A History of Kinematics from Zeno to Einstein. On the Role of Motion in the Development of Mathematics 9783031398711, 9783031398728

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History of Mechanism and Machine Science 46

Teun Koetsier

A History of Kinematics from Zeno to Einstein On the Role of Motion in the Development of Mathematics

History of Mechanism and Machine Science Volume 46

Series Editor Marco Ceccarelli , Department of Industrial Engineering, University of Rome Tor Vergata, Rome, Italy Advisory Editors Juan Ignacio Cuadrado Iglesias, Technical University of Valencia, Valencia, Spain Teun Koetsier, Vrije University of Amsterdam, Amsterdam, The Netherlands Francis C. Moon, Cornell University, Ithaca, USA Agamenon R. E. Oliveira, Technical University of Rio de Janeiro, Rio de Janeiro, Brazil Baichun Zhang, Chinese Academy of Sciences, Beijing, China Hong-Sen Yan, National Cheng Kung University, Tainan, Taiwan

This bookseries establishes a well-defined forum for Monographs and Proceedings on the History of Mechanism and Machine Science (MMS). The series publishes works that give an overview of the historical developments, from the earliest times up to and including the recent past, of MMS in all its technical aspects. This technical approach is an essential characteristic of the series. By discussing technical details and formulations and even reformulating those in terms of modern formalisms the possibility is created not only to track the historical technical developments but also to use past experiences in technical teaching and research today. In order to do so, the emphasis must be on technical aspects rather than a purely historical focus, although the latter has its place too. Furthermore, the series will consider the republication of out-of-print older works with English translation and comments. The book series is intended to collect technical views on historical developments of the broad field of MMS in a unique frame that can be seen in its totality as an Encyclopaedia of the History of MMS but with the additional purpose of archiving and teaching the History of MMS. Therefore. the book series is intended not only for researchers of the History of Engineering but also for professionals and students who are interested in obtaining a clear perspective of the past for their future technical works. The books will be written in general by engineers but not only for engineers. The series is promoted under the auspices of International Federation for the Promotion of Mechanism and Machine Science (IFToMM). Prospective authors and editors can contact Mr. Pierpaolo Riva (publishing editor, Springer) at: [email protected] Indexed by SCOPUS and Google Scholar.

Teun Koetsier

A History of Kinematics from Zeno to Einstein On the Role of Motion in the Development of Mathematics

Teun Koetsier Department of Mathematics Faculty of Science Vrije Universiteit Amsterdam Amsterdam, The Netherlands

ISSN 1875-3442 ISSN 1875-3426 (electronic) History of Mechanism and Machine Science ISBN 978-3-031-39871-1 ISBN 978-3-031-39872-8 (eBook) https://doi.org/10.1007/978-3-031-39872-8 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Dedicated to two great kinematicians, Oene Bottema and Bernard Roth. They had a decisive influence on the course of my life.

Preface

The ideas of the Ionian philosopher Heraclitus are often summarized with the phrase panta rhei, meaning ‘Everything flows’. In Plato’s Cratylus, Socrates says: “Heraclitus is supposed to have said that all things are in motion and nothing at rest; he compares them to the stream of a river, and says that you cannot go into the same water twice”.1 Of course, Heraclitus had a point. Everything is permanently in a state of flux, changing and moving. On the other hand, one of the characteristics of science is that it tries to transcend the permanent flux that Heraclitus observed. Science aims for permanent validity, for generality; its goal is to describe the aspects of reality that do not change. Undoubtedly, kinematical observations had been made before the rise of Greek civilization. Observations on the way in which smoke moves or birds fly are kinematical. So is the observation that circles can be created by rotating a stretched cord about one of its extremities. In Greek culture, some of such observations became part of coherent scientific doctrines. The history of kinematics started with the Greeks, and it started inside Greek mathematics. The word kinematics was coined by the French mathematician and physicist Ampère in the first half of the nineteenth century. He defined kinematics as a subdiscipline of mechanics dealing with motion independently of its causes, i.e. without taking masses and forces into consideration. Ampère’s definition has been interpreted in various ways, but we will use the word mainly as referring to all mathematical properties of motion that are independent of the cause of the motion. This includes kinematical definitions of curves and surfaces. It includes next to the time-independent geometrical properties of motion also the notions of velocity and acceleration and the properties of these notions as well. It also includes the transfer of motion in a machine related to the geometrical structure of the machine.

1

[Plato,1973], Cratylus 402a. vii

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In Greek mathematics and in seventeenth-century mathematics, kinematical definitions of curves played a central role. Remarkable is also Napier’s kinematical definition of the logarithm. In these cases, motion played an important but in the end a subordinate role. In the course of time, slowly motion itself became a subject of investigation. The Greeks gave a definition of uniform motion. Instantaneous velocity was only defined in the late Middle Ages, making the rigid investigation of acceleration possible. The parallelogram of instantaneous velocities did not appear until the seventeenth century. Slowly, the idea of velocity as a vector appeared: velocity has a direction and a magnitude. At the time, force and acceleration were not yet clearly distinguished. Only in the first half of the nineteenth century, acceleration got vector-like properties. Then also the parallelogram law for accelerations was introduced. We will restrict ourselves to Euclidean space, and we will not consider nonEuclidean kinematics. Moreover, in general the motions that we will consider will be smooth that is defined by smooth functions with continuous derivatives. We will distinguish kinematics of mechanisms from theoretical kinematics. A mechanism is roughly a system of parts moving together, for example in a machine. The geometrical properties of the motion and the curves described by the parts belong to kinematics of mechanisms. And so do kinematical definitions of curves, because they can be viewed as defining idealized mechanisms. The general theories of idealized motion without taking forces and masses into consideration belong to theoretical kinematics. The parallelogram of instantaneous velocities belongs to theoretical kinematics. And so does the insight that at a particular moment, the motion of a moving plane in a fixed plane is either an instantaneous rotation of a translation. It was an eighteenth-century discovery. Until the eighteenth century, our story is part of the history of mathematics and in particular the history of geometry. After the eighteenth century, in which mathematics had been dominated by analysis, Monge and Poncelet initiated the rebirth of geometry. The development of kinematics in the nineteenth century can only be understood against the background of the new interest in geometry of that century. In the seventeenth and eighteenth centuries, a new theory was born in which kinematical ideas play an important role: theoretical mechanics. Major contributors to the theory were Isaac Newton, Leonhard Euler and Joseph-Louis Lagrange. Moreover at the end of the eighteenth century, the first steps are taken towards a theory of machines that takes the motion of the machines into account. Also here kinematical ideas started to play an important role. Ampère coined the word ‘kinematics’ probably in particular because of developments pertaining to machines. Yet kinematics seems to have three sources: geometry, theoretical mechanics and mechanical engineering. In each of the three areas, the new word was widely accepted. As for kinematics, the nineteenth century was a golden age and, in particular, the second half of the century. However, two men stand out. They had the most influence on the direction in which the discipline developed. One of them was an engineer, Franz Reuleaux. The other one was a mathematician, Ludwig Burmester. In this book, we will concentrate on them.

Preface

ix

Many nineteenth-century engineers contributed to kinematics. Truly remarkable is, however, Franz Reuleaux’s influential Theoretische Kinematik of 1875, also written for mechanical engineers. Reuleaux used the word kinematics in his own way. Reuleaux, a great proponent of the view that the science of machines ought to be an independent science, used the word kinematics for the study of the mutual motions of the parts of a machine, considered as changes of position. For Reuleaux, kinematics was the core discipline of mechanical engineering. Many nineteenth-century mathematicians as well contributed to kinematics in one way or another. Here Ludwig Burmester’s Lehrbuch der Kinematik of 1888 is important. It contains the first serious attempt to give a synthesis of theoretical kinematics and kinematics of mechanisms. Burmester’s book was meant to be used by mechanical engineers. Burmester viewed kinematics as part of geometry. At the very end of the nineteenth century, the so solid looking building of Newtonian theoretical mechanics started to show cracks. Remarkably enough, Albert Einstein’s theory of special relativity, which meant a fatal blow to the illusion of the universal validity of Newtonian mechanics, including classical kinematics, is a kinematical theory. Because, on the one hand, it is based on classical kinematical notions, but, on the other hand, it is radically different; this book ends with it. In the nineteenth century, the subject of this book, kinematics, got its name. It was defined, although in different ways by different authors, as a separate area of investigation. Books entirely devoted to kinematics started appearing. At universities, chairs for ‘descriptive geometry and kinematics’ were created, and many mathematicians, physicists and mechanical engineers were publishing kinematical results or results related to kinematics. Yet kinematics never succeeded in releasing itself completely from other areas of research. In a sense, kinematics never got rid of its subordinate position. It was always perpetrated as part of some other more-including discipline like mathematics, theoretical mechanics or mechanical engineering. In the nineteenth century, kinematics received so much attention that some were inclined to give kinematics the status of an independent discipline. Yet the subject did not develop enough momentum to gain independence. The literature on the history of kinematics is extensive. I limit myself to a few remarks. In the period 1962–64, Pascal Dupont wrote four excellent articles on the history of planar instantaneous kinematics. And in the period 1974–75, H. Nolle published three papers on the history of linkage coupler curve synthesis. In 2007, Francis Moon wrote an excellent book on Franz Reuleaux. See the bibliography in this book. In particular, the Springer book series History of Mechanism and Machine Science edited by Marco Ceccarelli contains numerous contributions on aspects of the field or individuals who have made significant contributions. A book that attempts to give an overview of the entire field did not yet exist. When I was finishing this book, I discovered that in 2009 Alberto A. Martínez published a book entitled Kinematics, The Lost Origins of Einstein’s Relativity. That book is also devoted to the history of kinematics. Martínez describes kinematics as an initially neglected or even rejected science that was brought to the fore by Ampère. He is primarily interested in the role of kinematical ideas in the background of Einstein’s special theory of relativity. Although there is overlap here and there with the present

x

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book, the two books are very different, both in terms of content and method. In fact, Martínez concentrates on the nineteenth century, while half of this book deals with the period before that. Apart from Einstein’s 1905 paper, Martínez generally avoids theorems and proofs. This book focuses on just that and follows the mathematical chronology as closely as possible. In Martínez’s book, the chapters represent a variety of distinct approaches to the development of kinematics in the nineteenth century. This book contains many mathematical arguments from very different periods in history. An attempt has been made to follow the original reasoning as much as possible. This implies the presence of considerable differences in method and precision of argumentation and has inevitably led to variation in style. While preparing and writing the book, I have benefitted from many people: Gerard Alberts, Jorge Angeles, Luc Bergmans, Oene Bottema, Marco Ceccarelli, Evert Dijksman, Elisabeth Filemon, Alexander Golovin, Rien Kaashoek, Rainer Kaenders, Carlos Lopez Cajun, Klaus Mauersberger, Maarten Maurice, Jan van Mill, Francis Moon, Gerhard Rammer, Bernard Roth, Detlef Spalt, Rüdiger Thiele, Federico Thomas, Taco Visser and Baichun Zhang. Amsterdam, The Netherlands

Teun Koetsier

Contents

1

Philosophers, Mathematics and Motion . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motion Does Not Exist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Mathematics and the Idealist Tradition in Greek Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Mathematics and Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Aristotle Refutes Zeno . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Zeno’s Trick: Motion Is Interpreted as a Super-Task . . . . . . . . . . 1.6 The Neo-platonist Ontological Hierarchy . . . . . . . . . . . . . . . . . . . 1.7 The Postulates 1 Through 3 in Neo-platonism: Proclus Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Zeuthen’s Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 5 7 11 13 15

2

Motion Beyond the Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Euclidean Construction Game . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Incompleteness of the Euclidean Construction Game . . . . . 2.3 Archytas of Tarente . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 A Solution from Plato’s Academy . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Menaechmus and Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 A Remarkable Application and Heron’s Solution . . . . . . . . . . . . 2.7 The Doubling of the Cube: Eratosthenes’ Instrument . . . . . . . . . 2.8 The Neusis-Construction and the Conchoids . . . . . . . . . . . . . . . . 2.9 Diocles’ Cissoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 21 24 26 28 31 34 37 39 42

3

General Considerations and Kinematical Aspects of Motion . . . . . . . 3.1 Pappus’ Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Composition of Different Uniform Motions: The Quadratrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Time-Dependent Kinematical Aspects of Motion . . . . . . . . . . . . 3.4 Composition of Uniform Motions and Paradoxes of Motion in Mechanical Problems . . . . . . . . . . . . . . . . . . . . . . . .

45 45

17 19

46 49 51

xi

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3.5 3.6 4

A Remark on Methodology and a Theorem by Archimedes on Uniform Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Archimedes: Motion in Geometry . . . . . . . . . . . . . . . . . . . . . . . . .

53 57

Kinematical Models in Astronomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Plato and Astronomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Model in Plato’s Timaeus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Eudoxus’ Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Apollonius’ Epicycle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Hipparchus’ Theory of the Motion of the Sun (About 150 BCE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Ptolemy’ Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Ptolemy’s Contributions Continued . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Astronomy in the Islamic World: The Tusi-Couple . . . . . . . . . . .

63 63 66 69 72

5

The Birth of Instantaneous Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Velocity Distributions in Space and Time . . . . . . . . . . . . . . . . . . . 5.3 The Average Velocity of a Rotating Radius . . . . . . . . . . . . . . . . . . 5.4 The Average Velocity of a Rotating Disc . . . . . . . . . . . . . . . . . . . . 5.5 Bradwardine: Towards Instantaneous Velocity . . . . . . . . . . . . . . . 5.6 Dumbleton and the Merton Theorem . . . . . . . . . . . . . . . . . . . . . . . 5.7 Giovanni Casali and Nicole Oresme . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Acceleration: Euler and Newton’s Second Law . . . . . . . . . . . . . .

87 87 88 89 91 92 94 95 96

6

The Parallelogram of Instantaneous Velocities . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Gilles Personne de Roberval: The Tangent as the Line of Instantaneous Advance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Isaac Newton on Tangents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 D’Alembert on the Parallelogram of Instantaneous Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 A Philosophical Aside and Kant on the Parallelogram of Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101 101

Napier, Fermat, Descartes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 John Napier’s Kinematical Definition of the Logarithm and Torricelli’s ‘Logarithmica’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Pierre de Fermat and Motion in His Introduction to Plane and Solid Loci . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 René Descartes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Descartes’ Ambitions and His New Compasses . . . . . . . . . . . . . . 7.6 Algebra Comes In . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Pappus’ Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 An Example: The Turning Ruler and Moving Curve Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113 113

7

76 80 82 84

102 104 108 109

114 118 121 122 126 127 129

Contents

7.9 7.10 7.11 8

xiii

Descartes’ Solution of Pappus’ 5-Line Problem . . . . . . . . . . . . . . 130 The Use of Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 The Final Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

De Witt, van Schooten, Newton and Huygens . . . . . . . . . . . . . . . . . . . . 8.1 Frans van Schooten Junior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Jan de Witt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Frans van Schooten Junior: Mechanisms to Draw a Parabola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Frans van Schooten Junior: Mechanisms to Draw an Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Frans van Schooten Junior: Mechanisms to Draw a Hyperbola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Isaac Newton, Motion and the Fundamental Theorem of the Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 The Method of Fluxions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Circular Motion in the Work of Huygens and Newton . . . . . . . . 8.9 Huygens and Gear Trains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9.1 Leibniz and Transcendental Curves . . . . . . . . . . . . . . . . . .

135 135 136

Towards Theoretical Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 The Instantaneous Center of Rotation, Descartes and Johann Bernoulli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 The Cycloid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 The Inflexion Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 De La Hire’s Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Elliptic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Epicycloidal Gearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 The Euler-Savary Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8 Euler and the Euler-Savary Formula . . . . . . . . . . . . . . . . . . . . . . . . 9.9 The Instantaneous Axis of Rotation in Spherical Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10 Giulio Mozzi and the Instantaneous Screw Axis . . . . . . . . . . . . .

155

10 Theoretical Kinematics as a Subject in Its Own Right . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Augustin Louis Cauchy’s 1827 Paper . . . . . . . . . . . . . . . . . . . . . . 10.3 Michel Chasles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Bobillier’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Jacques Antoine Charles Bresse . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 The Ball Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

175 175 176 179 183 186 189

11 Towards a New Theory of Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Lazare Carnot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Collisions of Hard Bodies and Geometrical Movements . . . . . . . 11.4 The First Fundamental Equation . . . . . . . . . . . . . . . . . . . . . . . . . . .

191 191 193 194 195

9

137 139 141 143 146 147 151 152

155 157 159 160 162 163 166 169 171 173

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11.5 11.6 11.7

The Second Fundamental Equation . . . . . . . . . . . . . . . . . . . . . . . . Gaspard Monge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Theory of Machines in France in the First Half of the Nineteenth Century . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8 Coriolis’ View of Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.9 An Example of a Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.10 The Coriolis Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.11 Riccioli and Grimaldi Noticed the Coriolis-Effect in 1651 . . . . .

197 198

12 The New Science Is Given a Name: Kinematics . . . . . . . . . . . . . . . . . . . 12.1 A New Classification of the Sciences . . . . . . . . . . . . . . . . . . . . . . . 12.2 Robert Willis’ Principles of Mechanism . . . . . . . . . . . . . . . . . . . . 12.3 Henri Résal’s Traité de Cinématique Pure . . . . . . . . . . . . . . . . . . 12.4 Kinematics as the Essence of Theoretical Mechanics . . . . . . . . .

211 211 216 220 223

13 Developments in Kinematics of Mechanisms . . . . . . . . . . . . . . . . . . . . . 13.1 Scheiner’s Pantograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 The Year 1784 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Sweet Simplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Early Theoretical Interest in Watts Linkages . . . . . . . . . . . . . . . . 13.5 Peaucellier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6 Lipman Lipkin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

225 225 226 230 231 234 236

14 The Work of English Mathematicians on Linkages during the Period 1869–1878 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Chebyshev’s Role . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Roberts’ Work in Kinematics Before Sylvester’s Lecture . . . . . . 14.3 Kempe’s First Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Sylvester’s Role . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Roberts’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6 Some Remarks About Further Work . . . . . . . . . . . . . . . . . . . . . . . 14.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

239 239 241 244 245 247 249 251

15 Franz Reuleaux, Kinematics as the Essence of Mechanical Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Franz Reuleaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 The Central Idea: The Kinematical Chain . . . . . . . . . . . . . . . . . . . 15.4 Incomplete Pairs and Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Higher Kinematical Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 Equivalent Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7 Equivalent Rotary Engines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.8 Analysis Versus Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

253 253 254 254 256 257 258 260 260

201 203 204 205 209

Contents

xv

16 Ludwig Burmester, Kinematics as Part of Geometry . . . . . . . . . . . . . . 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Burmester’s Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 The Lehrbuch der Kinematik: Its Contents . . . . . . . . . . . . . . . . . . 16.4 An Example: Stephenson’s Motion . . . . . . . . . . . . . . . . . . . . . . . . 16.5 Martin Grübler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6 A Note on Chebyshev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.7 Grübler on Classifying Kinematical Chains . . . . . . . . . . . . . . . . . 16.8 The Burmester Theory and the Burmester Points . . . . . . . . . . . . . 16.9 On the Reception of Burmester’s Work . . . . . . . . . . . . . . . . . . . . . 16.10 Reuleaux’ Criticism of Burmester . . . . . . . . . . . . . . . . . . . . . . . . . 16.11 Some Nineteenth Century Developments Elsewhere . . . . . . . . . .

265 265 266 266 268 271 273 274 275 279 281 284

17 Albert Einstein, the Kinematics of Special Relativity . . . . . . . . . . . . . . 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 The Principle of Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3 The Principle of the Constancy of Light and the Paradox . . . . . . 17.4 The Willingness to Give Up the Axiom of the Absoluteness of Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.5 Checking the Inspiration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.6 The Technical Development in the 1905 Paper . . . . . . . . . . . . . . . 17.7 Derivation of the Differential Equation for τ = τ (x ' , y, z, t) . . . . 17.8 The Determination of ξ (x ' , y, z, t), η(x ' , y, z, t) and ζ (x ' , y, z, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.9 Towards the Formulae of the Lorentz Transformation . . . . . . . . . 17.10 The Twin Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

287 287 289 290

18 Minkowski: The Universe Is a 4-Dimensional Manifold . . . . . . . . . . . 18.1 Empiricists and Rationalists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Developments in Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 Hilbert’s Influence and Minkowski’s Rationalism . . . . . . . . . . . . 18.4 Minkowski and Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.5 A 4-Dimensional Interpretation of Newtonian Mechanics . . . . . 18.6 Special Relativity Deduced a Priori . . . . . . . . . . . . . . . . . . . . . . . . 18.7 The Twin Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

303 303 305 306 308 309 312 316

19 Kinematics in the 20th Century . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1 The Twentieth Century . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 Institutionalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3 Twentieth Century Mathematicians Working in Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

319 319 322

291 294 295 297 299 300 301

323

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

Chapter 1

Philosophers, Mathematics and Motion

Abstract Kinematics studies motion without considering masses and forces. It deals with geometric objects such as points, lines or planes that move in a fixed Euclidean space. The philosopher Zeno of Elea took an extreme position on this point. He held that motion does not exist, implying that kinematics makes no sense. Zeno had no followers among mathematicians. On the contrary, in Greek geometry motion played an essential role. Yet against the background of the Eleatic philosophy, motion seemed a foreign element in a realm of eternal truths, and many centuries later Proclus still wrestled with the problem.

1.1 Motion Does Not Exist Plato wrote in his dialogue Parmenides the following: “Zeno and Parmenides once came to Athens for the Great Panathenaea. Parmenides was a man of distinguished appearance. By that time he was well advanced in years, with hair almost white; he may have been sixty-five. Zeno was nearing forty. A tall and attractive figure. It was said that he had been Parmenides’ favorite. They were staying with Pythodorus outside the walls in the Ceramicus. Socrates and a few others came there, anxious to hear a reading of Zeno’s treatise, which the two visitors had brought for the first time to Athens.”1 The Great Panathenaea were not as important as the Olympic Games, but they were prestigious games, held every four years in Athens, attracting many spectators. See Palagia et al. (2007). We don’t know whether the meeting between Socrates, Zeno and Parmenides that Plato describes, ever really took place. If it did, it happened in the middle of the fifth century BCE. Let us assume that it took place. Then Parmenides and his pupil Zeno took the boat in their home town Eleain Southern Italyand sailed to Piraeus, the port of Athens, in order to watch wrestling, boxing, chariot racing and other entertainment. True, there was also a religious festival, but it is hard to imagine that the two philosophers did not watch the famous torch race from Piraeus to the 1

Plato (1973), Parmenides, 127 b-c.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Koetsier, A History of Kinematics from Zeno to Einstein, History of Mechanism and Machine Science 46, https://doi.org/10.1007/978-3-031-39872-8_1

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Acropolis or that they did not at all visit the Panathenaeaic stadium where the athletes were competing. This is remarkable, because Zeno brought a treatise with him, containing forty paradoxes. Zeno had invented two kinds of paradoxes: paradoxes of plurality and paradoxes of motion. The purpose of the paradoxes of plurality was to show that plurality does not exist and the paradoxes of motion were meant to show that motion does not exist. In Plato’s story the treatise contained many paradoxes of plurality. It may have contained the paradoxes of motion or kinematical paradoxes as well. After all, motion implies plurality (the plurality of different moments in time, different positions, etc.) and the kinematical paradoxes serve the same purpose as the paradoxes of plurality. Basically Zeno defended in the treatise Parmenides’ view that sailing from Elea to Piraeus and visiting games in Athens, although entertaining, is at heart an illusion and of no interest to the true philosopher. The view that the world of sense-perception is illusory, temporary, perishable and that beyond it there exists an eternal ideal world, accessible only through the mind, was held in classical antiquity by several major philosophers. From this point of view knowledge of the world of the senses is inferior and true knowledge concerns the ideal divine world. Most of the philosophers who thought along these lines viewed mathematical knowledge as related to knowledge of the divine. Although Pythagoras remains a rather mythical figure, he seems to have been the first philosopher of this kind. Parmenides is a representative of this position as well. After him Plato held similar views and the same ideas return in the works of neo-platonic philosophers like Plotinus and Proclus. The Greeks were polytheists. Many more gods than the twelve on the Olympus mentioned by Homer existed for them. The gods were potentially permanently present. Festivals always had a religious character too and on such occasions the statues of the gods were carried out of their temples. In particular in times of crisis, personal or collective, or when major political decisions had to be made, the religiousness of the Greeks became active. Next to the public cults families observed their domestic cults and there were personal vows to the gods by individuals. The gods communicated in dreams and at oracular shrines. The intentions of the gods could also be read from the flight of birds or from the entrails of an animal. For almost all Greeks the gods played an important role in making sense of the world. Pythagoras, Parmenides, Plato and the neo-platonic philosophers represent a mystical version of Greek popular religion. Parmenides wrote a famous poem, On Nature, of which a part is still extant, in which he describes how a goddess revealed the truth to him: Come now, I will tell you (and do you preserve my story, when you have heard it) about those ways of enquiry which are alone conceivable. The one, that a thing is, and that it is not for non–being, is the journey of persuasion, for persuasion attends on reality; the other, that a thing is not, and that it must needs not be, this I tell you is a path wholly without report, for you can neither know what is not (for it is impossible) nor tell of it […].2

2

Coxon (1986), p. 52.

1.1 Motion Does Not Exist

3

The core of Parmenides’ views is in this obscure quotation. In my mind there is little doubt that it is the description of a special experience that Parmenides must have had. Suddenly it will have become clear to him that something is or it is not. This is the logical principle of the excluded third. However, Parmenides turned it into a metaphysical truth. Being and non-being completely exclude each other. Being, true thinking and true knowledge are the same. They are eternal and compared by Parmenides to a perfect sphere. Being is also One and not plural. Every change and every imperfection implies non-being and ergo does not exist. The visible world of the senses is plural and permanently in a state of flux and consequently must be classified as non-being. Parmenides’ pupil Zeno defended Parmenides’ views by means of his paradoxes. I will briefly mention one paradox of plurality. Unfortunately Zeno’s treatise is not extant. However, in Plato’s Parmenides we find a description of Socrates questioning Zeno about the treatise: When Zeno had finished, Socrates asked him to read once more the first hypothesis of the first argument. He did so and Socrates asked, What does this statement mean, Zeno? ‘If things are many,’ you say, ‘they must be both like and unlike. But that is impossible; unlike things cannot be like, nor like things unlike.’ That is what you say isn’t it? Yes, replied Zeno. And so, if unlike things cannot be like or like things unlike, it is also impossible that things should be a plurality; if many things did exist they would have impossible attributes. Is this the precise purpose of your arguments – to maintain, against everything that is commonly said, that things are not a plurality? Do you regard every one of your arguments as evidence of exactly that conclusion, and so hold that, in each argument in your treatise, you are giving just one more proof that plurality does not exist? Is that what you mean, or am I understanding you wrongly? No, said Zeno, you have quite rightly understood the purpose of the whole treatise.3

The structure of the argument is a proof by contradiction of the non-existence of plurality. Actually the Eleatics may have been the first who identified the structure of such reductio ad absurdum arguments. Let there be plurality and let A, B and C be different things. Then A and B are like each other because they both differ from C. At the same time A is unlike B because they are different. So if A is like B, the predicate ‘like’ applies to A. If A is unlike B, the predicate ‘unlike’ applies to A as well. So A is both like and unlike, which seems to contradict the principle of contradiction, which says that a statement and its negation cannot both be true at the same time. Although half a century later Plato’s pupil Aristotle would turn logic into a subject of investigation in its own right, at the time logical principles like the principle of contradiction had not yet been clearly defined. Zeno’s argument is no longer acceptable as soon as one treats ‘like’ as a relative term and no longer as a predicate. Many of the problems that Plato discusses in his dialogues are logical problems and they reflect the state of the art in logic at the time. For example, elsewhere Plato describes a discussion between sophists, where one of them argues as follows: This 3

Plato (1973), Parmenides, 127e.

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dog is yours and it is a father, so it is your father and your father is a dog!4 The solution requires that we accept that ‘this is my father’ has logically a different structure from ‘this is my house’. Plato was struggling with matters of this kind. As for this particular paradox of plurality Socrates was on the right track: Why is it impossible, he asked, that something is like something else in one respect and unlike something else in another respect? Not all of Zeno’s paradoxes went away easily. Plurality, or the problem of the one and the many, constituted a major challenge for platonic philosophers. As we will see below, they attempted to solve the opposition between, on the one hand, a unique, immobile, eternal metaphysical origin of everything, Parmenides’ perfect spherical Being, say GOD, and, on the other hand, the manifestation of GOD, the word of the senses, plural, perishable and permanently in a state of flux. They had to, because, unlike Parmenides, they allotted a degree of reality to the world of the senses, albeit inferior to the divine reality. Zeno’s paradoxes of motion are kinematical paradoxes. They are meant to show that the assumption of the existence of motion leads to contradictions. They do not involve forces or masses. I will mention two of them. The first one, called the Dichotomy, involves repeated division into two parts of an interval. In order to cover the whole interval a runner must pass in a finite period of time infinitely many points, which is impossible according to Zeno. The second paradox, called the Achilles, involves Achilles chasing a tortoise. In order to overtake the tortoise Achilles must pass in a finite time infinitely many points where the tortoise arrives before Achilles. These two paradoxes were described by Aristotle as follows: The first is the one which declares movement to be impossible because, however near the mobile is to any given point, it will always have to cover the half, and then the half of that, and so on without limit before it gets there. […] The second is what is known as ‘the Achilles’, which purports to show that the slowest will never be overtaken in its course by the swiftest, inasmuch as, reckoning from any given instant, the pursuer, before he can catch the pursued, must reach the point from which the pursued started at that instant, and so the slower will always be some distance in advance of the swifter.5

Both paradoxes can be seen as an attempt to prove by contradiction the nonexistence of motion. This was certainly Aristotle’s view. Let us consider the Dichotomy. The structure of the argument is the following. Suppose motion exists. Then a motion from point A to point B along a straight line segment AB is possible in a finite period of time T. It is important to realize that Zeno does not consider real motion. He considers idealized motion: the motion takes place along a straight line segment, which possesses infinite divisibility. From a modern point of view Zeno discusses a mathematical model of motion. The crucial argument then is: the thing that moves passes over or comes in contact with infinitely many different points in a finite time, which is then allegedly impossible. By contradiction then, motion does not exist. Below we will discuss Zeno’s paradoxes in more detail. 4 5

Plato (1973), Euthydemus, 298d-e. Aristotle (1929), Physics, 239b10-239b29.

1.2 Mathematics and the Idealist Tradition in Greek Philosophy

5

1.2 Mathematics and the Idealist Tradition in Greek Philosophy Parmenides and Zeno are extreme representatives of the idealist tradition in Greek philosophy that started with Pythagoras in the sixth century. The successes of Greek mathematics undoubtedly contributed to the emergence of this tradition. Pythagoras must have been the first who noticed the peculiar nature of mathematics in this respect. He noticed that unlike the knowledge of the ever changing world of the senses, mathematics seemed to consist of eternal truths. Nowadays a considerable number of mathematicians still feel that what they are studying transcends physical reality. Those who hold such a ‘platonist’ philosophy of mathematics believe that what they are studying exists independently of the human mind. From this absolute character of mathematical truth Pythagoras drew the conclusion that mathematical knowledge was divine and that involvement with mathematics ought to be a way to reach God. It is often assumed that Pythagoras founded a sect on the basis of this idea. Actually this was quite a discovery. Let us consider an example and we will see that as soon as we understand the mathematical truth involved, it is as if we look through the figure and through the accompanying argument and observe an eternal transcendent truth. Pythagoras’ theorem The truth expressed in the theorem attributed to Pythagoras was known to the Babylonians more than a thousand years before Pythagoras lived. Because the Babylonians had no tradition of giving proofs of the mathematical properties that they used in their calculations, it is possible that Pythagoras gave the first proof of the theorem. The theorem says that the areas of the squares erected on the three sides of a right-angled triangle satisfy the following relation (see Fig. 1.1): Area square III = Area square I + Area square II. Pythagoras undoubtedly would have liked the following simple proof. Given a right-angled triangle we draw to congruent squares with sides equal to the sum of the two sides adjacent to the right angle of given right-angled triangle ABC. See Fig. 1.2. In the square on the left four copies of triangle ABC are positioned in the vertices of the square. In the square on the right the same four copies of the triangle ABC are positioned in such a way that they form two congruent rectangles. Because Area square III, and Area square I + Area square II are both equal to the area of the big square minus 4 times the area of triangle ABC, the truth of Pythagoras’ theorem is obvious. This is a very convincing proof. And, of course, it is not about the physical triangles and squares. It is about non-material ideal triangles and squares. In the interpretation of the Pythagoreans the theorem represents knowledge of an absolute divine reality which lies behind the visible world.

6 Fig. 1.1 Pythagoras’ theorem

1 Philosophers, Mathematics and Motion

Area square III = Area square I + Area square II.

C

C III I

A

B

A

B II

Fig. 1.2 Proof of Pythagoras’ theorem

1

2

II

1

III 3

2

4

I

4 3

Parmenides’ views and their defence by Zeno are theoretically interesting. The possibility of knowledge of the world of the senses is in fact denied completely. In the end such extremism is not satisfactory, because it leaves too many questions unanswered. Plato obviously appreciated Parmenides’ position. Yet, in his dialogues in different ways he attempted to develop a more complete metaphysical hierarchy. In the dialogue Timaeus Plato describes the creation of the world by a benevolent god who attempts to create the world as much as possible in accordance with the ideal forms. In different ways mathematics plays an important role. The soul of the cosmos is created mathematically in a rather obscure way and the four elements, earth, fire, air and water turn out to correspond to four of the five regular polyhedra: respectively, the cube, the tetrahedron, the octahedron and the icosahedron. In the Timaeus Plato describes the hierarchy top-down. In the Republic the emphasis is on a bottom-up movement. The Republic contains Plato’s description of an ideal state. In his view such a state ought not to be a democracy but it ought to be ruled by exceptionally wise men. He argues that the future philosopher-rulers of the republic ought to study mathematics for ten years, because it will lead their minds away from the perceptible world to an intelligible reality. Once their mental eye is directed towards the ideal world of forms, they will get to know the form of the Good, which will turn them into good men.

1.3 Mathematics and Motion

7

In Plato’s philosophy in the metaphysical hierarchy mathematics is in different ways an essential link between the world of the senses and the mysterious divine source of everything, the One or the Good. On the one hand, the visible world is created on the basis of mathematical principles and, on the other hand, the human mind can reach the divine through mathematics. This last point is elaborated in a different way in Plato’s dialogue the Menon. In this dialogue Socrates demonstrates that learning mathematics is in fact remembering mathematics. The human soul is a divine spark imprisoned in a material body. Yet, it can remember its divine origin. This manifests itself in the fact that mathematical knowledge is so different from the uncertain ever-changing opinions that people have about everything that is going on in the world. In the dialogue Socrates talks to a slave and makes a drawing that demonstrates Pythagoras’ theorem for an isosceles right angled triangle in a way similar to the above-given proof of the general theorem. Socrates’ point is that the slave does not accept the theorem because Socrates says that it is true. He accepts it because he can literally see that it is true and Socrates explains this insight as remembering.

1.3 Mathematics and Motion The perceptible world is in a state of permanent motion. Mathematical knowledge, however, is immutable. Plato and his followers carefully distinguished mathematics from the visible world. It is in this respect remarkable that Plato discussed Zeno’s paradoxes of plurality but, as far as we know, never his paradoxes of motion. A possible explanation is that he did not take the paradoxes of motion seriously enough in order to discuss them. This is not an absurd supposition. Maybe Plato had no special interest in the fact that inevitably motion is involved in the making of drawings that accompany a proof. Obviously the drawings are made in the material world and that requires some activity. Yet, what counts are the results and the insight in mathematical truth that it leads to. Essentially for Plato motion and mathematical knowledge had nothing in common. He may have felt that Zeno’s paradoxes of motion did not concern geometry. As for the paradoxes of plurality the situation was different. In geometry there are many different points, lines, triangles, etc. Plato had good reasons to discuss these paradoxes. Yet, in the fourth century BCE developments were taking place that would make the simple view that mathematics and motion could be completely distinguished untenable. In order to study mathematical relationships between figures, these figures must be drawn, really or in the imagination. The process of drawing, whether real or imaginary, determines the figure. We get access to the figure by means of this drawing process. The figure may pre-exist in some way, but in order to investigate it we must get hold of it. Once the figure is drawn we can study it. In Plato’s Menon Socrates draws the figure accompanying his proof without any methodological considerations. Yet, possibly at the same time Greek mathematicians were developing a great interest in precisely fixing their figures before they studied them. In this process

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circular motion and motion along a straight line would become essential constitutive elements. Constructions based on circular and linear motion became central in geometry. This created a problem for the later Platonists. As Thomas Heath wrote in 1921: Constructions, then, or the processes of squaring, adding, and so on, are not of the essence of geometry, but are actually antagonistic to it.6

The influence of the Pythagorean-Platonic tradition in classical antiquity was indeed considerable. It has influenced the image of Greek geometry until the present. Felix Klein wrote: History shows that motion was not always welcome in geometry. It was feared that motion would ‘bring into geometry an element foreign to it, namely the notion of time.7

In a more recent paper we read the following wild exaggeration: The Eleatics with their paradoxes of motion had shocked mathematics and led mathematicians to try to eliminate all motion from their discipline. Aristotle, for example, had forbidden the use of motion in geometry. Euclid avoided any explicit mention of it.8

This quotation expresses a familiar platonic image of Greek mathematics: motion only existed in the margin of Greek geometry. Yet, it is questionable whether mathematicians were really shocked by Zeno’s paradoxes of motion. Moreover, Euclid restricted the use of motion to the postulates 1–3 and the use of a few rotations in space geometry but it is an exaggeration to say that he avoided any explicit mention of motion. Finally, Aristotle indeed separated motion and geometry, but he took kinematics and geometry both very seriously. He merely felt that a science that abstracts from motion is more precise than one that deals with it. That is why, from his point of view, in geometry one ought to abstract from motion. The fourth century development culminated in Euclid’s Elements, written circa 300 BCE. Although Euclid compiled the Elements half a century after Plato’s death, most of its contents were known much earlier. The Elements is a text consisting of 13 books that contain a compilation of everything that from Euclid’s point of view belonged to the core mathematical knowledge of his time. It was a bestseller that became in the course of time one of the most influential books in the history of geometry. I will show that the possibility of certain idealized motions is an essential assumption in the book. Book I of the Elements starts with three kinds of statements that are not proven: Definitions, Postulates and Axioms. They are followed by 48 Propositions. There are two kinds of propositions: on the one hand, propositions stating the possibility of the construction of figures or additions to figures, and, on the other hand, propositions expressing properties of constructed figures. The propositions are always accompanied by a proof. In the case of a proposition stating the possibility of a construction the proof consists of a description of such a construction 6

Heath (1921B), p. 287. Klein (2016), p. 201. 8 Marchisotto (1992), p. 744. 7

1.3 Mathematics and Motion

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plus a proof of the correctness of the construction. It is important to remark that the constructions are fundamental in the sense that before properties of figures are proven, the figures involved must be constructed first. The book starts with the definitions. Some of the definitions seem to be attempts to characterise the ideal character of the objects of geometry. For example: Definition 1: A point is that which has no part. Definition 2: A line is breadthless length.

However, other definitions play a role in the proofs. For example: Definition 15: A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure equal one another. Definition 20: Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.

We will see below how Definitions 15 and 20 are used in the proof of Proposition 1. After the definitions, the second group of statements that are not proven are the postulates. The first three postulates concern the construction of figures: Postulate 1: To draw a straight line from any point to any point. Postulate 2: To produce a finite straight line continuously in a straight line Postulate 3: To describe a circle with any center and radius.

The three postulates define the three basic moves that are allowed when constructing geometrical figures. In fact the basic moves are the motions that one can execute with ruler and compass. One should, of course, interpret these motions as ideal motions generating ideal lines and circles. The lines drawn by material rulers and compasses are only indicating what is really meant. The propositions that express properties of figures, concern in particular equalities: equalities of lengths, of angles and of areas. In the proofs of these propositions the third group of statements that are not proven, the axioms, play a central role: Axiom 1: Things which equal the same thing also equal one another. Axiom 2: If equals are added to equals, then the wholes are equal. Axiom 3: If equals are subtracted from equals, then the remainders are equal. Axiom 4: Things which coincide with one another equal one another. Axiom 5: The whole is greater than the part.

Proposition 1 is an example of a proposition stating the possibility of a construction. The accompanying construction and the proof of its correctness nicely illustrate the deductive character of Euclid’s geometry (See Fig. 1.3). Proposition 1: To construct an equilateral triangle on a given finite straight line. The construction: Let AB be the given finite straight line. Construct the circle C1 with center A and radius AB. (Postulate 3)

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C

Fig. 1.3 Euclid’s Elements Proposition 1

A

B

C1

C2

Construct the circle C2 with center B and radius BA. (Postulate 3). The two circles intersect in point C. Construct the straight line AC. (Postulate 1) Construct the straight line BC. (Postulate 1). The proof of the correctness of the construction: AB=AC (Definition 15) AB=BC (Definition 15) Because AB=AC and AB=BC also AC=BC (Axiom 1) ABC is equilateral (Definition 20).

In Book I there are altogether 48 propositions. Proposition 47 is Pythagoras’ theorem: “In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle”. Proposition 48 is the converse of proposition 47: “If the square on a side of a triangle equals the sum of the squares on the other two sides, the triangle is right-angled.” In Book I, 14 of the 48 propositions state the possibility of constructions. The role of the constructions is essential. Without the constructions generating geometrical figures in a precisely defined way, Greek geometry is unimaginable. In the other geometrical books of the Elements we find the same pattern. Book II contains 14 propositions that deal with equalities of rectangular areas. Two of these propositions state the possibility of constructions. The 37 Propositions of Book III deal with the geometry of the circle; 6 of them concern constructions. Book IV contains 16 propositions and all of them concern constructions to inscribe or circumscribe figures. An example is Book IV, Proposition 9: “To circumscribe a circle about a given square”. All constructions are accompanied by a proof of their correctness. Postulate 1 and 2 refer to a moving point that generates or describes a straight line segment. Postulate 3 refers to a moving point describing a circle. But also in space geometry Euclid applies idealized rotation. In Book XI of the Elements Euclid defines the sphere as the figure comprehended by the revolution of a semi-circle about its diameter (Definition 14). In the same book Euclid defines a cone as the figure comprehended by the revolution of a right-angled triangle about one of its

1.4 Aristotle Refutes Zeno

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sides containing the right angle (Definition 18). The cylinder is defined by Euclid as the figure comprehended by the revolution of a rectangle (Definition 21). After each of these three definitions Euclid immediately defines the axis of, respectively, the sphere, the cone and the cylinder (Definitions 15, 19 and 22). Books XI, XII and XIII of the Elements deal with space geometry. Euclid must be able to generate planes. He does so, for example, by applying X.2: “When two straight lines cut each other, they are in a plane; and every triangle lies in a plane.” Although Euclid still first generates the figures before he proves their properties, the generation is less precise than in Book I. Remarkable is in Book XIII his method to show that the vertices of the regular solids are on a sphere. He constructs the solids relative to a straight line segment equal to the diameter of the sphere and afterwards he shows that the vertices of the figure are on the sphere by means of a rotation of a semi-circle. Also after Euclid rotation was repeatedly used in order to create a surface or a solid body. For example, as we will see below, Archimedes studied without hesitation paraboloids, hyperboloids and ellipsoids of revolution.

1.4 Aristotle Refutes Zeno We do not know of any refutations of Zeno’s paradoxes of motion by platonic philosophers. However, Plato’s pupil Aristotle discussed them. As we have seen in Plato’s view the visible world surrounding us possesses an inferior kind of existence. The visible world is temporary, perishable. Aristotle had different views. Aristotle was at heart a very down to earth philosopher who completely rejected the platonic view of the world. According to Aristotle there is no eternal unchangeable world of forms that is more real than the visible world. The visible world is all there is. Moreover, the visible world consists of two parts: the divine stars and the sublunar visible world on earth. According to Aristotle mathematicians do not study something more real than the visible world. Mathematicians abstract numbers and figures from visible reality and study them; this does not require any reality above the ordinary physical one. In mathematics Aristotle rejected actually infinite sets. According to Aristotle a mathematician can imagine bigger and bigger numbers; however, it is absurd to consider the totality of all natural numbers as an infinite given whole. Infinity is for Aristotle always potential infinity. The word ‘infinite’ means for Aristotle in fact ‘never ending’. For Aristotle motion was very much part of reality and he could not accept Zeno’s paradoxes. He wrote: Zeno’s argument makes a false assumption in asserting that it is impossible for a thing to traverse or severally come in contact with infinite things in a finite time.9

9

Aristotle (1952), Physics, 233a22-233a31.

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A paradox always involves a contradiction or something that can be interpreted as a contradiction. In this particular case Zeno apparently argued that coming in contact with infinitely many things requires an infinite time, while during the motion it would be done in a finite time, which is impossible. Aristotle does not buy it. He argues that there is no real contradiction. Although his solution to the paradoxes has not always been appreciated, it is logically not unattractive. He wrote: For there are two senses in which a distance or a period of time (or indeed any continuum) may be regarded as infinite: in respect to divisibility or in respect to its extension.10

The distance is with respect to divisibility potentially infinite, because we can go on pointing at half-way points as long as we wish. The process never ends; we can always go on. A distance, or a line segment, is also potentially infinite with respect to extension. This means that by applying Euclid’s Postulate 2 one can extend a given segment again and again, passing all bounds. The same holds for a period in time. It can be extended again and again passing all bounds. Of, course we can also extend a segment or an interval of time again and again without passing all bounds, for example, when each extension is half the previous one. However, this possibility plays no role in Aristotle’s reconstruction of Zeno’s argument. It would not yield a contradiction. According to Aristotle, we can look at the motion in Zeno’s Dichotomy in two ways. The distance and the period of time involved are both finite if we look at them with respect to their extensions: in a finite time a finite distance is covered. On the other hand, both are infinite if we look at them in respect of divisibility. According to Aristotle, Zeno’s mistake is not that he looks at the distance in respect of divisibility and concludes that it is infinite. The mistake is that he considers the infinity of the distance with respect to divisibility, but looks at the infinity of the corresponding time interval with respect to its extension. He can then conclude that an infinite extension of the period of time, passing all bounds, would be needed to cover the finite distance and he gets his contradiction. However, in the middle of the argument Zeno has erroneously replaced one kind of infinity by another kind of infinity which is not equivalent with it. It may have been Bertrand Russell who wrote that the runner can easily move from A to B in a finite period of time, however, while doing so in this finite period of time he cannot consciously distinguish all the midpoints, because this would require an unbounded period of time. Of course we then assume that being aware of the passing of a midpoint requires a minimum period of time. Aristotle would have liked this remark because it associates the potentially infinite number of midpoints with a potentially infinitely long period of time, passing alle bounds. It is my opinion that Aristotle’s solution is logically valid in the sense that he defined a consistent conceptual apparatus that made the contradictions disappear. However, the existence of a vast literature on Zeno’s paradoxes shows that Aristotle’s refutation did not convince everybody.

10

Loc. cit.

1.5 Zeno’s Trick: Motion Is Interpreted as a Super-Task

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1.5 Zeno’s Trick: Motion Is Interpreted as a Super-Task Let us look at the paradoxes in a different way. Suppose a point moves in a finite time from endpoint A of a (finite) line segment to the other endpoint of the segment B. This seems conceptually simple. However, by pointing out that during the motion the point first much reach the midpoint of the segment, then the midpoint of the remaining half, then the midpoint of the then remaining quarter, etc., an in itself ‘obvious and easy’ phenomenon is turned into something conceptually far from trivial. We call it nowadays a ‘supertask’. The word was coined by James Thomson in Thomson (1954–55). Definition. A super-task refers to the execution in a finite period of time of an infinite sequence of actions a1 , a2 , a3 , … such that for all n, after an has been executed an+1 can be executed.

The execution and outcome of the execution of such an infinite sequence of actions in a finite time is far from “something our thought grasps as obvious and easy”. The essence of Zeno’s paradoxes is that he succeeded in interpreting motion in terms of the execution of a supertask. The physical or the mental execution in a finite time of a super-task in the sense that each individual act in the infinite sequence is consciously executed, is of course impossible. This would require an actually infinite period of time. And this is precisely what Zeno would like us to see as necessarily following from the simple notion of a motion from A to B. It does not follow, though. But, the introduction of a super-task and the assumption that it can be executed in a finite time easily leads to paradoxes. That was Zeno’s trick: he interpreted something conceptually simple in terms of something conceptually complex and confusing: a super-task. In order to show this and shed some more light on Zeno’s paradoxes I will allow myself a brief digression. Let us consider another super-task: You have a countable infinity of balls labeled with the natural numbers 1, 2, 3, 4 … and an infinitely large urn which starts empty. At 1 minute before 12 p.m. you put in balls 1 through 10, and remove ball 1. At 1/2 minute before 12 p.m. you put in balls 11 through 20, and remove ball 2. At 1/4 minute before 12 p.m., you put in balls 21 through 30, and remove ball 3. And so on: at each instant (1/2)n minutes before 12 p.m. you add the next ten balls and you remove the ball with the lowest number. At 12 p.m. how many balls are there in the urn?

