we presume that master masons had rules, perhaps held secret,
that enabled them to turn architects' bold designs into reality. Everyone
knows that rational theories of strength and elasticity, created centuries
later, were influenced by the wondrous buildings that men of the sixteenth,
seventeenth, and eighteenth centuries saw daily. Theorists know that when,
at last, theories began to appear, architects distrusted them, partly because
they often disregarded details of importance in actual construction, partly
because nobody but a mathematician could understand the aim and func
tion of a mathematical theory designed to represent an aspect of nature.
This book is the first to show how statics, strength of materials, and
elasticity grew alongside existing architecture with its millenial traditions,
its host of successes, its ever-renewing styles, and its numerous problems
of maintenance and repair.
In connection with studies toward repair of the dome of St. Peter's by
Poleni in 1743, on p. 372 of Volume 2 Benvenuto writes
This may be the first case in this history of architecture where
statics and structural mechanics are successfully applied to a
real problem with maturity and full consciousness bf their im
plications. It marks a turning point between two eras: one in
which tradition and prejudice ruled the art of building, and
another in which the mathematicians' and physicists' new the
ories, elaborated in academies and laboratories, were allowed to
make their contribution. It is somehow pleasant to realize that
this anticipation of the great nineteenth-century synthesis of
viii Foreword
science and technology came not from an ordinary bit of build
ing but from one of the most daring and beautiful creations of
the Renaissance at the height of its splendor.
On p. xx of the introduction
The division between inspiration and technique is of very re
cent origin and is largely artificial. In building, science and art
have always been united in the creative act. Not even the most
narrow-minded aesthete or engineer can part the two without
losing something. To see Brunelleschi, Michelangelo, Guarini,
Wren, Mansart, Souffiot, a hundred others, merely as great
artists is to deprive them of credit for their brilliant engineering.
Their wonderful technical innovations, their perfect determina
tion of the weights that had to be balanced and the mechanisms
of collapse that had to be opposed-these give coherence and
splendor to their works.
The two paragraphs just quoted provide a kind of summary, indeed par
tial, of what Benvenuto wishes to tell us and to let us learn, step by step,
not as philosophy or by journalistic simplisms, but by reading expert ob
servations upon a gradual, not always direct history of the science of con
struction. The last paragraphs of his book read in part as follows:
The long, stormy commotion [about the ideas of Menabrea,
Castigliano, Crotti, and Mohr] enlivened scientific literature for
more than a century. Persuasive hypotheses, even more persua
sive confutations, fruitful but fallacious intuitions, sterile but
unexceptionable verdicts, agreements reached unexpectedly
all have been forgotten. What we remember today are the in
struments of engineers, the formulae in daily use. If we asked an
engineer about the origins of the equations he or she uses con
stantly, the reply would be disappointing. They exist; nothing
else matters. Why be curious about their derivation?
True, the authors with whom we conclude our historical out
line were able to supply such effective technical solutions that,
in their hands, the real meaning of the questions they tackled
seems to have been lost. But history has its uses ....
Indeed it does, as the reader will learn.
Not only is Benvenuto a man of astonishing erudition and breadth, but
also he loves his science and is humble before it. He thinks clearly, clearly
organizes his material, difficult and complicated as it seems, and writes
clearly with direct and masterly expression. In leafing over or reading his
book, we recognize a great work, one doubtlessly flawed by many small
errors among several grand truths. Parts of his matter, bit by bit or lacuna
by lacuna, may well be corrected or filled by historians in coming decades,
Foreword ix
but his book can never be replaced as a general, pioneering treatise, a
survey of a great field heretofore seen only dimly, from a distance, but
never trodden. Never before have I learned so much about the history of
mechanics from a single book.
As is often the case with books that start from the foundations of a sub
ject, the beginning of Benvenuto's is the part hardest to understand. The
reader accustomed to scientific works could well begin with Chapter 5 of
Volume 1, "Galileo and his 'Problem''', or with Chapter 8, "Early Theories
of the Strength of Materials". Perhaps, even, he might begin with Volume
2, which opens with "Knowledge and Prejudice before the Eighteenth Cen~
tury". Above all, to get an idea of the spread of the work, every reader
should study first of all and carefully the two tables of contents, for the
titles of the subsections are fascinating. He who is not already expert in
both architecture and mechanics will see there some names he has never
before encountered, associated to problems or structures or theories he is
unlikely to know. In fact, Benvenuto's clarity and directness are such that
a reader might start by fishing out some subsections. Any place you open
this book and read in it, you will be fascinated by what is there. Wherever
you start, for example at the passage first quoted above, I wager you will
end by studying the whole book.
