Galileo and the Equations of Motion

183 Pages·2016·2.046 MB·English

by  Dino Boccaletti (auth.)

Dino Boccaletti
Galileo
and the
Equations of
Motion
Galileo and the Equations of Motion
Dino Boccaletti
Galileo and the Equations
of Motion
With 26 Figures
1 3
Dino Boccaletti
Dipartimento di Matematica
University of Rome “La Sapienza”
Rome
Italy
ISBN 978-3-319-20133-7 ISBN 978-3-319-20134-4 (eBook)
DOI 10.1007/978-3-319-20134-4
Library of Congress Control Number: 2015943801
Springer Cham Heidelberg New York Dordrecht London
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To Ruben and Matilde
Preface
During my life as a researcher (a theoretical physicist) and as a teacher
(of celestial mechanics, for the last two decades), I always found that the vulgate
included in the textbooks of physics not always substantially coincided with what
affirmed by the historians of science.
What was taken for granted in the textbooks of physics actually was still a sub-
ject of discussion for the scholars of Galileo’s works, especially concerning the
equations of motion (those we know, and use, under the form of Newton’s laws).
From this remark, it came out the idea to “throw a bridge” between the world of
the scholars of history of science and the “users” of the textbooks of physics. This
bridge is limited to the equations of motion.
Acknowledgments
I wish to thank my wife Maria Grazia and my daughter Chiara for their invaluable
help in drawing up and completing this book.
vii
Contents
Introduction ................................................... xi
A Preamble .................................................... xi
Notice to the Reader ............................................. xiv
Part I The Problem of Motion Before Galileo
1 The Theories on the Motion of Bodies in the Classical Antiquity ..... 3
1.1 Kinematics Among the Greeks .............................. 4
1.2 Dynamics in the Opinion of Aristotle and His Continuators,
in Greece and in Rome .................................... 10
1.3 After Aristotle in Greece and in Rome ........................ 18
Bibliography ................................................ 22
2 The Theories of Motion in the Middle Ages and in the Renaissance ... 25
2.1 Preliminary Remarks ..................................... 25
2.2 The First Substantial Criticisms to Aristotelian
Mechanics—Philoponus and Avempace ...................... 29
2.3 The Medieval Kinematics .................................. 30
2.3.1 Gerard of Brussels and the Liber de Motu ............... 31
2.3.2 The Kinematics at Merton College .................... 32
2.3.3 The Kinematics of the Parisian School ................. 36
2.4 The Medieval Dynamics ................................... 38
2.4.1 Bradwardine’s Dynamics ............................ 38
2.4.2 Dynamics at the Parisian School
and the Impetus—Theory ............................ 39
2.5 The Diffusion in Italy of the Ideas of Mertonians
and of the Parisian Masters ................................ 42
2.6 The Theory of Motion in the XVI Century .................... 44
2.6.1 Niccolò Tartaglia (1500–1557)—His Life and His Works ... 45
2.6.2 The Mechanics of Giovan Battista Benedetti ............. 50
ix
x Contents
2.7 Galileo and the Engineers of the Renaissance .................. 58
Bibliography ................................................ 60
Part II Galileo and the Motion
3 The Young Galileo and the de Motu ............................ 63
3.1 On the Editions of the Dialogue and the Treatise ................ 63
3.2 The Vicissitudes of the Manuscripts .......................... 65
3.3 The Dialogue ........................................... 67
3.4 The Treatise ............................................ 74
3.4.1 The Natural Motions ............................... 76
3.4.2 The Circular Motion ................................ 83
3.4.3 The Motion of the Projectiles ......................... 84
3.5 Conclusions ............................................ 86
3.6 Additional Considerations ................................. 89
Bibliography ................................................ 91
4 The Inertia Principle ......................................... 93
4.1 The Lex I of Newton ..................................... 94
4.2 The Inertia in the De Motu ................................. 95
4.3 Inertia in Le Mechaniche and in Dimostrazioni
Intorno Alle Macchie Solari E Loro Accidenti .................. 98
4.4 The Inertia Principle in the Dialogue and in the Discourses ....... 102
4.5 The Inertia Principle in the “Sixth Day” ...................... 109
5 The Motion of Heavy Bodies and the Trajectory of Projectiles ...... 117
5.1 An Anticipation ......................................... 117
5.2 The Motion of Falling Bodies in the “Third Day”
of the Discourses ........................................ 122
5.2.1 Appendix—An “Unpleasant Incident” .................. 124
5.3 Velocity and Space in the Uniformly Accelerated Motion ......... 126
5.3.1 Appendix—From the Correspondence
of Paolo Sarpi and Galileo ........................... 137
5.4 The Historical Experiment and the Postulate ................... 137
5.5 The Motion of Projectiles and the Parabolic Trajectory ........... 148
5.6 A Gloss on the Pendulum .................................. 154
Bibliography ................................................ 162
6 Galileo and the Principle of Relativity .......................... 163