Time runs during the execution of the super-task from 1 min before 12 p.m. to 12 p.m. Then all acts have been executed. The number of balls in the urn develops as follows during the execution: 2 through 10, 3 through 20, 4 through 30, etc. At each step the number of balls in the urn increases with 9. So it seems as though there should be infinitely many balls in the urn at 12 p.m. However, on the other hand, every ball that enters the urn is taken out at some time before 12 p.m. More precisely: ball n is taken out at step n, that is at 12 p.m. minus (1/2)n−1 min. This suggests that the urn is empty at 12 p. m. I will call this paradox Littlewood’s paradox. The earliest occurrence, as far as I know, is in Littlewood (1953). What is happening here? The super-task involved is well-defined. And although the real physical or mental execution in a finite time of the super-task in the sense

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that each individual act in the infinite sequence is consciously executed is of course impossible, the execution in a finite time is not logically impossible and we can discuss its outcome. The clue to the understanding of such paradoxes is the following. The paradox disappears if one realizes that the definition of the super-task only defines an infinite sequence of acts. The definition does not imply anything about the situation in the world after the execution of all the acts in the sequence. In order to deduce anything about the situation after the execution of the entire super-task we need extra assumptions. The paradoxes appear if one adds carelessly, tacitly, such assumptions.11 Littlewood’s paradox is an example. On the one hand, in the course of the execution of the super-task the number of balls in the urn grows beyond all bounds. On the other hand, every ball that enters the urn before 12 p. m. also leaves the urn before 12 p. m. As for the extra assumptions that determine the situation at 12 p. m. we are completely free as long as we avoid inconsistency. We cannot on the one hand assume that there will be infinitely many balls in the urn and at the same time assume that there are no balls in the burn; this is what happens in Littlewood’s paradox. A good way to add consistent assumptions about the situation after the execution of a super-task is by mapping the task onto a consistent context. I will illustrate this by mapping Littlewood’s super-task on Zeno’s Achilles.12 Zeno extracted a supertask from a continuous motion and I will do the opposite; I will map a super-task on a combination of two continuous motions: Let us assume that two runners, Achilles and the Tortoise, hold a race on a path of length 2. Achilles starts at the beginning, while the Tortoise starts at point 1, halfway down the track. Achilles runs exactly twice as fast as the Tortoise (who runs 1 per minute). The race starts at 1 minute before 12 p. m. On the path, infinitely many small signs, numbered 1, 2, 3, and so on, have been posted, to show the race participants where they are. The signs have been placed in such a way, that they are passed in increasing order. Furthermore, given the speed of Achilles and the Tortoise, the following properties hold: at 1 minute to 12 p.m., the Tortoise is at sign 1, while at the same time Achilles has not passed any signs; at some point in time before 12 p.m. the Tortoise reaches sign 10, while at the same time Achilles reaches sign 1; at some point in time before 12 p.m. the Tortoise reaches sign 20, while at the same time Achilles reaches sign 2; at some point in time before 12 p.m. the Tortoise reaches sign 30, while at the same time Achilles reaches sign 3; at some point in time before 12 p.m. the Tortoise reaches sign 40, while at the same time Achilles reaches sign 4, and so on.

It is easy to see that it is possible to put the signs in the described way. Let sign n be put at a distance of F(n) from the finish. Because Achilles runs twice as fast as the Tortoise, when he still has to cover a distance F(n), the Tortoise still must run a distance (1/2)F(n). So in order to have the signs at the described positions we must have F(10n) = (1/2)F(n), which yields combined with F(1) = 1 that F(10 m ) = (1/ 2)m for all m. Putting n = 10 m , we get F(n) = (1/2)m with m =10 log n.

11 12

Victor Allis and I have shown this elsewhere: Allis&Koetsier (1991, 1995, 1997). I owe this idea to Victor Allis.

1.6 The Neo-platonist Ontological Hierarchy

15

If we assume that a sign is considered to be passed as soon as it is reached, then in this experiment the set of signs already passed by the Tortoise, but not yet passed by Achilles when Achilles passes sign n, corresponds to the set of balls in the urn in Littlewood’s super-task, when ball n is withdrawn. As Achilles and the Tortoise will at 12 p.m. both be at the finish, there are then no signs in between them, corresponding to an empty urn in the Littlewood’s paradox. Achilles overtakes the Tortoise at 12 p.m. because both move continuously. By mapping Littlewood’s supertask on the Achilles we have added this continuity assumption to its definition. With this extra assumption the urn is empty at 12 p.m.

1.6 The Neo-platonist Ontological Hierarchy Even if they did not take Zeno’s paradoxes seriously, for Neo-platonist philosophers the essential role of motion in geometry represented a problem. The fifth century philosopher Proclus discussed it. Proclus followed Plotinus (205–270 CE) in his interpretation of Plato. Plotinus had further developed the metaphysical hierarchy that is in some form already visible in Plato’s dialogues. Truly real is in Plato’s view the ideal world of the ideas, contemplation of which is the goal of philosophy. The ideas or forms are eternal intelligible objects. At the top of the hierarchy is the idea of the Good. The other ideas possess lower positions in the world of the ideas. For Plato the cosmos or the visible world is even lower in the hierarchy. Yet it related to the world of the ideas in the sense that the visible world is like a living being with a soul that gives the world its intelligent organisation and its goodness and beauty. This hierarchy does not describe a causal order in time, but rather a logical order. Plotinus developed this view as follows. See Fig. 1.4. At the bottom of the hierarchy there is the universe, the world of sense perception. The universe is very multiple. At the top there is the Ultimate One, which is not multiple at all. They are related as follows. The universe as we see it consists of many things; it possesses a very high degree of plurality. There is, however, in the universe a principle of unity, causally prior and ontologically superior to the universe: the Soul of the universe. According to Plotinus they are related as follows. The Soul unites the Universe and because of this it possesses much more unity than the universe. Yet, the Soul too is multiple. The Soul possesses in its turn a principle of unity, prior, superior and independent, the Intellect. This Divine Intellect has a maximum degree of unity, but yet, because it thinks many things, it is as a subject and multiple object of thought in itself still multiple. Above the Intellect, causally prior to it, superior to it, is the One, the ultimate principle of everything, in which subject and object are united. The One is absolutely non-multiple. Plotinus wrote: Many times it has happened: lifted out of the body into myself; becoming external to all other things and self-encentred; beholding a marvelous beauty; then, more than ever, assured of community with the loftiest order; enacting the noblest life; acquiring identity with the divine; stationing within It by having attained that activity; poised above whatsoever within

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ULTIMATE ONE (No multiplicity. Principle of unity of the Intellect.) ▼ INTELLECT (Thinks Plato’s unique transcendent Forms. Principle of unity of the Soul.) ▼ SOUL (Gives life to the Universe. Principle of unity of the Universe. Develops mathematics in order to understand knowledge belonging to the Intellect) ▼ UNIVERSE (The world of sense-perception.) Fig. 1.4 The Neo-platonist ontological hierarchy

the Intellectual is less than the Supreme: yet, there comes the moment of descent from intellection to reasoning, and after that sojourn in the divine, I ask myself how it happens that I can now be descending, and how did the Soul ever enter into my body, the Soul which, even within the body, is the high thing it has shown itself to be. (Plotinus (1917-1930)).

Plotinus here describes experiences that can only be qualified as mystical, in which the soul acquires identity with the divine. The experience is compared to being “lifted out of the body” and the return back to normal is described as a descent, as a reentering of the soul in the body. The hierarchy is a metaphysical hierarchy but it is at the same time a description of the way individual human beings are related to God and it offers a possibility to unite with God. One of Plotinus’ followers was Proclus. In Proclus’ view the hierarchy from the Ultimate One to the multiplicity of the Universe is in different ways related to mathematics. O’Meara wrote this about it: From this Ultimate devolve in a non-temporal, non-spatial successive order, levels of unified multiplicity, tending to ever greater multiplicity and lesser unity. […] For Proclus, the very structure of the domain of mathematical objects manifests these metaphysical principles, both the flow of numbers (in arithmetic) from 1 (the monad, limit) and 2 (the dyad, unlimited) to 3 (the first determinate number and the following members of the numerical series, and the flow (in geometry) from the point (limit) to line (unlimited in its tendency), plane figure (the line as limited) and solid. Euclid’s Elements (Book I) follow this metaphysical order, in Proclus view. See O’Meara (2005), p. 140.

Mathematics is according to Proclus a product of the soul. Mathematical concepts are derived by the individual soul through logical procedures from innate knowledge common to all souls. In the case of geometry this innate knowledge is developed in ‘extension’, in a more easily accessible dimension, in order to comprehend it more easily. This extension is provided by imagination (phantasia). The rules of this development are expressed by the first three postulates. According to Proclus we find in Euclid’s Elements the result of the following process: the soul, exercising her capacity to know, projects on the imagination, as on a mirror, the ideas of the figures; and the imagination, receiving in pictorial form these impressions of the

1.7 The Postulates 1 Through 3 in Neo-platonism: Proclus Solution

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ideas within the soul, by their means affords the soul an opportunity to turn inward from the pictures and attend to herself.13

This development of geometrical knowledge is, on the one hand, a process of self-discovery for the soul. In thinking geometrical truths the soul moves upward on the scale of divinity. On the other hand, the development of geometrical knowledge is similar to the way in which the visible world is an expression of the divine. The soul projects on the imagination “just as nature stands creatively above the visible figures.” Nature, the highly plural visible world, is in the final analysis an expression or a manifestation or an emanation of the ONE. By doing mathematics one can begin to understand how the one and the many are related.

1.7 The Postulates 1 Through 3 in Neo-platonism: Proclus Solution In his Commentary on Euclid, Book I, Proclus defended the status of the first three postulates as principles as follows. They are “simple, indemonstrable, and evident of themselves” because, For example, drawing a straight line from a point to a point is something our thought grasps as obvious and easy, for by following the uniform flowing of the point and by proceeding without deviation more to one side than to another, it reaches the other point. Again if one of the two ends of a straight line is stationary, the other end moving around it describes a circle without difficulty.14

The motions involved are simple; geometrical intelligence and simple reflection are enough to see their truth. However, the construction of, for example, an equilateral triangle is from Proclus’ point of view more complicated and it is necessary to follow a procedure and give a demonstration in order to grasp it. That is why the construction of an equilateral triangle is the subject of a proposition. It is the kinematical character of the postulates that makes them so evident of themselves. Proclus: The drawing of a line from any point to any point follows from the conception of a line as the flowing of a point and of the straight line as its uniform and undeviating flowing. For it we think of the point as moving uniformly over the shortest path, we shall come to the other point and so shall have got the first postulate without any complicated process of thought. And if we take straight line as limited by a point and similarly imagine its extremity as moving uniformly over the shortest route, the second postulate will have been established by a simple and facile reflection. And if we think of a finite line as having one extremity stationary and the other extremity moving about this stationary point, we shall have produced the third postulate; for the stationary point will be the center and the straight line the distance, and whatever length this line may have, such will be the distance that separates the center from all parts of the circumference.15 13

Proclus (1970), p. 113. Proclus (1970), p. 141. 15 Proclus (1970), p. 145. 14

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Proclus explicitly dealt with the question: “how we can introduce motions into immovable geometrical objects and move things that are without parts—operations that are altogether impossible […]?” Proclus answer is that the motion involved is not bodily, but in the imagination. The imagination occupies a middle position between sense-perception, on the one hand, and the realm of partless, un-extended beings, on the other hand. Sense-perception—corresponds to the visible world around us, and the realm of partless, un-extended beings corresponds to the Divine understanding. Proclus: […] the circle in the understanding is one and simple and unextended, and magnitude is without magnitude there and figure without shape; for such objects in the understanding are ideas devoid of matter. But the circle in imagination is divisible, formed, extended – not one only, but one and many, and not a form only, but a form in instances – whereas the circle in sensible things is inferior in precision, infected with straightness, and falls short of the purity of immaterial circles.16

Nota bene, the imagination is an active force: […] the imagination […] is moved by itself to put forth what it knows, but because it is not outside the body, when it draws its objects out of the undivided center of its life, it expresses them in the medium of division, extension, and figure. For this reason everything that it thinks is a picture or a shape of its thought. It thinks the circle as extended, and although this circle is free of external matter, it possesses an intelligible matter provided by the imagination itself.

Clearly, in a way, the idealized motions create the figures in our imagination. However, the figures are an expression of objects drawn out of the undivided center of the life of the imagination. Moreover, the unextended figures in the understanding clearly are not created. We have quoted Proclus extensively because it is very illustrative to see a platonic philosopher fully accept the essential role of the ideal motions expressed on the Postulates 1, 2 and 3. One notices, however, how Proclus in fact emphasizes the ideal and immaterial nature of the motions by distinguishing them sharply from sensible motions. It is remarkable that some 600 years after Proclus, also the Persian Omar Khayam saw movement as an alien element in geometry. Khayam wrote “What is the relationship between geometry and movement, and what is the meaning of movement? Or: it is obvious for specialists that the line is a length that can only exist in a surface and the surface in a body […] How then could movement of the line be allowed if one abstracts from its position? Or: how can a line result from the movement of a point if it precedes the point in its essence and in its existence.”17

16 17

Proclus (1970), pp. 42-43. Quoted by Bernard Vitrac in Vitrac (2005), p. 3.

1.8 Zeuthen’s Thesis

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1.8 Zeuthen’s Thesis In the first half of the twentieth century Cornford wrote about the Euclidean plane in antiquity: “In its full abstraction, as conceived by the mathematician, it was an immeasurable blank field, on which the mind could describe all the perfect figures of geometry, but which had no inherent shape of its own.”18 The remark that space has no inherent shape, seems to imply that the figures are created by the constructing mathematician. The situation reminds us of set theory. The axioms of Zermelo-Fraenkel determine the cumulative hierarchy of sets. For most mathematical structures isomorphic copies can be defined in this cumulative hierarchy. Yet this cumulative hierarchy is to a certain extent like Cornford’s completely empty space. It contains almost everything mathematicians would like to study like Euclidean space in antiquity contained everything worth the attention of the Greek geometers. Yet, without good definitions, we will never know what we are studying. What is precisely the role of idealized motion in Greek geometry? The essential part that the constructions play in the Elements has led the Danish mathematician and historian Hieronymus Georg Zeuthen (1839–1920) a century ago to defend the view that the constructions in the Elements together with their accompanying proofs of correctness were used to ‘ensure the existence’ of what was constructed.19 According to Zeuthen for Euclid the constructions are ‘existence proofs’ and he argued that this holds for the Greek mathematicians in general. The function of the constructions is not practical but purely theoretical. In an elegant and rigorous way they remove all doubt about the existence of the object and they make it available for further investigation. Zeuthen’s views have generated considerable discussion. Some have been inclined to interpret Zeuthen’s view from a modern constructivist perspective. Then the construction of an object not only ensures the existence, but it amounts to the creation of the object. In 1951 Frajese wrote in this vein that Zeuthen’s view cannot be combined with the influence of Plato in classical antiquity.20 The existence of the equilateral triangle that is constructed in Proposition 1 of Book I is a priori beyond doubt. The proposition merely gives us a means to get hold of such a triangle in a particular position, according to Frajese. Mueller has in this respect rightly pointed out that it is dangerous to approach Greek geometry with modern conceptions of geometry.21 I doubt whether Zeuthen really meant to argue that the Greek mathematician created the figures in the act of construction. It seems to me he merely meant to say that the constructions are existence proofs in the sense that they assure us of the existence of the figures without choosing a position on their ontological status. Some have read Zeuthen as arguing that a construction was for the Greeks the only way to show the existence of an object. That is incorrect. Knorr has, for example, 18

Cornford (1976), p. 5. Zeuthen (1896), pp. 222-228. Zeuthen uses the German word ‘sicherstellen’ on page 223. 20 Frajese (1951), pp. 383–392. 21 Mueller (1981), p. 15 and p. 27. 19

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pointed at cases where existence is established via theorems and not by means of a construction.22 Moreover, some have interpreted Zeuthen as if the constructions according to him merely served to establish existence of figures and have no further significance. According to Frajese, for example, Euclid of course needed the constructions, because, unlike a philosopher who can restrict himself to the mere contemplation of a given figure, the mathematician must study the figures. And that means more than merely constructing them: “Euclid must study the figures and not only contemplate them: that is why, first of all, he must link them with each other, otherwise they would stay isolated from each other, immobile, without the possibility of comparison […] By means of the constructions […] Euclid links the figures by the sudden appearance, as if by magic, of the lines and the circles.”23 Apparently unaware of Frajese’s paper Harari came to a similar conclusion. She argued that in the Elements geometrical constructions do not serve as a means of establishing the existence of geometrical objects, but rather as a means of exhibiting spatial relations between geometrical figures (See Harari (2003)). Clearly Zeuthen’s views must be amended. In Greek geometry idealized motions are used to generate straight line segments and circles, but also surfaces of revolution. The existence of these figures in particular positions yields points and curves of intersection. This process can be repeated, if necessary. The properties of the figures constructed in this way are proven deductively in the Greek way. For the Greeks the construction of the figures on the basis of idealized motions was a major way to assure the existence of such figures. At this point Zeuthen was right. Yet, proving the existence was not the final goal. The Greek mathematicians were problem solvers and the constructions were an intermediary step towards solutions of problems. Moreover they were not really interested in the motions. The motions were merely a means to reach a goal. The resulting figures and their properties, that is what it was all about.

22 23

Knorr (1986), p. 375, note 77. See also [Knorr,1983]. Knorr (1986), p. 391.

Chapter 2

Motion Beyond the Elements

Abstract Doing mathematics is often rather similar to playing a well-defined game. Book I of Euclid’s Elements gives us a beautiful example. I call it the Euclidean construction game. The Postulates 1 through 3 give us the rules of the game, the permitted ideal movements. The famous problems, the squaring of the circle, the doubling of the cube and the trisection of any angle, could not be solved using only the postulates 1 through 3. Trying to solve these anomalous problems anyway Greek mathematicians were experimenting with other means to execute constructions, all of them based on motion.

2.1 The Euclidean Construction Game “Empires die, but Euclid’s theorems keep their youth forever” wrote the Italian mathematician Vito Volterra on a postcard bearing his own picture, when he was asked in 1931 to take an oath calling for allegiance and devotion to the fascist regime. He refused to take the oath. See Goodstein (1984).

There is a difference between philosophers discussing mathematics and mathematicians discussing mathematics. In general the great Greek mathematicians were no philosophers and vice versa. Pythagoras may have been an exception, but there is not much that we know about him with certainty. As for the philosophers, Plato, one of the greatest Greek philosophers, was an enthusiastic supporter of mathematics but he was not a great mathematician. The same holds for Proclus, at the very end of classical antiquity. He was primarily a philosopher, although well-educated as a mathematician. As for the mathematicians the situation is even clearer. Most of them were no philosophers of any significance. Great mathematicians like Eudoxus, Archimedes and Apollonius were no philosophers. Philosophers reflect on the nature of mathematics which can lead to interest in the foundations of mathematics. Working mathematicians are problem solvers. And while foundational problems may become challenging for the working mathematician and difficulties in solving problems may lead to foundational interest, the two groups usually have different goals. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Koetsier, A History of Kinematics from Zeno to Einstein, History of Mechanism and Machine Science 46, https://doi.org/10.1007/978-3-031-39872-8_2

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2 Motion Beyond the Elements

Doing mathematics in a mature mathematical theory is for the working mathematician often rather similar to solving chess problems. Certain well-defined initial positions are allowed. There are a restricted number of well-defined moves that can be used. Solving a problem means finding the moves that lead from one position to another position. Book I of Euclid’s Elements gives us a beautiful example of such a mathematical game. I will call it the Euclidean construction game: As for constructions in the plane, the Postulates 1 through 3 are the rules of the game. As for the proofs the axioms represent an interesting other set of rules. The construction and proof of Proposition 1 that we discussed in the previous chapter illustrate how the game is played: a problem is given and the challenge is to solve it using only permitted rules. In the Elements there are two kinds of propositions: problems and theorems. A problem is a proposition that states the possibility of a particular construction. The proof consists of a description of the required construction and a proof of its correctness. A theorem is a proposition that states that a particular figure has a particular property. Sometimes the proof of a theorem requires a construction in order to introduce elements in a figure that are needed for the proof. Nathan Sidoli and Ken Saito have pointed out that it is important to distinguish between the use of constructions in the solution of a problem and in a proof of a theorem. See Sidoli&Sait (2009). The Euclidean construction game is only about problems and not about theorems. We do not really know exactly how the Euclidean construction game was invented. On the whole the mathematical texts from classical antiquity do not contain much information on how the constructions and proofs were found either. The extant texts contain syntheses and not the analyses that preceded the syntheses. Analysis is in antiquity a way to find a proof or a construction by assuming as given what one wants to prove or construct and create a chain of intermediary results that link the desired result to things already constructed or proven. By reversing the order of the analysis one obtains the synthesis. Pappus of Alexandria about 320 CE defined the analysis as follows in Book 7 of his Collection: “Now, analysis is the path from what one is seeking, as if it were established, by way of its consequences, to something that is established by synthesis. That is to say, in analysis we assume what is sought as if it has been achieved, and look for the thing from which it follows, and again what comes before that, until by regressing in this way we come upon some of the things that are already known, or that occupy the rank of first principle. We call this kind of method analysis, as if to say anapalin lysis (reduction backwards). In synthesis, by reversal, we assume what was obtained last in the analysis to have been achieved already, and, setting now in natural order, as precedents, what before were following, and fitting them to each other, we attain the end of the construction of what was sought. This is what we call synthesis.”1

1

Pappus (1986), p. 82.

2.1 The Euclidean Construction Game

23

Imre Lakatos has argued that the method of analysis also gives us some insight in the way in which in Greek mathematics theories obtained their deductive structure. Roughly this must have happened as follows. It started with isolated results. Some of them turned out to imply others or be implied by others. In repeated analyses and syntheses it turned out to be possible to construct and relate many figures using only the basic moves that can be executed by ruler and compass. Lakatos wrote: “It was only after hundreds of successful analyses and syntheses […] that certain lemmas kept cropping up […] and finally were turned into the hard core of a research programme (an ‘axiomatic system’) by Euclid.”2 Euclid’s Elements is a textbook; it is undoubtedly the end result of a long development. It is very probable that more than a century before Euclid the basic moves expressed by the Postulates 1–3 played already a central role in Greek mathematics. The earliest pre-Euclidean fragment that is extant contains work by Hippocrates of Chios. The text suggests that many of the elementary results that Euclid included in the Elements were known at the time. Mathematics, however, differs from chess. It occurs repeatedly that a mathematical theory generates problems that cannot be solved by the existing rules of the game. In such cases there is pressure to change or expand the rules. Such situations are interesting, because the rules cannot be changed arbitrarily. Since Pythagoras there is general agreement that the rules of the game must be such that the game leads to true and precise knowledge. However, what is true and precise knowledge? In periods in which the rules are changing different views of what true and precise knowledge is play a role. Conservatives will be inclined to refer to tradition. Philosophers develop for example views on mathematical intuition and working mathematicians look for rules that yield an exciting and fertile game. In practice mathematics turns out to be quite autonomous and the condition that the game should remain challenging prevails. Indeed Greek mathematics offers us a nice example of pressure to change or expand the rules. It turned out that the language of Euclid’s Elements allows the phrasing of construction problems that cannot be solved by using only the Postulates 1 through 3. Because of the existence of such problems new moves were introduced in the game and like the Postulates 1 through 3 they are directly or indirectly based on idealized motions of geometrical objects. The Elements does not contain any unsolved problems. As a textbook it offered established knowledge and it does not tell us what kinds of problems mathematicians were working on at the time of its appearance. Yet we have some idea about the problems the mathematicians were interested in. Three problems in particular profoundly influenced the course of Greek geometry: the squaring of the circle, the doubling of the cube and the trisection of any angle. They had all the characteristics of a challenging mathematical problem: easily understandable and difficult to solve.

2

Lakatos (1978), pp. 99-100.

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2 Motion Beyond the Elements

Before and after Euclid, driven by these challenging problems Greek mathematicians were experimenting with other means to execute constructions, all of them based on motion.

2.2 The Incompleteness of the Euclidean Construction Game The squaring of the circle is probably the oldest of the three problems and the most famous one. It is: construct a square with an area equal to the area of a given circle. Anaxagoras of Clazomenae (fifth century BCE), who was put in jail because he taught that the sun was not a god but a glowing stone and that the moon reflected the light of the sun, allegedly worked on the problem when in prison. The Pythagoreans claimed that they had solved the problem in their school.3 However, the first serious work on the problem that is extant was done by Hippocrates. The problem of squaring the circle is related to the problem of the rectification of the circle: construct a line segment that has a length equal to the circumference of a circle. The two problems are equivalent but in order to see that we need a theorem that was proved by Archimedes in The Measurement of the Circle: The area of a circle is equal to the area of the rectangular triangle in which the sides adjacent to the right angle are equal to the radius and the circumference of the circle. The doubling or duplication of the cube is: to construct the edge of a cube that has a volume twice as big as a given cube. How old the problem is we do not know. The first serious work on this problem that we know of was done by Hippocrates and Archytas. Below we will see that discussions about doubling a tomb or an altar may have played a role. On the other hand, it may simply have been suggested by the problem of the doubling of the square that is solved by Socrates in Plato’s dialogue Menon by constructing another square on the diagonal of the square. According to Heath the problem of the trisection of any angle—construct the two straight lines that divide a given arbitrary angle into three equal angles—arose from attempts to construct regular polygons.4 However, one cannot exclude that the problem was suggested by the Elements, Book I, Proposition 9, which tells us how to bisect a given angle by means of compass and ruler. After all, if the problem to bisect an arbitrary angle has been solved, in a context of pure mathematics the problem how to trisect an angle easily imposes itself. Already in antiquity it became clear that the problem of the duplication of the cube can be more generally rephrased as the problem to find two mean proportionals of two given line segments. We probably owe this insight to Hippocrates. So, given line segments A and B, find the two line segments X and Y such that A:X = X:Y = 3 4

Cf. Heath (1921A), p. 220. Cf. Heath ( 1921A), p. 235.

2.2 The Incompleteness of the Euclidean Construction Game

25

Y:B. The problem of the doubling of the cube is a special case of this problem. In a modern way one can check it easily: If 1: x = x: y = y: 2 then x2 = y and y2 = 2x. This yields x4 = 2x or x3 = 2. Clearly x is the edge of a cube with a volume double the volume of a cube with edge 1. Eutocius in his Commentary on Archimedes’ Sphere and Cylinder, quoted a letter, allegedly by Eratosthenes to King Ptolomy, as follows: They say that one of the ancient tragic poets represented Minos as preparing a tomb for Glaucus, and as declaring, when he learnt it was a hundred feet each way: ‘Small indeed is the tomb thou hast chosen for a royal burial. Let it be double, and thou shalt not miss that fair form if thou quickly doublest each side of the tomb’. He seems to have made a mistake. For when the sides are doubled, the surface becomes four times as great and the solid eight times. It became a subject of inquiry among geometers in what manner one might double the given solid, while it remained the same shape, and this problem was called the duplication of the cube; for, given a cube, they sought to double it. When all were for a long time at a loss, Hippocrates of Chios first conceived that, if two mean proportionals could be found in continued proportion between two straight lines, of which the greater was double the lesser, the cube would be doubled, so that the puzzle was by him turned into a lesser puzzle. After a time, it is related, certain Delians, when attempting to double a certain altar in accordance with an oracle, fell into the same quandary, and sent over to ask the geometers who were with Plato in the Academy to find what they sought. When these men applied themselves diligently and sought to find two mean proportionals between two given straight lines, Archytas of Tarente is said to have found them by the half-cylinders, and Eudoxus by the so-called curved lines […].5

This letter is considered to be a forgery and was not written by Eratosthenes but it dates from classical antiquity and the story illustrates that the problem of the doubling of the cube was taken seriously. After the discovery of incommensurable magnitudes, which we will discuss in Chap. 3 of this book, the Pythagoreans and others undoubtedly continued to apply their arithmetical theory of ratios to magnitudes. Hippocrates’ reduction of the problem of the doubling of the cube to finding two mean proportionals may have been based on this kind of Pythagorean reasoning. It is tempting to relate his discovery to the Pythagorean interest in ratios of numbers. The three problems were very challenging. They are very easy to understand and yet contumacious. In particular the problem of the trisection of an angle and the duplication of the cube seemed only slight modifications of the problems of the bisection of an angle and the duplication of the square that can be easily solved. The text by the pseudo-Eratosthenes shows that story of the problem of the duplication of the cube is exciting. It refers to Plato’s associate and contemporary Archytas, who came up with a wonderful solution based on a construction by means of idealized motions in space. The pseudo-Eratosthenes wrote that Eudoxus found a solution by means of the “so-called curves lines”. We do not know what that solution looked like.

5

Bulmer-Thomas (1939A), pp. 257-261.

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2 Motion Beyond the Elements

2.3 Archytas of Tarente A student of Greek mathematics may be inclined to focus on the figures that result from the constructions. Once they are constructed one can abstract from the motions involved in their construction and the theorems seem to have the eternal life that Volterra attributed to them in the quotation above. Yet, without the movements involved in the constructions we would not be aware of many of these beautiful eternally young theorems. In this respect an extant pre-Euclidean fragment, written by Archytas, contains a lovely illustration. This time objects in space are generated by means of rotations combined with intersection. Archytas was a Pythagorean and a contemporary and friend of Plato‘s. The fragment is a description by Eudemus of Archytas’ solution of the problem to find two mean proportionals between two given straight line segments, reproduced by Eutocius in his Commentary on Archimedes’ Sphere and Cylinder. The text starts as follows: Let the two given straight lines be AD, C; it is required to find two mean proportionals between AD, C

Archytas first gives a construction and afterwards a proof of its correctness. Archytas could have obtained his result as follows. Consider Fig. 2.1. K is an arbitrary point on a semicircle with diameter AD. KI is perpendicular to AD and IM is perpendicular to AK. Then we have AM: AI = AI: AK = AK: AD, because of the similarity of the triangles AMI, AIK and AKD. Clearly AI and AK are the two mean proportionals of AM and AD. It is obviously easy to construct such a figure when AK and AD are given, or when AM and AI are given. However, the problem is that AM and AD are given. The idea underlying the argument in Archytas’ fragment is the following. We start from AD and we construct a semi-circle on it. We can imagine that K moves from D to A along the semi-circle. In every position we have a triangle AKD and the points I and M. Obviously in one particular position of K the length of AM is precisely the given C. Archytas determines this position of K by means of the kinematical generation of a curve and a cone. The wanted point is the point of intersection of these two. Fig. 2.1 AM: AI = AI: AK = AK: AD

K M

A

T

I

D

2.3 Archytas of Tarente

27

Fig. 2.2 Archytas’ way find two mean proportionals

K

M

L P

B

A

T I D' E

D

Z O

The curve is generated as follows. See Fig. 2.2. Archytas draws a horizontal circle ABDZ with diameter AD and he erects a vertical half-cylinder on semi-circle ADB. He then takes the semicircle of Fig. 2.1 and starting from an initial vertical position in which AD coincides with the diameter AD of the horizontal circle he rotates it anticlockwise about its vertical tangent in A. Where the diameter of the rotating semicircle intersects the horizontal circle we have point I. The circumference of the vertical semi-circle cuts the vertical half-cylinder in point K. During the motion K describes a curve. Obviously this curve is the intersection of a torus and a cylinder. Now assume that the position of the semicircle drawn in Fig. 2.2 corresponds to the solution we are looking for. Then AM is equal to C. We draw a horizontal line through T perpendicular to AD, which intersects the horizontal circle in Z and B. Because T is the point of intersection of two cords of circle ABDZ, we have AT.IT = ZT.BT. Moreover, because in Fig. 2.1 triangles ATM and AMI are similar we have AT.IT = TM2 , which with AT.IT = ZT.BT implies that ZT.BT = TM2 . This implies that M is on the vertical circle with diameter ZB and when we connect A with the points of this circle we get a circular cone. Because the position of the semi-circle we are considering corresponds to the solution we are looking for, AM is equal to C. Then also AB is equal to C which enables us to construct the circular cone. That is what Archytas does. He determines B such that AB is equal to C. The extension of AB intersects the tangent in D to the horizontal circle in the point P. The cone can be obtained by rotating APD about AD. Let us briefly listen to Archytas himself: Let the circle ABDZ be described about the greater straight line AD, and let AB be inserted equal to C and let it be produced so as to meet at P the tangent to the circle at D. Let BEZ be drawn parallel to PDO, and let a right half-cylinder be conceived upon the semicircle ABD. And on AD a right semicircle lying in the parallelogram of the half–cylinder. When

28

2 Motion Beyond the Elements this semicircle is moved about from D to B, the end point A of the diameter remaining fixed, it will cut the cylindrical surface in its motion and will describe in it a certain curve.6

And he continues as follows: Again, if AD be kept stationary and the triangle APD be moved about with an opposite motion to that of the semicircle, it will make a conic surface by means of the straight line AP, which in its motion will meet the curve on the cylinder in a certain point; at the same time B will describe a semicircle on the surface of the cone. Corresponding to the point in which the curves meet let the moving semicircle take up a position D’KA, and the triangle moved in the opposite direction a position DLA.7

It is remarkable that Archytas reaches the point K corresponding to the solution by means of two opposite motions: the triangle generates a cone and the semicircle moving in the other direction traces a curve while the solution is found when the curve and the cone meet. If we look at this very ingenious construction one notices that it is based on a curve that is being generated by rotating another curve, a semicircle and intersecting this semicircle during its motion with a right half-cylinder. Moreover, the curve that we get in this way is intersected with the surface of a cone, obtained by rotating a triangle about a straight line. Archytas goes beyond what Euclidin Book I of the Elements in the Postulates 1 through 3 fixed as the permitted elementary ideal motions. He rotates not only points but lines and semicircles as well, which gives him surfaces. He does not only intersect circles and lines, but surfaces and curves as well. As we have seen, in the Books XI, XII and XIII Euclid applies methods similar to Archytas’ kinematical methods.

2.4 A Solution from Plato’s Academy In search of solutions the Greeks used a variety of well-defined motions. Archytas’ doubling of the cube is based on spatial motions. A semicircle is rotated about a tangent. Its intersection with a cylinder is a curve. This curve intersects a cone in a point that immediately gives the solution. The cylinder and cone can both be generated by rotation in accordance with Euclid’s definitions. It was inevitable that in this context the Greek mathematicians would discover instruments or mechanisms to generate particular figures. And like ruler and compass that generate ideal straight lines and ideal circles, these instruments are also used ideally, abstracting from their material and approximate nature. Such an instrument showed up in the context of Plato’s Academy, where also Menaechmus was active.

6 7

Bulmer-Thomas (1939A), pp. 284-287. Ibidem.

2.4 A Solution from Plato’s Academy

29

Our knowledge of how Greek mathematics developed is often fragmentary. Menaechmus doubled the cube sometime around 350 BCE and we know what his solution looks like. Menaechmus is credited with the discovery of the conic sections and his solution is based on them. However, we can only guess the way in which his research developed. This and the next section are a very speculative reconstruction. In Plato’s Academy the interest in doubling the cube was considerable. Plato learned from Archytas, Eudoxus was an associate of Plato and Menaechmus was Eudoxus’s pupil. The problem is given line segments A and B, find two line segments X and Y such that A:X = X:Y = Y:B. Menaechmus will have certainly realized that this boils down to three necessary conditions: X.X = A.Y, Y.Y = B.X and X.Y = A.B. Crucial is that he interpreted each of them as essentially the equation of a curve. That was new. And he probably also identified the curves as conic sections. Whether he was working on cones and intersections of cones first and later saw the applicability to the cube duplication, or whether the cube duplication led him to cones and conic sections, we don’t know. If he started with the problem of the duplication, which, given its challenging nature, is quite possible, Menaechmus may have been familiar with a method to find two mean proportionals that was later attributed to Plato by Eutocius.8 Eutocius gives us an abstract description, but he also describes an instrument that can be used to execute the construction in practice. Although this solution is ascribed to Plato by Eutocius and still often called ‘Platos’s solution’, the received view is that this cannot be correct. Heath supposes that the solution may have been found in the Academy by a contemporary of Menaechmus.9 Knorr agrees and suggests that the method was first merely a theoretical device and that only a century later Eratosthenes in Alexandria casted the method in the form of a mechanical device.10 Knorr also suggests that Eudoxus himself may be the inventor of the method. Consider Fig. 2.3. AB and BG are perpendicular. Now imagine we could construct lines through A and G that intersect the prolongations of BG and AB respectively in E and D such that the segments AE and GD are both perpendicular to ED. Then we would have. GB : BD = DB : BE = BE : BA. Clearly BD and BE are the two mean proportionals of GB and BA. Consider the instrument in Fig. 2.4. It makes it possible to construct Fig. 2.3 starting from the two perpendicular segments AB and BG. The part ZHTM of the instrument is fixed with ZH and TM perpendicular to part HT. The part KL is movable and during its motion parallel to HT. Eutocius literally says that we should move the instrument until four conditions are satisfied at the same time: HT goes through G, H is on the extension of AB (H 8

Bulmer-Thomas (1939A), p. 267. Heath (1921A), p. 255. 10 Knorr (1986), pp. 60-61. 9

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2 Motion Beyond the Elements

Fig. 2.3 GB: BD = DB: BE = BE: BA

A

B

E

G

D

Fig. 2.4 The instrument described by Eutocius

Z

M

L

K

H

T

then gives us D), KL goes through A and K is on the extension of GB (K gives us E). In order to see that this is really possible it is useful to order the conditions and use the instrument as follows. AB and BG are given and perpendicular. We extend AB and GB. We let HT of the instrument go through G in such a way that H moves on the extension of AB. In each position we move KL until it goes through A and we check whether K is on the extension of GB.When K is on the extension of GB, we have found the required figure. It is clear that the application of the instrument in this way, even if we consider the instrument made up of ideal elements, involves a process of trial and error. A process that may never end, even if we assume that we can indeed check whether the point K is finally on the extension of GB. There is, however, an argument that the Greeks may have used implicitly to accept the method. As Heath has shown when

2.5 Menaechmus and Conic Sections

31

we use the instrument in this way point K describes a curve of the third degree.11 When in Fig. 2.3 we take BG and BA as the ( positive x- )and y-axes and put BG = a and BA = b, the equation of the curve is x 2 + y 2 − by x + a(y − b)2 = 0. This way of viewing defines a curve that is continuous, because it is generated by means of a continuous motion. The point of intersection of this kinematically generated curve with the x-axis gives us then the point E. Knorr calls this curve an ophiuride with an apparently nineteenth century name. We saw that allegedly Eudoxus solved the problem of the two mean proportionals ‘via curved lines’. We do not know how he did this. Scholars have tried to suggest several reconstructions. Knorr has argued that it could have been this ophiuride that Eudoxus used. Other proposals for the curved lines that Eudoxus may have used, were made by Tannery and Riddell.12 Many of the solutions to the problems of trisection of the angle, the duplication of the cube and the quadrature of the circle require that one accepts certain new ways of kinematically generating curves as means of construction.

2.5 Menaechmus and Conic Sections In the reconstructions of Heath and Dijksterhuis, Menaechmus’ method to determine the two mean proportionals is accidental spin-off of an investigation of the properties of conic sections. The weakness of this reconstruction is that it is not clear which challenging problems the conic sections offered to begin with. It is interesting that Heath has argued that ‘Plato’s solution’ may have been invented by someone who wanted to realize Menaechmus’ solution mechanically13 and indeed Menaechmus’ solutions can be related to ‘Plato’s solution’. This suggests the possibility that the development may have been the other way round. It then all started with the duplication of the cube and Plato’s solution may have led to Menaechmus’ discovery and the investigation of the curves involved led to the insight that they are conic sections. This is in fact Knorr’s reconstruction. It boils down to the following. Two carpenter’s squares with one leg coinciding can do the same job as Plato’s instrument. In search of the solution of Fig. 2.3 we can start with the carpenter square GDE (See Fig. 2.5). We can also start with the carpenter square AED (See Fig. 2.6). Consider Fig. 2.5. Carpenter’s square GDE moves such that D moves on the extension of AB and one of its legs goes through G. The other leg cuts the extension of GB in the point E. During its motion BD (with length x) is the mean proportional between BG and BE. So we have b: x = x: y. This means: b.y = x2 . Clearly the point N of the rectangle EBDN describes a parabola during this motion.

11

Heath (1921), pp. 255-258. Knorr (1986), pp. 53-60. 13 Heath (1921A), p. 256. 12

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2 Motion Beyond the Elements

A

Fig. 2.5 Carpenter square GDE

a y

b

E

B

G

x N

D

A

Fig. 2.6 Carpenter square AED

a y

b

E

B

G

x N

D

Consider now in Fig. 2.6 the carpenter’s square AED moving such that E moves on the extension of GB and one of its legs goes through A. The other leg cuts the extension of AB in D. During this motion the point N of the rectangle EBDN describes another parabola. We have a: y = y: x or anachronistically a.x = y2 . It is easy to see that the point of intersection of the two parabolas corresponds to positions of the carpenter’s squares that give us Plato’s solution. This is exactly the second solution given by Menaechmus. It is easy to go from this solution to Menaechmus’s first solution. The point N is determined by the values of x and y. The two parabola’s intersect in this point. Once we approach the problem by means of curves it is obvious that this point is also on a curve determined by x.y = a.b, a hyperbola. In this solution the two parabola’s are kinematically generated. The hyperbola is not. In this context it is worth mentioning that there is a fragment in which Menaechmus is associated with mechanical devices in the context of the doubling of the cube. Plutarch of Chaeronea(first-second century A. D.) wrote in The Life of Marcellus about Plato’s involvement in the solution of the problem of the duplication of the cube the following:

2.5 Menaechmus and Conic Sections

A

A

B C

33

A

B

B C

C

Fig. 2.7 Obtuse angled cone, right-angled cone and acute-angled cone. Planes perpendicular to AB give, respectively, a hyperbolic, a parabolic and an elliptic section

[…] but geometry especially, being, as Philolaüs says, the source and mother-city of the rest, leads the understanding upward and turns it in a new direction, as it undergoes, so to speak, a complete purification and a gradual deliverance from sense-perception. It was for this reason that Plato himself reproached Eudoxus and Archytas and Menaechmus for setting out to remove the problem of doubling the cube into the realm of instruments and mechanical devices, as if they were trying to find mean proportionals not by the use of reason but in whatever way would work. In this way he thought, the advantage of geometry was dissipated and destroyed, since it slipped back into the realm of sense-perception instead of soaring upward and laying hold of the eternal and immaterial images in the presence of which God is always God.14

This was written roughly half a millennium after Plato’s death. Yet it may be true and it supports the hypothesis that Menaechmus may have played with mechanical solutions for the problem of the two mean proportionals, for example along the lines suggested by Knorr. If the curves to duplicate the cube came first and the insight that they are conic sections came later, the next question is how Menaechmus gained that insight. We saw that with Archytas a cone was obtained by rotating a rectangular triangle about one of the legs of the right angle. Cutting a cone by a plane perpendicular to one of the sides of the cone leads to three different situations, depending on the angle of the cone. See Fig. 2.7. Eutocius in his Commentary on Apollonius’s Conics wrote: But what Geminus says is correct: defining a cone as the figure formed by the revolution of a right-angled triangle about one of the sides containing the right angle, the ancients naturally took all cones to be right with one section in each- in the right-angled cone the section now called the parabola, in the obtuse-angled the hyperbola and in the acute-angled the ellipse; and in this may be found the reason for the names they gave to the sections. […] for the ancients investigated the so-called section of a right-angled cone in a right angled cone only, cutting it by a plane perpendicular to one side of the cone, and they demonstrated the section of an obtuse-angled cone in an obtuse-angled cone and the section of an acute-angled cone in the acute-angled cone, in the cases of all the cones drawing the planes in the same way perpendicular to one side of the cone; hence it is clear, the ancient names of the curves. But later Apollonius of Perga proved generally that all the sections can be obtained in any cone […].15 14 15

Plutarch (1961), pp. 120-123. Bulmer-Thomas (1951B), pp. 276-279.

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2 Motion Beyond the Elements

The names ortotome, amblytome and oxytome or, respectively, ‘section of a rightangled cone’ (= nowadays parabola), ‘section of an obtuse-angled cone’ (= nowadays hyperbola) and ‘section of an acute-angled cone’ (= nowadays ellipse) were still used by Euclid and Archimedes. Apollonius coined the modern names of the curves. Before Apollonius the three types of curves were seen as rather different. Apollonius showed that they are three special cases of a more general kind. If Menaechmus discovered the parabola and hyperbola as planar curves he may have wondered whether they might be generated by intersecting a plane with a surface. Merely on the basis of the shape of the curves he may have been led to the intersections of a cone and a plane. Heath and Dijksterhuis have given reconstructions of how Menaechmus possibly derived the ‘equations’ of the conic sections from their definition as the curves of intersection of a right-angled cone and a plane.16 They feel that Menaechmus may have discovered the equations in this way. I prefer Knorr’s view: he had the curves and the equations already but used such an argument to prove that they were conic sections. Eratosthenes also invented an instrument to determine two mean proportionals (as we will see below) and he argued that his instrument made the other methods superfluous. He wrote: Do not seek to do the difficult business of the cylinders of Archytas, or to cut the cone in the triads of Menaechmus, or to produce any such curved lines as is described by the divine Eudoxus17

This is a well-known quotation. The familiar view is that the ‘triad’ in the text refer to parabola, hyperbola and ellipse, suggesting that the conic sections were discovered before their application to the duplication. This makes some sense although the ellipse plays no role there. Yet I prefer Knorr’s view that the word ‘triad’ refers to the two parabolas and the hyperbola that Menaechmus used to find the two mean proportionals.

2.6 A Remarkable Application and Heron’s Solution Greek mathematics is not about solving real life problems. It was a noble game played for a variety of reasons. For the enlightenment of the mind, for the challenges and the honor of being the first to solve a problem. Not, however, to solve practical problems. Yet there were sometimes remarkable applications. After the death of Alexander the Great Alexandria became a major centre of scientific investigation, which included the investigation of machines. In this period astronomy, which had been very speculative became a serious empirical science as well. Although geometry applied to astronomy is undoubtedly applied mathematics, astronomy could still be viewed as a noble science in which the astronomer 16 17

Heath (1921B), pp. 110-116 and Dijksterhuis (1956), pp. 56-59. Bulmer-Thomas (1939A), pp. 296-297.

2.6 A Remarkable Application and Heron’s Solution

35

approached the divine. As for machines this was different. The beginning of Heron’s Archery (Belopoeica) leaves us no doubt with respect to Heron’s point of view. He wrote: The largest and most essential part of philosophical study deals with tranquility, about which a great many researches have been made and still are being made by those who concern themselves with learning; and I think the search for tranquility will never reach a definite conclusion through the argumentative method. But Mechanics, by means of one of its smallest branches – I mean of course the one dealing with what is called artillery-construction – has surpassed argumentative training on this score and taught mankind how to live a tranquil life. (Heron, Belopoeica W 71-72)

The word Belopoeica refers to the discipline of artillery construction. If you want to live a tranquil life, you must arm yourself. That is what Heron is saying. In the late Roman Empire Publius Flavius Vegetius would express the same thought in his On Military Matters (De Re Militari) as follows: Si vis pacem, para bellum (If you seek peace prepare for war). Until the discovery of gunpowder catapults were the most powerful artillery weapons, and they played an important role in antiquity. The early catapults will have been belly-bows, the very first piece of artillery ever invented.18 The early belly-bows consisted of a big composite hand-bow and a frame, put on a base and probably supplied with some pull-back mechanism. In order to be able to use the bow easily at some time a universal joint was put on top of the column of the base. The next generation of catapults was based upon the insight that animal sinew is strong and elastic. The basic idea is that one can plait sinew into cords and wrap the cords around two parallel beams. By twisting one of the beams the bundle of cords can be stretched considerably and a huge tension builds up. A lever pushed through the middle of such a stretched bundle of cylindrical form can exert an enormous force if pulled out of its position. The torsion catapult was based on two of such contraptions. Alexandrian engineers proceeded as follows. They acquired a small but good engine, possibly from Rhodes, and by “subtracting here and adding there” determined the optimal dimensions for this particular engine. The diameter of the cylindrical spring was for the engineers the fundamental basis and unit of measure for the construction of the engines.19 The engineers made a list that expressed the sizes of all the parts of a catapult in the diameter of the sinew. So, once the diameter of the hole was known, all other dimensions, the height of the spring, the dimensions of the hole-carriers, etc. could all be calculated and by an able craftsman a catapult of any size could be constructed. Let us suppose the Alexandrians experimented with an engine with a spring diameter of 11 dactyls (21 cm), hurling weights of 10 minae (4366 g) over a distance of several hundred meters. Clearly we can shoot a weight of 20 minae over the same distance by using two identical examples of this type of engine. When we want one engine to throw 20 minae we can imagine that we merge the two identical engines 18 19

Marsden (1969), p. 5. Marsden (1969), p. 107.