Part I of Volume I, although some may profit best from reading it last,
is of great value. Very few readers will know already all of the contents
of §1.2, "The Enigma of Force and the Foundations of Mechanics". It be
gins with a resume of what should now be regarded as vague meandering,
impotent struggles, foolish attempts at reduction, and justified doubt re
garding the nature of force, the first problem "against which science finds
itself powerless." It ends with "one of the most important events in the
history of mechanics," namely, Walter Noll's organization of the mechan
ics of continua as a mathematical science. There not only is "system of
forces" taken as a primitive term, but also it is clarified by a list of its
mathematical properties. The theory of systems of forces makes mathemat
ical sense, just as Hilbert's axiomatization of Euclidean geometry in terms
of the undefined objects "point", "line", and "plane" makes mathematical
sense. That will not stop philosophers from musing about force and his
torians of science from dilating upon old, obscure, unmathematical ideas
about force, but it does make "force" something a modern scientist, be he
mathematician or be he architect, can use as he does "point", "line", and
"plane" . The intuitive notions, both in geometry and in mechanics, remain;
not only that, they help both in applications and in creative thought; but
the precise concepts stand behind both.
Of course, Benvenuto makes use of secondary works, but also he studies
carefully and analyses meticulously the originals to which they refer. It is
not unusual-as I can vouch through reading his treatment of some sources
that I described too hastily some thirty years ago-not unusual, I say, that
in the end he silently corrects the secondary work he has studied.
x Foreword
Benvenuto rightly refers to many Italian sources which are largely un
mentioned in the general literature. As in many other fields, Italians were
the great leaders in architecture, structures, and remedies for the appar
ent beginnings of failure. Architects from other countries studied in Italy,
and Italian architects designed castles and palaces from Russia to Spain.
The Italians were also second to none in theoretical analyses of architec
tural members and assemblies. Failure to study Italian sources directly is
a general malady of the precise history of science.
Occasionally Benvenuto refers to a rule or solution of a problem as "cor
rect" or "incorrect". Even the sociological historians, with their belief that
the sciences are no more than ephemeral fads, much as history was called
by a famous and once powerful man "the lies that men agree to believe,"
can not justly cavil here, for in architecture the correctly designed arch
is one that does not fall except under conditions it was not intended to
withstand.
C. Truesdell
Contents of Part II
Foreword .. vii
Introduction. xvii
III Arches, Domes and Vaults 307
9 Knowledge and Prejudice before the Eighteenth Century. 309
9.1 "A Strength Caused by Two Weaknesses" . . . . 309
9.2 Viviani's "On the Formation and Size" of Vaults . . 311
9.3 Fr. Derand's Rule . . . . . . . . . . . . . . . . . . . 313
9.4 The First "Scientific" Treatment of the Statics of Arches 315
10 First Theories about the Statics of Arches and Domes . . . . . . 321
10.1 Philippe de la Hire. . . . . . . . . . . . . . . . . . . . . . . 321
10.2 Arches and Catenaries: David Gregory and Jakob Bernoulli 326
10.3 Philippe de la Hire's Memoir of 1712 . 331
10.4 Belidor's Variant . . . . . . . . . . . . . 336
10.5 Couplet's Two Memoirs. . . . . . . . . 338
10.6 Bouguer's First Static Theory of Domes 344
11 Architectonic Debates . . . . . . . . . . . . . . . 349
11.1 The Italians: An Introduction. . . . . . . . 349
11.2 The Case of S. Maria del Fiore in Florence 349
11.3 St. Peter's Dome and the Three Mathematicians 351
11.4 Giovanni Poleni's "Historical Memoirs" . . . 358
11.5 Poleni's Theoretical and Experimental Work 359
11.6 Boscovich and the Cathedral of Milan . . . . 371
12 Later Research . . . . . . . . . . . . . . . . . . . 375
12.1 The "Best Figure of Vaults": Abbe Bossut 375
12.2 Coulomb's Theory of Frictionless Vaults. 386
12.3 Coulomb's Theory: Friction and Cohesion. 394