6.1 The Relative Motion Before Galileo ......................... 164
6.2 How Galileo Expresses the Principle of Relativity .............. 165
Final Considerations ............................................ 169
Name Index ................................................... 173
Introduction
A Preamble
Currently, in the textbooks of physics (at any level of specialization—from the
texts for the secondary schools to university handbooks, to research books), a cer-
tain number of “results” are credited to Galileo.
Limiting ourselves to the field of dynamics, which we intend to be interested
in, one goes from “principles” (inertia principle, principle of relativity—that is
so Galilean) to “laws” and, anyhow, relations (motion with constant acceleration,
parabolic trajectory of projectiles, inclined planes, isochronism of the small oscil-
lations, etc.).
At present, we all know that in the textbooks, when one says “Tom or Dick
equation,” he does not want to assert that Tom or Dick had really obtained that
equation in that specific form, but that we are in the presence of an equation which
comes out from a work, possibly long and hard, unequivocally based on what Tom
or Dick had enunciated (guessed or demonstrated).
That is, what we find in the textbooks is just the last formulation, in the formal-
ism and with the notations adopted at present, of an “equation” or of a “law”
which is made “to go back” to Tom or Dick.1
1In this connection, it also exists the “Arnold’s principle” which states: «if a notion bears a
personal name, then this name is not the name of the discoverer».
xi
xii Introduction
Do not forget that Galileo was credited, in modern times, with the foundations
of dynamics,2 even if this opinion was questioned.3 Even if at present the thesis of
Pierre Duhem, who maintained that the Parisian School had anticipated Galileo, is
no more widely accepted, however, some effects still survive.
We shall discuss onward of this and of the work of the authors of Merton
College (Oxford), but we consider necessary here and now to make clear that the
assertions of Duhem, in our opinion, do form part of an attitude which should be
avoided when one is doing history of science.
One assumes the risk of inventing nonexistent priorities, by interpreting the
ancient manuscripts in a contemporaneous perspective, always having an eye to
elements which, just read looking at the present, support priorities of theories and
discoveries not recognized by the preceding historians.
Therefore, for each of those “results,” we must follow “historically” and criti-
cally Galileo’s path, by bringing out the difficulties, his dependence on forerun-
ners, but also his neat methodological detachment from the latter, he got also by
availing himself of experiments, both effectively realized and “ideal.”
Of course, we shall run up against still open questions on which the scholars
have not drawn generally accepted conclusions, for instance the case of the experi-
ments alluded above. A further difficulty in considering the conclusions reached
by Galileo comes from the fact that the terms he used for representing the kin-
ematic magnitudes do not correspond to the concepts and definitions used at
present.
Every book entails (or hopes) the existence of a possible type of readers. This
one wants to address the persons who are not satisfied with finding enunciations
of “laws,” “principles,” etc., credited to Galileo, but are curious to know what the
intellectual debate that culminated in their enunciation has been.
2On a European level, the clear assertion by Ernst Mach “the founder of dynamics is Galileo”
was very important. Mach’s work The Science of mechanics: a critical and historical account
of its development (which contains the quoted assertion) had nine editions (since 1883–1933)
and strongly influenced the debate on the subject. We can refer to the English translation by S.
MacCormack—The open court publishing, 1960.
3At the moment, we limit ourselves to report two excerpts from the well-known preface of Pierre
Duhem to the work Études sur Léonard de Vinci, troisième série—Les précurseurs par-
isiens de Galilée—p. VI and pp XIII, XIV. «Lorsque nous voyons la science d’un Galilée tri-
ompher du Péripatétisme buté d’un Cremonini, nous croyons, mal informés de l’histoire de la
pensée humaine, que nous assistons à la victoire de la jeune Science moderne sur la Philosophie
médiéval, obstinée dans son psittacisme; en vérité, nous contemplons le triomphe, longuement
préparé, de la science qui est née à Paris au XIV e siècle sur les doctrines d’Aristote et d’Averroès,
remises en honneur par la Renaissance italienne.» […] «Cette substitution de la Physique mod-
erne à la Physique d’Aristote a résultè d’un effort de longue durée et d’extraordinaire puis-
sance. Cet effort, il a pris appui sur la plus ancienne et la plus resplendissante des Universités
médiévales, sur l’Université de Paris. Comment un parisien n’en serait-il pas fier? Ses promoteurs
les plus éminents ont été le picard Jean Buridan et le normand Nicole Oresme. Comment un fran-
çais n’en éprouverait-il pas un légitime orgueil? Il a résulté de la lutte opiniâtre que l’Université
de Paris, véritable gardienne, en ce temp-là, de l’orthodoxie catholique, mena contre le pagan-
isme péripatéticien et néoplatonicien. Comment un chrétien n’en rendrait-il pas grâce a Dieu?»