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2 Motion Beyond the Elements

that each throw 10 minae by taking one of them and double its volume in every respect. In order to throw a weight that is twice as big we then need twice as much sinew and, of course, we double the volume of the sinew cylinders as well. This idea, expressed in Heron’s text, gives us a calibrating formula, because, more generally, if we want to throw a weight of λ.10 minae, then we need a sinew cylinder with a volume equal to λ time the volume of the original√cylinder. We can get a volume that is λ times as big by multiplying the diameter by 3 λ. Because the spring diameter of the catapult we started with is 11 dactyls, the weight of λ.10 minae corresponds to √ a diameter of 11. 3 λ, or, because λ = weight/10 we have in modern formulation: / Diameter = 11. ( 3

√ weight = 1.1 3 weight.100, 10

which is the calibrating formula that both Heron and Philon give. Their solutions for the determination of the cube root are equivalent. Heron does it as follows. Consider Fig. 2.8. A and B, the sides of the rectangular figure ECTZ, are two given line segments. We draw the circumscribed circle and we extend the sides CT and CE with D and G on the extensions. If by moving a ruler we succeed in drawing DZHG in such a way through Z that DZ is equal to HG we have solved the problem: DE and GT are the wanted two mean proportionals of A and B. Proof: D and G have symmetrical positions with respect to the circle and consequently D and G have equal powers with respect to the circle. This implies that DE. DC = GT.GC or DE:GT = GC:DC. However, we have GC:DC = EZ:DE, which means that EZ:DE = DE:GT = GT:TZ. Quod erat demonstrandum. The solution is based on a neusis-construction: one moves a straight line through a given point until certain points on the line satisfy a certain condition. Why would Greek mathematicians accept such a solution which seems to depend on trial and error? The reason may have been that also here the solution corresponds to a particular Fig. 2.8 Heron’s method to determine a cube root

G

H Z

T A

D E

K B

C

2.7 The Doubling of the Cube: Eratosthenes’ Instrument

37

point of intersection of a kinematically generated curve and a given straight line. Take the line through Z and D and let D move on the extension of CE. Then we can determine a point X on the line such that DZ = HX where H is the second point of intersection of the line with the circle. X describes a continuous curve which intersects the extension of CT in the point G we are looking for. We will see below that Nicomedes made this idea explicit for certain neusis-constructions.

2.7 The Doubling of the Cube: Eratosthenes’ Instrument Plato may have objected to mechanical solutions to geometrical problems, the texts on catapults show that in Alexandria a different attitude prevailed. The view that solutions to the duplication of the cube were viewed as having great practical significance is also clear from the fact that in the middle of the third century BCE Eratosthenes erected a column in Alexandria in honor of king Ptolomy. When Eratosthenes was about 40 years of age he became tutor to the son of King Ptolemy Euergetes, the third ruler of the Ptolemaic dynasty in Alexandria. Eratosthenes became at the same time librarian at the Museum in Alexandria. An epigram on the column said: If, good friend, you want to produce from a small cube one double thereof, or duly change any solid figure into another nature, this is in your power, and you can measure a byre or corn-pit or the broad basin of a hollow well by this method, when you take between two rulers means converging with their extreme ends. Do not seek to do the difficult business of the cylinders of Archytas, or to cut the cone in the triads of Menaechmus, or to produce any such curved form in lines as it is described by the divine Eudoxus. Indeed, on these tablets thou couldst easily find a thousand means, beginning from a small base. Happy art you, O. Ptolemy, a father who lives his son’s life in all things, in that you have given him such things as are dear to the Muses and kings; and in the future, O heavenly Zeus, may he also receive the sceptre from thy hands. May this prayer be fulfilled, and may anyone seeing this votive offering say: This is the gift of Eratosthenes of Cyrene.20

Just below the crown of the column an instrument made of bronze was fastened and below it a short proof of its functioning correctly together with a figure. Fortunately we know which instrument Eratosthenes had devised in order to double the cube. Consider Fig. 2.9. The three rectangular tables, I, II and III are congruent. The one in the middle is fixed, the other two can slide between the two parallel lines LF and ET. Rectangular table number III slides to the left under the one in the middle and rectangular table number I slides to the right above the one in the middle. While the rectangular tables are sliding G is the point of intersection of the diagonal of III and the right edge of II. Point B is the point of intersection of the diagonal of II and the right side of I. The point D is a point that we have marked on the right edge of III. The goal is to find the two mean proportionals of LE and TD. Figure 2.9 shows how we should slide the rectangular tables: in such a way that L, B, G and D are collinear. This can be realised by means of a ruler rotating about L. 20

Bulmer-Thomas (1939A), pp. 296-297.

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2 Motion Beyond the Elements

Fig. 2.9 Erathostenes’ instrument

L

F I

II

III D

E

Z

H

T

L

F B G

E

Z

H

D T

K

We have, because of proportionality in triangle LEK on the basis of the fact that the vertical edges of the tables are parallel and at the same time the diagonals of the tables are parallel: LE : BZ = LK : BK, LK : BK = ZK : HK and ZK : HK = BZ : GH. So we have LE:BZ = BZ:GH and similarly BZ:GH = GH:DT, which yields: LE : BZ = BZ : GH = GH : DT. Eratosthenes does not mention catapults, but one can easily imagine the background of the erection of the column. For the king weaponry was of great importance and the catapults had become indispensable in his army. In order to build efficient catapults that could throw rocks of given weight, cube roots had to be calculated, which was for the mathematicians equivalent to finding the two mean proportional of two given segments. In other words: Eratosthenes’ instrument seemed to be of great practical use.

2.8 The Neusis-Construction and the Conchoids

B

A

/

39

C

/

E

/

D

Fig. 2.10 Archimedes’ trisection

Eutocius in his Commentary on Archimedes’ Sphere and Cylinder wrote that Nicomedes in his On Conchoidal Lines greatly derided Eratosthenes’ discoveries as impracticable and lacking geometrical sense. Nicomedes was right. If one succeeds to move the two rectangular tables in such a position that the ruler connecting L and D goes through the points B and G we have a solution, but I see no idealized motions that will bring this about. Imagine we connect the points L and D. We can than move tablet I until B is on it. However, if we now move tablet III until G is on the line connecting L and D, B is no longer on that line. Constructions always give an approximation, but their idealized versions should give the exact result. In the case of Eratosthenes’ instrument, the idealized motion does not get further than an approximation either.

2.8 The Neusis-Construction and the Conchoids We saw above that Hero used a neusis-construction. There are many other examples of neusis-constructions.21 A well-known example is its use to solve the trisection problem. The solution is Proposition 8 of a book called Book of Lemmas (Liber Assumptorum), a book preserved in Arabic.22 The text is often attributed to Archimedes, but this may be accidental or caused by a cheating bookseller keen on making more money by attaching Archimedes’ name to the text. On the other hand the use of the neusis-construction is similar to Archimedes’ use of it in his text on spirals. A and E are two points on a circle with center D (See Fig. 2.10). The angle ADE or the arc AE is to be trisected. Archimedes extends ED. Then he draws a straight line through A such that it intersects the circle in B and the extension of ED in C such that BC equals the radius of the circle DB. Then angle 1 is equal to angle 2 (DBC isosceles). Then angle DBA equals twice angle 2. Angle BDA = 180 degrees minus four times angle 2. So angle 3 equals three times angle 2. 21 22

For more on the neusis-constructions in antiquity see e.g. Knorr (1986). For the text of the Book of Lemmas see volume 2 of Heiberg (1910-1915).

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2 Motion Beyond the Elements

Fig. 2.11 A ‘first’ conchoid: the points C and the pole E are on different sides of ADB. We get ‘secondary’ conchoids by marking off the given length towards the pole E

The construction used by Hippocrates and Archimedes assumes that it is possible to move a given line segment about in such a way that in the end its extension passes through a given point E and the given segment fits exactly between a straight line AB and a circle. If one ponders how one would go about in order to find this position the following procedure seems to be the most natural. One marks the given length on a ruler; this gives us, say, the points D and C on the ruler. We now put the ruler in such a position that it passes through E and the point D is on the given straight line AB. We then start moving the ruler such that it goes on passing through E while D moves on the given straight line. We stop moving when C is on the given circle. Then we have found the required position. Clearly during the motion of the ruler that we described the point C describes a curve and the position that we are looking for corresponds to a point of intersection of this curve and the given circle. When applying the neusis-construction in a case like this the early Greek geometers not explicitly introduced such a curve. Yet, they were sophisticated and we may assume that the early Greek geometers implicitly imagined such a mechanism of which one particular position, corresponding to a particular point of intersection or a line going through a given point, gave the desired solution. It is then not surprising that for neusis-constructions Nicomedes, probably in the third century BCE, made such a procedure explicit. See Fig. 2.11. Pappus in the Collection writes the following; For the duplication of the cube a certain line is drawn by Nicomedes and generated in this way. Let there be a straight line AB, with CDZ at right angles to it, and on CDZ let there be taken a certain given point E, and while the point E remains in the same position let the straight line CDEZ be drawn through the point E and moved about the straight line ADB in such a way that D always moves along the straight line AB […].23

This is an extract from a book that Nicomedes wrote On Conchoid Lines. Nicomedes must have been born about 270 BCE. He was younger than Apollonius and later than Eratosthenes, who was a contemporary of Archimedes. According to Pappus, Nicomedes proved that the conchoid can be described mechanically and that he introduced it to replace the existing neusis-constructions used to duplicate a cube. 23

Bulmer-Thomas (1939A), p. 299.

2.8 The Neusis-Construction and the Conchoids

41

Pappus then shows that such a construction can yield two mean proportionals of two given segments. According to Proclus, Nicomedes used the conchoids to trisect an arbitrary angle as well. As for the determination of two mean proportionals by means of a conchoid, consider Figs. 2.12 and 2.13. BGLA is a rectangle with center P. We would like the line MLK through L to intersect the extensions of BG and BA in the points K and M in such a way that LG : KG = KG : MA = MA : AL.

(2.1)

This occurs when M and K have the same distance to P. Then they have the same power with respect to the circumscribed circle of the rectangle ABGL, which implies: KG.KB = MA.MB or KG : MA = MB : KB.

(2.2)

Similarity of the triangles MBK, MAL and LGK yields together with (2.2) the desired relation (2.1). The desired position of the line through L can be found as follows. Connect L with the midpoint of AB; its extension cuts the extension of GB in the point H. Connect P with the midpoint of BG and find on it the point Z that has to G a distance equal to half of AB. Draw through G the line GT parallel to HZ. Now find on this line by means of a conchoid the point T such that ZT intersects the extension of BG in a point K such that TK is equal to half of AB. Fig. 2.12 P is center of rectangle BGLA

M L

A P B Fig. 2.13 LG: KG = KG: MA = MA: AL

G

K

M A

=

= B

P G

= Z

=

H

L

T

K

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2 Motion Beyond the Elements

Fig. 2.14 AB:BE = BE:BD = BD:BG

E

B

A

D G

X

Because HZ is parallel to GT we have ZT:TK = HG:GK, which implies ZT:TK = 2AL:GK or ZT:2TK = AL:GK or ZT:LG = AL:GK, which means that ZT:AL = LG:GK. Because we also have MA: AL = LG: KG this yields ZT = AM. By determining MP2 and KP2 by means of Pythagoras’ theorem it is easy to verify that MP = KP.24

2.9 Diocles’ Cissoid Apparently Diocles was a contemporary of Apollonius and lived early in de second century BCE.25 Diocles discovered a curve that was later called the cissoid. This curve is another example of a curve that is defined kinematically; one can easily imagine an instrument that draws it. Following a reconstruction by Knorr, I will first show how the cissoid is related to ‘Plato’s Method’. Consider Fig. 2.14, where we have applied Plato’s method to find the two mean proportionals between AB and BG. We then have: AB : BE = BE : BD = BD : BG. Consider the circumscribed circle of triangle ADE (See Fig. 2.15). The extension of DG cuts the circle in X. Clearly AEDX is a rectangle. Let us now execute a thought experiment. Consider a circle with AD as a fixed diameter. Imagine now that we move a point X on the circle and during its motion we draw XD and then a line in D perpendicular to XD, which intersects the circle in E. Then we drop a perpendicular line EB from E on AD and find the point G where it intersects XD. When X moves, the points E and B move on respectively the circle and the diameter AD. However, the point G moves on a curve. This curve is Diocles’ cissoid.

24 25

For a reconstruction of how Nicomedes possibly found this method see Knorr (1986), pp. 25-26. Knorr (1986), p. 234. Knorr refers to Toomer (1976), pp.1-2.

2.9 Diocles’ Cissoid

43

Fig. 2.15 If the circle with diameter AD and the corresponding cissoid are given the problem of the two means proportionals is solved by constructing a point G on the cissoid such that AB:BG is equal to a given ratio

During the motion we have: AB : BE = BE : BD = BD : BG. The ratio of AB and BG varies from arbitrarily big (when X is close to A) to 1 (when the arc AX is a quarter of the circumference). If the circle with diameter AD and the corresponding cissoid are given the problem of the two means proportionals is solved by constructing a point G on the cissoid such that AB:BG is equal to a given ratio (See Fig. 2.15). When we use this method the AB and BG that we find will differ from the AB and BG for which we want the mean proportionals. Their ratio however is correct and we can easily scale the four line segments. Diocles defined the curve differently. See Fig. 2.16. Fig. 2.16 Diocles’ definition of the cissoid

E

B

A

D G

X

Z

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2 Motion Beyond the Elements

Given a circle with a given diameter AD, we can mark off points X and Z such that arc AX equals arc DZ. Then the intersection G of XD and the perpendicular from Z on AD give a point on the curve.

Chapter 3

General Considerations and Kinematical Aspects of Motion

Abstract Pappus redefined the Euclidean construction game. Aristotle and Augustine pondered the nature of the phenomenon of time. Time-dependent properties of motion started to play a role. The quadratrix is one of the earliest curves involving time dependence. The notion of uniform motion was defined. In the Aristotelian corpus paradoxes of motion were discussed. In particular Archimedes did not hesitate to use motion in geometry when it led to interesting results.

3.1 Pappus’ Classification In the nineteenth century it was demonstrated rigorously that the quadrature of the circle, the trisection of the angle and the duplication of the cube cannot be solved with compass and ruler. Yet, solutions exist. The existence of the solutions follows from (implicit) mean value arguments based on continuity. For example: Take a cube with an edge of length 1 and imagine that the cube grows. When the edge has reached length 2, the volume of the cube equals 8. This implies that somewhere between 1 and 2 there is an edge length corresponding to a volume of the cube equal to 2. Although they could not prove it, mathematicians in classical antiquity were aware of the incompleteness of the Euclidean construction game. They accepted it as an empirical fact. When in the early fourth century CE, presumably in Alexandria, the mathematician Pappus looked back on a millennium of Greek mathematics, he came up with an intriguing classification of problems. He wrote: The ancients stated that there are three kinds of geometrical problems, and that some of them are called plane, others solid, and others curvilinear; and those that can be solved by straight lines and the circumference of a circle are rightly called plane, because the lines by means of which these problems are solved have their origin in the plane. But such problems that must be solved by assuming one or more conic sections in the construction are called solid because for their construction it is necessary to use the surfaces of solid figures, namely cones. There remains a third kind that is called curvilinear. For in their construction other

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Koetsier, A History of Kinematics from Zeno to Einstein, History of Mechanism and Machine Science 46, https://doi.org/10.1007/978-3-031-39872-8_3

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46

3 General Considerations and Kinematical Aspects of Motion lines than the ones just mentioned are assumed, having an inconstant and changeable origin, such as spirals, and the quadratrixes and conchoids, and cissoids, which have many amazing properties.1

The classification is based upon a hierarchy of curves: straight lines and circles, conic sections, and other curves. Problems that can be solved by means of ruler and compass or equivalent means are called plane. Problems that essentially require one or more conic sections are called solid, while the problems that require essentially more than straight line, circles and conic sections are called curvilinear. Elsewhere Pappus repeated the same classification and he continued: It appears to be no small error for geometers when a plane problem is solved by conics or other curved lines, and in general when any problem is solved by an inappropriate kind […].2

Pappus separates the solid problems from the curvilinear problems. This is remarkable. Why do the conic sections possess such a special position in Pappus’ classification? The only good reason I can think of is that, in particular after the appearance of Apollonius’ Conics, among the different curves that had been studied at the time the conic sections had acquired a special place. The theory of conic sections represented a coherent whole of results. Moreover, defining a curve as intersection of a wellestablished solid and a plane removes all movement from the pedigree of the curve, which is from a platonic point of view desirable. Also, although there is no evidence for it, because the set of conic sections includes circles of all sizes, the ancients may have felt that the conic sections were related to the circle and because of this more acceptable than other curves. Compared to what was known about the circle and the conic sections the knowledge of the curvilinear curves was an incoherent set of results. Pappus’ classification can be viewed as an attempt to redefine the Euclidean construction game. First one should try to solve a construction problem by means of compass and ruler. If this does not work, it should be tried by means of conic sections and if this is also impossible one can resort to other curved lines.

3.2 Composition of Different Uniform Motions: The Quadratrix So far we abstracted from the velocity of the motion in the motions that we considered. Whether the circles or straight lines were drawn in a uniform or an accelerated motion was irrelevant. The properties that we were considering are time-independent properties. The quadratrix is one of the earliest curves involving time dependence. It is defined by means of a combination of two uniform motions. The curve was allegedly invented by Hippias of Elis (fl. 420 BCE). For example, Proclus wrote: 1 2

The translation is based on Bos’ translation of this particular text in Bos (2001), p.38. Bulmer-Thomas (1939A), pp. 350-351.

3.2 Composition of Different Uniform Motions: The Quadratrix

47

B

Fig. 3.1 The quadratrix BZH

C E Z M

L

A

T

H

D

Appollonius for instance shows for each of his conic lines what its property is, and Nicomedes likewise for the conchoids, Hippias for the quadrices, and Perseus for the spiral curves.3

Most students of Greek geometry assume that the quadratrix was indeed invented by Hippias. Knorr is an exception. He argues that Hippias cannot have used it to square the circle, because that would foreshadow a sophistication that we only meet much later with Eudoxus and Archimedes. Pappus defined it as follows. See Fig. 3.1. We draw a square ABCD and AB rotates uniformly about A from its original position until it coincides with AD In the same period of time BC is moved uniformly while always remaining parallel to AD from its original position BC until it as well coincides with AD. Pappus: While the motion is in progress the straight lines BC and AB will cut one another in their movement at a certain point which continually changes place with them, and by this point there is described in the space between the straight lines BA, AD and the arc BED a concave curve, such as BZH, which appears to be serviceable for the discovery of a square equal to the given circle. Its principle property is this. If any straight line, such as AZE, be drawn to the circumference, the ratio of the whole arc to ED will be the same as the ratio of the straight line BA to ZT; for this is clear from the manner in which the line was generated.4

AB rotates uniformly about A until it reaches AD. At the same time segment BC uniformly moves parallel to itself from its original position BC until it coincides with AD. Clearly both the rotating AB and the translating BC reach the position AD at the same time. During the movement the point of intersection of the two segments describes the curve BZH: the quadratrix. It is easy to see that the quadratrix can be used to solve the problem of the trisection of an arbitrary angle. It can in fact be used to divide any angle in any integer number of parts. The quadratrix can, however, also be used to rectify a circle, because in Fig. 3.1 Arc BED : AB = AB : AH. 3 4

Proclus (1970), p. 277. Bulmer-Thomas (1939A), pp. 336-339.

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3 General Considerations and Kinematical Aspects of Motion

H is the endpoint of the quadratrix on AD. The proof given by Pappus is by contradiction. If Arc BED: AB /= AB: AH then Arc BED : AB = AB : AT with T either between A and H or between H and D. Suppose T is between A and H. Then we consider the circular Arc LMT. Because of similarity we have Arc BED : Arc LMT = AB : AT, These two relations imply that Arc LMT = AB. Moreover, because of the definition of the quadratrix, we have BA: TZ = Arc BED: Arc EAD = Arc LMT: Arc MAT. So, with Arc LMT = AB we have TZ = ArcMAT, which is absurd. Pappus here assumes the result that in any arc less than a quadrant the portion of the tangent cut off by the extension of the radius is longer than the arc itself. The other half of the proof is similar. Because of this property which gives a solution to the problem of the quadrature of the circle, the curve was given de name ‘quadratrix’. It seems to have been Christoph Clavius (1538–1612) who first used that word.5 Proposition 1 of Archimedes’ The Measurement of the Circle says: The area of any circle is equal to the area of a right-angled triangle of which the sides adjacent to the right angle are equal to the radius and the circumference of the circle. The proof is very rigid. Archimedes approximates the circle by means of regular polygons and shows that the supposition that the proposition is not true leads to a contradiction. This result implies that the rectification of the circle is equivalent to the quadrature, because once a straight line segment equal to the circumference has been constructed we have the right-angled triangle mentioned by Archimedes in this proposition and we can easily transform it into a rectangular quadrilateral or a square. It is essential that the two motions defining the quadratrix are uniform. Of particular interest is the fact that Pappus mentions two critical remarks by Sporus on the quadratrix and the possibility to execute a rectification if a circle by means of it. Sporus was not much older than Pappus himself. He lived towards the end of the 3d century. Firstly Sporus argued that the quadratrix can only be generated if we possess in advance a solution to the problem of the rectification of the circle. Because the velocities of the two uniform motions must have the right ratio, this requires that we 5

The Greeks called the curves ‘tetragonizousas’, which means ‘square-making’, used for the quadrature of the circle. Commandino called them in Latin ‘quadrantes’. Clavius called it ‘the quadratrix’ (Bos (2001), p.38, footnote 4).

3.3 Time-Dependent Kinematical Aspects of Motion

49

know the ratio of the arc BED and the segment BA. The second criticism concerns the point H which is needed for the rectification of the circle. According to Sporus the point H cannot be constructed because in the final position of BA and BC the two segments coincide and do not define a point of intersection. These are very acute criticisms. Pappus wrote: “With this [definition of the quadratrix] Sporus is rightly displeased.”6 It is highly remarkable that the first criticism, although justified, can be circumvented. After more than two millennia Henk Bos has pointed out that one may define the quadratrix differently.7 Without specifying the square in advance, one defines the curve as traced by the point of intersection of a half-line rotating counter-clockwise about a fixed point starting from a horizontal position and another straight line that translates in a vertical direction starting from the same position as the rotating line. Both motions are uniform but no in advance specification of the ratio of the motions is made. The ratio of the uniform velocities, of course, determines when the two lines are perpendicular to each other and the ‘top’ of the quadratrix is reached.

3.3 Time-Dependent Kinematical Aspects of Motion When we apply Euclid’s Postulates 1–3 we abstract from a precise characterization of the velocity. The only thing that counts is that the circles or straight line segments can be generated; it is unimportant whether this is done by means of a uniform or non-uniform motion. In a sense the aspects of motion that are being studied in such a way are ‘time-independent’, although time is still needed in an abstract sense. It is understandable that platonic philosophers wished to exclude the notion of time from mathematics. In Book XI of his Confessions St.Augustine (345–430 CE) discussed the notion of time. He started with the question “What was God doing before he made heaven and earth?”, and he pointed out that he did not appreciate silly answers like “He was preparing hell for those who pry too deep.“8 According to Augustine there was no time before the Creation, because time was created by God. So obviously it does not even make sense to ask the question, because there was no ‘before’, before God made heaven and earth. The next question is: What is time? Augustine struggles with the notion. One should distinguish the past time, the present time and the future time. But the past no longer exists, the future does not yet exist and the present owes its being to the fact that it will cease to exist. In what sense then does time exist? Augustine did not succeed in finding a satisfactory answer. He wrote: I once heard a learned man say that the motions of the sun, moon, and stars constituted time; and I did not agree. For why should not the motions of all bodies constitute time? What if the lights of heaven should cease, and a potter’s wheel still turn round: would there be no time 6

Bulmer-Thomas (1939A), pp. 338-339. Bos (2001), pp. 42-43, footnote 15. 8 Augustine (1955), Chapter 12. 7

50

3 General Considerations and Kinematical Aspects of Motion by which we might measure those rotations and say either that it turned at equal intervals, or, if it moved now more slowly and now more quickly, that some rotations were longer and others shorter? And while we were saying this, would we not also be speaking in time?

Clearly, one particular motion does not constitute time, although it can be used to measure it. Augustine does not succeed in satisfactorily explaining what time is. And yet, O Lord, we do perceive intervals of time, and we compare them with each other, and we say that some are longer and others are shorter. We even measure how much longer or shorter this time may be than that time. And we say that this time is twice as long, or three times as long, while this other time is only just as long as that other.9

Here Augustine proceeds to the fact that a more pragmatic approach to time is possible. As we have seen for Aristotle empirical reality was the only reality. Many centuries before Augustine Aristotle had discussed time and motion. Aristotle’s definition of time in fact boils down to the statement that time is continuous in the same way in which a line is continuous. Aristotle considers time as a magnitude, which possesses all the properties that other magnitudes like distance, area, volume and mass possess: continuity and divisibility. This implies that we can calculate with periods of time in precisely the way in which we deal with lengths and areas. This is the basis of the rigid approach to ‘time-dependent’ kinematical aspects of motion that we find in classical antiquity. In Aristotle’s Physics we find the first extant remarks on uniform motion from this point of view. By way of illustration I will give some quotations10 from Aristotle’s lengthy discussion of motion: And not only do we measure the length of uniform movement by the time, but also the length of time by uniform movement, since they mutually determine each other; for the time taken determines the length moved over […]. And the length moved over determines the time taken.

There is no circularity involved—time being measured by means of a distance and the distance by means of a time interval—because Aristotle considers a given uniform motion and he points out correctly: the time passed can be measured by means of a distance and the distance traversed can be measured by means of a period of time. Essentially Aristotle describes uniform motion as a linear correspondence between two continua: a straight line segment with a unit of length and a directed period of time with a unit of time. Because time is considered to be a magnitude, as for calculations the directed period of time is treated in every respect analogously to directed straight line segment. Aristotle: It is by reference to the standard unit of time that we determine the relative velocity of two several motions. For we ask what distance either motion has covered during the lapse of the standard unit of time, and pronounce the motion itself fast or slow in proportion as that distance is great or small. 9

Augustine (1955), Chapter 16. Aristotle (1952), pp. 398-401.

10

3.4 Composition of Uniform Motions and Paradoxes of Motion …

51

Fig. 3.2 The composition of uniform motions

The quotation contains a definition of ‘faster’: ‘A is faster than B if the distance covered by A in the unit of time is greater than the distance covered by B in that period of time’. Elsewhere Aristotle wrote: it follows that if P is quicker than Q it will (i) cover a greater distance in the same time; (ii) cover the same distance in a lesser time; (iii) cover a greater distance in a lesser time. ‘Quicker’ has been defined in this way.11

We will not investigate the logical relations between Aristotle’s statements about ‘faster’.12 We leave the considerations of philosophers instead and we enter the area of the rigid investigation of uniform motion. Aristotle wrote it down, but, undoubtedly, long before Aristotle, the Greeks knew what uniform motion is: a motion is uniform when equal distances are described in equal times and long before Aristotle they will have been inclined to treat periods of time as magnitudes.

3.4 Composition of Uniform Motions and Paradoxes of Motion in Mechanical Problems The oldest extant book about mechanics is Mechanical Problems usually included in the Aristotelian corpus. It is often assumed that it was written by a pupil of Aristotle in the time of Strato, who was a contemporary of Euclid. Yet Krafft has argued that the text was probably written by the young Aristotle and he traces part of its contents back to Archytas [Krafft,1970]. Recently Thomas Nelson Winter has given an argument that identifies Archytas as the most likely author. See Winter (2007). It is remarkable that the author discusses the composition of two uniform motions. See Fig. 3.2. The author expresses himself as follows: “Let the ratio according to which the body moves be represented by the ratio of AB to AC. Let AC move towards B while AB be moved towards the position CE; now let A travel to D, and let AB travel a distance determined by the point F. Then if the ratio of the movement is that of AB to AC, then the line AD must bear the same ratio to AF. Then the small parallelogram has the same proportions as the larger, so that its diagonal is the same, and the body 11

Aristotle (1952), pp. 102-103. The numbers (i), (ii) and (iii) were added by me. Wicksteed and Cornford wrote with respect to (iii): “Though not so much greater as in case (i)”. 12 For a logical analysis of Aristotle’s definition of ‘faster’ see Mendell (2007).

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3 General Considerations and Kinematical Aspects of Motion

Fig. 3.3 A paradox?

G A

D B

will move to Z. It can be shown that it will behave in the same way at whatever point its movement be interrupted; it will always be on the diagonal. Conversely it is obvious that an object travelling with two movements along a diagonal will always move in the ratio of the sides of the parallelogram.”13 The author discusses the composition of two linear uniform motions because he is interested in explaining circular motion. His conclusion is that circular motion cannot be explained as a composition of two uniform motions. The author of the Mechanical Problems also discusses two paradoxes that are worth mentioning. Problem 22 concerns Fig. 3.3. Imagine A moves towards B and B towards A with the same velocity. At the same time AB moves towards GD. What happens to A and to B? The answer is that A moves to D, while B moves to G. Now although in both cases two velocities of equal size and direction are combined the distance covered by A is much less than the distance covered by B. Why is that? The author of the Mechanical Problems knows why. It is a consequence of the parallelogram law for uniform velocities; the more two velocities that are combined are opposite, the more they neutralize each other. More interesting is Problem 23. It concerns Fig. 3.4: “How it is that a greater circle when it revolves traces out a path of the same length as a smaller circle, if the two are concentric?” Yet, “when they are revolved separately, then the paths along which they travel are in the same ratio as their respective sizes.” It is remarkable how the author struggles with this problem. Let ZI be the path traced in, say, a complete revolution by the circumference of the larger circle, when it travels independently, and HK the path traced in a complete revolution by the smaller circle when it travels independently. HK is equal to ZL and smaller than ZI. The problem occurs when the two circles are connected and do not travel independently. Which paths do the circumferences trace? The author distinguishes two cases: either the small one follows the larger one or the other way around. If the small one moves and the larger one follows, the paths traced are equal to HK for both circumferences. If the large one moves and the small one follows, the paths are both equal to ZI.

13

Aristotle (1936), p. 339.

3.5 A Remark on Methodology and a Theorem by Archimedes on Uniform … Fig. 3.4 The paradox of the two rolling circles

53

D E A

B

G K

T

L

I

H Z

What is the problem here? Take the case where the small one follows and we consider one rotation of both wheels. Of course, the author was aware of the difference between slipping and rolling. Slipping, however, meant for him: one particular point is dragged along a line segment. In this particular case there is no slipping in this simple sense of the small wheel. During the motion each of the points of the circumference of the small circle coincides at a particular moment with exactly one point of the path HT. Yet HT is longer than the circumference! This particular paradox has a long history.14 The attempts by Galilei and others contributed to the investigation of the cycloid.15 Galilei’s discussion of the paradox shows that great minds took it seriously. Galilei considers the wheels as polygons with an actually infinite number of sides and in the case of the small wheel following the big wheel he assumes that HT is longer than the circumference because during the motion vacua are interspersed between the infinitely small sides of the small wheel. And indeed when we approximate the circumferences with polygons with a large number of sides, during the motion the sides of the big polygon are continuously in touch with the line ZI, while there are empty spaces between the consecutive positions of the sides of the small polygon on line HT.

3.5 A Remark on Methodology and a Theorem by Archimedes on Uniform Motion Pythagoras’ theorem expresses equality: The three areas are magnitudes and one of them is equal to the sum of the two others. In classical antiquity in many cases the only way to discuss magnitudes required the use of ratios. We do not possess any details, but it is beyond doubt that the earliest Greek mathematicians were already interested in ratios of lengths, areas and volumes. A simple example of a theorem involving such ratios from Euclid’s Elements is.

14 15

Drabkin (1950), pp. 162–198. Mahoney (1973), p. 74.

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3 General Considerations and Kinematical Aspects of Motion Book VI, Proposition 1: “Triangles and parallelograms which have the same height are to one another as their bases.”

The early Greek mathematicians assumed that such ratios could always be represented by ratios of natural numbers. This assumption is equivalent to the supposition that if we have two magnitudes, like two lengths, two areas or two volumes, we can always find ‘common measures’, a unit of length, a unit of area and a unit of volume, by means of which we can measure the quantities. For example, if the ratio of the lengths of two segments corresponds to the ratio 5:9, there must exist a unit of length that is contained exactly 5 times in the first segment and exactly 9 times in the second segment. The assumption that two quantities of the same kind always possess a common measure makes life easy: you know exactly what you are talking about when you use ratios in geometry, you can calculate with such ratios in the way you calculate with ratios of natural numbers and arithmetical results often have immediate geometrical meaning. As we have seen in classical antiquity mean proportionals of numbers or magnitudes played an important role. If A X = , X B X is by definition the mean proportional of A and B. If X Y A = = , X Y B X and Y are the two mean proportionals of A and B. Let us consider two propositions on mean proportionals from Book VIII of the Elements, a book on arithmetic: Book VIII, Proposition 11: “Between two square numbers there is one mean proportional number, and the square has to the square the ratio duplicate of that which the side has to the side”.

We can illustrate this proposition as follows. Let the two squares be 9 and 64. Then the mean proportional of these two numbers is 24, because 9/24 = 24/64. The proof-idea is the following: Let a = c.c and b = d.d. Then the mean proportional is c.d. The second part of the proposition refers to the fact that a/b = (c/d)(c/ d). Euclid has a notion of multiplying ratios with themselves, which he calls duplication, or triplication. In this case a/b equals the duplicate of c/d, that is the ratio of the side (of the first square) to the side (of the other square). In his books on arithmetic Euclid interprets numbers as sets of units in a geometric way: then, for example, square 9 has side 3. The following proposition from the Elements concerns two mean proportionals: Book VIII, Proposition 12: “Between two cube numbers there are two mean proportional numbers, and the cube has to the cube the ratio triplicate of that which the side has to the side”.

3.5 A Remark on Methodology and a Theorem by Archimedes on Uniform …

55

Fig. √ 3.5 The irrationality of 2

For example, let the two cubes be 8 and 125. Then the two mean proportionals are 20 and 50, because 8/20 = 20/50 = 50/125. Proof-idea: Let a = c3 and b = d3 . Euclid constructs h = c2 d and k = cd2 and shows: a/h = h/k = k/b. He also shows that a/b = (c/d).(c/d).(c/d), so a/b is the triplicate of c/d. Although we quoted the two propositions from Euclid, they certainly illustrate early Pythagorean arithmetic with respect to ratios. Unfortunately the Pythagorean assumption that ratios of geometrical magnitudes can always be represented by ratios of natural numbers turned out to be false. We do not know how, and we do not know when, but it must have been a Pythagorean who discovered that in a square the ratio of diagonal and side cannot be equal to a ratio of natural numbers.16 This was the discovery of incommensurable quantities or magnitudes, or the discovery of the irrationals. Knorr has given a simple proof that we will reproduce here. Consider Fig. 3.5. AGFE is a square; D, B, H and I are the mid-points of the sides of the square. Suppose that DB and DC possess a common measure, such that there exists a unit of length that makes it possible to measure the lengths of DB and DC by means of natural numbers. Then also DB and DH possess a common measure and we have DB:DH = p:q. Suppose that the greatest common divisor of those natural numbers is equal to 1: gcd(p,q) = 1. This means that we have chosen the greatest unit of length that measures DB and DH. Because the area of the square AGFE, which equals q2 , is twice the area of the square DBHI, which equals p2 , we have q2 = 2p2 .

16

Knorr (1975), pp. 26-28.

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3 General Considerations and Kinematical Aspects of Motion

We now need a result from Pythagorean number theory, because “odd times odd is odd”, if q2 is even, q is even.17 This implies that q is even, say q = 2 s. This means that the area of square ABCD equals s2 . Moreover, because the diagonals in the drawing bisect the squares we have: p2 = 2s2 . So p is even as well. The fact that both p and q are even contradicts the assumption that gcd(p,q) = 1. This discovery has been depicted as extremely dramatic and as the basis of a foundational crisis in Greek mathematics. In the second half of the twentieth century historians have criticised this interpretation of the events. From my point of view one should indeed not exaggerate the dramatic consequences. Undoubtedly the Greek mathematicians went on applying ratios in geometry also after this discovery. Never ever in the history of mathematics have mathematicians discarded a coherent and fertile theory because of certain foundational problems. The Greeks will not have behaved differently. In the end the result of the discovery was very positive. The discovery of incommensurable magnitudes led to new mathematics. Book V of the Elements is completely devoted to the investigation of the phenomenon of incommensurability. Book V contains the final version of a highly original abstract theory of ratios, called the theory of proportions or ratios. In the development of this theory, more than half a century before Euclid, Eudoxus will have played a central role. It enabled the Greek mathematicians to handle ratios of all kinds of magnitudes, including incommensurable magnitudes, without considering them as ratios of natural numbers. Book V, Definition 5: Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equal multiples whatever are taken of the first and third, and any equal multiples whatever of the second and fourth, the former equal multiples alike exceed, are alike equal to, or alike fall short of, the latter equal multiples respectively taken in corresponding order.

This means in somewhat modern notation that x:y and z:w are equal when for all numbers n and m it is the case that, if nx > my, then nz > mw, if nx = my, then nz = mw, if nx < my, then nz < mw. From a modern point of view this definition is based on a comparison of ratios of magnitudes with rational numbers: two ratios are equal if in the comparison with respect to all rational numbers their ‘position’ is the same. On the basis of this definition Euclid derives many rules for the ‘calculation’ with ratios. 17

Pythagorean number theory was called ‘the theory of odd and even’. Odd and even were fundamental notions.

3.6 Archimedes: Motion in Geometry

57

A

Fig. 3.6 On Spirals, proposition 1

L

C F

D E

B

GH K

We will now see how Archimedes proved a theorem on uniform motion by means of the theory of proportions. In the treatise On Spirals, Archimedes defines a spiral line as follows; If in a plane a straight line is drawn and, while one of its extremities remains fixed, is made to rotate with a constant velocity, returning to the position it started from after as many rotations as one wants, while at the same time a point moves at a uniform velocity along the straight line starting from the fixed extremity, the point will describe a spiral in the plane.

The definition follows a number of introductory propositions. Proposition 1 says: If a point is moved on a straight line with constant velocity and if one takes on this line two segments. The ratio of the lengths of the segments is equal to the ratio of the periods of time in which the segments are described.

The proof is directly based on the definition of uniform motion (equal distances are covered in equal times) and on Eudoxus definition of the equality of ratios of magnitudes. See Fig. 3.6. We start with the segments CE and FH with the points D and G on them. Suppose that distance CD is covered in time FG and distance DE in time GH. We must prove that CD : DE = FG : GH. We now extend the segments. Let the points A, L, B and K are such that, for arbitrary natural numbers n and m, AD = n.CD and LG = n.FG, while DB = m.DE and GK = m.GH. Because the motion is a uniform motion, AD is covered in the time LG, while DB is covered in time GK. Then, because the motion is uniform, if AD > DB then LG > GK and similarly, if AD < DB then LG < GK, and, AD = DB then LG = GK. Because this is true for arbitrary multiples of CD and FG, we have proved the theorem, using Elements, Book V, Definition 5.

3.6 Archimedes: Motion in Geometry Archimedes did not hesitate to use motion in geometry when it led to interesting results. In his work one text in particular stands out: On Spirals. According to Pappus, Conon initiated the investigation of the Archimedean spiral.18 Archimedes’ On Spirals is entirely devoted to this planar curve. 18

Pappus (1933), Book VI, Section 21.

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3 General Considerations and Kinematical Aspects of Motion

Fig. 3.7 Rectifying circles with a spiral

We saw above how Archimedes defined the spiral. It is obvious that the spiral can be used to reduce the problem of the trisection of an arbitrary angle to the problem of the trisection of an arbitrary straight line segment. Yet Archimedes studied the spiral because he was interested in the quadrature of the circle. The link between the quadrature and rectification of the circle and the spiral becomes clear if one considers Fig. 3.7. Suppose the point P moves away from O with velocity v m/s and the rotational velocity is ω radians per second. Suppose P’T is tangent to the spiral in P’ and TO is perpendicular to OP’ in O. We will not reproduce Archimedes’ reasoning. From a modern point of view the instantaneous velocity of P’ has the components v and ω.OP’ and the triangle formed by these two velocities is similar to P‘OT. So ωO P' ωO P '2 OT = , and this implies O T = O P' v v O P' . = ωO P ' t = ar c P ' S, because v = t Clearly OT is equal to the circular arc P’S. This means that if we have a (segment of a) spiral and we can draw tangents to it, we can rectify circles. Archimedes devoted a considerable part of On Spirals to this result. The effective result was that Archimedes had reduced the problem of the quadrature and the rectification of the circle to the problem of drawing a tangent to a spiral. A pessimist might say that Archimedes had succeeded in replacing an insoluble problem by another insoluble problem, but it would be unfair. Attempts to solve the trisection of the angle, the duplication of the cube and the quadrature of the circle were important but it was only part of mathematics. Many of Archimedes’ results are theorems on areas and volumes. The questions that Archimedes was answering were easy to phrase but very difficult to answer. Moreover, his answers were usually beautiful. For example, the last part of On Spirals is devoted to a proof of the fact that the area under a segment of a spiral equals one third of the corresponding circle sector. We will not discuss Archimedes’ text On Spirals. Instead we will look at some results given by Pappus in his Collection. They are possibly based on an early text by Archimedes.19 19

Knorr (1978), p. 48.

3.6 Archimedes: Motion in Geometry

59

P

R

Q

O

Fig. 3.8 The spiral arc OP cuts off one third of the circle segment OQP

U

A B

V

E

C

A

B

C

D

D

T

P Q

S

O

Fig. 3.9 Comparing the ratio of two areas with the ratio of two volumes

In Fig. 3.8 the segment bounded by the spiral segment OP and the straight line segment OP is one third of the circle segment OQP. I will briefly consider Pappus’ proof of this result. See Fig. 3.9. The spiral segment ODCE on the right side of Fig. 3.9 is part of the circular segment with radius OE. The situation is compared to what we get when we rotate a rectangular figure TSVU with diagonal US about TS. See the left side of Fig. 3.9. This yields a cylinder with a cone inside. S is the top of the cone. Eudoxus, and after him Euclid had already proved that the ratio of the volumes of cylinder and cone is 3:1. We can now divide the circular segment on the right side of Fig. 3.9 and the cylinder on the left side in n equal parts. By comparing two corresponding parts, for example circular segment OBA and cylinder slice PQBA we can conclude the following: OC 2 PC 2 spiralsegment O DC Q D2 coneslice P Q DC O D2 < < < and < . 2 2 2 cir clesegment O B A cylinder slice P Q B A OB OB QB Q B2

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3 General Considerations and Kinematical Aspects of Motion

It is easy to see that the upper and lower bounds for the two ratios are equal. Moreover when n grows the difference between the lower and upper bounds can be made arbitrarily small. From this Pappus draws the conclusion that the ratio of the areas of the spiral segment and the circle segment is equal to the ratio of the volumes of the cone and the cylinder, which yields the ratio 1:3. The plane spiral is only one of several spirals that were studied by the ancient geometers. Pappus considers spirals on cylinders, on cones and on spheres. He also has an interesting theorem concerning the relation between the cylindrical helix and the quadratrix. The cylindrical helix can be generated as follows. On the one hand, a line segment BC rotates with a uniform velocity about the line BL. BL is perpendicular to the plane in which the rotation takes place. We combine this movement with a uniform translation in the direction of the line BL. The result is that the line segment BC describes a helicoid and the point C describes a cylindrical helix. In Fig. 3.10 ABC is a quadrant of a circle. It corresponds to a rotation of BC over an angle of 90 degrees. Pappus’ theorem says that the projection on the horizontal plane of the curve of intersection of the helicoid and the plane through BC and G is a quadratrix. Consider an arbitrary position LH of the moving line segment BC during the generation of the cylindrical helix. H stays below the plane through BC and G and let I be the point of intersection of LH in this position with the plane through BC and G is I. Such a point exists because H covers horizontally a bigger distance than vertically, so. Let E be the projection of I on the horizontal plane and let EF be perpendicular to BC. Then triangle IEF is similar to triangle GAB. This similarity implies that when during the motion IE = DH grows uniformly from 0 to AG, EF grows uniformly from 0 to AB. However, this implies that E describes a quadratrix. Pappus presents the theorem as an argument to make the quadratrix ‘less mechanical’. The theorem makes it possible to define the quadratrix by means of the curve of intersection of two surfaces. Fig. 3.10 A different way to generate a quadratrix

G

A

H I L B

E

D F

C

3.6 Archimedes: Motion in Geometry

61

G

A

H L B

I K

L

I

H

K

D

D C

B

Fig. 3.11 The plane spiral related to the cylindrical helix

Pappus gives us another method to generate the above-mentioned helicoid. Its main interest lies in the fact that the plane spiral can be related to the cylindrical helix via a spiral on a cone. See Fig. 3.11. If during the generation of a cylindrical helix we project the generating point H on a right-angled cone of which the axis coincides with the axis of the helix we get a conical spiral described by the point I. Because BI grows uniformly I describes on the cone the combination of a uniform rotation and a uniform rectilinear movement away from the top B. The projection of this conical spiral on the horizontal plane is an Archimedean spiral, described by the point K.

Chapter 4

Kinematical Models in Astronomy

Abstract Early on, the Greeks explained the motion of the stars by assuming that they are all attached to a sphere that rotates uniformly about a fixed axis, while the immovable earth is at the center of the sphere. However, explaining the motion of the Sun, the Moon, and the five planets posed a difficult problem because their motions are not uniform. By combining uniform circular motions in a sophisticated way, the Greek astronomers succeeded in defining kinematic models that were consistent with the observations.

4.1 Plato and Astronomy1 In the period that we are considering there is one other area in Greek science where idealized motion played a particularly important role: astronomy. As for the early history of Greek astronomy Aristotle wrote in Metaphysics A5, 986 a I, the following: They (the Pythagoreans) conceived that the whole heaven is harmony and number; thus, whatever admitted facts they were in a position to prove in the domain of numbers and harmonies, they put these together and adapted them to the properties and parts of the heaven and its whole arrangement. And if there was anything wanting anywhere, they left no stone unturned to make their whole system coherent. For example, regarding as they do the number ten as perfect and as embracing the whole nature of numbers, they say that the bodies moving in the heaven are also ten in number, and, as those which we see are only nine, they make the counter-earth the tenth.2

The assumption of the existence of a counter-earth, mentioned by Aristotle, is a puzzling feature of the Pythagorean system. Ancient astronomy was based on naked eye observations. In the sky there are two different kinds of phenomena. There are the numerous ‘fixed stars’. They move every night from east to west, but they are called fixed because they have fixed positions with respect to each other. Then there are other heavenly bodies. They move with the fixed stars, but their position with respect to the fixed stars changes slowly in the course of time; this movement is from west 1 2

A part of this chapter was published as Koetsier (2022). Heath (1932), p. 34.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Koetsier, A History of Kinematics from Zeno to Einstein, History of Mechanism and Machine Science 46, https://doi.org/10.1007/978-3-031-39872-8_4

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to east and is rather capricious. In antiquity these other heavenly bodies were seven in number: the Moon, the Sun and the 5 so-called wandering stars Mercury, Venus, Mars, Jupiter and Saturn. The word ‘planet’ comes from the Greek word ‘wanderer’ or ‘vagabond’. If Aristotle is right the Pythagoreans counted the fixed stars as one body, the wandering stars plus the Sun and the Moon as 7 bodies and the Earth as a one body. They are the 9 heavenly bodies that we see. The postulate of the harmony of the heaven associated with the number 10 implied for the Pythagoreans the necessity to introduce a permanently invisible 10th heavenly body: the counter-earth. This is certainly remarkable, and because the heavenly bodies are sometimes invisible, the supposition of the existence of a body that is always invisible is not absurd. According to Thomas Heath Diogenes Laërtius wrote: “Further we are told that Pythagoras was the first to call the heaven the universe and the earth round (i.e. spherical), though according to Theophrastus it was Parmenides, and according to Zeno it was Hesiod”.3 And Aëtius wrote: “The Pythagoreans held the Sun to be spherical”.4 These quotations all show the tendency that existed in early Greek thinking to approach the universe, and in particular the bodies that are moving in the heaven in a mathematical way. We do not know who came up with the idea to explain the motion of the fixed stars by assuming that they are fixed to a spherical surface rotating uniformly about a fixed axis. However, since Parmenides the existence of such a sphere of the fixed stars was generally accepted in Greek astronomy. Its center coincided with the center of the spherical earth. The development of kinematic modeling in Greek astronomy is reflected in Plato’s dialogues. In the Republic Socrates tells the so-called myth of Er in which Er, the son of Armenius, after having been slain in battle, revives after twelve days and, after having returned from death, relates about the trip that his soul has made. While traveling in a large company of other souls Er’s soul had seen the functioning of the universe: The staff turned as a whole in a circle with the same movement, but within the whole as it revolved, the seven inner circles revolved gently in the opposite direction to the whole, and of these seven the eighth moved most swiftly, and next and together with one another the seventh, sixth, and fifth, and third in swiftness, as it appeared to them, moved the fourth with returns upon itself, and fourth the third and fifth the second. And the spindle turned on the knees of Necessity[…].5

The spindle that is “turned on the knees of Necessity” is the axis of the universe, about which the sphere of the fixed stars completes a full rotation every 24 h. In the center is the immovable earth. The seven inner circles correspond to the Moon, the Sun and the planets that are also carried around in the revolution of the celestial sphere. However, they have, in addition, their own circular movements opposite to that of the daily rotation of the celestial sphere. 3

Heath (1932), p. 11. Heath (1932), p. 35. 5 Plato (1973), p. 841. 4

4.1 Plato and Astronomy 8th

Moon Smallest circle 1st in swiftness

7th

Sun 2nd in swiftness

65 6th

Venus 2nd in swiftness

5th

4th

Mercury

Mars

2nd in swiftness

3d in swiftness

3d

Jupiter 4th in swiftness

2nd

Saturn

5th in swiftness

1st

Fixed stars Biggest circle

Fig. 4.1 The universe as it is represented in the myth of Er

This text requires some explanation. See Fig. 4.1. The seven inner circles are called by Plato the 8th, 7th, 6th, 5th, 4th, 3d and 2nd. They are ordered on the basis of their diameters from small to large. The sphere of the fixed stars, which has the largest diameter, is then the 1st. The Moon, closest to the earth, is attached to the 8th circle. It is the heavenly body that describes its orbit the quickest of the seven. The 7th, 6th and 5th circle correspond to the Sun, Venus and Mercury; they are close to each other and take roughly a year to describe their orbit. They are second in swiftness of the seven. The third in swiftness is the 4th, the planet Mars. The fourth in swiftness is the 3d, Jupiter. Then implicitly the fifth in swiftness is the last planet, the 2nd, Saturn. As for the mathematics corresponding to this model one of the oldest extant works in Greek astronomy is On the moving sphere, written by Autolycus of Pitane around 320 BCE.6 Autolycus wrote in the second half of the fourth century BCE, but the contents of his books are undoubtedly much older. The book starts with the following text: It is said that points move along uniformly, when in equal time (intervals) they trace equal or similar lengths (of arc). If then on a certain line a certain point, moving uniformly, goes through two line (segments), the ratio of the time intervals in which the point has gone through either of the lines will equal the ratio of the line segments. Axis of a sphere is the diameter of the sphere which remains fixed while the sphere rotates about it. Poles of the sphere are the ends of the axis.7

The second sentence is identical with Proposition 1 of Archimedes’ On Spirals (Cf. Chap. 3). Archimedes gave a proof; Autolycus seems not to have felt a need to do so. Autolykus’ On the moving sphere deals with a sphere rotating uniformly about an axis. In Propositions 5 and 6 of the book a fixed large circle of the sphere (obviously fixed in the sense that it does not share the rotational motion of the sphere) is introduced. This circle defines the visible and the invisible part of the sphere. This circle is the horizon. Let us consider Proposition 6 of On the moving sphere. Prop. 6. “If on the sphere the large fixed circle defining the visible and the invisible (part) of the sphere is inclined to the axis, it will be tangential to two circles which are equal and parallel to another. Of these the one towards the visible pole is always visible, and the one towards the invisible (pole) always invisible.”8

6

Bruin&Vondjijdis (1971). Bruin&Vondjijdis (1971), p. 1. 8 Bruin&Vondjijdis (1971), p. 10. 7

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H

Fig. 4.2 Autolycus’ moving sphere

D

Z

A

C B F E K

See Fig. 4.2. Autolycus’ proof runs as follows: Let the large circle ABC define the visible and the invisible. Let the visible pole be D. Draw the large circle ADC. Let point E on circle ADC be such that arc CE is equal to arc AD. E is obviously the invisible pole. It is then clear that the circle AZH drawn around the pole D and the circle CFK drawn around pole E are both tangent to the horizon and they represent the areas on the rotating sphere that are respectively always visible and invisible.

4.2 The Model in Plato’s Timaeus In the myth of Er, the Sun, the Moon and the planets move in circles that seem to be parallel to the circles that are described by the fixed stars. This would imply that the Sun, the Moon and the wandering stars would always rise and set at the same spot on the horizon. This is not what really happens. In the Timaeus Plato describes a better model of the universe. The Sun, the Moon and the planets move in a plane which is not perpendicular to the axis of the sphere of the fixed stars; it intersects this sphere in the circle called Ecliptic which is the center of the belt of constellations called the Zodiac. Possibly the Greeks became acquainted with the signs of the Zodiac, defined earlier by the Babylonians, through Cleostratus of Tenedos about 550 BCE.9 At the time of Plato the Zodiac was known. Look at this part of Plato’s description of how God created the world: This entire compound he divided lengthwise into two parts which he joined to one another at the center like the letter X, and bent them into a circular form, connecting them with themselves and each other at the point opposite to their original meeting point, and, comprehending them in a uniform revolution upon the same axis, he made the one the outer and the other the inner circle. Now the motion of the outer circle he called the motion of the same, and the motion of the inner circle the motion of the other or diverse. The motion of 9

Waerden (1961), p. 84.

4.2 The Model in Plato’s Timaeus

67

E

Fig. 4.3 The Greek model of the universe

P

H D

I

A

B C

Q

G F

the same he carried round by the side to the right, and the motion of the diverse diagonally to the left. And he gave dominion to the motion of the same and like, for that he left single and undivided but the inner motion he divided in six places and made seven unequal circles having their intervals in ratios of two and three, three of each, and bade the orbits proceed in a direction opposite to one another. And three [Sun, Mercury, Venus] he made to move with equal swiftness, and the remaining four [Moon, Saturn. Mars, Jupiter] to move with unequal swiftness to the three and to one another, but in due proportion.10

The letter X in the quotation corresponds to the intersection of the ecliptic, the circle that carries the Zodiac, and the equator of the sphere of the fixed stars. Those two circles are created by connecting the end points of the legs of the letter X in such a way that those legs become the two circles. See Fig. 4.3. The two legs of the letter X in Plato’s Timaeus are HCGD, the celestial equator, and ECFD, the Ecliptic. The Sun, the Moon and the planets move in the plane of the Ecliptic. The celestial sphere rotates about the axis PQ perpendicular to the plane of the celestial equator. Of course, next to these two circles we have the horizon ABI. Elements of the geometry of this more sophisticated model of the universe are also treated in Autolycus’ On the moving sphere. Consider, for example, Prop. 11: “If on a sphere the large circle, defining the visible and invisible (part) of the sphere, is oblique to the axis, and if another oblique large circle is tangent to (parallel circles) larger than those which the horizon touches, it makes its risings and settings on the part of the arc of the horizon that is between the parallel circles to which it is tangential.”11

See Fig. 4.4 where we are looking at the horizon from a point high above the sphere of the fixed stars on the line connecting the center of the Earth with the Zenith. The big circle ABC is the horizon. It is tangent to the small circle AD, which is fixed, i.e. during the rotation of the celestial sphere it coincides with itself. This means that its center on the sphere is one of the poles of the rotation of the sphere. The large circle CZ is tangent to the circles ZB and CH.These two circles 10 11

Plato (1973), p. 1166. Bruin&Vondjijdis (1971), pp. 20–21.

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4 Kinematical Models in Astronomy

A

Fig. 4.4 The rising and setting of the points on circle CZ

D B L N C

F K

Z M Q H

are parallel to the equator of the celestial sphere, and, of course, also parallel to the circle AD. Autolycus defines the arc ZH as the rising side and arc BC as the setting side. During the motion of the celestial sphere all points of the semicircle CZ rise on arc ZH and set on arc BC. Clearly, if circle CZ is the ecliptic, the signs of the Zodiac will rise on arc ZH and set on arc BC. The Sun moving in the plane of the ecliptic will behave similarly. The endpoints of the two arcs ZH and BC correspond to the longest and the shortest day (the summer and the winter solstices); their midpoints correspond to the two days in the year for which day and night last equally long (the beginning of spring or the vernal equinox and the beginning of the autumn or the autumnal equinox). The model of the universe that we find in the Timaeus is based on the recognition that the plane of the annual motions of the Sun, the Moon and the planets must be inclined to that of the daily rotation of the sphere of the fixed stars. The latter rotation is dominant and accounts for the daily rising and setting of all heavenly bodies. However, in the course of time it became clear that this model is too simple. The Sun, the Moon and the planets exhibit anomalous deviations from the uniform circular movement in both longitude and latitude. It is said that Plato set it as a problem to all earnest students to find the uniform and ordered movements by means of which the apparent movements of the planets could be explained. Plato’s motives may have been very different from the motives of modern astronomers.12 Plato’s primary intention may have been to save his astral theology from these vagabonds. Why worship the stars if these divine beings cannot do better than set an example of disorderly behavior?13

12 13

For a balanced view on Plato’s attitude with respect to astronomy see Thomas (1984). Farrington (1944), p. 97.

4.3 Eudoxus’ Models

69

4.3 Eudoxus’ Models Eudoxus may have been the first to come up with a solution to the problem. According to Diogenes Laertius, Eudoxes of Cnidus in Asia Minor was an astronomer, geometrician, physician and legislator. He went from Cnidus to study geometry with Archytas. At the age of 23 he traveled to Athens where he listened to the lectures given by sophists. He was a pupil of Plato for some time. He spent a year and 4 months in Egypt. From there he went to Cyzicus in Asia Minor. Finally he returned to Athens. In mathematics he introduced or improved the method of exhaustion and the theory of proportions. As for kinematics his astronomical theory of homocentric spheres is a major contribution. The results given by Autolycus were certainly known to Eudoxus. We may even owe them to him, but they may be older. Eudoxus’ complex system has been reconstructed in the nineteenth century by Schiaparelli on the basis of very modest evidence. Others have elaborately commented on Schiaparelli’s reconstruction.14 The reconstruction is too lovely not to describe its basic structure. Eudoxus’s system is an attempt to refine the simple model described in the Timaeus based on the following idea, already implicit in that model. The movement of the fixed stars is modeled by means of one sphere, the celestial sphere, rotating about its axis. In one day it completes a full rotation. As we have seen, if one considers a longer period of time it becomes clear that the Sun moves with respect to the celestial sphere. This anomaly is taken care of by Eudoxus by adding another sphere coinciding with the celestial sphere, which rotates about an axis fixed inside the celestial sphere. This second sphere takes a year to complete a full rotation. Clearly the ‘equator’ of this second sphere, to which the Sun is attached, coincides with the Ecliptic in the celestial sphere. The angle between the axes of the two spheres would have to be some 24 degrees, which is the obliquity of the Ecliptic. The same idea is applied to the Moon and the planets. In a first approximation of their movement spheres are introduced that rotate about axes fixed inside the celestial sphere perpendicular to the ecliptic. The invention of the model with two coinciding spheres marked the discovery of the first qualitative results in spherical kinematics. Consider two coinciding spheres S1 and S2 . S1 rotates uniformly about an axis fixed in space and S2 rotates uniformly about an axis fixed in S1 . In Eudoxus’ two-sphere model the heavenly body is attached to the equator of the second sphere. During the motion points on S2 describe spherical curves Although most of the properties of such curves are not needed to understand the two-sphere model that we just sketched, Eudoxus was interested in the curves as we will see below. Unfortunately for none of the heavenly bodies the two-sphere model was sufficiently in accordance with the phenomena. In order to correct the anomalies further rotating spheres were added by Eudoxus. In the case of the Moon the two-sphere model combines the uniform rotation of the celestial sphere with a rotation in the 14

For example Yavetz (199)].

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opposite direction about the axis of the ecliptic that corresponds to the lunar month. However, the Moon possesses also a movement in latitude, north and south of the Ecliptic.15 In order to obtain this deviation in the model Eudoxus added a third sphere carrying the Moon on its equator rotating around an axis fixed inside the second sphere and inclined at a small angle (presumably some 5 degrees) to that of the axis of the second sphere. The third sphere rotated in a direction opposite to the direction of rotation of the second sphere and at a much slower speed. For the Sun Eudoxus also used a three-sphere model. For the five planets Eudoxus had to solve a more difficult problem. Their orbits not only show deviations in latitude. Sometimes for several days they don’t seem to move at all with respect to the fixed stars. Then suddenly they retrace their paths backwards for some time, are stationary again and then resume their eastward motion along the Zodiac. Eudoxus brilliant solution of this problem shows that he must have carefully studied the two-sphere model. Consider Fig. 4.5a. S1 rotates about a vertical axis from west to east. S2 rotates with the same but opposite velocity about an axis fixed in S1 inclined at an angle β. The point P starts in point Q in Fig. 4.5a. In Fig. 4.5b we consider the horizontal circle from Fig. 4.5a, the equator of S1 , with center M, through Q. The ellipse is the projection of the equator of S2 . This ellipse rotates counterclockwise about M with the angular velocity of the rotation of S0 . In Fig. 4.5b we have the position of the ellipse after both spheres have rotated over an angle α. Remember, their velocities are equal but opposite. P is the moving point we are interested in. We first Rotate S1 . Then P moves from position Q to the position T. The arc QT in Fig. 4.5 represents the real distanced that P covers. Then we rotate S2 over the angle α in the opposite direction. The arc TP of the ellipse is the projection of the distance covered by P in this second rotation. We will now show that during the motion of these two spheres the projection of the position of the planet on the plane of the horizontal circle in Fig. 4.5a is a circle. This means that the curve described by the planet is the intersection of a relatively small vertical cylinder and a fixed sphere coinciding with the moving spheres in Eudoxus’ model. See Fig. 4.6. This curve was called in antiquity the hippopede, because it has the shape of the ideal training course for a horse (on a circular course the legs on the outside cover a greater distance). The proof that the hippopede is the intersection of a cylinder and a sphere is rather simple. Consider the projection MP’ of MP on the long axis of the ellipse. For a moment we only consider the second rotation over an angle α. We get MP’ = R.cos α. Now we consider the first rotation. The long axis of the ellipse has rotated over an angle α, which means that the distance between M and the projection P’ of Q on its long axis is R.cos α = MP’. This means that after the two rotations P is on QP’. Moreover, the ratio PP’/QP’ does not depend on α. We have QP’ = Rsinα and PP’ = Rsinα.sinβ, because it is the horizontal projection of a segment equal to QP’. So the 15

Dicks (1970), p. 178.

4.3 Eudoxus’ Models

71

Fig. 4.5 Eudoxus’ two-sphere model Fig. 4.6 The hippopede

ratio PP’/QP’ equals sin β, a constant value. We can draw QM and through P a line parallel to MT. This line intersects MQ in a fixed point U, for which triangle UPQ is rectangular. The fixed line segment UQ is always seen from P under a right angle. This implies that the projection of moving point on the horizontal plane moves on a circle.16 Eudoxus realized that we could combine the hippopede of Fig. 4.6 with a rotation around the ecliptic. If we choose the right velocities we get the loop-shaped paths of Fig. 4.7. This means that in order to account for the stationary points and retrograde 16

I owe this elementary way of proving that the hippopede is the intersection of a cylinder and a sphere to Aad Goddijn.

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Fig. 4.7 Loop-shaped path of planet (Eudoxus)

motion Eudoxus used four spheres for each planet. For all planets the first sphere is the rotating celestial sphere and the axis of the second sphere is at an angle equal to the obliquity of the ecliptic to the axis of the first sphere. The period of the second rotation is equal to the sidereal period of the planet. The third and fourth spheres bring about the retrograde motion. In the course of time Eudoxos’ system was unable to satisfy astronomers. In his model of concentric spheres the distance of the Sun, the Moon and the planets to the earth is constant, which is not in accordance with the fact that the Sun and the Moon have a variable apparent diameter. Simplicius wrote: “[…] the theories of Eudoxus and his followers fail to save the phenomena, and not only those which were first noticed at a later date, but even those which were before known and actually accepted by the authors themselves.”17 Simplicius mentions a pupil of Eudoxos, Polemarchos of Cyzocos (c. 340 BCE), who knew the discrepancy between the model and the phenomena as for the apparent size of the heavenly bodies, but minimized its importance. Others like Calippus (about 370–300 BCE) and Aristotle, added spheres to Eudoxus’ model. Calippus did this to account for the unequal motion of the Sun in longitude and Aristotle found it necessary to add spheres when he attempted to envisage a working material mechanical system that could replace the set of seven separate abstract systems that Eudoxus had defined.18

4.4 Apollonius’ Epicycle Model The Greeks did not possess the notion of instantaneous velocity and a general treatment of non-uniform motion was for them impossible. Yet this does not mean that they could not at all handle non-uniform motion in a quantitative way. The motion of the Sun, the Moon and the planets is non-uniform. By combining uniform circular motions Eudoxus succeeded in creating precisely defined non-uniform motions, although the result turned out to be only roughly in accordance with the observed phenomena. The next ideas along these lines led to the final Greek solution with respect to the motion of the heavenly bodies, the stars, the planets and the Sun and the Moon. This solution has come to us in a book written by Claudius Ptolemaeus or Ptolemy 17 18

Heath (1932), p. 68. Heath (1932), pp. xlvi-xlix.

4.4 Apollonius’ Epicycle Model

73

Z

Fig. 4.8 The epicycle model

H A

K B

B

E

D

G

in English (circa CE 100—circa CE 175), an astronomer working in Alexandria, the main city in Greco-Roman Egypt at the time. The original title of the text was Mathematical Systematic Treatise. The modern title is Almagest derived from the Arabic ‘Almagisti’ which in its turn comes from the Greek word ‘greatest’ (μεγ´ιστη). At the end of classical antiquity the Almagest had become the standard textbook on astronomy, which it would remain for more than a millennium. The success of the book contributed much to the loss of earlier works on astronomy. In Almagest III.3 Ptolemy writes with respect to the irregularities or anomalies in the motions of some of the heavenly bodies the following: The reason for the appearance of irregularity can be explained by two hypotheses, which are the most basic and simple. When their motion is viewed with respect to a circle imagined to be in the plane of the Ecliptic, the centre of which coincides with the centre of the universe (thus its centre can be considered to coincide with our point of view), then we can suppose, either that the uniform motion of each [body] takes place on a circle which is not concentric with the universe, or that they have such a concentric circle, but that their uniform motion takes place, not actually on that circle, but on another circle, which is carried by the first circle, and [hence] is known as the ‘epicycle’. It will be shown that either of these hypotheses will enable [the planets] to appear to our eyes, to traverse unequal arcs of the Ecliptic (which is concentric to the universe) in equal times.19

We do not know who invented these hypotheses or models. It is, however, quite possible that Apollonius invented the second hypothesis, the epicycle model (see Fig. 4.8). As we will see he studied this model. Two circles are involved: the epicycle and the deferent. The epicycle ZHBK, or rather its center, moves anticlockwise on the circle ABGD with center E. This circle ABGD is called the deferent. The body moves anticlockwise on the epicycle ZHBK. It is quite obvious that seen from the point of view of the terrestrial observer, positioned in E, the motion is non-uniform. Suppose the epicycle moves in the direction from A to B. If the body on the epicycle is in the position K it appears to lag behind, if it is in H it appears to move faster than the epicycle.

19

Ptolemy (1984).

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4 Kinematical Models in Astronomy

There is a nice theorem on epicycle motion that we definitely owe to Apollonius. The theorem tells us exactly when, seen from the point of view of the terrestrial observer, in the epicycle motion a planet will be in a stationary point. Consider in Fig. 4.9 the epicycle with diameter AG. E is the center of the epicycle and Z is the position of the terrestrial observer, the center of rotation of the deferent. Imagine both rotations as anticlockwise. Let Ω be the angular speed with which the epicycle rotates and ω the angular speed of the planet. When the planet is in position G its velocity as a result if the rotation about E equals ω.EG and fully opposes the velocity Ω.GZ as a result of the rotation of the epicycle. So in order to have stationary points we must have Ω EG ≥ GZ ω If this is not the case there are no stationary points. If we rotate a line about Z from the initial position ZGEA anticlockwise it will cut the circle in points H and B. ZH grows during the rotation and HB diminishes. Clearly there is a line through Z cutting the circle in H and B such that 1 BH 2

HZ

=

Ω ω

Apollonius now proves that in this position the point H on the epicycle corresponds to a stationary point in the orbit of the planet. This means that when the planet moves on the epicycle from K to H the observer in Z will see the planet move to the left, while when the planet moves on the epicycle from H towards G the observer will see it move to the right. Fig. 4.9 Apollonius’ theorem

A B

E

K

H G

Point P Z

4.4 Apollonius’ Epicycle Model

75

Fig. 4.10 Apollonius’ lemma

V

W K

Z

H

U

B

This is a remarkable result, which the reader will be able to verify easily by drawing the velocity vectors that result from the two rotations. Apollonius’ proof is more complex. He first gave a proof of a lemma. See Fig. 4.10. Suppose that in triangle ZBK we have BH ≥ BK. Then the lemma says that in Δ Z B K we have BH > HZ

KZH . K BH

Draw the parallelogram HBUK and extend ZK and BU until they intersect in V. It is enough to consider the case that BH = BK. If BH > BK, the left hand side of the inequality only grows. Then we can draw a part of the circle with center K through B and U. It intersects ZV in W. We can write: VK VU Δ V KU BH = = = . HZ KZ UB Δ UKB We also have ΔV K U > Sector W K U and ΔU K B < SectorU K B. This means that Sector W K U BH > = HZ SectorU K B

W KU , UKB

which concludes the proof. We can now prove Apollonius’s theorem. We will reproduce a sketch of Apollonius’ proof as Ptolemy gives it in the Almagest. Consider Fig. 4.9 again. 1 BH . Moreover, BK < BH, so the lemma tells us that We have 2H Z = Ω ω BH HZK > = 2H Z 2 H BK

HZK . HEK

This yields Ω > HH EZ KK . Suppose the planet moves from K to H on the epicycle ω and suppose that H E K = ω.t, then Ω.t > H Z K and obviously for the observer in Z the planet moves forward, that is in the direction of the rotation on the deferent.

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Fig. 4.11 Apollonius’ theorem, the modern approach

r R.Ω

r.ω

R

Towards terrestrial observer

Similarly Apollonius proves that when the planet moves from H on the epicycle in the direction of G to a point P for the observer the planet moves retrograde, that is against the direction of the rotation of the deferent. One wonders how Apollonius found this theorem. We cannot exclude the possibility that he on a heuristic level did something akin to a modern approach. See Fig. 4.11. In a stationary point the displacement of the planet in a direction perpendicular to the line towards the terrestrial observer as a result of a small increase in the two angles of rotation will be zero. In modern terminology this means that vector R.Ω is neutralized by the projection of vector r.ω on the line that carries R.Ω. This easily leads to Apollonius’ theorem. If Apollonius indeed considered the effect of small increases in the angles of rotation—and his proof suggests it—he may have found the result in such a way.

4.5 Hipparchus’ Theory of the Motion of the Sun (About 150 BCE) As we have seen Ptolemy also mentions another hypothesis: the eccentric hypothesis or the eccentric model. We do not know who invented the model, but it may have been Hipparchus.

4.5 Hipparchus’ Theory of the Motion of the Sun (About 150 BCE)

77

From the period between 350 BCE and the year 0 many Babylonian astronomical cuneiform tablets are extant. They contain calculations and predictions concerning the lunar phases, positions of planets etc. This was an arithmetic tradition in astronomy which kept a close track of the phenomena and differed from the more speculative tradition in which Eudoxus and Apollonius were working. The two great Greek mathematicians were developing kinematical models for astronomical phenomena, but they did not go beyond a qualitative comparison between the model and the phenomena. In this respect Hipparchus (about 150 BCE) represents a revolution. Hipparchus was familiar with Greek geometry and with Babylonian observations. As far as we know he was the first to try to really adapt the models to the actual observed data. He is famous for his investigation of the motion of the Sun. It led to the first sinetable. Consider a circle and suppose that a cord is seen under an angle α from the center of the circle. Hipparchus’ table differed from the modern sine-tables in the sense that it gave the lengths of the cords corresponding to a sequence of angles α. The lengths were given in terms of the Babylonian sexagesimal system. They were denoted using the Greek alphabet. How different Hipparchus approach is from the approach of Eudoxus and Apollonius, is clear from the fact that many of the entries in the table are approximations, inevitably so because the cord and any ‘unit of length’ are often incommensurable. The Babylonians had divided the yearly circular path of the Sun against the background of the Zodiac, in twelve parts of 30 degrees. According to Neugebauer the Sumerians had a unit of distance, the danna, equal to approximately 7 English miles. The danna was subdivided in 30 UŠ or 30 ‘lengths’. These units were also used to measure time. A period of one danna covered the time needed to travel a distance of one danna. One day or one revolution of the sky took twelve danna. This is where, according to Neugebauer, the subdivision of the ecliptic in 12 times 30 degrees comes from. Later all other circles were divided in 360 degrees.20 The twelve parts of the ecliptic got the names of twelve constellations of stars with which the twelve parts more or less coincided.21 At a certain moment the Babylonians realized that the velocity of the Sun in the course of a year is not constant. The number of days elapsing between the vernal solstice (longest day) and the equinoxes is not the same as the number of days between the equinoxes (daytime and nighttime are of approximately equal duration) and the winter solstice (shortest day). The equinoxes, autumnal and vernal, and the two solstices divide the year in unequal parts. We know that earlier than 200 BCE a system was created in which the Sun moved in two parts of the year with a different constant velocity. Later another system was used in which the velocity of the Sun increased linearly for half a year and afterwards decreased linearly for half a year. 20

Cf. Otto Neugebauer, Some fundamental concepts in astronomy, in Neugebauer (1983), pp. 5-21. The modern signs in unicode are: Ariesà, Taurusá, Gemini â, Cancerã, Leo ä, Virgo å, Libra æ, Scorpio ç, Sagittarius è, Capricorn é, Aquarius ê and Pisces ë. 21

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Fig. 4.12 The eccentric model

A B

E Z G D

It is quite possible that Hipparchus discovered the eccentric model while pondering these problems. The Sun never moves backward with respect to the ecliptic, so this particular aspect of the epicycle model is not needed her. The basic idea of the eccentric model is that if we observe a uniform circular motion from an eccentric point we get a non-uniform motion in which for half a year the speed decreases and then increases for half a year. This is how Ptolemy described the motion. Consider Fig. 4.12. The body travels with uniform motion on the circle ABGD with center E. Let the point Z on the diameter AED represent the observer. Let the angle BEA be equal to the angle GED. Ptolemy writes “the body will traverse the arcs AB and GD in equal times, but will [in doing so] appear to have traversed unequal arcs of a circle drawn on center Z”. In this eccentric motion A is the apogee, by definition the point farthest from the observer, and D is the perigee, by definition the point nearest to the observer. Clearly, to the observer in Z the least speed occurs at the apogee A and the greatest at the perigee D. If in an epicycle model the two angular velocities are equal but have an opposite direction, it is equivalent to an eccentric model. We do not know who discovered this property. It may have been Hipparchus. Consider in Fig. 4.13 the two parallel line segments DE and AB. When DE and AB rotate with equal velocity, say counterclockwise, in the same direction about, respectively, D and A, the distance BE remains equal to AD and ADEB is in every position a parallelogram.22 Moreover, during this motion, the rotational velocity of BE about AB is equal to the velocity of DE and AB about respectively D and A, but has the opposite direction. Clearly one and the same motion of point E can be realized by means of an eccentric motion about D and an epicyclic motion composed of a motion of B about A and a motion of E about B. Let us briefly discuss Hipparchus’ theory of the Sun. Hipparchus’ model describes the motion of the Sun with respect to the Zodiac. See Fig. 4.14. The earth T is in the center of the universe. D is a fixed point at a distance e from the center. The Sun rotates uniformly about D on a circle with radius

22

Excluding the two special positions in which AB and DE coincide with the same straight line.

4.5 Hipparchus’ Theory of the Motion of the Sun (About 150 BCE) Fig. 4.13 The equivalence of eccentric and epicyclic model

79

E β B

β β D A

R. P1 is the position of the Sun at the vernal equinox in the sign Aries. The unknown parameters in the model are e, R and the angle DTP1 = λa . Let us see how Hipparchus determined the values of these parameters. As for e and R he determined the ratio e/R leaving here aside the problem of the distance between the Sun and the earth. A solar year lasted according to Hipparchus 365¼—1/300 days (Almagest III.1). This implies that the mean angular velocity of the Sun ω = 360°/ 365¼ = 59' 8” per day. This is the speed of the rotation of DP about D. Moreover Hipparchus used the lengths of two seasons: he calculated from the data that he possessed that from P1 to P2 it takes 94½ days and from P2 to the autumnal equinox, when the Sun is in the sign Libra, it takes 92½ days (Almagest III.4). Ptolemy relates that Hipparchus found: e/R = 1/24 and λa = 24◦ 30'. Cancer

Fig. 4.14 Hipparchus’ theory of the Sun

P2

Libra

T

e

D

R

Capricorn

P P1

λa Aries

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4 Kinematical Models in Astronomy

One of Hipparchus’ major discoveries is the need to distinguish the Zodiac, the yearly path of the Sun divided in twelve equal signs, from the Ecliptic, the circle on the sphere of the fixed stars with which the Zodiac coincides. In this way the Zodiac is fixed with respect to the seasons. Whenever spring starts the Sun is in the vernal equinox and in the sign Aries. This means that signs of the Zodiac move with respect to the constellations of stars that they originally represented: the solar year moves slowly with respect to the ecliptic. Every hundred years the signs of the Zodiac rotate one degree backwards with respect to the ecliptic. This phenomenon is called the precession.

4.6 Ptolemy’ Contributions Ptolemy followed in the footsteps of Hipparchus and other Greek astronomers who attempted to come up with models that were fully in accordance with the observations. This asked for a lot of ingenuity. By definition the orbit of the Sun is in the plane of the Ecliptic: it only possesses a rotation in longitude. Next to the anomalies in the rotation in longitude the orbits of the Moon and the planets possess deviations in latitude that must be taken into account as well. Ptolemy essentially follows Hipparchus as for the model of the motion of the Sun. Hipparchus had given an epicyclic model for the motion of the Moon. In the model the epicycle takes care of the (first) anomaly in the motion of the Moon: the deviations from the uniform rotation. The angular velocity of the Moon varies from 10° to 14° per day. Ptolemy doesn’t follow Hipparchus because he discovered that the motion of the Moon deviates from the predictions that result from the epicyclic model: there exists a second anomaly. The model he comes up with to take care of this second anomaly is highly original. As for the calculations it is important to realize that the Moon is big and has a considerable daily parallax of the size of almost twice the diameter of the Moon: observations by different observers are different. The most suitable observations to be used as data are observations of lunar eclipses. They take place at Full Moon when the Sun and the Moon are diametrically opposed. One takes the longitude of the Sun at the exact moment of the eclipse from the table of the longitude of the Sun (made on the basis of the theory of the Sun) and one adds 180°. Ptolemy started in Book IV of the Almagest with an epicycle model of which he determined the parameters using data on three eclipses. He possessed a list of observations covering a total of 15 eclipses spread over 900 years. The first one was observed in Babylon in 721 BCE.23 A discussion of the way in which Ptolemy exactly determined the parameters of the epicycle model falls beyond the scope of this book. Once he had determined the parameters, he discovered and this is seen as one of his 23

Petersen (1974), p. 169.

4.6 Ptolemy’ Contributions Fig. 4.15 The epicycle model for the Moon

81

Position quadrature

Moon

μ

Mean Sun

R ΩS

Aries

Position quadrature

Position Full Moon

major contributions, that the predictions deviated from the measurements when the Moon and the Sun are in quadrature with respect to the Earth. This second anomaly was later called the evection. It occurs when the Moon is in its exact first quarter or last quarter phases. In such positions the angle between the Sun and the Moon as seen from the Earth is 90°. In these positions the apparent size of the epicycle is larger. Consider the epicycle model for the Moon in Fig. 4.15. The arrow pointing at the Mean Sun doesn’t point at the actual position of the Sun; it points at the position that the Sun would have if it were executing a uniform rotation about the earth. However in order to couple the motion of other heavenly bodies to the motion of the Sun, the Mean Sun can be used, as we will see. The position of the Mean Sun is determined by means of the angle ΩS, starting from a fixed point on the ecliptic, for example, Aries. The position of the epicycle of the Moon is determined by means of the angle ΩS + μ, also measured from Aries. Obviously ΩS, μ and ΩS + μ are linear functions of time. The epicycle model for the Moon was fine except for the fact that Ptolemy needed a mechanism to pull the Moon somewhat towards the Earth in positions of quadrature. Ptolemy did this by adding to the simple epicycle model what is from a modern point of view an abstract crank and slider mechanism. See Fig. 4.16. The connections A, B and C are hinges. Point A is a fixed and so is the line AC along which C can slide during the rotating input of crank AB. B

Fig. 4.16 Crank and slider mechanism

C

R

Range of C

e A

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Fig. 4.17 Ptolomy’s epicycle model combined with a slider-crank

Moon

C

Position quadrature

R e A

Position Full Moon

Mean Sun

μ -μ B

ΩS

Aries

Position quadrature

The basic idea is that we start from the crank slider motion. A is the position of a terrestrial observer and C is the center of the epicycle of the Moon. A is fixed but AC rotates about A with the corresponding angular velocity. We also maintain the angular velocity of the Moon about the center of the epicycle from the first model. The distance of the epicycle from the earth is regulated by means of the crank AB; the crank moves in such a way that its direction, measured from Aries, is given by the angle ΩS + μ. The result is visible in Fig. 4.17. Suppose AB has a length e and the length of BC equals R-e. The value of μ is a linear function of time. When μ = 0 or 180° we have either Full Moon or New Moon and the center of the epicycle is at a distance R from the earth. The model behaves exactly as the epicycle model at these moments. However, when μ = 90° or μ = 270° the ‘slider’ of the crank-slider mechanism is its other opposite position, we have positions of quadrature and the center of the epicycle is at a distance R-2e from the earth.

4.7 Ptolemy’s Contributions Continued In his theory of the outer planets, Saturn, Mars and Jupiter, Ptolemy in a sense applies the crank-slider mechanism in a different way. See Fig. 4.18. The point D, E and C are hinges. C slides on the line CE. Now the link ED is fixed. A rotation of EC about E (in the case of Ptolemy a uniform rotation) yields a non-uniform rotation of C about D. The mechanism is applied by Ptolemy as follows (see Fig. 4.19).

4.7 Ptolemy’s Contributions Continued

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input

Fig. 4.18 The crank-slider applied differently

E e

C

ED fixed D

R

output

e T

Fig. 4.19 Ptolemy’s model for the outer planets

C P planet R

E e D e T

Mean Sun

T is the earth. Point E is fixed as well. It is the centre of uniform motion, because EC rotates uniformly about E. The position of C on EC is determined by the constant length R of CD. C is in Ptolemy’s model the centre of an epicycle on which the planet P rotates with a constant angular velocity relative to the direction of EC. The retrograde motions of the planets Saturn, Jupiter and Mars always take place when the planets are in opposition to the Sun. We will not discuss Ptolemy’s models for the planets Venus and Mercury. In the models the idea of the slider crank mechanism is used also; a point moves on a circle, but its rotation is uniform with respect to another point which is not the centre of the circle. This particular point has been the source of much debate. Aristotle had defended a very strict position with respect to the uniform circular motions that were allowed in mathematical astronomy: i) They had to be uniform as seen from their own centre, and ii) they had to be concentric with the whole universe.24 Eudoxus’ theory was in accordance with these requirements, but not able to account for the varying distances of the planets. Apollonius and Hipparchus were forced to 24

Cf. Petersen (1974), pp. 34-35.

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B

Fig. 4.20 Slider-crank model

C

R

e A

Range of C

abandon the second requirement. Ptolemy abandoned the second requirement as well. Technically there were no objections; the models that he uses are well defined. Yet later astronomers have criticized him for applying a uniform rotation with respect to a point that is not the centre of the circle: for example, Ibn al-Haitham and Copernicus.

4.8 Astronomy in the Islamic World: The Tusi-Couple Led by the grandson of Genghis Khan, Hulagu, the Mongols destroyed Baghdad in 1258. In Iranian Azerbaijan, this same man Hulagu built an astronomical observatory at Maragha in 1259. Scientific leadership at Maragha was in the hands of Nasir al-Din al-Tusi (1201–1274). The Maragha school is associated with criticism of Ptolemy’s Almagest. In the Maragha school attempts were made to purge Ptolemy’s kinematical models of an alleged flaw. As we have seen in Ptolemy’s models for the Moon, the outer and the inner planets the motions are no longer merely combinations of circular uniform motions. We saw that Ptolemy used for example the slider-crank model of Fig. 4.20 to generate the motion of the Moon. A is the Earth. BA rotates uniformly about A and C slides to and fro on AC. Moreover, the whole slider-crank rotates uniformly about A. The resulting motion of C is the motion of the center of the epicycle of the Moon. Clearly BC does not describe a uniform rotation about a fixed center. This contradicted the conviction, widely held in antiquity and medieval times that any celestial motion must be uniform and circular or a combination of only such motions. The astronomers of the Maghara school succeeded in improving Ptolemy in this respect. Nasir al-Din al-Tusi wrote a text that was widely admired, his Al-Tadhkira. Several commentaries on it were written by later scholars. The second of the four treatises the text consists of, contains the non-Ptolemaic proposals. Chap. 13 of Treatise 2 commences with a simple theorem: If a circle rolls inside the periphery a circle that is twice as big, all points on the rolling circle describe a to and fro motion on a diameter of the second circle. This is in modern kinematics a very well-known motion. It is the so-called elliptic motion, because the points inside and outside the rolling circle describe ellipses. See Chap. 9 of this book. E. S. Kennedy called the device a Tusi-couple.25

25

Kennedy (1966).

4.8 Astronomy in the Islamic World: The Tusi-Couple Fig. 4.21 A Tusi-couple. Point P on the inner circle moves up and down on the vertical diameter of the outer circle

85

P1

P2

P3

P4

The precise way in which the astronomers in Maragha applied the Tusi-couple falls beyond the scope of this book. The Tusi-couple is based exclusively upon uniform circular motion, while it can at the same time generate linear to and fro motions of the kind Ptolemy used slider-crank mechanisms for. The work of the Maragha school represents a very important chapter in the history of astronomy. The astronomers clearly went with great ingenuity beyond Ptolemy’s work. It is in this respect worth mentioning however, that the resemblances between the non-Ptolemaic models of the Islamic astronomers of the Maragha school and Copernicus ‘ later work are considerable. Kennedy wrote: “the conclusion seems inescapable that, somehow or other, Copernicus was strongly influenced by the work of these people.” (Fig. 4.21).26

26

Kennedy (1983), p. 377. See also, for example, Saliba (1979).

Chapter 5

The Birth of Instantaneous Velocity

Abstract The Greeks could only handle accelerated or decelerated motion indirectly. The introduction of a quantitative concept of instantaneous velocity was a major step forward. This required a fresh start. A crucial role was played by the so-called Oxford calculators in the fourteenth century. They did prove the Mertontheorem: In the case of uniform acceleration the motion is, as for the distance covered, equivalent to a uniform motion over the same period with the ‘medial velocity’. This medial velocity is an instantaneous velocity. Soon Casali and Oresme represented the results by means of geometrical graphs and instantaneous velocity had become a quantity.

5.1 Introduction The fall of the Western Roman Empire in 476 CE is usually taken as the beginning of the Middle Ages. The Middle Ages are followed in 1450 by the Renaissance. The Renaissance lasted until 1600. The years 476, 1450 and 1600 are rather arbitrarily chosen. In 476 CE Rome fell to the Germanic king Odoacer. In 1450 the impact of the invention of book printing became clearly tangible and 1600 is a nice round number. Yet the periodization makes sense. The Middle Ages in the West used to be called the Dark Ages because after the fall of Rome a cultural decline set in. Access to the literature of classical antiquity was severely limited and only in the later Middle Ages and the Renaissance, via texts directly from Byzantium or via Arab translations, more sophisticated Greek texts became slowly available. Although it is out of fashion, the term Dark Ages makes some sense in particular with respect to the early Middle Ages. However, the West recovered and, for example, the Catholic Church had a positive development on the cultural development. In the Carolingian Empire (800–888) the clergy were back on track and serious theological research took place. This intensified in the Late Middle Ages when the interest of the clergy also shifted to mathematics. It is in this context that something really new was born: the notion of instantaneous velocity.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Koetsier, A History of Kinematics from Zeno to Einstein, History of Mechanism and Machine Science 46, https://doi.org/10.1007/978-3-031-39872-8_5

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It is said often that in the background of Greek astronomy there is the metaphysical assumption that as a result of their perfect nature the celestial bodies move eternally with uniform circular movement. This assumption is sometimes attributed to Plato (427–347), although Geminus attributed it to the Pythagoreans. Yet it seems worth remarking that the Greeks had no quantitative way to handle non-uniform velocities. They simply had no choice but to try to explain the planetary motion in terms of uniform circular motion. The mathematicians of classical antiquity handled accelerated and decelerated heavenly motion by combining uniform circular motions. It was the only way until in the late Middle Ages scholastic scholars discovered the possibility of quantitatively handling instantaneous velocity.

5.2 Velocity Distributions in Space and Time We saw that in classical antiquity idealized motion played a crucial role in mathematics. Moreover, some general properties of uniform motion were discovered. When in the late Middle Ages in the West medieval philosophers and theologians developed an interest in kinematics, they started from a rather incomplete knowledge of what had been done in classical antiquity. Their work is also less sophisticated. They viewed the theory of proportions “through the distorted lenses of Boethius’ Arithmetica”.1 Below, when we get to Bradwardine, we will briefly discuss the medieval use of this theory. And as for kinematics On the rotating sphere or, in medieval Latin De spera mota (N. B. the medieval’spera’ instead of ‘sphaera’) by Autolycus of Pitane (fl. 320 or 310 BCE), which was translated as early as the twelfth century by Gerard of Cremona, was an important source for the medieval authors. The concept of uniform velocity is, as we have seen, in this work a central concept. The interest in kinematics in medieval Europe started in the thirteenth and fourteenth centuries. Two somewhat similar problems were studied. Problem I: Consider a moving body with a non-uniform velocity distribution. That is, not all points have the same velocity. What is the ’average velocity’? Problem II: Consider a non-uniform motion of a point. Question: Which uniform velocity covers the same distance in the same period of time? In the manuscript On motion by John of Holland (De motu Johannis de Hollandia), possibly written at the beginning of the second half of the fourteenth century, these two problems are clearly distinguished: “Difform motion is of two kinds. Some is difform as to magnitude and some is difform as to time. Motion difform as to magnitude is the motion of a magnitude where some part of the magnitude is moved more slowly than the whole magnitude […] Movement difform as to time is the motion of a mobile body whereby the body traverses more space in one part of the time than in some other equal part of the time.”2

1 2

Mahoney (1978), p. 163. Clagett (1959), p. 247.

5.3 The Average Velocity of a Rotating Radius

89

Remember, initially velocity is in these problems not a magnitude or quantity. The velocity of a uniform motion is measured by the distance that is covered in a fixed period of time. Only in the course of the fourteenth century instantaneous velocity comes to be represented as a quantity. We will briefly discuss the work of Gerard of Brussels, a geometer of the first half of the thirteenth century, who wrote the first treatise on kinematics in the Latin West, On Motion (Liber de motu).3 Euclid and Archimedes are cited and it is definitely in the geometric tradition revived in the last half of the twelfth century. Gerard deals with what we called Problem I: the velocity distribution in space. Problem II is not mentioned. After Gerard’s work we will turn to some of the work of the members of the so-called Merton College group working in Oxford in the first half of the fourteenth century. This work represents a different tradition, started by Thomas Bradwardine, whose primary goal was the application of the theory of proportions to Aristotle’s physics. Quite naturally kinematical problems attracted the interest of the Merton College Group. Its members dealt primarily with Problem II: the velocity distribution in time. Finally we will briefly discuss the work of the fourteenth century Italian Casali and the Frenchman, Nicole Oresme.

5.3 The Average Velocity of a Rotating Radius In, On Motion, Gerard of Brussels deals with Problem I. The text was, according to Clagett, composed between the late twelfth and the mid-thirteenth centuries.4 It consists of thirteen kinematic propositions in three books. Gerard of Brussels studies in particular rotation about a fixed axis of three kinds of objects: line segments (Book 1), two-dimensional figures (Book 2) and solids (Book 3). Gerard considers always a full rotation of an object and what he in fact determines is the average trajectory. Let us first consider some parts of Book 1. It starts with 8 postulates. Postulate 1.1 expresses the kind of situations that Gerard is interested in: “Those which are farther from the center or immobile axis are moved more: those which are less far are moved less”. Postulate 1.2 says: “When a line is moved equally, uniformly, and equidistantly, it is moved equally in all of its parts and points”.5 Gerard seems to mean that when a straight line segment is translated, the motion of each part of the segment represents the entire motion. The Postulates 1.4 through 1.7 are related. Postulate 1.4 says: “Of equal straight lines moved in equal times, that which traverses greater space and to more distant termini, is moved more.” Postulate 1.5 says: “Of equal straight lines moved in equal times, that which traverses less space and to less distant termini, is moved less.” Postulate 1.6 says: “Of equal straight lines moved in equal times, 3

Clagett (1956). Contains an interpretation of the text that Clagett later rejected. See Clagett (1984). Clagett (1959), p. 185. 5 I base my translations on those in Clagett (1984). 4

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Fig. 5.1 The rotating radius

that which does not traverse greater space, nor to more distant termini, is not moved more.” Postulate 1.7 says: “Of equal straight lines moved in equal times, that which does not traverse less space, nor to less distant termini, is not moved less.” These postulates express, I think, that having covered equal areas and equal distances is a sufficient condition for two line segments of equal length to have moved equally. Gerard must have been aware of the fact that the space traversed by a segment and the distance covered by its points, are different things. Indeed, when a straight line segment is translated in a direction not perpendicular to the segment, a huge distance can correspond to a very small area. Finally Postulate 1.8 says: “The proportion of the movements of points is that of the lines described in the same time.” It expresses that (uniform) movements can be compared by looking at the distances described in a fixed period of time. The first theorem of the text is Proposition 1.1: “Any part as large as you wish of a radius describing a circle, which part is not terminated at the center, is moved equally as its middle point. Hence the radius is moved equally as its middle point. From this it is clear that the radii and the speed are in the same proportion.” This proposition deals with a complete rotation of part of a radius of a circle. Gerard’s proof of the first part of the theorem runs as follows. Consider Fig. 5.1 (similar to a figure in Gerard’s text). He considers a radius OF rotating in a plane about a center O. C is a point on OF. Gerard considers a complete rotation. Using proposition 1 of Archimedes’ The Measurement of the Circle (which says that the area of a circle is equal to a rectangular triangle which has sides adjacent to the right angle equal to the radius and the circumference of the circle), Gerard proves that in one complete rotation the subsegment CF (of OF) describes an area between two concentric circles, which is equal to the area of the trapezium SLNQ. Nota bene: SL = CF and SQ and LN have the lengths of the circumferences of the two concentric circles. If, moreover, we choose O on QN such that QO = ON, the area of trapezium SLNQ equals the area of rectangle SLMP. Firstly, Gerard concludes that in the rotation CF describes an area equal to the area described by SL when it describes rectangle SLMP in a motion parallel to itself. In the proof Gerard says: “Therefore let SL be moved through surface SLMP and line CF through the difference between circles OF and OC. I say therefore that the lines SL and CF are equally moved, for they traverse equal spaces and to equal termini, as is clear from the things already said.” As for the equal termini that are allegedly reached (Cf. Postulates 1.4–1.7) one should remember, that the equality of the triangles QPO and NMO, which is used in the proof, also implies that MN = PQ. In the proof Gerard uses this fact and remarks, undoubtedly with respect to the termini, that SP = 1/2(LN + SQ). Because SL moves equally as any of

5.4 The Average Velocity of a Rotating Disc

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Fig. 5.2 The rotating disc

its points (Postulate 1.2), it moves equally as its middle point. However, the middle point of CF describes in the same time a line of the same length as the middle point of SL, so CF and SL reach equal termini.

5.4 The Average Velocity of a Rotating Disc Gerard extends this approach to other situations. The first theorem of Book 2 is Proposition 2.1: “An equinoctial circle is moved in 4/3 ratio to its diameter […]”.In this case Gerard considers the motion a circular disc rotating about a line through its center perpendicular to the plane of the disc. The disc makes a full rotation. However, because during the rotation the disc constantly coincides with itself and does not traverse any space, Gerard must apply a trick. He transforms the moving object and interprets the motion differently. He views the rotating disc as consisting of concentric circles, which he straightens by turning the disc into a triangle (analogous to the transformation in the proof of Proposition 1.1). This triangle is then rotated about a line in such a way that the points describe exactly the same circles that they describe in their original position in the disc. The set of trajectories now has a volume and Gerard uses this fact to find the average trajectory. We will briefly indicate the argument. The rotating disc has the same area as a rectangular triangle BDF with the right angle at D, BD equal to the radius of the disc and DF is equal to its circumference. Consider Fig. 5.2 (similar to Gerard’s figure). We consider a complete rotation about BH. During this rotation each point of the triangle describes a circle equal to the circle described by the corresponding point in the disc during its rotation. C is the midpoint of BD. Let us quote Gerard: “Therefore the movement of the superior triangle is twice that of the inferior triangle. Since, therefore, the movement of the triangles (together) and the movement of the rectangles (together are equal and the movement of the superior rectangle is triple the movement of the inferior rectangle and the movement of the superior triangle is double the motion of the inferior triangle, these movements are related as 9/3 and 8/4. Hence the movement of the superior rectangle is to the movement of the superior triangle as 9–8, i.e. in 9/8 ratio. And the movement of the inferior triangle is to that of the inferior rectangle as 4–3, i.e. in 4/3 ratio.[…] Since therefore the superior rectangle is moved in 3/2 proportion to BD and is moved in 9/ 8 proportion to the superior triangle, the superior triangle is moved in 4/3 proportion to BD.”6 6

Clagett (1959), p. 193.

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The ratios are the ratio’s of the volumes described by the triangles and he rectangles when they are rotated about BH. The superior triangle describes 2/3 of the volume of cylinder, the inferior triangle describes 1/3 of the volume of the cylinder. The superior rectangle describes ¾ of the volume of the cylinder. The inferior rectangle describes 1/4 of the volume of the cylinder. When the volume of the cylinder equals 12 units, the superior triangle, the inferior triangle, the superior rectangle and the inferior rectangle describe volumes of respectively 8, 4, 9, 3 units. Clearly the superior triangle is moved with respect to the superior rectangle in the ratio 8/9. The superior rectangle consists of line segments equal to CD and they all move in an identical way. The movement of the superior rectangle equals he movement of CD is equal to the movement of its midpoint according to Proposition 1.1. Moreover, the movement of the midpoint of CD is 3/2 times the movement of the midpoint of BD. Conclusion: the superior triangle is moved in 4/3 proportion to BD.

5.5 Bradwardine: Towards Instantaneous Velocity Thomas Bradwardine died in 1349, roughly a century after Gerard of Brussels. He was the first member of the so-called Merton College group in Oxford. This group studied problems of mechanics. In 1328 he composed his Treatise on proportions (Tractatus de proportionibus). The book consists of four chapters. In Chap. 3 of the text Bradwardine refers to “the author of On the proportionality of motions and magnitudes (De proportionalitate motuum et magnitudinum)” and he discusses what we called Problem I. The name of the author is not mentioned, but clearly Gerard of Brussels is meant and his Liber de motu. Bradwardine calls Gerard of Brussels “truly much more penetrating than the others”, but yet he did not really appreciate his work. He argued that “The speed of any local motion is to be understood as referring to the greatest linear interval described by any point of the body in motion”.7 So instead of looking at an average velocity, Bradwardine looks at the point with the maximum speed in order to characterize the velocity of a whole body. Referring to Euclid’s Elements, book 12, where we have the theorems that say: “The proportion between (the areas of) any two circles is equal to the square of the proportion between their respective diameters” and “The proportion between any two spheres is equal to the cube of the proportion between their respective diameters”, Bradwardine derives theorems like: “The proportion between any two spherical surfaces, revolving uniformly and in the same time on their unmoving axes, is equal to the square of the proportion between their speeds” and “The proportion between any two spheres, revolving uniformly and in the same time on their unmoving axes, is equal to the cube of the proportion between their speeds”.8 The speeds of the spherical surfaces and the spheres are the greatest lengths described by any point of the body in motion. The reader will be able to check Bradwardine’s theorems. 7 8

Crosby (1955), p. 131. Crosby (1955), p. 133.

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So as for Problem I, Bradwardine rejects the most interesting parts of Gerard’s work and he concentrates on the more trivial aspects. Yet the other chapters of Bradwardine’s book are more interesting. By the year 1250 the works of Aristotle were well understood and diffused in the Latin West. Scholasticism flowered in thirteenth century in Paris and Oxford. Bradwardine’s main purpose in his Tractatus de Proportionibus is to apply the theory of proportions to the relation of force and speed in Aristotle’s physics. This was a highly original move. In his text he first treats the theory of proportions. In Book V of the Elements Euclid writes in Definition 9: “When three magnitudes are proportional, the first is said to have to the third the duplicate ratio of that which it has to the second”. Illustration: when A/B = B/C then A/C = (A/B)2 . And in Definition 10: “When four magnitudes are continuously proportional, the first is said to have to the fourth the triplicate ratio of that which it has to the second, and so on continually, whatever be the proportion”. Illustration: when A/B = B/C = C/D then A/D = (A/B)3 . For example:1/2 = 2/4 = 4/8 implies 1/ 8 = (1/2)3 . On the basis of these definitions tripling or doubling a ratio of quantities means for Bradwardine exponentiation. Then Bradwardine turns to the relation between force and speed. For Aristotle speed is proportional to motive force, and inversely proportional to resistance. Bradwardine has a different approach. In Chap. 3 of his book he gives Theorem 1: “The proportion of the speeds of motions varies in accordance with the proportion of motive to resistive forces, and conversely”.9 Theorem 1 is immediately followed by Theorem 2: “If the proportion of the power of the mover to that of its mobile is that of two to one, double the motive power will move the same mobile exactly twice as fast”. Bradwardine illustrates Theorem 2 as follows: “Let A be a motive power that is twice B (its resistance), and let C be a motive power that is twice A. Then the proportion of C to B is exactly double that of A to B. Therefore, (by the immediately preceding theorem) C will move B exactly twice as fast as A does. This what was to be proved.” This is what Bradwardine’s law means: If the proportion A (motive power)/ B(motive power) = 2/1 brings about a certain speed V and the proportion C(motive power)/A(motive power) = 2/1, then C/A = A/B, so C/B (the double of A/B) = 4/1 and it brings about a speed 2 V. One notices that doubling proportions means exponentiation and doubling velocities means multiplication. And doubling speed does not mean doubling an instantaneous velocity but doubling the distance that is covered in a certain period of time. Bradwardine restricts himself to simple cases involving doubling and halving. Let us, however, look at his law in a modern way. Suppose a force F0 and a resistance R0 make a mobile cover a distance D0 in the unit of time. We change F, such that F/R0 = (F0 /R0 )x . Then the law says that the mobile covers a distance x.D0 . Here F and x are related by log(F/R0 ) = x.log (F0 /R0 ) or x = log(F/R0 )/ log (F0 /R0 ), so the resulting speed measured by the distance x.D0 covered, is a logarithmic function of F.

9

Crosby (1955), p. 113.

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Obviously when Gerard of Brussels wrote his text, he did kinematics and he considered motion independently of its causes. Bradwardine also clearly distinguishes between speed as a kinematic notion and dynamics, dealing with the causes of motion. In Bradwardine’s law the two aspects are related. For both speed is measured by a distance covered in a certain period of time. When Bradwardine is further developing his theory he introduces some distinctions. For our purposes the major distinction is the one between qualitative equality of motions and quantitative equality of motions. Quantitative equality means that equal distances are actually covered in equal times. Qualitative equality means that potentially equal distances are covered in equal times. He writes: “acting powers may be considered as proportionate to the things upon which they act, either qualitatively (i.e. by virtue of their capacity to act), or quantitatively (i.e., with respect to action upon the entire quantity of what undergoes the action). It is from a qualitative proportion that qualitative equality of motions may arise (i.e. equality with respect to fastness and slowness); from a quantitative proportion there correspondingly arises quantitative equality of motions (i.e. equality with respect to the temporal length or brevity of the motion). This is the proper interpretation of the authorities cited.”10 Bradwardine here distinguishes qualitative velocity (fastness and slowness) from quantitative velocity (temporal length or brevity). In fact the concept of instantaneous velocity shows up here, called “qualitative velocity”.

5.6 Dumbleton and the Merton Theorem The word ‘latitude’ comes originally from ‘latus’ referring to the breadth of geometrical figures. Later a derived meaning developed. Latitude could also apply to qualities, for example with Galen where we have the concept of the latitude of human health. The human constitution can range from perfectly healthy via different degrees of sickness to dead at the other extreme. In pharmacology in particular, a clear concept of latitude or the range of degrees of qualities developed. Often numerical values were assigned. Some of the earlier theorists distinguished between degrees, which corresponded to a discrete non-divisible scale, and latitude, on the other hand, which corresponded to a continuum, a divisible scale. For the fourteenth century Oxford Calculators of Merton College latitudes and degrees came to be virtually equated. See Murdoch&Sylla (1978) and Sylla (1973). Next to Thomas Bradwardine, three others at Merton College dealt with kinematical concepts in the period from about 1328 to about 1350: William Heytesbury, Richard Swineshead and John Dumbleton. Bradwardine was the first, but immediately after him in the work of the other three the notion of instantaneous velocity is present. Heytesbury wrote for example: “From this it clearly follows that such a non-uniform or instantaneous velocity (velocitas 10

Crosby (1955), p. 119.

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instantanea) is not measured by the distance traversed, but by the distance which would be traversed by such a point if it were moved uniformly over such or such a period of time at that degree of velocity with which it moves at that assigned instant”.11 Moreover, Heytesbury, Swineshead and Dumbleton all have the solution to Problem II we referred to in the above section on velocity distributions. It is the so-called Merton-theorem: In the case of uniform acceleration over a period of time the motion is as for the distance covered equivalent to a uniform motion over the same period of time with the medial velocity, that is the velocity in the accelerated motion at the middle instant of the period time. We will consider the proof given by Dumbleton in his Summa of logical and natural things (Summa Johannis Dumbletonis de logicis et naturalibus). Dumbleton talks about degrees of velocity. We will concentrate on the essential element of the proof and use modern notation. We have a motion that is such that over a period of time [0,t] the degree of velocity increases uniformly from 0 at time 0 to a latitude of velocity V at time t. The proof is by contradiction. We want to prove that MV, the medial velocity, equals V/2. Suppose that the medial velocity MV = V/2 + Δ. We now look at the intervals [0,t/2] and [t/2,t]. Let MV half be the medial velocity corresponding to [0,t/2]. Dumbleton then applies the following trick: The motion in the second half of the period of time [t/2,t] is a combination of two motions: a uniform motion with velocity V/2 and an accelerated motion with medial velocity MV half ! Because the total distance covered by a uniform motion with velocity V/ 2 over this second half of the period of time can be replaced by a uniform motion over the entire interval with velocity V/4 and the motion on the first half has medial velocity MV half , we can replace the entire motion by a uniform motion with velocity V/4 + MV half . Then MV = V/2 + Δ = V/4 + MV half , so MV half = V/4 + Δ. This means that if the medial velocity over the entire time interval is Δ bigger than the degree of velocity in the middle, the same holds for the medial velocity over the first half of the time interval and the same holds for the first quarter of the time interval, etc. This leads to a time interval [0, (t/2)n ] for which for n large enough the mean velocity is larger than the velocity at the end of the time interval. Contradiction!

5.7 Giovanni Casali and Nicole Oresme Before 1350 the ideas of the Merton College Group reached Italy and Paris. In 1346 Giovanni Casali (c.1320 after 1374) writes On the velocity of the motion of change (De velocitas motus alterationis). Probably a few years later Nicole Oresme (ca.1325– 1382) writes Configurations of Qualities (De configurationibus qualitatum). Let us quote Casali: “The third conclusion is that any uniformly difform latitude is precisely as much as a latitude uniformly intense at the mean degree. Or, anything at all uniformly nonuniform in hotness is precisely as hot as something uniformly hot at the mean degree of the uniformly nonuniform hotness […]. One can exemplify these 11

Grant (1974), p. 238.

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5 The Birth of Instantaneous Velocity

Fig. 5.3 Oresme’s proof of the Merton theorem. Source: Clagett (1959), Plate 7

qualitative things. For something uniformly hot is throughout like a rectangular parallelogram constructed between two parallel lines. Then any part of such a rectangle is equally wide as any other part, because the latitude of any such part is measured by the base. Similarly a uniformly difform hotness is in every way like a right triangle. This would be a uniformly difform hotness terminated in one extreme at zero. This holds because one quarter of such a triangle has a line which is just as distant from one extreme of the quarter as from the other and it is just as distant from either extreme as the middle line of another small quarter is distant from its extremes. [..] Therefore, because any triangle having two equal sides is equal to some rectangle, it is evident that the latitude of the rectangle will be uniform and measurable by a line passing through the midpoint of the same triangle”.12 Clearly we have here the Merton theorem. However, the proof is new. It is based upon the idea to represent the degrees of velocity (i. e. the instantaneous velocities) by means of line segments perpendicular to a line segment representing a time interval. Oresme develops this idea much more extensively than does Casali. Moreover, Oresme seems to have understood that the areas of the figures represent the distances traversed (Fig. 5.3).13

5.8 Acceleration: Euler and Newton’s Second Law14 The Merton theorem returns in Galilei’s work. In the Dialogue Concerning the Two Chief World Systems (Dialogo sopra i due massimi sistemi del mondo) of 1632 and in the Discourses and Mathematical Demonstrations Relating to Two New Sciences (Discorsi e dimostrazioni matematiche intorno a due nuove scienze) of 1638 Galilei applies the Merton theorem to understand the motion of a falling object. Yet, there is a big difference between the thirteenth and fourteenth century work of the Merton 12

Clagett (1959), pp. 383-384. Clagett (1959]) p. 343. 14 This section owns a lot to Koetsier (2007). 13

5.8 Acceleration: Euler and Newton’s Second Law

97

College group and Casali and Oresme, on the one hand, and Galilei’s work, on the other hand. The medieval schoolmen were engaged in completely theoretical considerations, while Galilei was executing experiments as well. With Galilei the Middle Ages were definitely over. In his Discorsi Galileo Galilei did show how the distance covered by a body dropped from a rest position is related to the course of time: it is proportional to the square of the time passed. Proposition 2 of the discussions on Day Three in the Discorsi says: AD 2 HL = . HM AE 2 Nota bene: Also Galilei is still working within the framework of Eudoxus’ theory of proportions. The body is dropped in point H. HL and HM are the distances covered in, respectively, the periods AD and AE. This still concerns a linear motion, but on Day Four of the Discorsi Galilei discusses the trajectory of a bullet with a horizontal initial velocity, by combining two linear motions. Galilei seems to have been one of the first who applied the principle of superposition to earthly phenomena. Clearly the Greek astronomers had systematically applied this principle while studying the motion of the heavenly bodies. The opposition between natural and artificial movement in Aristotelian physics on earth, however, seems incompatible with the idea that the uniform horizontal inertial movement, which is artificial, and the vertical accelerated movement, which is natural, both contribute independently to the final parabolic movement of the bullet. Galilei had brushed Aristotelian physics aside and, on the one hand, based his model for the trajectory of a bullet on the superposition of these two independent movements. In 1644 Evangelista Torricelli (1608–1647) published On the motion of naturally descending objects and projectiles, book 2 (De motu gravium naturaliter descendentium et projectorum, libro duo). It contains the treatment of the trajectory of a bullet shot away under an angle in vacuum. The resulting velocity is directed along the tangent to the curve. He showed also that the maximum range corresponds to an angle of projection of 45 degrees. He established for the first time the idea of an envelope. Projectiles fired out at the same speed at all different angles in a fixed vertical plane describe parabolas which are all tangent to a common parabola, the parabola of safety (parabola di sicurezza). When we fire at the same speed in all directions the parabolic trajectories are all tangent to a common paraboloid that one obtains by rotating one of the parabola’s of safety about the vertical axis.15 Also in other ways Torricelli continued where Galilei had stopped. Galilei still silently distinguished between, on the one hand, the finite cosmos, and, on the other hand, the infinite space of Euclidean geometry. Torricelli identified Euclidean 3dimensional space with physical space.

15

Robinson (1994).

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5 The Birth of Instantaneous Velocity

The notion of instantaneous velocity as a directed quantity was readily accepted in the seventeenth century. It took considerably longer before acceleration as a kinematical quantity acquired a status comparable to the status of velocity. Consider for example a paper by Varignon from 1700 in which he applied the differential calculus to linear motion. He distinguished force, velocity, distance and time, denoted by y, v, x and t, respectively, but he does not mention acceleration [Varignon,1700]. Why? Varignon gives two general rules for linear motion. Rule 1 : v =

dx dv and Rule 2 : y = dt dt

Because Varignon equates what we nowadays call acceleration with force, he does not need a separate notion of acceleration. The same happens in D’Alembert’s Treatise on dynamics (Traité de dynamique) of 1743. The ‘accelerating force’ ϕ is defined as follows: ϕdt = du, where u and t are velocity and time respectively. And also when Euler in 1736 published his two volume Mechanics or the science of motion exposed in an analytical way (Mechanica sive motus scientia analytice exposita) on the motion of a moving mass point, he expressed a similar view: a force can be characterized by the change that it brings about in the motion of a point. Euler and D’Alembert did not agree about the interpretation of the equation. For D’Alembert the notion of force was a derived notion, while it was for Euler a primary notion and consequently ϕdt = du was for Euler a law and not merely definition of the notion of force. See Le Ru (1994). If the motion of only one mass point is considered, we do not need next to the notion of force a separate notion of mass. We can apply Varignon’s formula. However, as soon as we consider several masses simultaneously, we must be able to distinguish them and we can no longer “hide” them in a notion of force. So instantaneous acceleration only became a notion that was clearly separated from the notion of force when rigid body dynamics was being developed. Euler played a crucial role in this development. The early forms of Newton’s law can only be understood if one realizes how the early physicists were measuring the different quantities that occur in their equations. Time was measured by pendulums, instantaneous velocity was measured by a length of fall. Forces, masses and weights were all measured by weights. Accelerations could not be measured directly and, consequently, “an examination of the literature shows a marked reluctance to speak of accelerations more than necessary.” See Ravetz (1961). We will briefly consider two versions of Newton’s second law in Euler’s papers. Euler knew very well that the precise form of Newton’s second law depends on the units of measurement. In his Guide to natural science (Anleitung zur Naturlehre) Euler expressed Newton’s second law is as follows16 :

16

E842: Leonhard Euler, Anleitung zur Naturlehre, Originally published in Opera Postuma 2, 1862, pp. 449-560 (also in Opera Omnia III, Vol. 1, pp. 16-180).

5.8 Acceleration: Euler and Newton’s Second Law

dv = n.

99

p.dt . M

dv is the increment of velocity, p is the force, t is time and M is the mass of the object. Euler pointed out that we are free in the way we measure v, p, t and M. However, the way we measure these quantities determines the value of n. Once we made up our mind in a specific case how we want to measure v, p, t and M, we must determine n. Studies on the movement of celestial bodies in general (Recherches sur le mouvement des corps célestes en général)17 is the first paper in which Euler uses rectangular Cartesian coordinates and decomposes Newton’s second law with respect to the three axes. He considers the motion of a mass M. The ‘instantaneous change’ of the motion of the body is then expressed with respect to each of the coordinates by means of the equation: 2d x x X = 2 dt M X is the ‘accelerating force’. Euler points out that the X is the ‘moving force’, and M dx 2 square of the velocity ( dt ) expresses the height corresponding to this velocity and that is why, he writes, the factor 2 occurs in the formula. It was common to measure instantaneous velocity by means of a length of fall. In the case of an object falling along the x-axes X = M, so Euler’s formula yields 2ddx = dt2 and this yields dx/dt = t/2 and x = t2 /4. Clearly the height is the square of the velocity at the end of the fall. This version of Newton’s second law occurs frequently in Euler’s work. Although, for example, Lagrange18 in his Analytical mechanics (Mécanique analytique) of 1788 applied Newton’s second law in the form F = m. ddtx 2x , only in the first half of the nineteenth century acceleration defined in a purely kinematic way became a separate object of investigation.

17

E112: Originally published in Mémoires de l’académie des sciences de Berlin 3, 1749, pp. 93–143 (also in Opera Omnia II, Vol. 25, pp. 1–44). 18 Lagrange (1788), p. 232.

Chapter 6

The Parallelogram of Instantaneous Velocities

Abstract The kinematical generation of a curve is based on the definition of the motion of a point that describes the curve. During the movement the point at each moment has a directed instantaneous velocity. The seventeenth century geometers realized that the line drawn in the direction of the instantaneous velocity is the tangent to the curve in the position of the point at that instant. Both Evangelista Torricelli and Gilles Personne de Roberval came up with the idea to use the parallelogram of instantaneous velocities to determine tangents. The young Isaac Newton had the same idea. Yet, even in the eighteenth century the parallelogram of instantaneous velocities was still not seen as conceptually unproblematic. A proof given by D’Alembert illustrates this.

6.1 Introduction The parallelogram rule for the addition of instantaneous velocities is well-known, but it was not easily found. The notion of instantaneous velocity in itself was not trivial and the problem of addition of velocities represented an extra difficulty. In 2022 Marius Stan wrote: “Adding velocities (by the Parallelogram Rule) was the gateway insight of early-modern science, on a par with the Law of Inertia but far more useful than it. And yet, crucial and indispensable as it was, velocity addition was a foundational enigma for early-modern theorists. There were two sources of difficulty, and they required much skill and insight to navigate safely past them. Briefly, the difficulties were: • Velocities have directions too, not just sizes. And, direction makes a difference to the size of the sum, or resultant, of their addition. • But the algebraic framework of classical mathematics had no way to determine the result of adding velocities. It lacked rules for adding directions.” See Stan (2022). Stan is right. As we will see in Chap. 12 of this book only in the nineteenth century velocities and accelerations were actually treated as vectors.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Koetsier, A History of Kinematics from Zeno to Einstein, History of Mechanism and Machine Science 46, https://doi.org/10.1007/978-3-031-39872-8_6

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102 Fig. 6.1 Composition of uniform movements

6 The Parallelogram of Instantaneous Velocities

A F

E

B

G C

D

6.2 Gilles Personne de Roberval: The Tangent as the Line of Instantaneous Advance For the Greeks the tangent to a circle was a line touching the curve but once. In the seventeenth century a new way to view and determine tangents was born. The kinematical generation of a curve is based on the fact that we have a moving point that describes the curve. During the movement the point at each moment has a directed instantaneous velocity. The seventeenth century geometers realized that at each instant the line through the point drawn in the direction of the instantaneous velocity is the tangent to the curve in the position of the point at that instant. Because in general seventeenth century mathematicians looked at curves in terms of their kinematical generation, it is not surprising that they used the insight to determine tangents. In this context both Evangelista Torricelli and Gilles Personne de Roberval came up with the idea of the parallelogram of instantaneous velocities to determine the directed instantaneous velocity. Gilles Personne was born in Roberval, a village in North-Western France, in a family of peasants. Thanks to a local priest, who noticed the boy’s intelligence, he got an education. Roberval went to Paris and lectured there first in the 1630 s at the Collège de maître Gervais and afterwards for 41 years until his death in 1675 at the Collège Royal de France where he held the chair of Ramus.1 One of his pupils, a gentleman from the area of Bourdelois, François de Verdus, wrote down some of Roberval’s lectures. The title of the text was Observations on the composition of movements and on the means to find the tangents to curved lines (Observations sur la composition des mouvemens et sur le moyen de trouver les touchantes des lignes courbes). A few years before his death Roberval presented the text to the Académie with a few remarks added. Much later the text was printed in the Mémoires de l’Académie Royale, Roberval (1699). Roberval starts with some general considerations on movement. He shows that the direct composition of two uniform rectilinear movements is a uniform rectilinear movement. See Fig. 6.1. AB moves parallel to itself with a constant velocity to position CD. At the same time AC moves parallel to itself with a constant velocity to position BD. The intersection of the two segments moves with a constant velocity from A to D. This was not new; as we have seen, the Greek mathematicians knew it. New is that Roberval assumes that this result for the composition of uniform rectilinear motions can be 1

Auger (1692), pp. 9–10.

6.2 Gilles Personne de Roberval: The Tangent as the Line of Instantaneous …

D

Fig. 6.2 The tangent to the cycloid

E

A

103

H

F C

applied also to instantaneous velocities. Roberval: “By the specific properties, (which will be given to you) examine the various movements which the point has at the place where you want to draw the tangent; compose all these movements into one and draw the direction line of the compound movement, you will have the tangent.” See (Roberval, 1699). By constructing the diagonal in a parallelogram of which the sides are instantaneous velocities he determines the tangents to conic sections, the conchoids, Archimedes’ spiral, the cissoid, the cycloid, the quadratrix and to the third degree curve called Descartes’ parabola. All of Roberval’s results are correct, although his reasoning is sometimes somewhat dubious. Consider first an example which is entirely correct: the way he determines the tangent to the cycloid. The cycloid may have been noticed by Cusanus. Charles de Bovelles (1479–1553) referred to it at the beginning of the sixteenth century and so did Galilei at the end of the century. Yet they did not make headway in determining its properties.2 See Fig. 6.2 for Roberval’s construction of the tangent to the cycloid. The instantaneous movement of a point on the circumference of a rolling circle is decomposed into two independent movements: a linear horizontal movement of the center of the circle and a rotation about the center. The circle with radius R rolls without slipping on the horizontal line. Suppose in time T it rotates about an angle 2π. One full rotation corresponds to a horizontal movement of the center equal to 2πR. This implies that the instantaneous rotational velocity of point E on the circumference equals the translational instantaneous velocity of the center. Roberval draws EF horizontal through E and ED equal to EF along the tangent to the circle in E. Application of the parallelogram of instantaneous velocities yields the direction of the tangent EH. Let us consider another example. If a fixed straight line BH and a fixed point A are given, the locus of all points that have an equal distance to the line BH and the point A are on a parabola. In Fig. 6.3 point E is on the parabola because EA equals EH. Roberval determines the tangent as follows. Imagine E moves in the position it has in Fig. 6.3 along the parabola upward. Point E moves towards A with a certain velocity. This velocity is equal to the velocity with which E moves towards the line BH. These velocities can be represented by EA and EH respectively. According to 2

Boyer (1945), pp. 300–301.

104 Fig. 6.3 The tangent to the parabola

6 The Parallelogram of Instantaneous Velocities

C

B

H

A

E

Roberval the direction of the tangent is the direction of the diagonal of parallelogram AEHC. Roberval’s conclusion is correct, but his argument is unsatisfactory. Roberval behaves as if in general when we have the instantaneous velocities in two directions of a point on a kinematically generated curve the parallelogram gives the total instantaneous velocity and then also of course the direction of the tangent. Yet, the sum of the projections of an instantaneous velocity on two arbitrary directions in general does not yield the original instantaneous velocity.

6.3 Isaac Newton on Tangents In 1661 Isaac Newton (1642–1727) entered Trinity College in Cambridge as an undergraduate. It is remarkable that he spent these years in considerable isolation. Although he attended the lectures of Isaac Barrow, at the time one of Britain’s leading mathematicians, he largely taught himself mathematics through extensive reading. In 1665–1666, when the university was closed because of the plague, Newton returned to the family farm in Woolsthorpe, a village near Grantham. There, still only a student at Cambridge, he developed his version of the calculus. He also discovered the rule that in uniform circular motion the acceleration towards the center is proportional to v2 /r. Very probably unaware of Roberval’s work Isaac Newton wrote in October 1665 a text called “How to draw tangents to Mechanicall lines”.3 So far he had dealt with algebraic curves. He now turned his attention to the curves that were at the time called ‘mechanical’ until Leibniz coined the word ‘transcendental’ for them. 3

Whiteside (1967), p. 369.

6.3 Isaac Newton on Tangents

105

The principle that is applied by Newton is the following: “In the description of any Mechanicall line whatever, there may be found two such motions which compound or make up the motion of the point describing it, and by those motions may the motion of the point be found whose ‘determination’ is a tangent to the crooked line.” The ‘determination’ of the motion is the instantaneous direction of the tangent. In the example of the spiral (See Fig. 6.4) a half line is rotated clockwise from an initial position CB with constant velocity about C and at the same time on the line a point moves away from C with constant velocity. Newton constructed the tangent in point A of the spiral as follows. He drew the line CA from the center C to A and the tangent to the circle through A with center C. The points F and E on these two lines are such that AF : AE = CA : Arc BMA. The ratio of CA and Arc BMA is equal to the ratio of the two instantaneous velocities and Newton’s argument is correct. Yet, AF can be taken arbitrarily, but an actual construction of AE, of course, requires a rectification of Arc BMA. Subsequently Newton applied the principle to the quadratrix and the ellipse. It is remarkable that in this case both constructions are flawed, although in the case of the ellipse the method gives a correct answer. See Fig. 6.5. As for the quadratrix BAQ he drew AF perpendicular to CG, and AE on the tangent to the circle HAP such that AF : AE = HC : arc HAP = BC : arc BNR = CQ : BC. The last equality is a well-known theorem that was as we have seen in Chap. 3 of Fig. 6.4 The tangent to the spiral

F D

A E

C

M

B

106

6 The Parallelogram of Instantaneous Velocities

E D B

A F

C

Fig. 6.5 Tangents to quadratrix and ellipse

this book already known in antiquity. Newton assumed that then the diagonal in the parallelogram AFDE would be the tangent to the quadratrix. As for an ellipse with foci B and C, when point A moves on the ellipse, clearly AB and AC get shorter or longer with the same speed. Newton took AE equal to AF assuming that then the diagonal of the parallelogram AFDE would be the tangent to the ellipse. Yet it did not take long before Newton realized his errors. In a note dated November 8, 1665, Newton started all over again.4 In the case of the quadratrix the ratio of the lengths of AF and AE is the ratio of the perpendicular projections of the instantaneous velocity on a vertical line and on the tangent to arc HAP. Newton’s mistake was that he briefly assumed that the sum of the projections of an instantaneous velocity on two arbitrary directions in general yields the original instantaneous velocity. If the projection of the instantaneous velocity of a point on a line is known, the endpoint of that instantaneous velocity lies on the line perpendicular to that line through the endpoint of the projection. The line through E and F perpendicular to respectively AE and AF intersect in the endpoint of the instantaneous velocity of the point A describing the quadratrix. Yet, also here AF can be taken arbitrarily, but an actual construction of AE requires a rectification of Arc BNR. In the same way the tangent to the ellipse can be constructed. See Fig. 6.5. Draw in E and B lines perpendicular to, respectively, AE and AB. The point of intersection is on the tangent. Half a year later, in May 1666 N explicitly wrote down the parallelogram law. His reasoning seems to have been the following.5 On the Third Day of the Discorsi Galilei considers the motion of a falling body along planes of different inclination. See Fig. 6.6i. It turns out that if the body falls vertically along the diameter ac of a circle the body would be moved in equal times along the cords ad and ae of the 4 5

Whiteside (1967), p. 377. Whiteside (1967), p. 390, footnote 5.

6.3 Isaac Newton on Tangents

107

Fig. 6.6 Newton proves the parallelogram law

circle. This means that the average velocities along the cords are proportional to the lengths of the cords. This means that the average velocity along the inclined planes can be obtained by projecting the diameter on them. For Newton this is a theorem on instantaneous velocities. Newton generalizes the result: not only in the case of a falling mass point but in general whenever we have a point moving along the diameter ac of a circle, the instantaneous velocities corresponding to the motion along the diameter ac and cords ad and ae are proportional to the lengths of the cords. See Fig. 6.6i. Newton clearly realizes that the projection of the instantaneous velocity of a point on a line gives the instantaneous velocity in the direction of that particular line. It gives him immediately a proof of the parallelogram law. See Fig. 6.6ii. He assumes that the triangles aec and cda are similar. When three bodies move uniformly from a, the first one to d, the second one to e, and the third one to c, then the motion of the third body is compounded of the other two. Newton’s proof is this: he projects the motions on ad and ae on the line ac and shows that their sum is equal to ac. In the same note Newton argues that in plane motion at a particular instant there are only three possibilities: (i) All points in the plane have the same instantaneous velocity, (ii) All points in the plane rotate about some axis, and (iii) The motion is a mix of instantaneous translation and instantaneous rotation. In case iii. Any axis may be taken as the axis of rotation. It is remarkable that Newton seems not to have realized that the third possibility can always be reduced to the second. In the case of a mix of a translation and a rotation, there is always a point where the translational velocity and the rotational velocity cancel each other and the mix turns out to be a rotation about this point.

108

6 The Parallelogram of Instantaneous Velocities

Fig. 6.7 D’Alembert on the parallelogram of velocities

6.4 D’Alembert on the Parallelogram of Instantaneous Velocities Although we will return to the seventeenth century in the next chapter, we will jump forward for a moment and look at two 18th century treatments of the parallelogram of velocities, a proof by Jean le Rond d’Alembert (1717–1783) and a discussion by the philosopher Immanuel Kant. In his Traité de dynamique of 1843 D’Alembert defines  the velocity of a point by means of the differential calculus. He writes u = de dt when u is the velocity and e and t are respectively the distance and the time.6 Eight pages hereafter he discusses the parallelogram law. He writes that, given a quadrilateral ABDC, one usually did prove the parallelogram law by considering a body A moving from A to B while at the same time AB moves in the direction of AC. Then it is easy to see that when the two movements are both uniform A follows the diagonal AD of the quadrilateral ABDC. However, that is not what D’Alembert wants to prove. At the end he will use this result, but he first wants to know what the body A does when it at the beginning has both the tendency to move to B and to C and these tendencies are composed. From his point of view that is an entirely different situation. The tendencies are brought about by two different powers. He wants to know the direction and the velocity of the body after the two powers are combined. That is what he means, referring to Fig. 6.7, when he writes “Theorem: If two arbitrary powers act at the same time on a body or point A to move it, one from A to B uniformly in a certain time, the other from A to C uniformly during the same time, and if one constructs the parallelogram ABCD; I say that the body A will move along the diagonal AD uniformly in the same time it would have moved along AB or AC.”7 D’Alembert is not combining two uniform velocities, he is in fact combining instantaneous velocities. D’Alembert’s proof is interesting. The two powers bring about uniform motions along the sides AB and AC of the parallelogram ABDC. The two powers combined 6 7

D’Alembert (1743), p. 14. D’Alembert (1743), p. 22.

6.5 A Philosophical Aside and Kant on the Parallelogram of Velocities

109

result in one composed power, so A moves in a straight line. D’Alembert refers here to Article 6 of Chap. 1 in his book, which is the law of inertia. He assumes that the point reaches a point g in the period of time we are talking about. D’Alembert now introduces the parallelogram goac, which is equal to ACDB, and he argues that if at g the moving point A would be confronted by two powers that bring about uniform motions defined by the sides go and gc of the parallelogram goac, it would stop moving. That is because equal velocities with opposite directions cancel each other. D’Alembert must prove that g coincides with D. He does it as follows, referring to Fig. 6.7. D’Alembert imagines that the body A, that describes the line Ag is on a plane KLMH that is subjected to two uniform motions. On the one hand, it can glide along KL and IM, parallel to AC. As this plane glides all points g describe lines gc equal and parallel to CA, at the same time that the point A describes the line CA. On the other hand, in the same time KL and IM move taking along with them the plane in a manner parallel to AB but in the opposite sense, with speed equal to the one which the body A would have along AB. It is clear that all the points g on the plane would uniformly describe lines ga equal and parallel to the diagonal AD of the parallelogram ACDB. It is also evident that the point A, pushed continuously in this state by four powers that are equal and opposite when taken in pairs, must remain at rest in absolute space. It thus follows that when the body has arrived in g on the plane, this point must be at the place it had when it began to move. Finally, d’Alembert concludes that this cannot happen unless the line Ag lies upon the diagonal AD and thus if gc = AC and go = AB it will be seen that the point g must lay upon the point D as ga = AD and that the point a must coincide with A. Briefly this seems like a complicated way of saying that two powers capable of producing uniform motion combined have the same effect as the two powers working simultaneously without combining them.

6.5 A Philosophical Aside and Kant on the Parallelogram of Velocities In 1786 the German philosopher Immanuel Kant (1724–1804) also wrote about the parallelogram of velocities. Yet his interest was different from the interest of the working mathematicians. In 1781 he had published his Critique of Pure Reason (Kritik der reinen Vernunft) in which he investigated the possibility of a priori knowledge, that is knowledge of which the justification does not depend on experience. Mathematical knowledge is from his point of view a priori, like logical knowledge. Yet mathematical truths differ from logical truths in the sense that they are not analytical but synthetic. Analytic means being derived purely from concepts. Mathematical truths are synthetic in the sense that they essentially depend on mental constructions. Mathematical concepts are constructed in the pure intuition of space and time. At the same time space and time are a priori forms of intuition that structure all experience.

110

6 The Parallelogram of Instantaneous Velocities

Bertrand Russell illustrated Kant’s ideas with the metaphor of the spectacles. If one wears red spectacles everything is seen red. In Kant’s philosophy human beings as it were wear space and time spectacles in their mind and they see the outside world always in space and time and the inner world in time. When we do mathematics we quasi study our space and time spectacles without input. In our mind’s eye we construct geometrical figures or numbers and we see from the construction what their properties are. Kinematics deals with space and time and nothing else and it is understandable that Kant when dealing with the status of mechanics would write about it. He even coined his own name for the subject. What we call kinematics Kant called phoronomy. In Chap. 15 of this book we will see that a century later the German engineer Franz Reuleaux also used the word phoronomy for kinematics. The result of Kant’s considerations can be found in his Metaphysical Foundations of Natural Science (Metaphysische Anfangsgründe der Naturwissenschaft) of 1786.8 The first of the four chapters of the book is devoted to phoronomy. The second chapter is about dynamics. For Kant phoronomy deals with “motion considered as pure quantum—portions of which can be combined in various ways—with no attention being paid to any quality of the matter that moves”. The word ‘phoronomy’ comes from the Greek ϕoρα´ (motion) and ν´oμoς (law). The term was apparently used earlier by Leibniz in a paper called The theory of abstract motion of 1671.9 Also Hermann used it in 1716 in the title of his book Phoronomy, or on the forces and motions of solid bodies and fluids.10 Yet unlike the others Kant used it where we would use the word kinematics. Phoronomical truths are synthetic a priori, like other mathematical truths. Of course Kant is not interested in developing phoronomy. He feels that the crucial idea he needs to understand the applicability of mathematics to experience is composite motion. To get an idea of Kant’s approach let us quote the 5 definitions Kant starts with: Definition 1. I call something ‘material’ if and only if it is movable in space. Any space that is movable is what we call ‘material’ or ‘relative’ space. What we think of as the space in which all motion occurs—space that is therefore absolutely immovable—is called ‘pure’ space or ‘absolute’ space. Kant added:” The whole topic of phoronomy is motion; so the only property that is here attributed to the subject of motion, i.e. matter, is its movability.” Kant is clearly not at all thinking of a physical substance. Only in the chapter on dynamics Kant defines ‘matter’ as something that is movable and fills a space. Definition 2. The motion of a thing is the change of its external relations to a given space. One of the remarks he made after this definition is “In any motion we have just two factors to think about—speed and direction—once we have set aside all the other properties of the moving thing.” 8

Kant (1786). The translation is by Jonathan Bennett. See Palter (1972), p. 111, footnote 4. 10 See Hermann (1716). 9

6.5 A Philosophical Aside and Kant on the Parallelogram of Velocities

111

Definition 3. Rest is time-taking presence in the same place; for something to be time-taking is for it to exist throughout a time. Definition 4. To construct the concept of a composite motion means to present a priori in intuition a motion as the result of two or more given motions united in one movable thing. Here Kant added a Principle:”Every motion that could be an object of experience can be viewed either as the motion of a body in a space that is at rest or as the rest of a body in a space that is moving in the opposite direction with equal speed. It’s a free choice.” Definition 5. The composition of motion is the representation of the motion of a point as identical with two or more motions of the point combined. After these definitions Kant is ready to prove the one and only proposition in the first chapter. Proposition: The only way to think of two motions as composing the motion of a single point is by representing one of the two as occurring in absolute space, and the other as consisting in the movement of a relative space in the opposite direction. Kant’s proof consists of the consideration of three cases. First case: A single point undergoes two motions in the same direction along the same line at one time. Second case: Two motions in exactly opposite directions are to be combined at one and the same point. Third case: Two motions of a single point go in different directions—not opposite directions but different ones that enclose an angle. Clearly Kant’s point is in each of the three cases that it is impossible that one single point has at the same time two different motions. It would violate the principle of contradiction which says that it is impossible to deny and at the same time affirm the same predicate of the same subject. The only way to understand a point undergoing two motions is by introducing a relative space. For example, one motion of a point can take place in absolute space. If we have two motions we add a relative space that moves with respect to the absolute space and at the same time a point moves with respect to the relative space. In this way the point gets two motions: one with respect to the relative space and one via de relative space with respect to the absolute space. We then look at the composed motion from the point of view of the absolute space (Fig. 6.8). Let us consider the third case. Kant writes: “On the hand, that our moving point x undergoes motion AC in absolute space while—instead of x’s undergoing motion AB—some relative space that x is in undergoes motion BA. Then while x moves AE in absolute space, the relative space moves Ee, that is, moves to the left, so that x’s position in the relative space is m. And the same story holds for x’s absolute move AF while the relative space moves Ff ; and for x’s entire absolute move AC while the relative space moves Cc. From the standpoint of the relative space, therefore, x moves smoothly down the diagonal, through m and n·and of course all the intermediate positions to D, which is exactly the same result as if it had undergone movements

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6 The Parallelogram of Instantaneous Velocities

Fig. 6.8 Kant’s proof of the a priori possibility of the composition of motions

AB and AC. So we get the result we want without having to postulate two motions that affect one another.” Kant looks at the composed motion from the point of view of the relative space. At first sight Kant’s argument does not concern instantaneous velocities. It seems to concern only the composition of two uniform velocities. Yet, consider the following remark that Kant made after he discussed the three cases of the composition of velocities: “But if a ‘double speed’ is explained as ‘a motion whereby a doubly great space is traversed in the same time’, then something is being assumed that shouldn’t be taken for granted, namely that two equal speeds can be combined in the same way as two equal spaces. It isn’t obvious that a given speed consists of smaller speeds—that a speed is made up of slownesses!—in the way that a space consists of smaller spaces. The parts of the speed aren’t external to one another, as the parts of the space are; and if a speed is to be considered as an amount, then the concept of its amount (‘How fast?’) can’t be constructed in the same way as the concept of the size of a space (‘How big?’), because the former is intensive and the latter extensive.” (italics mine—TK). So Kant viewed velocity as intensive and that suggests that he saw velocities as instantaneous. Some Kant experts feel that “Kant thought of speed not as explained as the ratio of extended intervals of space and time, but as fundamentally instantaneous. And this in turn suggests that an aim of the Phoronomy is to directly construct instantaneous velocities, thereby legitimating their mathematical treatment.”11 Kant then allegedly thought of instantaneous motion in terms of ever-diminishing intervals . In this way Kant’s proof yields a of space and time.12 In other words: v = lim s t t→0

proof of the parallelogram of instantaneous velocities. Possibly Kant felt no need to point this out because in his argument the emphasis is not on the mathematical proof but on understanding how a point can have two motions at the same time.

11 12

Sutherland (2014), p. 207. Sutherland (2014), p. 688 and p. 714.

Chapter 7

Napier, Fermat, Descartes

Abstract In seventeenth century mathematics motion is everywhere. Napier’s definition of the logarithm is based on an exponentially decelerating motion. When Fermat and Descartes discovered analytical geometry motion played a crucial constitutive role.

7.1 Introduction In the first chapters of this book we discussed motion in Greek mathematics. One of our conclusions was that in classical antiquity in geometry motion played a major role; without idealized motion Greek geometry as we know it would not have been possible. Against this background it is interesting to consider the role of motion in geometry when geometric research revived during the Scientific Revolution. We will see that in the work of nearly all seventeenth century mathematicians motion played an essential role as well. Kinematical definitions created new curves and often the investigation of the curves was also based on their kinematical properties. For most seventeenth century mathematicians the possibility of idealized motion going beyond the motions of compass and ruler in geometry was a natural starting point. In doing so they combined great admiration for the Greek achievements with the willingness to violate the canons of Greek mathematics.1 In this chapter we will discuss the role of motion in the work of Napier, Fermat and Descartes. We will start with Napier’s kinematical definition of the logarithm. Napier was the first to show how tables of logarithms could make calculations easier. The definition is based on an exponentially decelerating motion. Napier finds its existence apparently unproblematic.

1

Mahoney (1973), pp. 8–9.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Koetsier, A History of Kinematics from Zeno to Einstein, History of Mechanism and Machine Science 46, https://doi.org/10.1007/978-3-031-39872-8_7

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Fermat and Descartes introduced algebraic methods in geometry. In particular Descartes had a great interest in methodology. We saw above that Pappus redefined the Euclidean construction game. Descartes continued on that road, but went further. He introduced algebraic methods in geometry and limited the object of geometry to a precisely definable class of kinematically generatable curves. Only in the eighteenth century a completely algebraic treatment of geometry with the full exclusion of motion became acceptable. The game was redefined again and it was no longer a construction game.

7.2 John Napier’s Kinematical Definition of the Logarithm and Torricelli’s ‘Logarithmica’ In the sixteenth century ephemerides were tables giving the apparent positions of astronomical objects over time. They were used by astronomers and for navigation. Preparing the tables took a lot of work. In particular the multiplications took much time even though a clever method called prosthaphaeresis was used. This method used formulae like, in modern notation, sin a · sin b =

cos(a − b) − cos(a + b) 2

To use the method one needed sine- and cosine-tables. In order to do a multiplication of say, p and q, one shifted the decimal points to get the numbers sin a and sin b between − 1 and + 1. Using the sine-table one determined the corresponding a and b. By means of the sum and difference of a and b and a cosine-table the value of the product of sin a and sin b was determined. Finally one shifted the decimal point to the right position. In 1614 the Scottish nobleman Sir John Napier (1550–1617) published a booklet with the title A description of the miraculous canon of the logarithm (Mirifici logarithmorum canonis descriptio). The English translation by Edward Wright appeared 2 years later: J. Napier, A description of the admirable table of logarithms, London: Nicholas Okes (1616). Napier had a brilliant idea which would bring him everlasting fame: if I want to multiply the numbers in a geometrical sequence, I can do this by adding the exponents. Because addition is easier than multiplication, this is advantageous. So, if I cover an interval with a geometrical sequence, all I need to apply the idea, is a table of the exponents. I determine the exponents of two numbers sufficiently close to the numbers I want to multiply, I add the exponents and again by means of the table I determine the product with sufficient accuracy. Napier did neither possess our modern concept of function, nor our modern algebraic symbolism. He introduced the arithmetical sequence and the geometrical sequence in a kinematical way by means of two points that are describing straight lines. The two movements are coupled. He applied the theory of proportions for his calculations. By means of the two synchronously moving points he in fact defined a

7.2 John Napier’s Kinematical Definition of the Logarithm and Torricelli’s …

115

Fig. 7.1 The point B moving uniformly from A to C, D, etc. covering equal distances in equal periods of time T along a potentially infinite half line

Fig. 7.2 The point B moves from a to c, d, e, etc. in such a way that in each period of time T the remaining segment is shortened in the ratio QR/QS

function on the continuum in which the geometrical sequence is embedded, mapping it on the continuum in which the arithmetical sequence is embedded: Definition 1: “A line is said to increase equally, when the poynt describing the same, goeth forward equall spaces, in equall times, or moments.” (See Fig. 7.1)

The motion is uniform in the sense of the Greek definition of uniform motion (Fig. 7.1). During the first uniform movement a point B starting from A covers a sequence of intervals of equal lengths. The second movement takes place on a given line segment AZ and the point B starting from A ‘shortens’ AZ according to a fixed ratio. Definition 2: “A line is said to decrease proportionally into a shorter, when the poynt describing the same in æquall times, cutteth off parts continually of the same proportion to the lines from which they are cut off”. (See Fig. 7.2)

This is an interesting definition. In equal times equal proportions are cut off. Napier’s definition of the logarithm, which we will denote with LN, is the following: The logarithm of the distance remaining in the decelerating motion is equal to the distance covered in the uniform motion (Fig. 7.2). We will use a modern approach to analyze Napier’s treatment of the two motions and his definition of the logarithm. Although this is obviously anachronistic, it is the easy way to understand what Napier does.2

2

For a thorough treatment of Napier’s work on logarithms I refer to Roegel (2012).

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Assume the two motions start at t = 0 with velocity v0 . Let the distance covered in the two motions be x(t) and y(t) respectively for the uniform and the decelerating motion. Then x(t) = v0 · t. Assuming the length of the segment covered in the decelerating motion to be SQ = a and the ratio QR/QS = c we have y(T ) = a − c · a y(2T ) = a − c2 a y(3T ) = a − c3 a . . . .. y(nT ) = a − cn a. t

This yields for t equal to multiples of T: y(t) = a − c T a. We can, however, generalize and vthen we get y(t) = a − λt a with y' (0) = v0 . With y'(t) = −λt ln(λ)a 0 we get λ = e− a . From a modern point of view Napier’s decelerating motion can be described as v0

y(t) = a − e− a t a. Napier’s definition of the logarithm, which we will denote with LN, is the following: The logarithm of the distance remaining in the motion y(t) is equal to the distance covered in motion x(t). This means: ( v0 ) L N e− a t a = v0 t. Putting v0

z = e− a t a yields the relation between Napier’s logarithm LN and the natural logarithm: L N (z) = a · ln

(a ) . x

Napier uses a = 107 , which means that we get ( L N (z) = 107 · ln

) 107 . x

It is easy to see that in this way multiplication is reduced to addition, because L N ( p · q) = L N ( p) + L N (q) − L N (1)

7.2 John Napier’s Kinematical Definition of the Logarithm and Torricelli’s … Fig. 7.3 When the motion of B is uniform and the motion of A is geometric, point C describes the logarithmica

A

117

C

D

B

This involves the subtraction of the constant term LN(1) which Napier takes as 161,180,896.38. It is interesting that probably Napier introduced with his Definition 2 the first example of continuous (negative) exponential growth in history. A few decades later the combination of the two movements led to a new kinematically defined curve, the logarithmica.3 The curve was treated by Evangelista Torricelli (1608–1647) in the 1640 s.4 Torricelli called the curve because of its form and generation the hemhyperbola logarithmica. Torricelli’s text was only published in the twentieth century and it is not clear how much influence it exerted.5 He defines the curve by its property that whenever you choose equidistant vertical parallel lines the segments cut off from these lines by the horizontal axis and the curve form a geometrical sequence. One of his conclusions is that the sub-tangent has a constant value (DB in Fig. 7.3; CD is tangent to the curve in C). This follows from Torricelli’s definition.6 Torricelli concludes: “data unica tangente dantur omnes”, if you have one tangent you have them all. Obviously the logarithmica corresponds to an exponential function of the form y = ax if A and B move respectively on an y- and x-axis. Torricelli and Huygens, whose names are connected to the early history of this curve, in a sense were dealing with the exponential function. In Huygens work the curve shows up, for example, when Huygens decided to study the motion of a mass-point in a medium in which the resistance is proportional to the velocity.7 In 1694 Berhard Niewentijt noticed that at the time Leibniz’ differential calculus did not work for such transcendental8 curves as the logarithmica. A year earlier, L’Hôpital had written to Johann Bernoulli that he had no idea what an expression 3

Boyer (1945), p. 303. Evangelista Torricelli, De hemhyperbola logaritmica, In Loria and Vassura (1919), pp. 335–347. 5 For references to the full story see footnote 47 of p. 376 of (Whiteside, 1967). 6 The subtangent can be approximated by f (x) · Δx/Δy. Because Δy = (r − 1) · f (x), where r is the ratio of the geometrical progression corresponding to Δx, the subtangent does not depend on x. 7 Huygens (1937), p. 114. 8 The word ‘transcendental’ was coined by Leibniz. 4

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like mn meant in general. He could not imagine the significance of one line segment m raised to the power n representing another line segment. In 1697, Johann Bernoulli solved the problems by publishing his extension of Leibniz’ differential calculus to exponential functions in the Acta Eruditorum.9

7.3 Pierre de Fermat and Motion in His Introduction to Plane and Solid Loci Pierre de Fermat (1601–1665) is rightly seen as one of the fathers of analytical geometry. Yet although Fermat’s first text circulated in handwritten form since 1629, it was only printed in 1679. Analytical geometry is based on the application of algebraic methods in geometry. With hindsight the fact that in the seventeenth century algebraic methods would be applied in geometry is not surprising. The sixteenth century discoveries by Del Ferro, Tartaglia, Cardano and Ferrari of methods to solve third and fourth degree equations made algebra a hot topic. Moreover, with his analytical art Viète introduced a well-defined algebraic formalism and his definition of algebra as dealing with abstract magnitudes gave a foundation to its applicability to geometry. Fermat seems to have realized quickly that basically all interesting geometrical objects, curves and surfaces can be represented by equations and that, moreover, the equations can give us insight in the nature of these objects. This insight, however, needed elaboration. In 1637 Pierre de Fermat sent to his correspondents in Paris a manuscript entitled Introduction to Plane and Solid Loci (Ad locos planos et solidos isagoge). At about the same time the galley proofs of René Descartes’ Discourse on the Method of Rightly Conducting One’s Reason and of Seeking Truth (Discours de la méthode pour bien conduire sa raison et chercher la vérité dans les sciences) with its three appendices were ready. Independently the two mathematicians had arrived at similar results. In the appendix entitled Geometry Descartes wrote: “all points of those curves which we may call ‘geometric’, that is, those which admit of precise and exact measurement, must bear a definite relation to all points of a straight line, and […] this relation must be expressed by means of a single equation.”10 Descartes argued that in geometry all curves that can be defined precisely can be represented by means of equations. Fermat’s message was similar: equations in two unknowns represent curves. He pointed out that if the algebraic solution of a geometrical problem is an equation in two unknowns; the solution in the plane is a straight line or curve that can be described by one endpoint of a variable line segment of which the other 9

For the whole story see Bos (1996), pp. 1–19. Descartes (1954), pp. 48–49.

10

7.3 Pierre de Fermat and Motion in His Introduction to Plane and Solid Loci

119

endpoint moves along a fixed straight line. The line segment has a fixed direction. In Fermat’s words: “Whenever two unknown quantities are found in final equality, there results a locus [fixed] in place, and the endpoint of one of these [unknown] quantities, describes a straight line or a curve.”11 Descartes and Fermat express here the same important idea. Boyer called Fermat’s statement one of the most significant statements in the history of mathematics.12 Grootendorst, possibly unconsciously echoing Boyer, called Descartes’ statement one of the most important sentences in the mathematical literature.13 And indeed, Fermat and Descartes both discovered analytical geometry. This was truly a major discovery with immense consequences. Analytical geometry in itself was very fruitful, but it led to the discovery of the calculus by Leibniz and Newton as well. We will see below that this discovery was directly linked to the generation of geometrical objects through motion. In early analytical geometry the equations representing geometrical objects had no significance independently of the objects that they represented. The equations needed the objects they referred to in order to mean something. As for these geometrical objects Fermat and Descartes did exactly what the Ancients had also done: geometrical objects were created by imagining a motion that generates them. I’ll first discuss the role of motion in Fermat’s manuscript. The final equality that Fermat refers to in the quoted sentence, is for Fermat the simplest form that an equation can be given.14 The quoted sentence represents something essentially new: algebraic objects, indeterminate equations in two unknowns, are related to geometrical objects, curves. Fermat continued as follows: “The equations can easily be set up, if we arrange the two unknown quantities at a given angle—which we will usually take as a right angle—and if one endpoint of one of these [quantities] given in position is given. Provided that neither of the unknown quantities exceeds the square, the locus will be plane or solid, as will be made clear from what is said.”15 The first sentence shows how for Fermat the equations are related to the curves. The last sentence contains Fermat’s central result: if the degree of the equation is not higher than 2, the resulting locus is a straight line or a circle (plane loci), or one of the three conic sections (solid loci). Fermat uses here Pappus’ classification. See Fig. 7.4. The line NZM is fixed with the points N and M fixed on it. Z moves on NZM from N towards the right. The two unknown quantities are the lengths of NZ = A and ZI = E. The angle NZI is equal to a given angle. The first case that Fermat treats concerns the equation D · A = B · E.

11

Translation by Mahoney (1973), p. 78. Boyer (1956), p. 75. 13 Grootendorst (1999), p. 419. 14 Mahoney (1973), p. 78. 15 Mahoney (1973), p. 80. 12

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7 Napier, Fermat, Descartes

Fig. 7.4 Fermat relates equations to curves

L

K I

N

Z

M

D and B are constant values. Fermat wrote: If D · A = B · E then the point I describes a straight line (NK in Fig. 7.4). Clearly Fermat is working in the first quadrant only. Moreover, there is only one axis, the line NZM, plus a given angle. The y-axis was introduced at the end of the seventeenth century by Claude Rabuel (1669–1728).16 Mahoney17 called Fermat’s system is uniaxial and that is what it is: there is only one axis, the x-axis, and the x-coordinate leads. In order to find a point of the curve, one takes a value of A, which determines the position of Z. Then one constructs the angle NZL equal to a given angle. One constructs a segment of length E such that D · A = B · E is satisfied and one determines I on ZL such that ZI = E. Repeatedly in his text Fermat gives a kinematic description of the way the curve is found on the basis of the equation: Z moves along NM and ZI moves with it, always pointing in the same direction, while the length of ZI depends on the position of Z. In this process the curve involved is literally described. Let us consider another example: the equation A · E = Z2 . Z 2 is a constant. A and E are again the unknown quantities. Fermat had not yet liberated himself from the Greek dogma that magnitudes can only be compared if they are of the same kind. A · E is the product of two lengths. It can be interpreted as an area and can only be compared to another area. That is why in the equation we have Z2 , instead of merely Z. See Fig. 7.5 NZ = A and ZI = E. If the point Z moves along the fixed axis, the point I describes the hyperbola. Do not confuse the line NR with a y-axis; NR is merely one of the two asymptotes of the hyperbola. In the first text Fermat only considered straight lines, circles and conic sections. It was followed by a second text, Introduction to surface loci (Isagoge ad locus ad superficiem), written before 1643. In it Fermat attempted to generalize what he had done for curves to surfaces in space, restricting himself to surfaces of revolution and translation. It is highly remarkable that Fermat was in a sense close to modern 16 17

According to Grootendorst (1999), p. 420. Mahoney (1973), p. 82.

7.4 René Descartes Fig. 7.5 Point I describes a hyperbola

121

R

I O

N

Z

M

two-dimensional analytical geometry, but his uniaxial approach kept him far away from modern three-dimensional analytical geometry. In the Introduction to surface loci Fermat attempts to characterize planes, spheres, ellipsoids, paraboloids, hyperboloids, cones and cylinders in terms of properties that imply properties of their intersections with an arbitrary plane. The correctness of the characterization then follows from results that he obtained in the Introduction to Plane and Solid Loci. There is hardly three-dimensional analytical geometry in the book. As Mahoney has pointed out the uniaxial system constituted a conceptual hindrance to its generalization to three dimensions. Mahoney wrote: “While one moving ordinate may rest comfortably in the plane while sliding along the axis, a second ordinate jotting out into space and moving simultaneously in two directions would be in a precarious situation”.18

7.4 René Descartes We now turn to Descartes. It is remarkable that among non-mathematicians the fame of René Descartes (1596–1650) is primarily based on his philosophical work. Yet his mathematical work also had an immense influence. Allegedly his father wanted him to become a lawyer and he studied law. However, after his studies he became a military officer, a mercenary. In 1618 he joined for some time the army of the Dutch Republic. The Dutch Republic was small but had an extensive trade network. It was rich and had an excellent army that could compete with the major military powers. He also enlisted in other armies. Later he wrote that he was very much driven by curiosity and in search of knowledge. In Part I of his major philosophical work, the Discourse on the Method of Rightly Conducting One’s Reason and of Seeking Truth (Discours de la méthode Pour bien conduire sa raison, et chercher la vérité dans les sciences) of 1637, he wrote that he intended: “to 18

Mahoney (1973), p. 119.

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7 Napier, Fermat, Descartes

seek no knowledge other than that of which could be found in myself or else in the great book of the world”. His approach differed fundamentally from the traditional medieval way to find truth which consisted in studying the works of the great minds of the past. After returning to the Dutch Republic in 1628 Descartes renounced a military career and devoted himself to his studies. He had an exceptional mathematical talent, but his mathematical work was embedded in far reaching ambitions. Descartes is rightly seen as the first modern philosopher who represents the transition from a style in which one invokes authorities to a style in which the only thing that really counts is personal insight. In a sense this was not completely new. In a religious context Calvin and Luther preceded Descartes. In Descartes’ Discourse on the method systematic doubt concerning everything is the starting point. One of the things that are beyond doubt is the fact that one exists. I think and so I exist: Cogito ergo sum! Next to the cogito the existence of God can be deduced beyond doubt from the idea of God that we have. God guarantees the criterion of clarity and distinctness that gives us certain knowledge. And finally mathematics is in particular the domain where knowledge that satisfies this criterion is possible. For Descartes his philosophical work and his work in mathematics were intimately related. The very high standards that he applied in mathematics reflect his highly ambitious goals in philosophy. The Discours on the method has three appendices. One of them is called the Geometry (La Géométrie). It is about analytical geometry but idealized motions are its backbone. We will discuss its genesis.

7.5 Descartes’ Ambitions and His New Compasses In 1618 Descartes had met Isaac Beeckman in Breda in the Netherlands. A year later he wrote Beeckman five letters that contain valuable information on the development of Descartes’ thought. Descartes’ starting point was the Greek tradition of problem solving in geometry. In his second letter to Beeckman Descartes referred to new instruments for tracing curves that he had invented. From another text we know what these instruments were like. It is the text known as Private Reflections (Cogitationes Privatae). Descartes called the instruments ‘new compasses’ meant for angular trisection, the construction of mean proportionals and the solution of third degree equations. The first two applications illustrate his working in the Greek tradition. Yet his willingness to purposely design complex instruments also shows his own idiosyncratic approach. Moreover the third application shows that the sixteenth century developments had influenced him.

7.5 Descartes’ Ambitions and His New Compasses Fig. 7.6 Descartes’ trisector

123

D C M J L

B

H G

I F K

O

E

A

Descartes was, moreover, a child of the Renaissance. He shared its optimism and already when he wrote to Beeckman he had the plan to develop a “general art to solve all problems” and “a completely new science by which all questions in general may be solved that can be proposed about any kind of quantity.”19 The new instruments to trace curves that he had designed suggested to him that he was on the way to a very powerful method to manipulate and answer questions concerning quantities. In 1618 and 1619 Descartes imagined he could reach generality by means of controlled idealized motion. He imagined he was on his way to a universal problem solving method. In the course of time he would make his ideas more precise. Below I will first discuss Descartes’ new compasses. Then I will discuss the further developments that led to the Geometry. The Trisector See Fig. 7.6. Four rigid rods, OA, OB, OC and OD, are joined in the point O about which they can freely rotate. EI, GI, FJ and HJ are equal rigid rods of length a. They are connected to the rods OA, OB, OC and OD by means of hinges at E, F, G, H and I and J. The hinges E, F, G, H are fixed at distance a from O. The hinges I and J can move freely along OB and OC. It is easily seen that OA and OD can form any angle within a considerable range. Moreover, because the quadrangles OEIG and OFJH are both equilateral, rhombi, that are bisected by, respectively, OB and OC, the angles AOB, BOC and COD are equal in every position. Descartes uses the instrument as follows. If we keep OA fixed and rotate OD about O, the point J describes a sixth-degree curve KLM. In order to trisect the angle A' OD' we put the instrument such that OA coincides with OA' (Fig. 7.7). We draw the curve KLM by means of the instrument. Then we construct H' on OD' such that 19

Bos (2001), p. 232.

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7 Napier, Fermat, Descartes

D'

Fig. 7.7 Trisecting an angle

M J'

a

L

H' a K A'

O

OH' is equal to the defining length a of the trisector. Then we construct the circle with radius a and center H' . This circle intersects the curve KLM in J' . Then OJ' is one of the trisectors of the angle OD' . One notices that the instrument is used to draw a curve and in its turn the curve is used to find the solution to the trisection problem. The Mesolabum In Chap. 2 of this book we discussed Archytas’ kinematical solution to the problem of the two mean proportionals. See Fig. 7.8. The solution is based on the fact that when AI and AD are diameters of the circles AIM and ADK, A, M, K and L are collinear, and MT and KI are perpendicular to ATID we have: AM : AI = AI : AK = AK : AD.

Fig. 7.8 AM:AI = AI:AK = AK:AD

E K M

A

T

I

D

7.5 Descartes’ Ambitions and His New Compasses

125

Fig. 7.9 Descartes’ mesolabum

AI and AK are the two mean proportionals of AM and AD. Archytas determined these two mean proportionals, given AM and AD, by means of a highly ingenious kinematical construction in space. The lengths of the segments AM, AI, AK and AD represent the beginning of a geometrical sequence. Indeed if we have 1:x = x:y = y:z = z:u = u:v etc. we have y = x 2 , z = x 3 , u = x 4 , v = x 5 , … This idea is at the basis of an instrument devised by Descartes. Henk Bos, following a remark by Descartes, calls this instrument the mesolabum.20 The mesolabum is shown in Fig. 7.9, copied from Descartes’ Geometry. YZ and YX are rulers movable around Y. The ruler BC is fixed perpendicularly to YX. When YX rotates anti-clockwise about Y, the ruler BC pushes the ruler CD, which is perpendicular to YZ while C can slide along YZ. In its turn ruler CD pushes ruler DE which is perpendicular to YX while D can slide along YX. Similarly DE pushes EF and FG pushes GH. We can use the mesolabum to trace curves. The dotted lines in Fig. 7.9 are the curves described by the points D, F and H when we rotate YX about Y starting from the position YZ. These curves can be used to find mean proportionals as follows. See Fig. 7.10. Segment e is given by the instrument. Suppose another length a is given as well and we want two segments x and y such that e:x = x:y = y:a. We draw the curve AD by means of the instrument. Then we construct on YZ the point E such that YE = a. We construct the semi-circle EDY which intersects the curve CD in the point D. Then y = YD and x = YC, if C is the projection of D on YZ. The segment e is fixed, but it is easy to find the two mean proportionals between two arbitrary segments f and g by means of the same construction. Determine a such that f :g = e:a. Then apply the construction to find x and y and determine u and v such that e:f = x:u and y:v = a:g. Then u and v are the two lengths we are looking for. Segments x and y are then in fact multiplied with the factor f/e (or g/a). 20

Bos (2001), p. 239.

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7 Napier, Fermat, Descartes

Fig. 7.10 Finding mean proportionals

X y

B

e Y

A

x a

D

C E

Z

Also in the case of the mesolabum the instrument is used to draw curves which are in their turn used to find a solution to the problem of finding mean proportionals.

7.6 Algebra Comes In The idea of mesolabum was also used by Descartes to solve third degree equations. Again the instrument is used to trace a curve and the curve is used to find the solution. Yet in 1619 algebra not yet played an important role in Descartes’ view of geometry. However, this changed when he succeeded in the 1620s in finding the general solution of the third- and fourth degree equations by the intersection of a parabola and a circle. Descartes considers the equation x 4 = px 2 + q x + r. This represents a general fourth degree equation after the third degree term has been eliminated. On the basis of the values and the sign of p, q and r Descartes constructs the parabola and the circle. The intersections determine the roots. The construction also works for third degree equations. Then r = 0. This means that the construction can also be used to find two mean proportionals. Descartes shows essentially that the roots are the x-coordinates of the points of intersection of the parabola with equation y = x 2 and the circle with equation (x − a)2 + (y − b)2 = c2 . It is easy to express a, b and c in p, q, and r. Such a modern proof makes the result look rather trivial, but that is misleading. Descartes was very happy with it and Mersenne pointed out that Menaechmus still needed two conic sections to duplicate the cube, while Descartes needed only one. It may have been this particular result that convinced Descartes of the importance of algebra. The Greek geometrical tradition remained Descartes’ focus, but algebra started to play a role. The situation was not without complications. Descartes was familiar with the work of Viète. If an unknown quantity x corresponded to a line segment for Viète x2 and x3 corresponded respectively to a square and a cube. From such a point of view an equation like x = x2 is meaningless. In Viète’s algebra we

7.7 Pappus’ Problem

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have a hierarchy of quantities of different dimensions. In Viète’s algebra a segment of unknown length can be denoted with the letter E and then E quad, E cub. and E quad.quad. are magnitudes of respectively dimensions 2, 3 and 4. It is fascinating that he transcends geometry by allowing higher dimensions. However, only quantities of the same dimension can be compared, added or subtracted from each other. This is the homogeneity condition. The dimension of a product or a quotient is equal to respectively the sum or the difference of the dimensions of the quantities involved. Viète calculates with ratios of quantities using Eudoxus’ theory of proportions. Ratios of quantities are no quantities for Viète. He has no unit of multiplication either. In the Rules for the direction of the mind (Regulae ad directionem ingenii) written circa 1628 Descartes started to develop a methodology of science. Geometry was to play a central role, but this time supported by algebra. In his Rules Descartes introduces the unit of length and in Rule 16 he explicitly says that a3 is generated in the sequence 1:a = a:a2 = a2 :a3 , but that we need not view a3 as a cube. Descartes abolishes the hierarchy of quantities that Viète was still working with. Yet, in the Rules Descartes ran into problems. In Descartes’ view notions had to be “clear and distinct”. Moreover, our imagination enables us to represent continuous magnitudes in geometry. Descartes was struggling with the legitimacy of constructions going beyond compass and ruler. Pappus had distinguished (i) problems soluble by means of circles and straight lines (plane problems), (ii) problems requiring conic sections (solid problems), and (iii) problems requiring more complex curves (curvilinear problems). Descartes’ new compasses were based on the use of more complex curves, curves that were rejected by Pappus because of their “inconstant and changeable origin”. Descartes disagreed with Pappus. The application of the new compasses had to be justified, but in 1628 Descartes did not yet see how exactly.

7.7 Pappus’ Problem In 1631 the Dutch mathematician Jacob Van Gool (Golius) suggested to Descartes that he might try his new method on a problem mentioned by Pappus. It has become famous as the “Pappus problem”. Descartes was attracted by the problem. In the Geometry he emphasized that according to Pappus neither Euclid nor Apollonius had been able to solve the problem completely. Descartes himself in the end did not completely solve the problem either, but in several respects his work represents a major step forward. It must have been Pappus’ problem that led Descartes to his final position published in the Geometry. The Pappus problem in its full generality is the following: n straight lines L i and n angles θ i are given. For any point P in the plane the oblique distances di to the lines Li are defined as the lengths of the line segments drawn from P towards the lines Li such that the angle between the line segment with length di and Li is equal to θi .21 21

I follow Bos and use modern notation. See Bos (2001), pp. 272–273. Descartes describes the problem in the Geometry without any symbolic notation.

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Fig. 7.11 The Pappus’ problem with four lines. Source Bos (2001), p. 273

These distances should satisfy a relation: a ratio depending on the number of lines should be equal to a given ratio δ:1. The relations are in the notation used by Bos. For 3 lines: d 1 2 : d 2 d 3 = δ:1 For n = 2k lines with n > 3: d 1 …d k :d k+1 …d 2k = δ:1 For n = 2k + 1 lines with n > 3: d 1 …d k+1 :ad k+2 …d 2k+1 = δ:1 In this scheme the 3-line problem is actually a 4-line problem with two line coinciding. One would expect d 1 d 2: ad 3 = δ:1. Actually this 3-line problem plays a role in Descartes’ work, as we will see. In the plane a line segment of length a is given. This segment only plays a role when n is odd. The reason is that the Greeks could only compare quantities of the same dimension. For example, if n = 5 the quantities d 1 d 2 d 3 and d 4 d 5 have, respectively, the dimensions 3 and 2. In order to get a problem that makes sense from a Greek point of view we compare, for example, d 1 d 2 d 3 and ad 4 d 5 . Figure 7.11 shows the situation of the general Pappus’ problem with four lines. The points in the plane for which d 1 d 2 :d 3 d 4 = δ:1 or d 1 d 2 = δ d 3 d 4 have to be determined. Applying modern analytical geometry we could introduce a coordinate system and put P = (x, y). The lines would be represented by equations ai x + bi y + c = 0 for i = 1, 2, 3, 4. Then the distances di turn out to be linear expressions in x and y. We would subsequently substitute these expressions in the equation d1d2 = δ d 3 d 4 and finally we would analyze the resulting equation in order to find out what curve we are dealing with. In the Rules 19–21 Descartes argued that the first step in solving a problem is to derive an equation. That is exactly what he did in this case. Descartes determined the equation of the curve. From a modern point of view this equation would represent a solution to the problem. However, he had a different goal. He wanted a construction of the curve and his goal was kinematical. He wanted an idealized mechanism to construct the curve.

7.8 An Example: The Turning Ruler and Moving Curve Procedure

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Fig. 7.12 The turning ruler and moving curve procedure

N C

G

K L B

A

In Descartes’ work the algebraic equations and their manipulation always play a subordinate role. And although the algebraic techniques were quite sophisticated, algebra remained the servant of geometry. In the Geometry he first completely solves the general 3-line and 4-line problem and then he proceeds to two special cases of the 5-line problem. We will discuss one case, but we will first discuss a kinematical procedure that Bos called the “turning ruler and moving curve procedure.”22

7.8 An Example: The Turning Ruler and Moving Curve Procedure Consider an instrument (Fig. 7.12) consisting of a ruler GL fastened to a pivot G. The rectangular triangle KNL slides along another fixed ruler KA. In L, fixed to the plane of KNL, there is a hinge that can slide along the ruler GL. The point of intersection C of (the extension of) KN with GL describes a curve. Descartes first determines the equation of the curve. He draws CB parallel to GA. Descartes says: “Since CB and BA are unknown and indeterminate quantities, I shall call one of them y and the other x”. In fact AB = x and CB = y. He calls GA, KL and NL, that are known quantities, respectively, a, b and c. The resulting equation is y 2 = cy −

22

Bos (2001), p. 278.

cx y + ay − ac. b

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C

Fig. 7.13 Descartes’ procedure generates a conchoid

C' =

K

L

L'

=

L''

C''

= A

G

This is the equation of a hyperbola, a fact that is stated by Descartes without further ado. Later Van Schooten gave a proof that the equation corresponds to a hyperbola. See Descartes (1683), pp. 171–172. Van Schooten started with a hyperbola and showed that it can be represented by means of this equation. Descartes immediately considers the possibility that we modify the mechanism of Fig. 7.12 by replacing in the plane of triangle KLN the extended straight line KN by some other curve: a circle with center L or a parabola with axis KB. Replacing the line KN by a circle with center L means that we generate a conchoid. See Fig. 7.13. Descartes generates the conchoid in a way that differs from the way in which it was generated by Nicomachus. In both cases L is a point on GC that slides along KA. However, Nicomachus fixes L on the line GLC and the conchoids is generated by the point C on the line GLC that has a fixed distance CL to L. The result is that GLC rotates about L and slides through G during the generation of the conchoid. In Descartes’ case GLC rotates about G and L is the endpoint of an extra link LC of constant length of which the endpoint C rotates about L. Point L slides on both GLC and on KA. The two different ways to generate the same conchoid suggest different generalizations. Nicomachus generation suggests replacing the line KA by other curves. Descartes way of generating suggests replacing the circle by other curves. Actually the curve that we generate if we use instead of the circle a parabola with axis KB (See Fig. 7.12) possesses a special significance for Descartes, because as he says in The Geometry: “[it] is the first and simplest of the curves required in the problem of Pappus, that is, the one which furnishes the solution when five lines are given in position”. According to Bos this particular ‘Pappus curve’ must have played an important role in the development of Descartes’ ideas in 1632.23

7.9 Descartes’ Solution of Pappus’ 5-Line Problem It seems almost certain that Descartes studied in 1632 first some simple special cases of Pappus’ problem. For example, the case with two parallel lines L1 and L2 and a line L3 perpendicular to L1 and L2 (See Fig. 7.14) If θ i ’s are all 90° it is very easy to see, for example analytically, that the requirement d1 · d2 = δ · d3 defines a parabola. 23

E. g. Bos (2001), p. 275.

7.9 Descartes’ Solution of Pappus’ 5-Line Problem

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Fig. 7.14 A simple case of Pappus’ problem

L3

L1

L2

Let us consider a special case of the five-line problem. Assume that L1 through L4 are equidistant parallel lines and L5 is perpendicular to them. If θ i ’s are all 90° it is not very difficult to derive an equation for the Pappus-curve. Let us take δ = 1. The defining relation then is: d1 d2 d3 = a d4 d5 . Consider Fig. 7.15 P is a point on the locus that we intend to determine. We draw OP, which intersects L4 in point Q. LQ is a line parallel to L5 . Let z be the (perpendicular) distance from P to LQ. On the basis of similarity we can say that z : d4 = d5 : d1 or z = d4 d5 /d1 . Combination of the two equations yields d2 · d3 = a · z. Fig. 7.15 A special case of the five-line problem

L1

L2 U

L4 Q

P

L3 V

z

d5

O d1 a

L5

d4

d2

d3 a

LQ

a

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Descartes must have realized that this is the defining relation in a three line problem. The three corresponding lines are L1 , L2 and LQ. As we have seen the Pappus curve is a parabola with axis L4 . The points U and V are on the parabola. It is remarkable that for all points P on the five line Pappus curve the defining relation of this parabola is the same, this means that when P moves on the five line Pappus curve the parabola moves up or down with the line LQ. This simple argument24 shows that we can easily generate the five line Pappus curve by means of a line rotating about O and a parabola sliding along a straight line.

7.10 The Use of Strings In The Geometry Descartes argued that only curves that can be traced by acceptable motion were permissabele in geometry. This excluded curves like the spiral and the quadratrix, because they cannot be traced by means of a single continuous motion. He accepted all curves that can be described by linkages with one degree of freedom consisting of rigid elements. Although he excluded the use of a string, for example, in order to determine the equality of a straight and a curved line segment, he did not completely exclude the use of strings. The use of strings was acceptable in order to guarantee the equality or difference of one, two or more straight line segments drawn from a point of the curve we want to describe, to other points. Here Descartes referred to the well-known constructions of an ellipse or a hyperbola. The ellipse and the hyperbola are special Cartesian ovals. What is nowadays called a Cartesian oval is the locus of all points P with the property that a particular linear combination of the distances between P and two given points F and G has a constant value. The defining relation is: λPF + μPG = Constant. Clearly, if λ = − μ we get a hyperbola and if λ = μ we get an ellipse. When we exclude these cases the Cartesian ovals are quartic curves. Descartes discovered them in an optical context. If a Cartesian oval is the border between two media and the index of refraction is λ/μ, a bundle of light rays generated at F is by refraction turned into a bundle of light rays through G. The hyperbola and the ellipse can be generated by means of strings. There are, however, other Cartesian ovals that can be generated by means of string-mechanisms. Descartes gave an example in the Geometry. Consider Fig. 7.16. The ruler FE rotates about the fixed point F. C is a point that can slide along FE. K and G are fixed points on the horizontal line. A string is stretched from E to C, passes to K, goes back to C 24

Cf. Bos (2001), pp. 275–276.

7.11 The Final Results

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Fig. 7.16 Generating Cartesian ovals

E C

F

K

G

and then goes from C to G where the other end is fastened. So we have EC + 2CK + CG = L, where L and CK have a constant value. We move C while we keep the string stretched. While we do so C describes an oval curve with foci F and G.

7.11 The Final Results In the Geometry Descartes exhibited the final results of his geometrical research. In Book I of the Geometry Descartes explained the use of algebra to solve geometrical problems. He went further than Fermat in the sense that he removed the dimensional aspect. Descartes showed how the arithmetical operations could be interpreted as concerning line segments. He showed how a geometrical problem could be translated into a set of equations. In book II of the Geometry Descartes criticized Pappus’ classification. As we have seen Pappus had distinguished (i) circles and straight lines, (ii) conic sections, and (iii) more complex curves. Descartes first pointed out that if a curve is rejected because drawing it requires some kind of instrument, then we must also reject circles and straight lines. Other instruments cannot be rejected because they are less accurate in practice. The fact that some of them are used in practice implies that they can be quite accurate. In geometry however, practical accuracy is not the point; in geometry exactness of reasoning is what it is all about. In fact, elsewhere, Descartes, like Proclus, argued that the generation of the curves takes place in the imagination.25 25

Descartes view of the relation between mathematics and physical reality differs from the ideas of the Neoplatonists. Proclus was not interested in the application of mathematics. Descartes was. See Nikulin (2002).

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Descartes saw the imagination as a screen on which the mind could create figures, manipulate them, and inspect the results of these manipulations. Neither can we, according to Descartes, reject other instruments because their use would require extra assumptions. The Greeks did introduce extra assumptions when they needed them. In the Greek treatment of conic sections they introduced “without hesitation” the assumption that any given cone can be cut by a given plane. So Descartes argued that all curves are acceptable “provided they can be conceived of as described by a continuous motion or by several successive motions, each motion being completely determined by those which precede; for in this way an exact knowledge of the magnitude of each is always obtainable”. Descartes rejected the spiral, the quadratrix and similar curves, “since they must be conceived of as described by two separate movements whose relation does not admit of exact determination”. Descartes replaced Pappus’ classification by a new one: (i) Plane problems are the ones solvable by straight lines and circles, (ii) Non-plane problems are the ones solvable by curves that can be traced by one single motion, (iii) Problems solvable only by certain special curves that cannot be traced by one single motion.

Chapter 8

De Witt, van Schooten, Newton and Huygens

Abstract Descartes lived from 1628 to 1649 in The Netherlands and his stay had quite an impact there. The Dutchman Frans van Schooten junior translated Descartes’ Geometry into Latin. Van Schooten also wrote a book on mechanisms to draw conic sections. His pupil and friend, the Dutch statesman Johan de Witt, wrote a book on conic sections on the basis of Descartes’ ideas. Through Van Schooten’s translation Newton got to know Descartes work. Thinking about curves in terms of their kinematical generation led Newton to his calculus. Motion also played a role in Leibniz work on the calculus. We also briefly discuss the work of Huygens and Newton on circular motion and Huygens calculations on gear trains.

8.1 Frans van Schooten Junior In 1637 Descartes’ Discourse on Method with its three appendices was published in Leiden. In particular the third appendix, the Geometry, was read eagerly by Frans van Schooten junior (1615–1660). At the time his father, Frans van Schooten senior (1581?–1645), was professor of mathematics at the engineering school that was part of the University of Leiden. In 1646 Van Schooten junior would succeed his father in this position. Van Schooten junior found the Geometry a masterpiece but difficult to read. Descartes’ text was ambitious and programmatic but the author left out many details and Van Schooten decided to make it more accessible by to translating it into Latin. The Latin translation appeared in 1649, Descartes (1649). We have seen that for Descartes in geometry only such curves are acceptable that can be conceived of as “described by a continuous motion or by several successive motions, each motion being completely determined by those which precede”. Descartes did assume, without being able to give a proof, that all acceptable curves have algebraic equations. He moreover did assume that the acceptable curves could

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Koetsier, A History of Kinematics from Zeno to Einstein, History of Mechanism and Machine Science 46, https://doi.org/10.1007/978-3-031-39872-8_8

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be classified on the basis of (the degree) of their equations.1 Another obvious assumption, although Descartes did not write about it, is that every algebraic equation in two unknowns describes an acceptable curve. In fact Descartes left the general problem of the relation of algebraic equations and idealized mechanisms for his successors. While reading the Geometry Van Schooten and the men around him developed an interest in this problem. Van Schooten himself and the Dutch statesman Jan de Witt (1625–1672) concentrated on the simplest case: the conic sections.

8.2 Jan de Witt In 1660 Frans van Schooten published a second Latin edition of the Geometry, Descartes (1659–1661). This edition and its accompanying commentaries were very influential. In an appendix Van Schooten included the two books of a text entitled Foundations of Curved Lines (Elementa Curvarum Linearum), written by the Dutch statesman Jan de Witt (1625–1672). In Book I of the Elementa De Witt proves most of the theorems on conic sections of Books I and II of Apollonius Conics. He did this because he was not satisfied with Apollonius’ way of defining the conic sections as plane intersections of a cone. De Witt wrote: “But when I had carefully studied the textbooks on the other curves, so far as they have been handed down by the Ancients and explained by the younger ones, I thought it utterly contrary to the natural order, which must be taken into account as much as possible, that one seeks the origin of these curves in a spatial body and then transfers it to a flat surface.”2 De Witt defined the curves kinematically by means of planar mechanisms. De Witt gives for example, the following kinematical definitions of the parabola and the hyperbola. See Fig. 8.1. Take a fixed line r and a fixed point T and draw through T a fixed line m that intersects r in a point A. We now let an angle α with legs s and w rotate about T. The leg s cuts r in a point P. Draw through P a line b parallel to AT, this line intersects the other leg in a point S. When α is equal to the angle TAP = β, the point S describes a parabola. When α /= β the point S describes a hyperbola. As for the ellipse, De Witt has the definition in which a fixed point of a line segment describes an ellipse if the endpoints of the segment move on to fixed intersecting lines. Descartes laid the foundation of analytical geometry, but De Witt’s Book II of the Elementa can be considered as the first textbook on the subject. The book is about the correspondence between conic sections and algebraic equations in x and y. De Witt studied the problem of how to reduce a given second-degree equation to the simplest form. Jan de Witt starts out with a linear or quadratic equation in two variables and proves that it represents a straight line or a conic section. De Witt uses a coordinate system with one axis. The y-axis was only introduced near the end of the seventeenth century by Claude Rabuel (1669–1728). 1 2

Bos (2001), pp. 340–341. Witt (1997), pp. 40–41.

8.3 Frans van Schooten Junior: Mechanisms to Draw a Parabola

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Fig. 8.1 De Witt’s kinematical definition of hyperbola and parabola

Basically De Witt shows that the following relation holds: Equations y = ax + b, x = ay + b represent a line Equations y2 = ax + b, x2 = ay + b represent a parabola Equations a2 x2 − b2 y2 = 1, a2 y2 − b2 x2 = 1, xy = 1 represent a hyperbola Equation a2 x2 + b2 y2 = 1 represents an ellipse. He, moreover, shows that more complex second degree equations can be reduced to these cases. Yet, for De Witt life was less easy than this suggests. He restricted himself to positive coefficients and he stuck to the condition of homogeneity.

8.3 Frans van Schooten Junior: Mechanisms to Draw a Parabola The constructive kinematic approach of De Witt to conic sections is completely in line with the work of Van Schooten junior. In 1659/1660 Van Schooten published Mathematical exercises in five books (Exercitationum mathematicarum libri quinque), which was also published in Dutch with the title Mathematische Oeffeningen. Book 4

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of the five is called Fourth book of the mathematical exercises covering the description of conic sections in a plane by means of a true mechanism (Vierde Bouck der Mathematische Oeffeningen begrijpende de Tuychwerckelycke beschrijving der kegel-sneden op een vlack). However, this fourth book was written much earlier, possibly around 1646 at the time Van Schooten became a professor. The book made clear that the conic sections can be generated with planar mechanisms in different ways. At the same time, Van Schooten also had applications in mind, for example, in optics and the construction of sundials. Van Schooten gives us two different methods that can be used to describe parabolas when the focus and the directrix of the parabola are given: (i) A method based on a linkage of six elements, four of which form a rhombus, and (ii) A string-mechanism. In Fig. 8.2 the horizontal line EG, the directrix, and point B, the focus, are fixed while the rhombus BHGF rotates about point B. G moves along EG in such a way that GL is perpendicular to EG. In GL and in the extension of FH there are slits. Because GD = DB and during the motion point D has equal distances to point B and line EG, a pin through the intersection D of GL and the extension of FH describes a parabola. In Fig. 8.3 again directrix GE and focus B are given. Point G moves along GE in such a way that GI remains perpendicular to GE. Point D moves on GI. A rope fixed to B passes through D, goes upwards to I, then downwards and is fixed in G. We move D on GI in such a way that the rope remains stretched. During the motion DG equals DB so that D describes part of a parabola. Fig. 8.2 Point D describes a parabola

8.4 Frans van Schooten Junior: Mechanisms to Draw an Ellipse

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Fig. 8.3 D describes a parabola

Fig. 8.4 Van Schooten’s elliptic motion

8.4 Frans van Schooten Junior: Mechanisms to Draw an Ellipse Essentially Van Schooten gives us four different methods that can be used to describe ellipses when the two foci are given: (i) By means of mechanisms based on what is nowadays called the elliptic motion.3 (ii) The ‘gardener’s construction’ by means of a string, (iii) A method based on an anti-parallelogram, (iv) A method based on a linkage of six elements, four of which form a rhombus. Van Schooten starts with a systematic investigation of the mechanism of Fig. 8.4. A is a fixed point and AB can rotate about A. D slides along a line through A. The triangle BCD can rotate about D. Triangle BCD and AB are connected by means of a hinge in B. AB = BD = BC. He proves that C moves on a straight line through A such that ∠ DAC = ½ ∠DBC. Moreover he proves that the points in the plane of triangle BCD that are not on the circle with radius AB and center B describe ellipses. His first way to generate ellipses is based on this result. See Fig. 8.5. With a reference to Proposition 52 of Book 3 of Apollonius’ Conics, which says that the sum of the distances from a point on an ellipse to the two foci has a constant value (and vice versa) Van Schooten also gives the gardener’s construction of the ellipse by means of a string. 3

See Chaps. 4 and 9 of this book.

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Fig. 8.5 Van Schooten’s ellipse generator

The next method by means of which Van Schooten generates an ellipse is remarkable as well. See Fig. 8.6. H and I are fixed points. HG rotates about H and IF rotates about I. HG = IF. F and G are connected by means of a link FG equal in length to HI. In both FI and HG there is a slit. By means of a pin coinciding with the point of intersection of the Fig. 8.6 Van Schooten generates an ellipse with an antiparallelogram

8.5 Frans van Schooten Junior: Mechanisms to Draw a Hyperbola

141

Fig. 8.7 E describes an ellipse with foci H and I

two slits an ellipse can be described. The reader easily verifies after drawing FH that ΔFIH ∼ = ΔHGF which implies that ΔHIE ∼ = ΔGEF. We find HE + IE = HG and E apparently describes an ellipse with foci H and I. The mechanism of Fig. 8.7 consists of a rhombus OIPG that can rotate freely about a fixed point I. The vertices of the rhombus are hinges. The bar OP is fixed to the rhombus in such a way that it hinges about O and permanently coincides with the diagonal AP of the rhombus. Vertex P of the rhombus slides in a slit in OP. Another bar HG which is provided with a slit rotates about the fixed point H and it is connected to hinge G of the rhombus. A pin that follows the point of intersection of HG and OP describes an ellipse. This is obvious because PIEG is a kite, which means that IE = EG. Conclusion: HE + EI = HE + EG = HG and E describes an ellipse with foci H and I.

8.5 Frans van Schooten Junior: Mechanisms to Draw a Hyperbola Van Schooten gives us four different methods that can be used to describe hyperbolas: (i) By means of mechanisms based on a ‘sliding angle’ and a turning ruler. (ii) A string-mechanism, (iii) A method based on an anti-parallelogram, (iv) A method based on a linkage of six elements, four of which form a rhombus. See Fig. 8.8. An asymptote (the diagonal AD of rectangle IAED) and the ‘top’ E of the hyperbola are given. The hand in the drawing covers the point e. The ange dbe with legs bd and be is rigid and slides with leg bd along the (extension of) the diagonal DA of the parallelogram IAED. By means of a hinge in d a straight bar is fixed to leg bd of the angle dbe. While the angle dbe slides, this bar is kept in such a position that it goes through E. During this motion the point of intersection e of the leg be and this bar describes a hyperbola. The reader will be able to check easily

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Fig. 8.8 A hyperbola by means of a ‘sliding angle’ and a turning ruler

that the curve is of the second degree and because the extension of DA is obviously an asymptote, the curve must be a hyperbola. As we saw in Chap. 7 in the Geometry Descartes had given the same mechanism. See Fig. 7.12. The string-mechanism of Fig. 8.9 requires that the foci of the hyperbola are given. NF is a rigid bar that rotates about F. A string of fixed length is connected to F, runs to N, returns to a point P on FN from where it is connected to a fixed point C. Keeping the string stretched against FN with a pin P, this pin describes a hyperbola with foci C and F. Van Schooten refers here to Proposition 5 of Book 3 of Apollonius’ Conics. And indeed it is easy to see that FP + 2PN + CP = constant (length of string) and FP + PN = constant (length of FN), which means after elimination of PN that FP-CP = Constant. Also for the mechanism in Fig. 8.10 the foci C and F of the hyperbola must be given. CD = GF and GD = CF. So GDCF is an anti-parallelogram. The point of intersection P of the extensions of FG and DC describes a hyperbola with foci F and C because PF-PC = FG. Also the mechanism in Fig. 8.11 for the construction of a hyperbola requires that the foci C and F are given. In C and F there are hinges so that the bars can rotate about them. There is a slit in NO so that P, L and M can slide in it. There is also a slit in the extension of DC, so that P can slide in it. Clearly PF is equal to DP and this means that PF-PC = PD-PC = DC is a constant length.

8.6 Isaac Newton, Motion and the Fundamental Theorem of the Calculus

143

Fig. 8.9 String mechanism to draw a hyperbola

Fig. 8.10 A hyperbola generated by means of an anti-parallelogram

8.6 Isaac Newton, Motion and the Fundamental Theorem of the Calculus Van Schooten’s Latin translation of Descartes’ geometry had a particularly great influence on Newton. It is striking that Newton succeeded in moving to the level of research of his contemporaries Huygens and Leibniz in less than two years, largely

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Fig. 8.11 Van Schooten’s fourth linkage to draw a hyperbola

on his own. Newton read Descartes’ Geometry in Van Schooten’s annotated second Latin edition, probably in 1664. The analytical representation of curves interested him, but he collected also from Van Schooten’s edition the different ways by means of which a conic section could be constructed.4 As we have seen in Chap. 6 of this book in the middle of 1665 Newton thought about kinematically generated curves and the generation of curves by motion was one of the elements that led Newton at that time to the fundamental theorem of the calculus. He had studied the existing literature concerning tangents and he had developed a method to determine the subnormal from the equation of the curve. See Fig. 8.12. If CM is the normal to the curve and P the projection of C on the axis, MP is the subnormal. As for integration or quadrature, the result ∫ xndx =

1 x n+1 (n + 1)

was known in the 1660 s in various forms, although it was not yet written down in this form. In the winter of 1664–1665 N discovered the binomial theorem which states that the binomial series expansion for (a + x)n for natural numbers n can be 4

Westfall (1980), p. 107.

8.6 Isaac Newton, Motion and the Fundamental Theorem of the Calculus

145

Fig. 8.12 CM is the normal to the curve and MP is the subnormal

C

M

P

Fig. 8.13 Newton derives the fundamental theorem of the calculus

generalized to fractional powers n = p/q. Newton was studying the quadrature of the circle. The binomial theorem enabled him to write y = (1 + x 2 )1/2 , which corresponds to the circle, as an infinite series. By treating such series as polynomials he could determine the quadrature of such functions by determining the quadrature of each term of the series separately and adding them. Yet, although Newton could determine tangents to certain curves and areas under certain curves he had not yet related these methods. This changed in 1665. Consider Fig. 8.13. The figure is Newton’s, but I use other letters to identify the points.5 The argument is also Newton’s; only my use of y(x) and z(x) instead of y and z is anachronistic. Possibly Newton wrote the text in the middle of 1665. The horizontal axis is an x-axis directed to the left. There is a vertical y-axis and the curve y = y(x) is given. st(x) is the subtangent corresponding to the point (x,y). There is also a vertical z-axis pointing downward and the curve z = z(x) is defined as Q S. Newton starts with a curve y = y(x) that is a parabola and follows: z(x) = st(x) y(x) QS corresponds to a point x0 where st(x0 ) = y(x0 ). This means that z(x0 ) = QS. It is clear that z(x) represents the derivative of y(x) times QS. After having defined z(x) Newton takes VW equal to SQ and he considers the area SQPR and he compares it with the rectangular area VWUT. Now suppose that R moves along the curve z(x) starting at S. At the same time P, which is the projection of R on the x-axis moves on the x-axis starting at Q. PR then while decreasing in length moves uniformly to the left until it reaches a final position corresponding to x1 . While we move PR we move VW as well, in such a way that during the motion the extensions of VW and 5

Whiteside (1967), p. 299.

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8 De Witt, van Schooten, Newton and Huygens

QS intersect on the curve y = y(x). VW keeps its length but its velocity decreases. Newton actually says: “If VW = QS then area VTUW = area PRSQ for PR moves uniformly from SQ and TU moves from VW with motion decreasing in the same proportion that PR does shorten”. Is this correct? It is, because for infinitesimal increments when x increases to x + Δx, the moving segment SQ covers an area z(x)·Δx = QS·Δy. At the same time the moving segment VW moves over a distance of Δy and it covers an area equal to VW·Δy = QS·Δy. The area PRSQ ‘under’ the curve of z(x) is equal to the area VTUW = VW·(QT − QV) = QS·(QT − QV), or in the modern notation we owe to Leibniz: ∫x1 z(x)d x = Q S · (y(x1 ) − y(x0 )). x0

Because z(x) equals QS times the derivative of y(x) this is the fundamental theorem of the calculus. Newton discussed a special situation in which y(x) is a parabola, but the argument is generally valid.

8.7 The Method of Fluxions Probably in the autumn of 1665 Newton extended the kinematic approach he used for areas to tangents. It led him to the method of fluxions, Newton’s version of the calculus. The role that motion plays in it is clear from the first sentences of Newton’s Treatise of the quadrature of curves (De quadratura curvarum), written in 1676 and published in Latin in 1704. He wrote: “I consider mathematical Quantities in this Place not as composed of very small Parts; but as describ’d by a continued Motion. Lines are describ’d and thereby generated, not by the Apposition of Parts, but by the continued Motion of Points; Superficies’s by the Motion of Lines; Solids by the Motion of Superficies’s, Angles by the Rotation of the Sides, Portions of Time by a continual Flux: and so in other Quantities. These Geneses really take place in the Nature of Things, and are daily seen in the motion of Bodies. And after this Manner the Ancients by drawing moveable right Lines along immoveable right Lines, taught the Genesis of Rectangles. Therefore considering that Quantities, which increase in equal times, and by increasing are generated; become greater or less according to the greater or less Velocity with which they increase and are generated; I sought a Method of determining Quantities from the Velocities of their Motions or Increments, with which they are generated; and calling these Velocities of the Motions or Increments Fluxions, and the generated Quantities Fluents, I fell by degrees upon the Method of Fluxions, which I have make use here in the Quadrature of Curves, in the Years 1665 and 1666.” See Newton (1745).

8.8 Circular Motion in the Work of Huygens and Newton

147

In this quotation Newton says that he discovered the method of fluxions (flux = rate of change) by degrees in the years 1665 and 1666. One notices that Newton’s investigation of curves and their properties is in every respect related to the kinematical generation of the curves. Let us look at a simple example of the method of fluxions in its final form. Determining fluxions is essentially differentiation or the determination of tangents. When a point describes the curve defined by the equation x3 − ax2 + axy − y3 = 0 during the motion the two fluent quantities x and y change their value in the course of time. When the fluxions or speeds of x and y are x˙ and y˙ , the increases of x and y ˙ and y˙ o. We can substitute x + xo ˙ and y + y˙ o during an infinitesimal time o are xo in the equation x3 − ax2 + axy − y3 = 0. Using the fact that (x, y) is on the curve, a division by o, followed by the substitution o = 0, yields ( 2 ) ) ( 3x − 2ax + ay x˙ + ax − 3y 2 y˙ = 0. What we have here could be written nowadays as ∂ f dy ∂ f dx + = 0, ∂ x dt ∂ y dt with f(x, y) = x3 − ax2 + axy − y3 = 0. In the Treatise of the quadrature of curves Newton obviously, given the title of the text, also considers the inverse problem: determine the area under a curve or find the fluents from the fluxions.

8.8 Circular Motion in the Work of Huygens and Newton In Mechanical Problems Aristotle or, if he did not write the text, the pupil who did, could not explain circular motion. It obviously could not be a combination of two uniform motions, so it had to be a combination of two motions that are not ‘in a fixed ratio’. The problem was solved in the seventeenth century, independently by Christiaan Huygens and Newton. They showed that the acceleration required to produce a linear radial deviation from the tangent in uniform circular motion is directly proportional to the square of the tangential velocity and inversely proportional to the radius of the circle. Huygens’ result is in a treatise now called On the centrifugal force (De vis centrifuga),6 on which he started working in 1659. Huygens considers a small ball connected to the circumference of a rotating wheel with center A that rotates with constant speed. See Fig. 8.14. In equal intervals of time Δt the ball moves from 6

Huygens (1929), pp. 255–301. Mahoney published a translation. See: https://www.princeton.edu/ ~hos/mike/texts/huygens/centriforce/huyforce.htm.

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8 De Witt, van Schooten, Newton and Huygens

Fig. 8.14 Huygens on centrifugal force

position B to E, from E to F and from F to M. If the ball were released from the wheel in position B keeping its velocity it would move in those intervals of time along the tangent BS to the positions K, L and N, respectively, where BK, KL and LN are equal to the arc EB. The ball connected to the wheel moves from B to E, but it has a tendency to move to K along the arc EK of an involute of the circle. This tendency is the vis centrifuga, the centrifugal force, that Huygens is investigating. Without proof Huygens argues that these arcs EK, FL and MN increase as the series of squares from unity 1, 4, 9, 16 … This is true when the arcs are extremely small and they coincide with the segments EC, FD and NS, and also with the perpendiculars from E, F and M on the tangent BS.7 Huygens: “But the tendency of which we have been speaking is clearly similar to that by which heavy bodies hanging on a string tend to fall.[… But, while the same ball always has the same tendency to fall whenever it is suspended on a string, the tendency of the ball carried around on a wheel is, on the contrary, greater or less according as the wheel turns more quickly or more slowly.”8 Huygens then proves a number of propositions. Proposition I says if two equal moving bodies traverse unequal circumferences in equal times, the centrifugal force in the greater circumference will be to that in the smaller as these circumferences, or their diameters, are to each other. Proof: See Fig. 8.15. Let the first body move from B to D and the second body in the same time from C to E. The ratio of the centrifugal forces is by definition the ratio of the tendencies to move away from the circle and equal to the ratio of DF and EG, which is equal to the ratio of AB and AC. 7

The parabola y = x 2 /2 is for small x an excellent approximation of the semicircle y = 1 −

x2 ). 8

The translation is Mahoney’s.

√ (1 −

8.8 Circular Motion in the Work of Huygens and Newton

149

Fig. 8.15 Huygens on centrifugal force

Proposition II says that the centrifugal force is proportional to the square of the velocity. The argument is simple. See Fig. 8.14. Equal bodies move in equal time intervals on the same circle. Yet the first body moves from B to E and the other one from B to F. In the two cases the tendency to move away from the circle is measured by arcs EK and FL. These arcs grow proportionally to the squares of de distance moved on the circle which is in itself proportional to the velocity. Proposition III says that the centrifugal force is inversely proportional to the radius of the circle. Huygens proves the proposition as follows. See Fig. 8.16. Equal bodies move on concentric circles. Body 1 moves from B to D, while in the same time interval Body 2 moves from C to F. Let DA cut the inner circle in E. Huygens introduces a third Body 3 equal in weight to the two others which moves slower such that in the time interval it covers the arc CE. Let the centrifugal forces corresponding to the three bodies be Force 1, Force 2 and Force 3. Then on the basis of Proposition I AB For ce 1 = For ce 3 AC and on the basis of proposition II For ce 3 = For ce 2

(

AC AB

)2

and this yields AC For ce 1 = For ce 2 AB

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8 De Witt, van Schooten, Newton and Huygens

Fig. 8.16 Huygens on centrifugal force

This proves that the centrifugal force is inversely proportional to the radius of the circle. Propositions II and II together imply that the centrifugal force is proportional to v2 /r. Let us briefly look at a sketch of a proof of the same result by Newton. It was published in a manuscript written before 1699, now called On Circular Motion. Brackenridge and Nauenberg describe Newton’s solution as follows. See Fig. 8.17. When the rotation is uniform the centrifugal force is constant. As Galilei has demonstrated we then have QR = s.t 2 . Here s represents the centripetal force. Newton does the same in Lemma 10 of the Principia. Moreover, from Proposition 36 of Book 3 of Euclid’s Elements follows: RP2 = RQ · RU R

P

Fig. 8.17 Newton on centrifugal force

Q

S

U

8.9 Huygens and Gear Trains

151

or RU/PR = PR/QR. When SR deviates very little from SP we have RU = QU and PR = PQ. Then QU/PQ = PQ/QR or 2r/vt = vt/s.t2 . The result is the well-known formula: s = v2 /r.

8.9 Huygens and Gear Trains A kinematical subject of a completely different nature concerns the design of gear trains in order to bring about a specific ratio between the number of rotations of the input wheel and the output wheel. Consider a center axle with two gear wheels on it, one with 10 teeth and one with 17 teeth. The one with 10 teeth is driven by a wheel with 13 teeth on an input axle and the one with 17 teeth drives a wheel with 5 teeth on an output axle. Then one rotation of the input axle corresponds to 13/10 rotations of the center axle and one rotation of the center axle corresponds to 17/5 rotations ∗ 17 = 221 of the output axle. This means that one rotation of the input axle yields 13 10 ∗ 5 50 rotations of the output axle. Already in classical antiquity such trains were designed, for example, for the Antikythera mechanism. We do not know, however, which methods the Greeks used for the design. We do know how Christiaan Huygens in 1680 designed the gear trains for a planetarium that he was building.9 He first calculated for Mercury, Venus, Mars, Jupiter and Saturn the ratio of their orbital period to the earth’ year.

9

Planet

Mercury

Venus

Earth

Ratio

25,335/105,190

64,725/105,190

1/1

Huygens (1703), pp. 431–460. Also in Huygens (1944). Huygens text is not easy to read. For an accessible presentation of Huygens calculations see Rockett and Szüsz (1992), pp. 59–60.

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8 De Witt, van Schooten, Newton and Huygens

Planet

Earth

Mars

Jupiter

Saturn

Ratio

1/1

197,836/105,190

1,247,057/105,190

3,095,277/105,190

We want to realize these ratios by means of trains of gear wheels. The ratio for Mercury yields the continued fraction [0, 4, 6, 1, 1, 2, 1, 1, 1.1, 7, 1, 2, …] or 1 4+

1 6+

1 1+

1+

1 1 1 2+ 1+...

The sequence of convergents of the ratio for Mercury is: 0, ¼, 6/25, 7/29, 13/54, 33/137, 46/191, 79/328, 125/519, 204/847, … Such convergents have the pleasant property that the sequence converges to the value of the original fraction, while fractions consisting of smaller natural numbers always give a worse approximation. ∗ 17 , which means that The 9th convergent of the ratio for Mercury 204/847 equals 12 7 ∗ 121 we need four gears with 12, 17, 7 and 121 teeth while the wheels with 17 and 7 teeth have the same axis. Huygens may have been the first to use convergents for the design of a gear train. He certainly did not invent the method. Bombelli explicitly used continued fractions in his Algebra of 1572 and in the seventeenth century the use of the convergents as approximations seems to have been common knowledge.10

8.9.1 Leibniz and Transcendental Curves Independently of Newton also Leibniz discovered the calculus. We saw that in Newton’s discovery of the calculus motion played an essential role. Leibniz followed a different route, which we will not discuss. Yet motion played a role in Leibniz’ work. First of all, like Descartes, Leibniz only accepted curves as legitimate objects to be studied in geometry if they could be constructed by motion. The question is of course: What kinds of motion? Descartes had argued that only algebraic curves were acceptable. Curves like the cycloid, the logarithmica, the spirals or the quadratrix were not acceptable. He called them derogatively ‘mechanical’. Leibniz called them ‘transcendental’, because they transcended the algebraic curves. Leibniz, however, felt they absolutely had to be included in geometry and, of course this meant that it ought it be possible to apply his calculus to them. In 1692 Leibniz wrote: “Descartes, in order to maintain the universality and sufficiency of his method, found it appropriate to exclude from geometry all the problems and all the curves which could not be subjected to this method, under the pretext that these things were only mechanical. Since, however, these problems and lines can 10

Fowler (1994), p. 736.

8.9 Huygens and Gear Trains

153

Fig. 8.18 The tractrix

be constructed or conceived by means of certain exact motions, and have important properties, and nature often uses them, one may say that he commits an error similar to that he reproached in certain ancients, which restricted themselves to the constructions for which one needs nothing but ruler and compass, as if all the rest was mechanical.”11 Leibniz was rightly proud of his invention of the calculus and also because it did for transcendental curves what Descartes’ method could only do for algebraic curves. Leibniz’ favorite example was the cycloid. Descartes had come up with a method to find tangents to algebraic curves. This is the idea: In a given point P on the curve, consider all circles through that point P with their center on the x-axis. Find the circle that has two coinciding points of intersection with the curve in the point we are interested in. The tangent to this circle in P is equal to the tangent to the curve in P. The circle is found by determining an equation that should have two equal roots. This does not work for the cycloid. Leibniz repeatedly used the example of the cycloid to show the superiority of his calculus. Because its equation can be found easily, we can differentiate and find the slope of the tangent.12 Leibniz’ calculus turned out to be indeed very powerful. After Descartes the mathematicians discovered that curves like the cycloid and the logarithmica offered challenging problems. Another curve that generated interesting research is the common tractrix. See Fig. 8.18. It is defined as follows: a point P is dragged in a plane by means of a cord PQ of fixed length. Q moves along a straight line and the velocity of the point is directed along the stretched cord. Allegedly the curve was born in the 1670 s in Paris. There Claude Perrault would put his watch on the table and move the end of the watch chain along a straight line and challenge the mathematicians to study the curve described by the watch.13

11

Quoted by Blåsjö (2016), pp. 17–18. Blåsjö (2016), pp. 64–65. 13 Bos (1988), p. 9. 12

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8 De Witt, van Schooten, Newton and Huygens

In the 1690s Leibniz’ calculus made it possible to write down equations for transcendental curves. The fact that the logarithmica (representing an exponential function) has a constant sub-tangent implies that its inverse function (representing a logarithm) is the quadrature of a hyperbola. In this case the equation involves an integral.14 As for the tractrix in 1693 Leibniz answered Perrault’s question. He first wrote down a differential equation based on a simple application of Pythagoras’ theorem in ΔPQR. √ a2 − y2 dx = − dy y Integration yielded: ∫ √ 2 a − y2 x =− dy y Yet in the seventeenth century an equation of a curve did not represent a solution of the problem, without an acceptable geometrical construction of the curve. If the curve was algebraic within the Cartesian framework the equation would indeed basically solve the problem, because the equation would in principle yield a method of point wise construction of the curve. However, the cycloid, the logarithmica and the tractrix are not algebraic. In such cases the seventeenth century mathematicians tried different ways out, none of them really satisfactory. One of them was to find other acceptable ways to pointwise construct the equation involving the quadrature. In his 1693 paper Leibniz came up with a construction of the equation, but it was not really satisfactory. At the end of the seventeenth century all major mathematicians worked on transcendental curves. Huygens did and so did Johann and Jacob Bernoulli. They all saw analytical methods merely as a means to study geometrical problems. In the end their research led to a situation in which a radical change of perspective was inevitable. It turned out that the world of analytical methods independently of geometry offered new results and challenges. With Leonhard Euler’s Introductio in analysin infinitorum of 1748 the calculus became the theory of analytical expressions and geometry merely an area of application.

The logarithmica corresponds to y =∫ax . It would take some time before this notation was accepted. The inverse function is x = lny/lna = (1/y)dy/lna.

14

Chapter 9

Towards Theoretical Kinematics

Abstract Theoretical kinematics deals with the general properties of motion. In the eighteenth century in particular work on cycloids, gear wheels and the motion of a rigid body in space yielded the first results. First we discuss the roots of the instantaneous center of rotation. We describe how De la Hire discovered what is nowadays called the inflexion circle and how Euler discovered the so-called EulerSavary formula. We also discuss the appearance of the instantaneous axis of rotation in spherical kinematics and the instantaneous screw axis in space kinematics. Pieces of theoretical kinematics that later became part of a coherent whole emerged in different areas.

9.1 The Instantaneous Center of Rotation, Descartes and Johann Bernoulli1 Theoretical kinematics is the general theory dealing with the motion of a Euclidean space with respect to another Euclidean space coinciding with it. This theory was developed in the nineteenth century but its roots go back much further. We will first restrict ourselves to instantaneous planar kinematics and we consider the general properties of the motion at a particular instant of a Euclidean plane moving with respect to another Euclidean plane coinciding with it. We will informally consider some central theorems and we will show how in the seventeenth and eighteenth centuries elements of this theory started to show up. The first major result concerns the velocity distribution at a particular instant of the points of a Euclidean plane moving with respect to a fixed Euclidean plane. We have either an instantaneous translation, all points move with the same velocity in the same direction, or we have an instantaneous rotation. In the case of an instantaneous rotation there is one point in the moving plane with velocity zero while all other points rotate about this point. The point with velocity zero is called the pole, or the 1

Parts of this chapter are based on Koetsier (Mech Machi Theory 21:489–498, 1986) and Koetsier (Euler and kinematics, 2007)

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Koetsier, A History of Kinematics from Zeno to Einstein, History of Mechanism and Machine Science 46, https://doi.org/10.1007/978-3-031-39872-8_9

155

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9 Towards Theoretical Kinematics

Fig. 9.1 Figures from Descartes’ letter to Mersenne, 23 August 1638

instantaneous center of rotation. The velocity of a point is in this case determined by the angular velocity and the distance of the point to the pole. The second result concerns the motion of the of the pole. During the motion of a moving Euclidean plane with respect to a fixed Euclidean plane in general the position of the pole describes a curve in the fixed plane. This curve is called the fixed polode. The word ‘polode’ was coined by Louis Poinsot (1777–1859) in 1834. Poinsot combined the word ‘pole’ with the Greek word Ðδ´oς (hodos) = route. The pole also describes a curve in the moving plane. This curve is called the moving polode. For the two polodes we have the lovely theorem that says: During the motion of a moving Euclidean plane with respect to a fixed Euclidean plane the motion of the moving polode with respect to the fixed polode is a pure rolling motion, while the point of contact is the pole corresponding to that position of the moving plane. It is remarkable that in the seventeenth and the eighteenth century in a development that started with the cycloid several results on rolling motion were derived. It seems to have been Cauchy who finally realized the generality of those results. In a sense in 1638 Descartes was close. Apparently the great letter writer among seventeenth century mathematicians, the Minimite friar Marin Mersenne (1588– 1648) suggested to Descartes that Roberval was unable to determine the tangents to the curves described by the motion of a circle that rolls on a straight line. Yet Roberval was about to find a solution. As we have seen, he soon solved the problem by means of a composition of motions. On 23 August, 1638 Descartes, who was at that time in Santpoort in The Netherlands, wrote Mersenne a letter in which he offered a solution. He wrote, referring to Fig. 9.1 left: “If one wants, for example, the straight line that touches in the point B the curve ABC, which is described on the basis AD by one of the points of the circumference of the wheel DNC then one should draw through this point the line BN parallel to the basis AD, then draw another line from the point N, where the parallel line meets the wheel, towards the point D, where the wheel touches the basis, and afterwards draw BO parallel to ND, and finally BL perpendicular to BQ because this line BL is the wanted tangent.”2 Descartes’ proof runs as follows. 2

Mersenne (1963), p. 35.

9.2 The Cycloid

157

He wrote, referring to Fig. 9.1 right: “So, if one lets the hexagon ABCD roll on the straight line EFGD, its point A describes the curved line EHIA, composed of the arc EH, which it described while the hexagon touches the basis in F, which is the center of this arc, of the arc HI whose center is G, of the arc IA whose center is D, etc. while through these centers pass all the lines that meet the tangents to the arcs at right angles. Now, the same happens in the case of a polygon with hundred thousand million sides, and consequently also in the case of the circle”.3 Descartes also gave similar constructions for the tangents to the trajectories of points attached to the wheel inside or outside the circumference. He added that everything he had written could also be said about non-circular wheels. With Descartes we have here the theorem: planar motion generated by rolling (without slipping) of a curve on a straight line, is at each moment an instantaneous rotation about the point where the rolling curve touches the straight line. In a certain sense Descartes discovered the instantaneous center of rotation, although he was not aware of the fact that planar motion is instantaneously always either a translation or a rotation. Almost a century later Johann Bernoulli got closer to that purely kinematical result, although he discovered it in a dynamical context. He considered the situation in which a rigid body in the plane is hit by an impulsive force of which the line of action does not pass through the centre of gravity. After he calculated the force needed to keep a particular point instantaneously at rest, he discovered that there is a point for which this force is zero. He called it the ‘spontaneous centre of rotation’.4

9.2 The Cycloid Another well-known result in instantaneous planar Euclidean kinematics is the following. We consider two coinciding Euclidean planes moving with respect to each other. The motion of one of the two planes, called the moving plane, is considered with respect to the other plane, called the fixed plane. During the movement the points of the moving plane describe curves in the fixed plane. Then at each instant, in general,5 the set of points in the moving plane that coincide with an inflexion point of the curve that they describe, is a circle. This circle is nowadays called the inflexion circle. On August 11, 1706, Philippe de La Hire (1614–1718) presented a paper to the Académie Royale des Sciences in Paris with the title A treatise on roulettes (Traité des Roulettes), (De La Hire 1706). The word roulette means here a curve generated by a point in the plane of a curve rolling on another curve. De La Hire was an influential 3

Mersenne (1963), p. 37. Bernoulli (1742). See also Cannon and Dostrovsky (2012), pp. 111–112 for a brief discussion of the text. 5 There are exceptions. For example, when the movement is a translation (or rotation) and all points in the moving plane describe straight lines (or circles) in the fixed plane. 4

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9 Towards Theoretical Kinematics

member of the Academy from 1678 until his death in 1718. The paper contains a result on the basis of which De La Hire is considered to be the discoverer of the inflexion circle. De La Hire‘s paper is part of the development that started with the investigation of the ordinary cycloid. It was Galileo Galilei who gave the curve its name and after Galilei, in the seventeenth century not only Roberval and Descartes, but also Pascal, Wren, Wallis, Huygens, Newton, Leibniz and others studied the properties of the curve. This great interest is understandable. The curve offered challenging problems to mathematicians. Well known is Pascal’s 1658 challenge to the mathematicians of his time. Using the pseudonym Amos Dettonville and offering 600 francs he proposed problems on the cycloid that he had recently solved himself. The problems concerned the area under the curve, its center of gravity and volume and surface area of the solid of revolution generated by rotating the cycloid about the horizontal axis. Yet, the cycloid was not only mathematically challenging. In the course of time the curve turned out to possess physical significance. This is immediately clear if one considers some aspects of Huygens‘ work. Christiaan Huygens (1629–1695) had a lifelong interest in the design of clocks. He is usually called the inventor of the pendulum clock. And indeed Huygens The clock (Horologium) of 1658 introduced features that guaranteed great accuracy in the clock and the book popularized the pendulum clock. Scientifically even more significant is, however, the text that Huygens published in 1673 under the title The Pendulum Clock (Horologium Oscillatorium). In the second part of the book Huygens showed that the cycloid is isochronous: a mass point falling along an inverted cycloid (the ordinary cycloid turned upside down) reaches the bottom in an amount of time irrespective of the point where the fall begins. In the third part of the book Huygens introduced his theory of evolutes by means of which he proved that if the bob of a pendulum moves between two cycloidal-shaped plates, the bob is forced to move along an inverted cycloid and ideally will keep time uniformly, no matter how wide it swings. Joella Yoder has shown in (Yoder 1988) that most of Huygens 1673 book was developed in 1659 in a period of three months. This means that Huygens is the father of the theory of curvature of planar curves. Let us briefly consider the central notions of Huygens‘ theory. The concept of the two curves related by “unrolling” was brand new. One of the two curves was called by Huygens the evolute (or evoluta = “the one that is unrolled”—from the Latin evolutus = unrolled) and the other one was called descripta ex evolutione (or “the one drawn by unrolling”). Modern mathematicians have kept the name evolute and they call the one drawn by unrolling the involute. The easiest way to define the relationship between evolute and involute is kinematical: Let the evolute be given. Fix a thread to the shape of the given evolute and then unwind the thread from one end keeping the thread always pulled taut. The end of the thread then describes the involute. Huygens also considered the inverse relationship. He derives the evolute from a given involute as follows. If P and Q are two infinitesimally close points on a given involute, the lines perpendicular to the tangents in P and Q intersect in a point on the evolute. From a modern point of view the evolute is the set of all centres of curvature of the involute.

9.3 The Inflexion Circle

159

9.3 The Inflexion Circle With a clear implicit reference to Huygens‘ work, De La Hire wrote in his paper: “All geometers know already that every given curve can be described by the unrolling of another curve, and the given curve will be generated by a straight line which rolls on the other curve.”6 De la Hire in his paper considered the most general situation of arbitrary curves rolling on each other without slipping. We have a fixed curve, which he calls the base curve. We have another curve rolling on the base curve, which he calls the generating curve. And then we have the curves described by arbitrary but fixed points in the plane of the generating curve. We will call them generalized cycloids. De la Hire uses the word roulette for such a curves. De La Hire‘s main objective in his paper is to prove two theorems. He wants to show that (i) All curves can be generated as generalized cycloids from an arbitrary given base curve, and that (ii) All curves can be generated as generalized cycloids from an arbitrary given generating curve. In the first case a suitable generating curve must be determined and in the second case a suitable base curve.7 However, before proving these theorems, De La Hire gave some results concerning the inflexion points of generalized cycloids and in this context, unaware of the generality of his result, he discovered the inflexion circle. We will restrict ourselves to this particular discovery. The notion ‘inflexion point’ was well known in the seventeenth century. De La Hire defined an inflexion point as a point where “a curved line turns in two opposite directions”. Let us consider the first figure of De La Hire‘s treatise. See Fig. 9.2a. The base curve is YA. The generating curve is EAB; it touches the base curve in A. Both base curve and generating curve are seen as involutes. YA is the involute of MO. So O is the centre of curvature of the base curve in the point A. The generating curve is the involute of NC and C is the centre of curvature of the generating curve in the point A. The generalized cycloid RPS is generated by point P attached to the plane of the generating curve. The question that De La Hire answered is the following: At this particular moment when the generating curve and the base curve touch in A, what is in the plane connected to the generating curve the set of points P that coincide at this moment with an inflexion point of the curve that they describe? De La Hire gave the following answer: It is the circle AXV with diameter AV with V on OAC such that CO : CA = AO : AV. This is a nice result. It relates the diameter of the inflexion circle to the radii of curvature of the base curve and the generating curve in the point where they touch each other at this particular moment. Once we have this result it is also immediately obvious that the tangents in these inflexion points all go through V, because the tangent to the trajectory in a point O is perpendicular to AP. 6

De La Hire (1706), p. 341 In 1707 François Nicole discusses the equation of the curve described by a point in the plane of an arbitrary curve rolling on another arbitrary curve. See Nicole (1707). Nicole’s approach is analytical.

7

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Fig. 9.2 The first figures in De la Hire’s treatise on roulettes

De La Hire pointed out that, of course, in other positions of the generating curve this circle will have a different size and position. De La Hire’s proof is not exactly easy reading and one cannot blame the reader for skipping it. It is quite characteristic of the geometrical reasoning with infinitesimals that one finds often before in the nineteenth and twentieth centuries more rigid reasoning was introduced. The infinitesimals are infinitely small and that is why they are sometimes equated to zero. On the other hand, they are treated as finite quantities.

9.4 De La Hire’s Proof Consider Fig. 9.2b De La Hire wants to find out what happens to P when the generating curve rolls over an infinitely small distance on the base curve. The basic idea is that in such situations the two curves can be considered locally as circles and a circle can be considered as a regular polygon with infinitesimal sides. B is a point on the generating curve ‘indefinitely close’ to A and E is a point on the base curve ‘indefinitely close’ to A. AE and AB can be considered to be straight and the points are chosen such that AE = AB. The lines perpendicular to the tangents in B and A intersect in C, while

9.4 De La Hire’s Proof

161

the lines perpendicular to the tangents in E and A intersect in O. Because we are treating the curves as circles, the triangles ABC and EAO are isosceles with equal bases. De La Hire now first rotates triangle CAB about A until AB coincides with AE. Then BC coincides with EM. Subsequently De La Hire rotates CAB from its new position about E until BC coincides with the prolongation of OE. While we execute these two rotations the point P in the plane of the generating curve moves first to Z and then to S. These two rotations together constitute one infinitesimal part of the rolling process. Subsequently De La Hire considers what happens to the arbitrary point P when it is subjected to this infinitesimally small rotation. See Fig. 9.2b. The line FP, perpendicular to AP, is the tangent to the generalized cycloid that P describes. De La Hire now argues that if S, that is the position of P after the infinitesimal rotation, is on FP, P is in an inflexion point of the curve that it describes. When is S on FP? This is the case if ES is parallel to AP and this is the case when the (infinitesimal) angle APB is equal to the (infinitesimal) angle about which the generating curve was rotated. At this point De La Hire points out that we can consider the points B and E as coinciding and he proceeds to Fig. 9.3a. In Fig. 9.3a the points B and E coincide. The complete infinitesimal angle of rotation is now ∠CEQ. Moreover, V is chosen such that CO:CA = AO:AV and the circle VXA with diameter AV is drawn. The goal is to show that points P on this circle have the property that they stay on the tangent to their trajectory after the infinitesimal rotation. This requires that ES is parallel to AP (Fig. 9.2b) and that means that we should prove that ∠APE = the complete infinitesimal angle of rotation = ∠CEQ. Nota bene: after the complete rotation BP coincides with ES. De La Hire now argues that if we have a triangle with two infinitesimal angles α and β, the ratio of the angles equals the ratio of their opposing sides a and b. This is correct. If we apply the sine-rule in a triangle we get a a:b = sinα: sin β. Moreover, when the two angles are infinitesimally small this gives: a:b = α: β. Moreover, when α and β are infinitesimally small, the third side of the triangle c = a + b and c:b = (a + b):b = a:b + 1 = α: β + β: β = (α + β): β. When applying this theorem in triangle ΔCEO we have CO : CE = ∠CEQ : ∠COE. Because CE = CA this yields CO:CA = ∠CEQ: ∠COE. Moreover, in triangle ΔVOE we have OE : EV = AO : AV = ∠OVE : ∠VOE. Because of the position of V we have CO:CA = AO:AV. So ∠CEQ : ∠COE = ∠OVE : ∠VOE.

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Fig. 9.3 More figures from De la Hire’s treatise on roulettes

And because ∠COE = ∠VOE this yields: ∠CEQ = ∠OVE. We also have ∠OVE = ∠APE because from P and V on the circle the arc EA is seen under the same angle. Conclusion: ∠CEQ = ∠APE. So the points P on the circle VXA satisfy the condition implied by the requirement that P is in an inflexion point of its trajectory. De La Hire’s conclusion is that in general the set of all points in the moving plane who are at a certain moment in an inflexion point of the generalized cycloid that they generate, is a circle. There is another nice property. The tangents in all these inflexion points all go through one particular point in the fixed plane: it is point V.

9.5 Elliptic Motion De La Hire considers also the case when the centres of curvature C and O, respectively, of the generating curve and of the base curve are both on the same side of the common tangent. The point V is determined by the same condition: CO:CA = AO:AV. A special case occurs when the generating curve is a circle that rolls inside a circle twice as big as the generating circle. In this case (See Fig. 9.3b) the inflexion circle is identical with the generating circle. De La Hire points out that this is a very remarkable situation because all points of the generating circle describe straight lines during the entire motion. Moreover, the centre of the generating circle

9.6 Epicycloidal Gearing

163

Fig. 9.4 The epicycloidal tooth DH on the wheel with center C pushes a pin on the wheel with center A from position D to the positions E. Source De la Hire (1694), p. 56

describes clearly a circle, while—and this is less obvious—all other points in the plane of the moving circle describe ellipses. This result is sometimes called De la Hire’s Theorem. Yet, it is the so-called elliptic motion, already known in antiquity. We know this because Proclus wrote in his commentary on Definition 4 of Book 1 of Euclid’s Elements: “When a straight line is moving thus, its extremities, moving nonuniformly, describe straight lines, whereas the middle point moving non-uniformly, describes a circle, and the other points ellipses.”8 Clearly Proclus only considered the points of the moving line segment, and not the points of an entire plane connected to the segment.9 Moreover he considered the case in which the two lines described by the two extremities are perpendicular. This is obvious because otherwise the middle point would not describe a circle.

9.6 Epicycloidal Gearing What should be the shape of gear teeth such that motion can be transferred smoothly? We know that Ole Rømer (1644–1710) worked on the problem. The first published work seems to have been by De la Hire. In 1794 he published a text on epicycloids and their applications in mechanics. Epicycloids are the curves described by the points of a circle rolling without friction on another circle. In the preface he writes that before him the shape of the gear teeth was considered to be a problem to be solved in practice. He writes that he worked on the problem for 20 years and that he has reached the conclusion that the teeth must have the shape of an epicycloid. A central result is that when a uniformly rotating wheel with teeth that have the shape of epicycloids the uniform rotation can be transferred smoothly to a wheel carrying pins. See Fig. 9.4, taken from De la Hire’s treatise. A wheel with center C and radius CR drives a wheel with center A and radius AD.

8

Morrow (1970), p. 86. Proclus may have gotten the result from Geminus. Much can be said about the history of the elliptic motion. For the contributions of Nasir Edwin, Copernicus, Cardan and others see Boyer (1947–48).

9

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Fig. 9.5 De la Hire’s epicycloidal drive

DH is part of an epicycloid generated by the circle with center A and radius AD rolling on the circle with center C and radius CR. We attach an epicycloidal tooth DH to CR. Then when we rotate the wheel with center C anticlockwise, the tooth DH pushes a pin on the circumference of the circle with center A from position D towards the positions E. Because of the way in which the epicycloidal tooth is generated during the motion arc RP = arc DE and the motion of the two circles with respect to each other is one of pure rolling. De la Hire’s argument is not purely kinematical. The mechanism should function smoothly. Because arc RP = arc DE, the resisting force in E does as much work as the force that drives the rotation of the wheel with center C.10 So as for those two forces there is equilibrium, which is necessary. See Fig. 9.5, taken from De la Hire’s treatise, for the resulting mechanism. Another important classical French text on epicycloidal gearing, written by Charles-Étienne-Louis Camus as part of his Cours de mathematique, (Camus 1766), dates from the 1750s. In 1806 it was translated into English by J.I. Hawkins and then reprinted in 1837 and 1868. See (Camus 1837). When the tooth of one gear wheel is pushing the tooth of another wheel, the two profiles touch each other. Camus proves that the normal to the common tangent of the profiles passes through a fixed point on the line connecting the centers of the wheels. This point divides the line segment connecting the centers in a ratio equal to the ratio of the angular velocities of the wheels.

10

De la Hire (1694), p. 53.

9.6 Epicycloidal Gearing

165

Fig. 9.6 Camus explains the principle of cycloidal gearing. Source Camus (1766), Pl.XXVI, Fig. 18.2

Camus describes the cycloidal drive as we know it now. See Fig. 9.6. Camus considers three circles.11 A circle with center F and radius FA (circle R), a circle with center B and radius BA (circle X) And a third circle Y with center G and radius GA. The three circles touch each other in A where they can roll upon each other without friction. The three centers are fixed. We rotate one of the three circles and it carries along the other two. Now consider a point E on the circumference of circle Y. This point E describes in the plane of circle X inside that circle a portion of an interior epicycloid and it describes in the plane of circle R outside of that circle a portion of an exterior epicycloid. When describing the two epicycloids E rotates instantaneously about A. In each position of E the two epicycloids touch each other in E and the common tangent bE is perpendicular to AE. When the three circles rotate uniformly the interior and exterior epicycloids touch each other. Cycloid gearing is based on the possibility to give the wheels X and R teeth that have the form of the epicycloids and use them to transfer the uniform motion from one of the wheels to the other.

11

Camus (1766), pp. 327–328.

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When the circle Y is given a diameter equal to half the diameter of circle X the interior epicycloids are straight line segments and they coincide with diameters of circle X. This has advantages in clockmaking, because such teeth are easy to shape and polish. The motion of Y with respect to X is clearly the elliptic motion that we met before.

9.7 The Euler-Savary Formula Nineteenth century kinematicians have extensively studied the relation between the points of the moving plane and the corresponding centers of curvature of their trajectories in the fixed plane. This particular relation has many properties. Consider Fig. 9.7, which represents the situation at a particular moment. The fixed polode is π1 . The moving polode is π2 . The point O in which the two polodes touch each other is the instantaneous centre of rotation or pole at the moment that we are considering. k 2 is a curve in the moving plane. k 1 is the envelope in the fixed plane of the set of positions of k 2 in the fixed plane. In the position that we are considering k 1 and k 2 touch in the point C. The points N1 , N2 , M1 and M2 are, respectively the centres of curvature of k 1 , k 2 , π1 and π2 corresponding to the points C and O. Let θ be the angle between the common tangent to the polodes and the common perpendicular in C to k 1 and k 2 . Then we have, in general, the following relation: (

1 1 − O N1 O N2

) · sin θ =

1 1 − O M1 O M2

This is the general Euler-Savary formula. The variables ON 1 , ON 2 , OM 1 and OM 2 correspond to directed line segments; they have a sign. O, the pole, is the origin of a Cartesian coordinate system with pr as positive x-axis and pn as positive y-axis. Similarly O is also the origin of a Cartesian coordinate system Oξη with directed line segment OC defining the positive direction of the ξ-axis. The two systems have the same orientation. As for the signs of the variables in the Euler-Savary formula, ON i is positive if moving from O to Ni is a move in the direction of the ξ-axis. OM i is positive if moving from O to Mi is a move in the direction of the y-axis. It is easy to deduce De la Hire’s result from this general Euler-Savary formula. k 2 becomes a point and ON2 = OC = ρ. In an inflexion point the center of curvature is at infinity. So ON1 = − ∞. This yields −

1 sin θ 1 = − . ρ O M1 O M2

This is the equation of the inflexion circle in polar coordinates. The diameter of the inflexion circle is ρ. By taking θ = π/2 it is also easy to verify De la Hire’s formula for the diameter of the inflexion circle.

9.7 The Euler-Savary Formula

167

Fig. 9.7 Fixed polode π1 and moving polode π2 . Curve k 1 is the envelope in the fixed plane of the set of positions of curve k 2 . Source of picture: Veldkamp (1970)

The following proof of the Euler-Savary formula was given in 1970 by G. R. Veldkamp.12 We first need a lemma. As simple as it is, it was only discovered in the second half of the nineteenth century. It is usually called the Aronhold-Kennedy theorem. Siegfried Heinrich Aronhold (1819–1884) was a German mathematician and Alexander Blackie William Kennedy (1847–1928) was a British engineer. The theorem says that when three coinciding Euclidean planes W1 , W2 and W3 are moving, in general, at each instant the three instantaneous centres of rotation or poles are collinear.13 Proof: Let Pij denote the pole for the motion of plane Wi with respect to plane Wj . Consider in a particular position of the three planes the line in W1 that connects P12 and P13 . It is easy to see that, in general, there will be a point on this line for which the velocity with respect to W2 is equal to its velocity with respect to W3 . This is P23 . We can now prove the Euler-Savary formula: W1 is the fixed plane. W2 is the moving plane. Consider again Fig. 9.7. We introduce a third plane W3 by means of a right angle which moves in such a way that the vertex always coincides with the point O, the pole of the motion of plane W2 with respect to plane W1 , while one of 12

Veldkamp (1970), pp. 70–72. Aronhold and Kennedy found the theorem independently. Cf. Aronhold (1872) and Kennedy (1887).

13

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the legs of the angle is the tangent to π1 in O. Nota bene the centres of curvature M1 and M2 are at each instant the instantaneous centres of rotation of respectively the motion of W3 with respect to W1 and the motion of W3 with respect to W2 . We now introduce a fourth plane W4 , which is such that its vertex coincides with the point C where k1 touches the envelope k 2 , while one of its legs coincides with the common tangent of k 1 and k 2 . Clearly N1 and N2 are at each instant the poles of respectively the motion of W4 with respect to W1 and W2 . Consider now the motion of W1 , W2 and W4 and apply the Aronhold-Kennedy theorem. This yields that the line connecting N1 , N2 and C runs through O. During the motion of W4 with respect to W3 the leg q of the right angle that defines the motion of W4 always goes through O, the vertex of the angle that defines the motion of W3 . So the velocity of O in W3 with respect to W4 is directed along q. This means that the pole P34 must be on line η perpendicular to q in O. Because P34 and P31 (= M1 ) and P41 (= N1 ) are collinear, and P34 and P32 (= M2 ) and P42 (= N2 ) are collinear as well, on the basis of the Aronhold-Kennedy theorem, the line N2 M2 and the line N1 M1 intersect in P34 (= H) on the line η perpendicular to q in O. The situation is correctly depicted in Fig. 9.7. In order to get the Euler-Savary formula we now return to the Cartesian coordinate system Oξη that we introduced above. We have the following coordinates: H = (0, h), N1 = (ON 1 , 0), N2 = (ON 2 , 0), M1 = (OM 1 sin θ, OM 1 cos θ ), M2 = (OM 2 sinθ, OM 2 cosθ ). The equations of the lines HN1 and HN2 with respect to the coordinate system Oξη are, respectively, ξ ξ η η + = 1 and + = 1. O N1 h O N2 h Because M1 is on HN1 and M2 on HN2 , we get 1 1 sin θ cos θ sin θ cos θ = = + and + . O N1 h O M1 O N2 h O M2 Subtraction of these two equations immediately yields the general Euler-Savary formula.14 The Euler-Savary formula is often applied in a the following form. We will call it the standard Euler-Savary formula. We get it when k 2 becomes a point and it gets the following form: (

14

1 1 − OC O N1

) sin θ = C.

In 1692 Jacob Bernoulli (1654–1705) published a result on the curvature of the epicycloid that is a precursor of the Euler-Savary equation. See Bernoulli (1692).

9.8 Euler and the Euler-Savary Formula

169

Fig. 9.8 BT = a, AT = b, a + b = c, AP = p, PO = r. Source Euler (1767)

In their classic textbook Kinematics of Mechanisms from 1967, Rosenauer and Willis give it in this form.15 The constant value C in the formula is the inverse of the diameter of the inflexion circle. Its value is C=

1 1 − . O M2 O M2

9.8 Euler and the Euler-Savary Formula In Euler’s first paper on gears, On finding the best shape for gear teeth (De aptissima figura rotarum dentibus tribuenda), (Euler 1754/5), was written in the first half of the 1750s. Euler’s approach is analytical and based on differential equations. In his paper Supplement on the shape of the teeth of wheels (), (Euler 1767), written presumably 10 years after his first paper on gearing, Euler returned to the problem.16 We will explain his approach using a figure from the second paper. See Fig. 9.8. AEM is rigid and rotates about A. Also BFN is rigid, rotates about B and is pushed by AEM. Profiles FON and EOM are the shapes of the teeth. Euler first phrases the condition that if the input rotation is uniform, the output rotation should be uniform as well. In the first paper Euler expresses the uniform character of the transfer of motion by means of the condition that ∠FBA = n·∠EAB or η = n·ζ. This means that one full rotation of the wheel with center B yields n rotations of the wheel with center A. If the two profiles touch each other in the point O and ω is a point on profile NF between N and O and o a point on profile ME between O and M, pure rolling or the absence of friction means that during the motion ωO + Oo = constant.

15

Rosenauer and Willis (1967), p. 41. In the preface of Euler (1982) Charles Blanc and Pierre de Haller give a nice discussion of the two papers.

16

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Euler shows in the first paper that if we add to the condition of the uniform transfer of motion the condition that there is no friction, i.e. that the motion of the teeth with respect to each other is one of pure rolling, the point of contact necessarily remains on the line connecting the centers. This does not lead to a practically useful mechanism. So friction is inevitable. In the second paper Euler started with Fig. 9.8. The points A and B are the centers of the two wheels. EOM and FON are the two profiles of the teeth of the wheels. O is the point where the two profiles touch and the line perpendicular to the tangent in O cuts AB in the point T. When the gear wheels are functioning, a moment MA about A yields a moment MB about B. It is easy to see that at the instant under consideration the ratio of these two moments equals BT/AT because it is equal to the ratio of the arms of the two moments. Euler argues that the condition of a constant velocity ratio implies that the ratio of these two moments must be constant. As we have seen this kinematical result was already known to Camus. From a modern point of view T is the pole of the motion of the two gear wheels with respect to each other. The two polodes of the motion of the two gear wheels with respect to each other are two circles, one with center A and one with center B. The two circles touch in T. After having established that the point T is fixed, Euler determined several relations between the parameters depicted in Fig. 9.8 and differentiated. He basically considered a slight change in the position of the two profiles with respect to each other, using the fact that the common normal intersects AB always in the fixed point , the ratio of the angular velocities, is T. After some calculations this yields that dη dξ TA equal to T B . Euler then derives expressions that enables him, in principle, to calculate in an arbitrary position the radius of curvature ρ’ of profile NOF out of the parameters of profile EOM. Euler then continues with some examples. He starts with the case in which the profile EOM is on a straight line through A. His second example is the case in which EOM is on a straight line segment not through A. His third example is important. He assumes that profile EOM is a circle with center P and that the center of curvature of profile NOF coincides with Q. The radius of curvature of the second profile turns out to be: O Q = c cos ω − h − f cos ϕ −

b2 f cos ϕ cos ω . c f cos ω − a 2 cos ω

If we, moreover, introduce the footpoints R and S of the perpendiculars from, respectively A and B, on the line PQ (See Fig. 9.9), Euler can show that RT · S Q · T P + ST · R P · T Q = 0 or RT · T P TS ·TQ + = 0. RP SQ The line segments are directed. AB is oriented from A to B and RS is oriented from R to S. These are Euler’s versions of the Euler-Savary formula. With TR = TA cosω and TS = TB cos ω one can show that the result is equivalent to the modern form of the Euler-Savary equation.

9.9 The Instantaneous Axis of Rotation in Spherical Kinematics

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Fig. 9.9 Euler discovers involute gearing

Félix Savary (1779–1841) was the first to derive the Euler-Savary formula in its modern form. Savary’s proof can be found in Leçons et cours autographiés, Notes sur les machines, par le professeur F. Savary, Ecole Polytechnique 1835–36, (Savary 1835–1836). Euler also gave a construction that enables the graphical determination of the center of curvature Q of NOF if the center of curvature P of EOM is given. The formula has an amazing interpretation. It turns out that when P coincides with R, then Q coincides with S. And naturally Euler considered the possibility that this is the case during the entire motion. The profiles then are involutes of the circles CB and CA . See Fig. 9.9 right. At this moment Euler discovered what is nowadays called involute gearing, It is the most commonly used system for gearing today. Euler did not study general planar motion at a particular instant; he studied the form of the teeth of gear wheels. The general validity of the formula that he discovered is accidental spin-off. The reason is the fact that in general as for first and second order properties a planar motion at a particular instant can be represented by a circle rolling without slipping on another circle, which is exactly what we are dealing with when we have planar circular gear wheels satisfying Euler’s condition of a constant velocity ratio.

9.9 The Instantaneous Axis of Rotation in Spherical Kinematics Spherical kinematics deals with the motion of a rigid body about a fixed point. It is not unusual to study such motion by considering two coinciding spheres. When one of the two spheres is fixed and the motion of the other sphere, the moving sphere, is studied with respect to the fixed sphere, the analogy with planar kinematics is clear. Some fundamental results in what is called spherical kinematics are the following. The velocity distribution of the moving sphere with respect to the fixed sphere is always a rotation about an instantaneous axis of rotation. During the motion the instantaneous axes of rotation describe conical ruled surfaces in both the fixed and

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Fig. 9.10 The rotation axis in spherical kinematics

the moving space. The motion of these two ruled surfaces is one of pure rolling while the surfaces touch in the instantaneous axis of rotation corresponding to that position of the surfaces. The analogy with planar kinematics is clear. Euler discovered the instantaneous axis of rotation but, as we will see, it was Cauchy who realized that all spherical motion can be generated by two conical surfaces rolling upon each other. As for the discovery of the instantaneous axis of rotation Euler shares the honor with D’Alembert who showed in 1749 that at each instant the locus of points of the earth that are at rest with respect to the center of gravity of the earth is an axis of rotation.17 Euler gave the following nice geometrical proof of a more general result: Given two different positions of a body with a fixed point, it is always possible to move the body from one of the two positions to the other one by means of a rotation about an axis through the fixed point.18 Euler considered in the moving rigid body a spherical surface of which the center coincides with the center of gravity. On this surface he considers an arc AB of a great circle that moves in a time dt to a position ab on another great circle. Clearly AB = ab. Now prolong BA and ba until they meet in the point C (Fig. 9.10). C moves in time dt to position c on the prolongation of ba. Now imagine a point M outside the great circle ABC. In time dt point M moves to a point m. cm and CM wil intersect in a point O. If triangle cCO is isosceles, point O will not move in time dt. Then point O is on the instantaneous axis of rotation. However, this is a situation that we can bring about by choosing M in the right way. Spherical triangle cCO is isosceles if ∠cCO = ∠CcO. Because we have ∠cCO = ∠CcO = 1800 − ∠acO = 1800 − ∠ACO = 1800 − ∠cCO − ∠ACc, 17 18

D’Alembert (1749), pp. 82–83. Euler (1752), pp. 185–217.

9.10 Giulio Mozzi and the Instantaneous Screw Axis

173

we can draw the conclusion that ∠cCO = 900 − ½∠ACc. M must be chosen such that CM is perpendicular to the angular bisector of the angle ACc. Then O is on the instantaneous axis of rotation. Although Euler only discussed two positions of the spherical surface that are infinitesimally close, the argument also holds for two arbitrary positions. Euler discovered here in fact the rotation axis in discrete spherical kinematics.

9.10 Giulio Mozzi and the Instantaneous Screw Axis Euler discovered the instantaneous axis of rotation in spherical kinematics. The theorem says that when a moving sphere is moving with respect to a fixed sphere with which it coincides, instantaneously the motion is a rotation about an axis. When we discusses the Euler-Savary equation we mentioned the Aronhold-Kennedy theorem which says that when three coinciding Euclidean planes W1 , W2 and W3 are moving, in general, at each instant the three instantaneous centres of rotation or poles are collinear. In spherical kinematics there is an analogous theorem which says that when three coinciding spheres are moving with respect to each other the three instantaneous axes of rotation are in a plane.19 It is interesting that the Italian priest Paolo Frisi (1728–17,840, who has been called the Italian D’Alembert, worked on the composition of instantaneous rotations about two intersecting axes. He studied the motion of a rigid body in several works, In 1759 he published a Various Dissertations (Dissertatiorum Variarum). One of the texts in the book is on the rotation of the axis of the earth. There he writes “two rotational movements are combined into one movement, in exactly the same way that two forces expressed on two sides of a parallelogram combine into a third force, which is expressed diagonally.”20 Clearly Frisi had the theorem that says that the resultant of two instantaneous rotations with intersecting axes is an instantaneous rotation about a third axis. See Ceccarelli (2022). In space kinematics one considers a Euclidean space moving with respect to a fixed Euclidean space. What is the velocity distribution at a particular instant? In general it is a screw motion and the velocity of an arbitrary point consists of two components: the component that results from the rotation of space about the instantaneous screw axis, and the component that is the velocity of the slide in the direction of the screw axis. The first one to be aware of this seems to have been Giulio Mozzi (1730–1813), a student of Frisi.21 In his Mathematical discourse on the instantaneous rotation of rigid bodies (Discorso matematico sopra il rotamento momentaneo dei corpi), Mozzi (1763), published in 1763, he studies the motion of a rigid body in space. He is aware 19

For a modern treatment see McCarthy (2000), pp. 134–137. See for more details Ceccarelli (2022), p. 11. 21 My discussion of Mozzi’s contribution is based on Ceccarelli (2000). 20

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Fig. 9.11 Mozzi’s figure and Ceccarelli’s interpretation

of the existence instantaneous axis of rotation when the motion of a rigid body with one fixed point is considered. Probably he got the result from D’Alembert, whose work on the precession he mentions. Following Euler he separates in the motion of a rigid body the motion about the centre of gravity C and the motion of this centre. The velocity distribution of the space connected to the moving body consists of two elements: a rotation about the axis CS through the centre of gravity C and a translation in which all points of the moving space have the instantaneous velocity CD (Fig. 9.11). Mozzi argues that the velocity of C can be split into two components: component PD parallel to the axis CS and the component CP perpendicular to the axis CS. Now let π be the plane through C perpendicular to CS. Mozzi argues that there is in this plane appoint H where the velocity HI as a result of the rotation about CS and CP cancel each other. It is easy to see that the velocity distribution of the moving space consists of a rotation about HE, parallel to CS and a translation PD in the direction of the axis. HE is the instantaneous screw axis. Mozi calls it the ‘spontaneous axis of rotation’, which is reminiscent of Johann Bernoulli’s terminology when he discovered that when a plane disc moving in a plane is his by a force not directed towards the center of gravity, this leads to an instantaneous rotation about what he called the spontaneous center of rotation. Obviously during the motion of a mobbing space in a fixed space the instantaneous screw axis describes both in the fixed space and in the moving space a ruled surface. The two ruled surfaces are called the fixed and the moving axode. During the motion the moving axode is sliding and rolling with respect to the fixed axode. In German this particular motion is called “schroten”. Mozzi does not discuss the axodes. As we will see below also in this case it seems to have been Cauchy who was the first to look at spatial motion in this way.

Chapter 10

Theoretical Kinematics as a Subject in Its Own Right

Abstract Slowly the contours appear of a general theory encompassing the special results found earlier. The central results concern velocity and acceleration, i.e. first and second order properties in instantaneous kinematics. Cauchy wrote an important paper but in particular Chasles, who is often called the father of the geometry of motion, played an important role. We will first discuss their fundamental work and then we will look at some specific results in plane instantaneous kinematics concerning curvature given by Bobillier, Bresse and Ball.

10.1 Introduction When in 1886 Arthur Moritz Schoenflies, also written as Schönflies, (1853–1928) wrote his Geometry of motion synthetically presented (Geometrie der Bewegung in synthetischer Darstellung), Schoenflies (1886), it was his intention to write the first coherent presentation of the geometry of motion, as he said. Schoenflies noted that in the publications on kinematics of his time the emphasis was usually on velocities and accelerations, while the form and properties of the objects generated by motion do not depend on the bigger or smaller velocity of the motion. So he abstracted from time-dependent properties. Yet, the book can be considered as the first comprehensive book on theoretical kinematics. Schoenflies’ book has three chapters corresponding to three major subjects in theoretical kinematics. Chapter 1 is on plane kinematics. Schoenflies considers the two position theory, the three position theory and the four and more position theories. Two position theory deals with two discrete position. It leads, for example, to the theorem that the simplest movement bringing about the transition from one position of a moving plane in a fixed plane to another position is either a translation of a rotation. Moving positions infinitely close together leads to results in instantaneous kinematics. In the case of two positions this leads to the instantaneous center of rotation and tangent constructions. Three positions of such a moving plane lead to

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Koetsier, A History of Kinematics from Zeno to Einstein, History of Mechanism and Machine Science 46, https://doi.org/10.1007/978-3-031-39872-8_10

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the investigation of the relation between a point in the moving plane and the center of the circle through its three positions. When the positions are infinitely close together, this yields insight in curvature. The Chaps. 2 and 3 are on spherical kinematics and space kinematics where analogous results are obtained. In the nineteenth century Cauchy was the first to prove some fundamental theorems in theoretical kinematics, but in particular Chasles defined the geometry of motion as a subject in its own right. Both not only considered plane motion but also spherical and spatial motion. We will discuss these results below. Yet in the nineteenth century in the further development of the kinematics the emphasis is very much on plane kinematics. At the end of this chapter we will turn to some early nineteenth century results from planar theoretical kinematics. We will show how Bobillier did prove an elegant theorem that can be used to construct centers of curvature and how Bresse studied accelerations in instantaneous kinematics. Finally we discuss a short paper by Ball in which he introduced what are nowadays called Ball points.

10.2 Augustin Louis Cauchy’s 1827 Paper We saw in Chap. 9 how the instantaneous center of rotation, the instantaneous axis of rotation and the instantaneous screw axis were discovered, respectively, in planar, spherical and space kinematics. In a 1827 paper with the title “On the movements that an invariable system, free, or subject to certain conditions, can make” (Sur les mouvements que peut prendre un système invariable, libre, ou assujetti à certaines conditions), Cauchy (1827), Augustin Louis Cauchy (1789–1857) not only summed up these general results but he also defined what would later be called the polodes and axodes. He derives these kinematical results using the principle of virtual velocities.1 This principle says that if P1 , P2 , P3 , etc. is a system of forces working on a system of mass points in space, this system is in a state of equilibrium if and only if the sum of the virtual work of the forces is equal to zero: n ∑

Pi ωi cos(Pi , ωi ) = 0.

i=1

In this formula ωi represents the velocity of the point where the force Pi applies. The expression ωi cos(Pi , ωi ) is the component of the velocity ωi in the direction of the force Pi . Cauchy calls the term Pi ωi cos(Pi , ωi ) the moment of the force Pi . The principle is also called the principle of virtual work. However, as we will see below, only Coriolis gave the term ‘work’ its modern technical meaning.

1

For the history of the principle see Capecchi (2012).

10.2 Augustin Louis Cauchy’s 1827 Paper

177

Cauchy observes that the principle is usually applied to determine forces starting from virtual velocities. However, as he says, we can reverse the question and use it to determine velocities. He applies the principle in the following form: two systems of forces are equivalent if and only if the sums of the virtual moments of the forces of the two systems are equal. Cauchy proves the following theorems with respect to the planar motion of a rigid system: Theorem I: If at a certain moments two points of a rigid system have zero velocities, the velocities of all other points reduce to zero. Theorem II: If at a certain moments all points of a rigid system have velocities different from zero, these velocities are all equal and are directed along parallel lines. Theorem III: If at a certain moment exactly one point of a rigid system has a zero velocity the velocity of an arbitrary other point will be perpendicular to the ‘radius vector’2 drawn from the first to the second point, and proportional to this radius vector. Cauchy calls the point with zero velocity the instantaneous center of rotation. (‘centre instantané de rotation’). To illustrate Cauchy’s approach let us consider his proof of Theorem I. Imagine the points A' and A'' of a rigid system have zero velocity. Consider a point A of the system different from A' and A'' and let a force P be applied in A. Now split the force P by means of the parallelogram law in two components, P' directed along AA' and P'' directed along AA'' . Application of P' in A' and application of P'' in A'' is equivalent to application of P in A. Because the sum of the moments of P' and P'' is zero, the moment of P is also zero, which means that A has velocity zero. The proof of the first part of Theorem III runs as follows. Suppose the point O has zero velocity and the point A has velocity ω. Then a force P applied in O in the direction of OA has moment zero. We can transport the point of application of the force from O to A. Its moment is then Pω cos(P, ω) = 0. Because P and ω are unequal to zero, we get cos(P,ω) = 0, which implies ω is perpendicular to OA. Cauchy adds some remarks on this instantaneous center of rotation. He abstracts completely from material rigid systems and considers the motion of a rigid plane surface extending indefinitely that moves in a certain plane. Suppose that the point O is at a certain moment instantaneous center of rotation. It is clear then that we can imagine two curves, one in the moving rigid plane and one in the fixed plane, that consist of the points that will become later instantaneous center of rotation. Suppose after a period of time Δt point A has become instantaneous center in the moving plane and B has become instantaneous center in the fixed plane. See Fig. 10.1.

2

Cauchy says ‘rayon vecteur’. Nota bene: this is long before the word vector acquired its modern meaning.

178 Fig. 10.1 The moving polode rolls on the fixed polode

10 Theoretical Kinematics as a Subject in Its Own Right

A

B

Δs O

Cauchy argues as follows. Suppose Δs is the length of the arc OA. Arc AB is the . This average velocity arc described by A in time Δt. The average velocity of A is AB Δt is equal to the product of, in Cauchy’s words, the “average angular velocity” and “the average infinitely small distance that separates the point A from the instantaneous = ω · Δs or AB = ω · Δs · Δt. The quantities center of rotation”. So we have AB Δt Δs and Δt are infinitely small quantities of the first order. AB is, however, says Cauchy, an infinitely small quantity of a higher order than one. This implies that on the first order level arcs OA and AB will coincide and will have equal lengths. Once we have this it is obvious that during the motion the locus of instantaneous centers in the moving plane rolls without slipping on the locus of instantaneous centers in the fixed plane. Cauchy now turns to the motion of a rigid system in space and he proves the following theorems: Theorem I: If at a certain moments three points of a rigid system that are noncollinear have zero velocities, the velocities of all other points reduce to zero. Theorem II: If at a certain moment two points of a rigid system have zero velocity and all points not situated on the line that connects these two point have a velocity different from zero one can state 1. that the velocities of all points situated on the mentioned straight line reduce to zero; 2. that the velocity of a point chosen arbitrarily outside the straight line is not only perpendicular to the plane that is determined by the point and the straight line, and consequently perpendicular to their shortest distance r, but also proportional to the length r. In the course of the proof Cauchy identifies the line of points with zero velocity as the instantaneous axis of rotation. He shows that the case in which only one point of a rigid system in space has zero velocity cannot occur. Proof of the last statement: Suppose O has zero velocity and another point A has velocity ω. Let a force P act in point A in the direction of AO. The virtual moment of this force does not change if we move its point of application from A to O. Applied in O its moment is zero. So Pω cos(P, ω) = 0, which means that either ω = 0 or the angle (P, ω) = π/2. Assume we have another point A' with velocity ω' perpendicular to OA' , such that ω' and ω are not parallel. We can then intersect the two planes through A' and A perpendicular to their velocities. The point O is on the line of intersection. By applying a force R in a point C on this line of intersection and by decomposing it in three forces along CO, CA and CA' we can show that the virtual moment of this force R is zero. This implies that C has zero velocity.

10.3 Michel Chasles

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Cauchy proceeds now to the consideration of the motion of a body with one fixed point in space. He considers the motion of a spherical surface inside the body moving about the fixed point and by means of an argument similar to the one he applied in the planar case shows that such a motion can be generated by means of a spherical curve in the moving sphere rolling without slipping on a spherical curve in the fixed sphere. Cauchy’s argument culminates in the following theorem on spherical motion: Theorem III: Suppose a solid body moves in an arbitrary way about a fixed point. Suppose, moreover, that at a given moment we trace: (1) in the body, (2) in space, the different straight lines with which the instantaneous axis of rotation successively coincides. While the conical surface of which the generators are the straight lines traced in the body will be entrained by the movement of the body, it will constantly touch the conical surface of which are the generators the straight lines traced in space, and, consequently the second surface will be nothing else than the envelope of the portion of space traversed by the first. After this Cauchy starts the investigation of the case in which all points of a moving body have a velocity different from zero. In a very cumbersome way, compared to Mozzi’s method, Cauchy derives the fundamental theorem: Theorem VI: Whatever the nature of the movement of a solid body, the relations existing between [the velocities] of the different points will be those that would occur if the body was restricted in such a way that it could only turn about a fixed axis and slide along the axis. This theorem leads Cauchy to the last important theorem: Theorem VII: Suppose a solid body moves in an arbitrary way in space. Suppose, moreover, that at a given moment we trace: (1) in the body, (2) in space, the different straight lines with which the instantaneous axis of rotation successively coincides. While the ruled surface of which the generators are the straight lines traced in the body will be entrained by the movement of the body, it will constantly touch the ruled surface of which are the generators the straight lines traced in space, and, consequently the second surface will be nothing else than the envelope of the portion of space traversed by the first.

10.3 Michel Chasles Although Cauchy’s paper contains fundamental results in theoretical kinematics, his proofs are based on what is in fact an alien element in kinematics, the mechanical principle of virtual velocities. It seems to have been Michel Chasles (1793–1880) who gave the first purely geometrical proofs of the same results. Chasles wrote several important papers on the geometry of motion (i.e. kinematics minus time-dependent properties). On the basis of the totality of this work, in which he also dealt with spatial motion, both considering instantaneous, discrete and continuous situations, he may be called the father of the geometry of motion.

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Fig. 10.2 Chasles’ proof of the existence of the-(instantaneous) center of rotation

In 1829 Chasles offered a long paper to the Société Philomatique in Paris. The paper, which was only published in 1877–1878, was called “A note in geometry on the construction of normals to several mechanical curves” (Mémoire de géométrie sur les constructions des normales à plusieurs courbes mécaniques)”, Chasles (1877–78). The paper contains the first geometrical proof of the existence of the instantaneous center of rotation. Chasles considers two discrete positions of a triangle in a plane (Fig. 10.2): ABC and A’B’C’. By means of elementary geometry he shows that, if N is the intersection of the perpendicular bisectors of AA' and BB' , then N is also the perpendicular bisector of CC’. By considering two infinitely close positions Chasles draws without further proof the conclusion that, in general, when a plane figure moves in its plane all normals to the trajectories of the points of the figure drawn at an arbitrary moment, intersect in one point, the instantaneous center of rotation. Chasles’ paper is very rich in content. He, for example, also proves that the common normal of a curve in the moving plane and its envelope in the fixed plane pass through the instantaneous center of rotation. Charles applies these general results for the construction of tangents to a large number of kinematically defined curves. Chasles’ proof of the theorem that the moving polode rolls on the fixed polode runs as follows. He considers both polodes as polygons with infinitely small sides. The motion then consists of successive rotations about the vertices N, N' , N'' ,… of the fixed polode, while v, v' , v'' ,… are the corresponding vertices of the moving polode. During the motion the sides NN' , N’N'' , … successively coincide with vv' , v' v'' which, according to Chasles, proves that the moving polode rolls without slipping on the fixed polode. In 1831 Chasles read another paper to the Société Philomatique. The title was “Note on the general properties of the system of two bodies similar to each other, placed in any way in space; and on the finite, or infinitely small, displacement of a solid body” (Note sur les propriétés générales du système de deux corps semblables entre eux, placés d’une manière quelconque dans l’èspace; et sur le déplacement fini, ou infiniment petit d’un corps solide), (Chasles 1830). The paper is strictly kinematical and it generalizes Cauchy’s results. Chasles considers in the plane and in space two discrete positions of a moving body. Cauchy’s results concern the limit case when the two positions are infinitesimally close. Chasles’ considerations are

10.3 Michel Chasles

181

more general. Moreover, the body is also subjected to a magnification or a downsizing that transforms a figure into a similar figure. Chasles lists the following theorems, leaving the proofs to the reader: Theorem I: When two equal polygons are placed in an arbitrary fashion in a plane, there always exists a point of the plane which is equally distant from two arbitrary corresponding vertices of the two polygons; this point is similarly positioned with reference to the two polygons. Theorem II: When one has in a plane two similar polygons, positioned arbitrarily, there is always a point of the plane such that its distances from two arbitrary corresponding vertices of the two polygons are in a constant ratio to each other. This point is similarly placed with regard to the two polygons; and this ratio is that of two corresponding sides of the two polygons. Chasles gives no proof. I’ll sketch one. Consider Fig. 10.3. A1 B1 and A2 B2 are two straight line segments. Q is the point of intersection of A1 A2 and B1 B2 . The circumscribed circles of the triangles QA1 B1 and Q A2 B2 intersect in Q, but also in P. Repeated application of the theorem that says that the locus of points from which a given straight line segment is seen under a given angle (or its complement) is a circle yields, that a rotation about P can bring A1 B1 in such a position that if it is followed by a multiplication with center P with a factor equal to A2 B2 /A1 B1 the result is that A1 B1 is mapped on A2 B2 . A1 B1 and A2 B2 are seen from P under the same angle ρ. Moreover, because A1 P is seen from B1 under the same angle as A1 P is seen from Q while A2 P is seen from Q under the same angle as A2 P is seen from B2 the triangles ΔB1 A1 P and ΔB2 A2 P are similar. Chasles passes on to three dimensions. Theorem IV: When one has in space two similar bodies, arbitrarily situated with respect to each other: (1) There exists always in space a certain point O for which the distances to two arbitrary corresponding vertices of the two bodies are in a constant ration; this point, which is unique, is similarly placed with respect to the two bodies; that is, that if one regards it as belonging to one of the two bodies, it is itself its homologue in the second; (2) there exists always a certain line D for which the distances to two arbitrary homologous points of the two bodies are in a constant ratio Fig. 10.3 Two similar figures in a plane

P ρ ρ A1 α

A2

α ρ B2

B1

Q

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to each other; this straight line, unique, is similarly placed with respect to the two bodies; that is, that considered as belonging to the first, it is its own homologue in the second; (3) there always exists a certain plane P such that the distances from two corresponding arbitrary points of the two bodies to the plane are in a constant ratio to each other; this plane, which is unique, is similarly located with respect to the two bodies; that is, if one regards it as belonging to one of them, it will be its own homologue in the second; (4) finally, this plane and the line D are mutually orthogonal, and they pass through the point O. One of the conclusions that Chasles draws from this theorem is that when the two bodies are not similar but identical the point O and the plane P move to infinity and we have: Theorem V: When one has in space a free solid body, if one causes it to experience an arbitrary displacement, there will always exist in the body a certain indefinite straight line which, after the displacement, will be located in the same place as previously. Chasles adds: Theorem VI: One can always move a free solid body from one position to another arbitrarily determined position by the continuous motion of a screw to which this body will be invariably fixed. Again Chasles gave no proofs. We will sketch a possible proof of the theorems IV and VI. We saw in Chap. 8 of this book that Euler showed that given two different positions of a solid body with a fixed point O it is always possible to move the body from one of the two positions to the other one by means of a rotation about an axis through O. Suppose now that we have two completely different positions P1 and P2 of a body. And suppose that A1 and A2 are two homologous points, i.e. two positions of the same point. We can then subject the body to a translation that moves A1 to A2. Let us call the position of the body after translation P3. Because A2 corresponds in the positions P3 and P2 to one and the same point, we can move from P3 to P2 through a rotation about an axis through A2. Clearly we can move from P1 to P2 by means of a translation A1A2 followed by a rotation about an axis through A2. By decomposing the translation in two translations, one parallel to the axis of rotation and one perpendicular to it and by realizing that the composition of the latter and the rotation yields one rotation about an axis parallel to the former translation it is clear that Theorem VI is correct. Suppose now that we have two similar bodies. We can subject one of the two to a multiplication with an arbitrarily chosen center and a well-chosen factor such that the two bodies become congruent. A proof of theorem IV requires that we show that the composition of a screw motion with a multiplication possesses the properties given in Theorem IV. We leave this to the reader.

10.4 Bobillier’s Theorem

183

That Chasles really deserves to be called the father of the geometry of motion is clear from a series of five papers that were published in 1860 and 1861. They consist of a series of 150 theorems without proofs, most of them on two positions of a rigid body in a plane or in space.3 Without proof I will list some of them. Theorem 2: Given two positions of a line in a fixed plane such that the second position can be reached by rotating the plane in the first position about a center of rotation P. We connect all points A, B, C… of the line in position one with their homologue points A' , B' , C' … on the line in position two. Then then midpoints of all the segments AA' , BB' , CC' … are on a line. This line goes through the footpoints of P on the two homologue lines. Theorem 11: Given two positions of a moving plane in a fixed plane such that the second position can be reached by rotating the plane in the first position about a center of rotation P. Given also a point O in the fixed plan different from P. The set of points X in the moving plane that are such that the line connecting X with its homologue X' goes through O, is a circle through P, O and the homologue point in the first position of the point that in the second position coincides with O. The converse also holds: Take a circle in the first position passing through P and the lines connecting the points of this circle with their homologues in the second position will all pass through the same point on the circle. Theorem 47: Consider two positions L and L' of a straight line in space. We connect all points A, B, C… of the line in position one with their homologue points A' , B' , C' … on the line in position two. Then the planes through the midpoints of the segments AA' , BB' , CC' …perpendicular to the segments all intersect in a line λ. By means of a rotation about λ we can make L coincide with L' such that all points coincide with their homologues. Theorem 119: The displacement of a body V to a position V' can be done by two successive rotations about two different straight lines X and L, of which L is taken arbitrarily. The first rotation (about X) brings the line L of the body onto its counterpart L' , that is to say into its final position; and the second rotation which takes place around this straight line L' , makes the two bodies coincide.

10.4 Bobillier’s Theorem There is a coincidence of interests and results between Chasles and another French mathematician, Etienne Bobillier (1798–1840). See Itard (1970).

3

Chasles (1860a) contains theorems 1–32, Chasles (1860b) contains theorems 33–62, Chasles (1861a) contains theorems 63–103, Chasles (1861b) contains theorems 104–146 theorems, Chasles (1861c) contains theorems 147–150.

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Fig. 10.4 Bobillier’s theorem

It is therefore possible that the results contained in the memoire “Lois géométriques du mouvement”, which Bobillier wrote in the 1830s were found independently. That memoire is lost,4 but we have an abstract of it in Bobillier’s Geometry lessons (Cours de Géométrie), Bobillier (1837) and Bobillier (1880), handwritten lecture notes that went through many identical lithographed editions, before appearing in print long after Bobillier’s death. Bobillier proved the existence of the instantaneous center of rotation and the theorem that the moving polode rolls without slipping on the fixed polode in the same way as Chasles did it. Chasles had restricted himself to first order instantaneous properties of general planar motion. Bobillier considered also second order properties of general planar motion. Bobillier’s theorem on centers of curvature was one of the first theorems concerning second order properties to be proved for general motion. The theorem says that when at a particular instant of a planar motion P is the instantaneous center of rotation, while the trajectories of the points A and B have respectively the centers of curvature O and O' , and Q is the point of intersection of AB and OO' , then the reflection of PQ across the bisector of angle ∠APB yields the tangent to the polodes in P. See Fig. 10.4. Bobillier gives his theorem in the form of a problem: “Being given the centers of curvature f, g corresponding to the points a, b of the trajectories described by the vertices a, b of the triangle abc, determine the corresponding center of curvature s of the trajectory of the third vertex c”. See Fig. 10.5.5 Bobillier reasoned as follows. Let fg and ab intersect in k (Fig. 10.5). A very small rotation about the instantaneous center of rotation o moves a to a' and b to b' . We obviously have aa' oa = bb' ob

4 5

Itard (1970), p. 215. Bobillier (1837), p. 166. Also in Bobillier (1880), pp. 232–233.

10.4 Bobillier’s Theorem

185

Fig. 10.5 Figure from Bobillier’s Géométrie6

Let a’f and b’g intersect in o' . Then oo' coincides with the tangent in o to the polodes. Let p and q be the footpoints of the perpendiculars from o' to af and bg respectively. Then, because pf and of , and also qg and og differ very little, of bb' bg o' p = and = aa' af o'q og Dropping kq' , kp' and oh perpendicular to respectively bg, af and ab we can write ak oh ob kp' = and = oh oa kq' bk Multiplication of the above five equalities yields o f · ak · bg o' p · kp' = o'q · kq' f a · kb · go Application of Menelaus’ theorem to the Δoab cut by the transversal fk we have o f · ak · bg =1 f a · kb · go

6

Bobillier (1880), p. 232.

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So kq' o' p = o'q kp' Or o' op = kob while we also have o' op = moc. Immediately we have kob = moc and it easy to see how s can be constructed. First one constructs m and then mf intersects oc in s. Bobillier’s proof is another example of pre-twentieth century reasoning with infinitesimals. In Chap. 19 of this book we will give a twentieth century proof of the same result.

10.5 Jacques Antoine Charles Bresse In the first half of the nineteenth century there is a growing interest in second order properties. From 1831 to 1840/1841 Felix Savary (1797–1841) taught the course on machines at the Ecole Polytechnique. Sometime before 1836, while dealing with gear wheels, he derived the general Euler-Savary formula (see Chap. 9 of this book) for epicycloidal motion. See Savary (1835–1836). The equation became a standard result in planar theoretical kinematics in 1845. In 1845 Abel Transon derives the formula for arbitrary polodes without mentioning Euler or Savary. See Transon (1845). This result, however, is less general than Euler’s or Savary’s because Transon only deals with the radius of curvature of the trajectory of a point in the moving plane. Chasles reacted to Transon’s paper, pointing out that Savary had proved earlier the more general equation for the radius of curvature of the envelope in the fixed plane of the collection of positions of a curve in the moving plane. See Chasles (1845). It seems that at this time Euler’s 1765 paper was forgotten. From 1837 until April 1848 Jean Victor Poncelet (1788–1867) taught classes on ‘physical and experimental mechanics’ at the Faculté des Sciences in Paris.7 Opposing Lagrange’s entirely analytical approach Poncelet defended the use of more geometrical methods in mechanics. In his classes he started to handle not only velocities but also accelerations as directed line segments. Under the influence of Ampère his approach was, moreover, kinematical. In Poncelet’s work the acceleration of a moving point in the plane is related to the radius of curvature of the trajectory of the point as follows. If a point has velocity v and ρ is the radius of curvature, then the projections of the acceleration vector on the tangent and the normal are, respectively, v2 dv and . dt ρ

7

Taton (1975) and the appendix of Résal (1862).

10.5 Jacques Antoine Charles Bresse

187

Fig. 10.6 OE OD, O' E OE, ϕ = ∠DOB, p = OD, ds = OO'

Jacques Antoine Charles Bresse (1822–1883) was influenced by Poncelet. Unaware of Bobillier’s construction (it remained unnoticed until in 1858 when Mannheim drew attention to it) he set out to determine the radii of curvature of the trajectories of points in a moving plane by means of the properties of their accelerations. See Bresse (1853). He knew the Euler-Savary equation but found its application cumbersome in situations where the radii of curvature of the polodes are not given. In his 1853 paper Bresse considers (Fig. 10.6) two consecutive instants t, t + dt of the motion of a moving plane. The corresponding instantaneous centers of rotation are O and O' and the corresponding angular velocities are ω and ω + dω. Let D be the position at instant t of an arbitrary point in the moving plane and D' its position at instant t + dt. Then the angle between the two consecutive normals OD and O' D' equals D D' − O E 1 = ( p · ω · dt − ds · cosϕ). p p This expression equals zero if p=

1 ds · · cosϕ. ω dt

This equation corresponds to the inflexion circle c1 , (Fig. 10.6), which was (re)discovered by Bresse. Except O all points of c1 are in an inflexion point of their trajectory and have, consequently, a normal acceleration equal to zero. In his 1853 paper Bresse also considers the tangential acceleration of the point D. We differentiate the velocity pω and we get dp dω dpω =ω +p . dt dt dt

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Because (Fig. 10.6) dp = −O’E = −ds·sinϕ this acceleration is zero if p = ds . This equation corresponds to the circle c2 in Fig. 10.6. ω · sin ϕ dω This circle is nowadays sometimes called the normal circle. The point of intersection C of c1 and c2 (different from O) has a total acceleration equal to zero. Bresse calls it the “second instantaneous center of rotation”, because the accelerations of all points in the moving plane are equal to the accelerations that they would have if the plane would rotate about C with angular velocity ω and angular acceleration dω/dt. The point C is nowadays usually called the acceleration pole. The similarity between the instantaneous center of rotation O and the acceleration pole C is considerable. Let ω be the angular velocity. Then the velocity OX·ω of a point X in the moving plane is proportional to OX and for all points X perpendicular to OX. The acceleration pole C has a velocity vC . When we subject the moving plane to an extra translation with velocity − vC , point C becomes the instantaneous center of rotation while the acceleration distribution does not change. This is an application of the classical principle of relativity. See Chap. 17 of this book. After having added the extra translation the tangential acceleration is CX·(dω/dt) and the normal acceleration is CX·ω2 . Clearly the total acceleration is proportional to CX and its angle with CX has for all X the same value, also during the original motion. Bresse applies these results in his paper to determine radii of curvature of trajectories of moving points. Because the centers of curvature are time-independent, we can assume that dω/dt = 0. If dω/dt = 0 then the tangential acceleration becomes zero, OA becomes infinite and C coincides with B. Then (Fig. 10.7) the total acceleration of D is directed along DB and equals ω2 ·DB. The component of the acceleration of D along DO then equals ω2 ·DF, which means that we have for the radius of curvature ρ of D’s trajectory ρ=

O D 2 ω2 O D2 = . D Fω2 DF

Given O, this relation can be used in two ways. If the radii of curvature of the trajectories of two points in the moving plane are known, the corresponding points F can be determined and B can be constructed. Conversely, if B is known for an arbitrary point D the corresponding F can be determined and then one has ρ. Fig. 10.7 When dω/dt = 0 the acceleration pole coincides with B

10.6 The Ball Points

189

Bresse shows in a number of examples (e.g. the planar four-bar linkage) how to apply these results.

10.6 The Ball Points Let’s take also here a step forward in time and consider a short paper from 1871 by Ball. See Ball (1870–74). In the paper Robert S. Ball (1840–1013) proves the following Theorem: “A plane figure is moving in a plane according to any law. There are always two points in the figure, or rigidly attached to it, so that the four consecutive positions of each of these points is in the same straight line”. He notes that we can always generate the motion of a moving plane by the rolling of one curve upon another curve. Then he notes that for his purposes the two curves can be replaced by a circle with radius R' with center B rolling on a circle with radius R with center A. He gives us Fig. 10.8. The instantaneous center of rotation is O. A point P in the moving plane is turning about O and reaches P' . The center of rotation has then moved to O' . Let the angle O’AO be ω. Ball writes that it is easy to see that P P' = ω · O P ·

Fig. 10.8 Ball’s argument

R + R' R'

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10 Theoretical Kinematics as a Subject in Its Own Right

Fig. 10.9 Two points of Ball or only one? Source Ball (1871)

And indeed we have O' O = Rω. This corresponds to an arc of the same length on the rolling circle seen from its center B under an angle Rω/R' . Clearly the moving circle rotated about O over an angle ω(R + R' )/R' . To see this: first rotate the rolling circle on the spot about its center over the angle Rω/R' and then rotate the circle about A over an angle ω to give it its second position. Addition of the two angles gives the required result. Ball now considers the case that O' P' is parallel to OP. This means that P' continues to move in the direction PP' which in its turn means that P is in an inflexion point of its trajectory. This is the case—see the drawing on the right—in Fig. 10.8—when O O' sin θ = Rω · sin θ = ω · O P ·

R + R' R'

This implies OP =

R R' sin θ. R + R'

This equation obviously represents the inflexion circle with diameter OQ = RR' / (R + R' ). The tangents to the trajectories of its point all pass through Q. Ball then gives the following interesting argument. See Fig. 10.9. Ball assumes that the circle with center B rolls until the point O'' is the point of contact. The inflexion circle OPQ gets a new position: circle LXY (See Fig. 10.9). Q is a point in the fixed plane that remains behind. Ball wants to find on this circle the points that continue to move on lines through Q. So he draws in the new position in the moving plane the circle with diameter O'' Q. In Ball’s words: “Let O'' be the new instantaneous center. Join O'' Q and upon O'' Q as diameter describe a circle, the points XY will still continue to move on lines through Q. Hence four consecutive positions of XY will lie upon the same straight line”. The tangents to the trajectories of the points on this last circle pass through Q which explains Ball’s argument. So Ball finds two points X and Y of which four consecutive positions are on a straight line. Such points are now called ‘Ball points’. Yet, as we will see in the last chapter of this book, there is in general only one point of Ball. Ball’s approach shows the weakness of pre-twentieth century reasoning with infinitesimals. The other point of intersection that Ball finds is the instantaneous center of rotation, which is in a cusp of its curve.

Chapter 11

Towards a New Theory of Machines

Abstract First we discuss Lazare Carnot’s theory of machines. Carnot realized that the ‘geometrical movements’ of the parts of a machine represent an independent important research area. Then we discuss Gaspard Monge’s kinematical classification of machine elements. He defined a machine element as a mechanism that transforms motion. Finally we discuss Coriolis’ discovery of the Coriolis’ acceleration while he studied the applicability of the principle of living force to relative movement. The centrifugal and Coriolis accelerations can be seen as purely kinematical phenomena.

11.1 Introduction Imagine three countries called A, B and C, inhabited respectively by Mechanians, Machinians and Geomonians. See Fig. 11.1. At a certain moment a prophet appears who teaches that in the area where the three countries meet, the inhabitants are not really Geomonians, Mechanians or Machinians, but something else: they are actually Kinamanians. The leader urges that all Kinamanians unite and form a new country: country D. In such a situation all kinds of interesting questions can be asked. What is going to happen in the future? How are the populations of A, B and C going to react? All kinds of scenarios are imaginable: the Kinamanians get restricted autonomy in the three countries, they are exterminated, they take over the whole world, etc. One also wonders what the position of these Kinamanians was in the past. Did they already view themselves as different? The prophet cannot have come out of the blue. Obviously in course of time the borders between A, B and C need not always have been the same and the names of the peoples may have been subject to change. This product of my fantasy gives us a way to look at the history of kinematics. As we will see in the next chapter André-Marie Ampère (1775–1836) was the prophet who realized that parts of rational mechanics, the theory of machines and geometry deserved a different name: kinematics. At least that is a way to understand his message. As we have seen in geometry motion has always played an important role. This was especially so in classical antiquity and in the seventeenth century. And although © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Koetsier, A History of Kinematics from Zeno to Einstein, History of Mechanism and Machine Science 46, https://doi.org/10.1007/978-3-031-39872-8_11

191

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11 Towards a New Theory of Machines

A B

Mechanics

D C

Theory of Kinematics machines Geometry

Fig. 11.1 Kinematics at the tripoint of three disciplines

repeatedly results were obtained that specifically referred to properties of motion, this was not enough to feel the need to combine these results in a special subscience of geometry. Initially mechanics was just the name of the theory of machines but in the seventeenth and eighteenth centuries theoretical or rational mechanics was gradually formed as a general theory dealing with the motion of mass points and rigid bodies. And as we have seen inevitably kinematical properties of motion were discovered. But also here no need was felt to create a new subscience of mechanics called kinematics. As for the theory of machines the situation is different. In the eighteenth century the theory of machines was basically identical with the theory of the simple machines like the lever, the wheel and axle, pulleys, the wedge or inclined plane, and the screw. This theory dates from classical antiquity and reached maturity with Galilei.1 Yet in the eighteenth century a lot happened. The Industrial Revolution led to many new mechanical inventions and a need arose for theoretical considerations. In this chapter we will concentrate on this development. As we will see it was the background of Ampère’s proposal. It is remarkable that in the 1770s the Frenchman Lazare Carnot (1753–1823)2 developed a unique and very general theory of machines and while he did so argued that a special science of geometrical movement in machines ought to be created. We will devote several sections of this chapter to Carnot’s theory. The foundation of the École Polytechnique in 1794 was in this respect also important, because Gaspard Monge (1746–1818) was in charge and he decided that a course on ‘elementary machines’ had to be included in the curriculum. No longer the simple machines but elementary machines became the building blocks of a complex machine. And, moreover, the elementary machines were classified on the basis of their geometrical characteristics. Also here we see a shift away from forces and masses towards the geometry of machines. Yet forces and masses undeniably play a role in the functioning of machines and in the first half of the nineteenth century there were other attempts to create a science of machines. We will briefly discuss the work of Jean-Victor Poncelet (1788–1867) and 1 2

Koetsier (2019), pp. 159–160. For more on Carnot see Gillespie (1971).

11.2 Lazare Carnot

193

Gaspard Gustave de Coriolis (1792–1843). Coriolis discovered the Coriolis force, which is based on a kinematical phenomenon: the Coriolis acceleration. The discussion creates some more context for Ampere’s proposal to define a new science called kinematics and its reception.

11.2 Lazare Carnot3 It is remarkable that the idea to create such a new science was born when the Frenchman Carnot decided to define what a machine essentially is. Lazare Carnot, born in Nolay in the Bourgogne, was trained at the military school in Mézières where Monge was one of his teachers. As a lieutenant in the Royal Corps of Engineers, after having graduated on January 1, 1773, Carnot wrote his An essay on machines in general (Essai sur les machines en general), Carnot (1783), stimulated by prize contests organized by the Academy of Sciences in Paris. Carnot did not win, but was awarded honorable mention in 1781. He turned his treatise for the Academy into a book which appeared in 1783. At the request of others, as he wrote himself, Carnot published in 1803 an elaborated version of his 1783 essay with the title Fundamental principles of equilibrium and movement (Principes fondamentaux de l’équilibre et du movement), Carnot (1803). This book does not contain anything new compared to the early version, but it makes the understanding of the early concise version easier. Before Carnot beyond the theory of simple machines, there was no coherent theory of machines. This is understandable because machines are complex mechanical entities and the only general theory dealing with moving masses, rational mechanics, was still very much under construction. For example, a famous eighteenth century issue concerned the question whether ‘momentum’ = mass times velocity, or ‘living force’ = mass times velocity squared, was the fundamental quantity conserved when motion was transferred. In the collision theory of both hard and elastic bodies momentum was considered to be conserved. In collision theory of perfectly elastic bodies the living force was conserved as well, but not in the collision theory of ‘hard bodies’. The notion ‘hard body’ is important in eighteenth century mechanics. See Scott (1959). The ultimate corpuscules in nature were imagined to be completely impenetrable. Hard solid matter was considered to consist of such corpuscules connected to each other by perfectly rigid rods. In elastic matter the rods were imagined as springs. Carnot’s approach to machines is unique in its generality. He argued that there are two ways to consider the principles of mechanics. The first one is to define mechanics as the theory of forces. The second one is to define mechanics as the theory of motion. The notion of force, however, is for Carnot a metaphysical notion unless it is defined in terms of empirical notions like mass and velocity. A theory of machines should be based upon a theory of the communication of movements, according to Carnot.

3

Parts of this chapter are based on Koetsier (2004).

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11 Towards a New Theory of Machines

For Carnot a machine is a connected system of hard bodies. The connections between the bodies constrain the movement of the bodies. The geometry of the system determines which motion is possible and which is not. The movement of the bodies is constrained in order to allow for only those movements that serve the purpose of the machine. This view of what a machine is led Carnot to two conclusions. The first one was that the world needed a theory dealing with the possible motions of machines. He called it a theory of geometrical movements. He left it to others to develop this theory. The second conclusion was that understanding a machine means that one understands how motion can be transferred from one hard body to another hard body. This is what Carnot concentrated on.

11.3 Collisions of Hard Bodies and Geometrical Movements Imagine that in a machine at a particular moment a hard body is, for example, in touch with another hard body which it pushes forward. Such interactions can take place smoothly (in Carnot’s words “by imperceptible degrees” (par dégrées insensibles) or by means of collisions. Because according to Carnot such smooth interaction can be viewed as the result of a sequence of infinitesimally small percussions, we need only one theory for both cases. At a particular moment the masses of the hard bodies in a machine all have a certain velocity. One moment later, after interaction between the bodies, the velocities have changed. In the context of these changing velocity distributions he introduced his theory of geometric movements. Consider the following quotation: “The theory of geometrical movements is very important; it is […] a kind of science intermediate between ordinary geometry and mechanics. It is the theory of the movements that an arbitrary system of bodies can make in such a way that they do not hinder each other or exert some action or reaction on each other. This science has never been dealt with in particular. This science must be created completely and it deserves for its beauty and its utility the full attention of the scholars; because the great analytical difficulties one meets in mechanics and especially in hydraulics are only caused by the fact that a theory of geometrical movements has not been created at all.”4 Carnot was proud of his discovery of the notion of geometric movement (mouvement géométrique). He defines geometric movements as movements that do not have any effect on the interaction of the bodies on each other; they do not violate the constraints of the system. From the definition he deduces that the inverse of a geometric movement is another geometric movement. He also points out that the combination of two geometric movements is also a geometric movement. One can obviously distinguish between discrete and infinitesimal geometric movements. Infinitesimal geometrical movements are virtual velocity distributions. 4

Carnot (1803), p. 116.

11.4 The First Fundamental Equation

195

Obviously not all velocity distributions are geometrical movements. A velocity distribution that would tear a system apart is not a geometrical movement.

11.4 The First Fundamental Equation According to Carnot the essence of the interaction of the rigid bodies in a machine can be understood completely by studying at a certain moment in time the collision of two of such hard bodies. Imagine we have a body of mass M, which possesses before the interaction velocity W and after the interaction velocity V. Then we have a body of mass M' , which has before the interaction velocity W' and after velocity V' . Nota bene: Carnot treats velocities in fact as vectors although he does not possess the full notational apparatus of the vector calculus. The body with mass M' hits the body with mass M. Their surfaces touch in a point where they have a common tangent. The collision only concerns the components of the velocities along the perpendicular to this common tangent. By definition U = W − V and U' = W' − V' (these are vector equations) are the velocities respectively lost or gained by the two masses in the collision. See Fig. 11.2. These velocities U an U' are directed along the normal to the touching surfaces. Z is the angle between respectively U and V; Z' is the angle between U' and V' . Carnot handles the interaction of the bodies in terms of what he calls the fundamental laws of the theory of collisions: (i) Action is Reaction directed along the normal to the touching surfaces. Action and reaction are determined by multiplying a mass with a velocity. (ii) After the shock the relative velocity of two touching surfaces in the direction of the normal is zero, in other words, in the direction of the normal the velocities of the two bodies are equal. Hard bodies do not bounce. Fig. 11.2 Interaction of two rigid bodies

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11 Towards a New Theory of Machines

Carnot’s law of action is reaction boils down to the conservation of momentum in the direction of the perpendicular to the common tangent. The idea to treat a system of interacting bodies in this way comes from d’Alembert’s Treatise on Dynamics (Traité de Dynamique).5 This leads to the following two equations: MU = M' U' , ) ( Vcos Z = V' cos 180◦ − Z' = −V' cos Z' . The first equation expresses that the loss of momentum by M' equals the momentum gained by M. The minus sign in the second equation is necessary because the angle Z' is obtuse. In these equations U, U' , V' and V are scalars and no longer vectors. The first equation is based on the principle Action is Reaction. The second equation is based on the principle that after the interaction the bodies have equal relative velocities in the direction perpendicular to the common tangent. These two equations yield by multiplying them the following scalar equation: MVUcosZ + M' V' U' cos Z' = 0. It is striking that in the Essai Carnot immediately generalizes to a multi-body system and writes his First fundamental equation6 ∑

MVU cos < U, V >= 0.

N. B. In ‘cos ’ U and V are vectors and is the angle between them; in ‘MVU’ the V and U are scalars. The fundamental equation is powerful. For example, we have for all vectors W, V, U with W = V + U, the scalar equation: W2 = V2 + U2 + 2VU cos < V, U >, which implies with the fundamental equation: ∑

MW2 =

∑

MV2 +

∑

MU2 .

This implies that ∑MU2 represents a loss of energy and on the basis of this scalar formula Carnot draws the conclusion that if we avoid collisions and the motion in a machine only changes smoothly, “par degrees insensibles”, ∑MU2 is so small (of the second order, he says) that we can ignore it and we get

5 6

Dugas (1988), pp. 248–253. Carnot (1783), pp. 15–22 and Carnot (1803), pp. 131–143.

11.5 The Second Fundamental Equation

∑

MW2 =

197

∑

MV2 .

The conclusion is that percussions or other sudden changes in the functioning of a machine should be avoided. They cause the loss of living force, “déperdition de forces vives”. The idea would survive him. J. A. Borgnis wrote in 1823, the year of Carnot’s death, in his Dictionary of mechanics applied to the arts (Dictionaire de mécanique appliqué aux arts) that Carnot had advocated to boost the efficiency of machines by avoiding to-and-fro motions and eliminating sudden impacts.7

11.5 The Second Fundamental Equation Carnot’s handling of his second fundamental equation nicely shows how he uses the notion geometric movement in order to get results. Consider the first one: ∑

MVU cos < U, V >= 0.

In this formula U = W − V is the velocity distribution lost when the interaction turned an initial velocity distribution W into a velocity distribution V. W was not necessarily a geometric movement. Yet V definitely is. Let u be an arbitrary geometric movement. Then u − V is a geometric movement as well (the inverse −V of V is one and the sum of u and −V is one). Because a geometric movement does not interfere with the interaction of the bodies, we can simply superpose it on the system before an interaction. Then we get an initial velocity distribution W + (u − V). The resulting velocity distribution after interaction will be V + (u − V) = u. The velocity distribution that is lost will be W + (u − V) − (V + (u − V)) = W − V = U. So application of the first fundamental equation yields the Second fundamental equation: ∑

MuU cos < U, u >= 0.

Valid for an arbitrary geometric movement u and all possible distributions U of velocities that can be lost because of interaction. This Second fundamental equation is a powerful formula as well. Carnot proves that after sudden change in the state of a system the geometrical movement ∑ that actually results is the geometrical movement that corresponds to a minimum of MU2 . Proof: Suppose that instead of the movement V actually occurring after the sudden change the movement V − u' (u' is an infinitesimal geometrical movement) occurs. We do not change W, the movement before the sudden interaction of the hard bodies. The new velocity lost is then, obviously, U' = W − (V − u' ) = V + U − V + u' = U + u' . So u' is the vector U increases with, as a result of the change in V. Then we 7

Scott (1959), p. 204.

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11 Towards a New Theory of Machines

have: ∑

Mu' U cos < U, u' >= 0 or

∑

MUu' cos < U, u' >= 0,

but then, u' cos 3 if the centers of (n − 2) systems with respect to the other two systems are given, all other centers with their corresponding velocity distributions can be constructed. Very often less centers are needed. Burmester showed that for n even, the position of a set of 23 n − 2 well-chosen centers (e.g. no three on one line) determines the position of the other centers. In his 1883 paper Grübler refers to this result and argued as follows. If all centers can be uniquely determined than the system has internally one degree of freedom. Burmester’s result implies that if we have a chain consisting of an even number n of links, connected by means of g = 23 n − 2 turning joints that do not have special positions, the chain has one internal degree of freedom. This is one of Grübler’s central results: a kinematical chain consisting of n links connected by g turning points has one degree of freedom if 2g − 3n + 4 = 0.

4

Schadwill (1876). Rotate the velocity vectors of all points on a line l through an angle of 90 degrees and the new endpoints will be on a line parallel to l.

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16 Ludwig Burmester, Kinematics as Part of Geometry

Let us look at Grübler’s own reasoning in his 1883 paper and in his book of 1907.5 The reasoning in his book is not essentially different from what he does in his 1883 paper. Suppose we have in a closed kinematical chain of n links, connected by means of g cylindrical joints. Grübler distinguishes binary, ternary, quaternary etc. links depending on the number of ways they are connected with cylindrical joints to other links. Let n2 denote the number of binary links, n3 the number of ternary links etc. This means that  ni = n i=2

where n denotes the total number of links. Initially Grübler’s approach is very general. The cylindrical joints may also join more than two links. Grübler calls them twofold, threefold, etc. g2 denotes the number of twofold joints, g3 the number of threefold joints etc. This means that g2 + g3 + · · · gi + · · · = g, where g is the total number of joints. Yet Grübler soon restricts himself to kinematical chains with twofold joints. So will we and below g = g2 . In chains without two-fold joints the number of hinges is related to the number of links as follows:  in i = 2g. i=2

Grübler introduces a coordinate system in which the coordinates of joint Ghp which joins the planes Eh and Ep are (x hp ,yhp ) and the coordinates of joint Ghq which joins the planes Eh and Eq are (x hq ,yhq ). The connection between the two hinges Ghp and Ghq is rigid, because they are both in plane Eh . This gives a rigidity condition of the form 

x hp − x hq

2

2  + yhp − yhq = constant.

The number ni is the number of rigid links connected to i hinges. To guarantee that in this link the position of the hinges with respect to each other does not change we need 2i − 3 of such rigidity conditions. For the whole chain this gives us  i=2

equations.

5

Grübler (1883) and Grübler (1907).

(2i − 3)n i

16.6 A Note on Chebyshev

273

Grübler differentiates the equations and gets 

     x hp − x hq δx hp − δx hq + yhp − yhq δyhp − δyhq = 0

The total number of variables and also the number of differentials involved equals the number of coordinates 2 g. We can fix one of the links of a kinematical chain by choosing three of the differentials equal to zero. When the chain is constrained fixing one more  differential should determine all the 2 g − 4 others. This means that by means of i=2 (2i − 3)n i independent equations 2 g − 4 unknowns ought to be determined. Conclusion:  (2i − 3)n i = 2g − 4. i=2

We saw above how the number of links is related to the number of hinges and we get Grübler’s formula. 2g − 3n + 4 = 0. It is Grübler’s necessary and sufficient condition for a closed kinematical chain with cylindrical joints to have one degree of freedom, i.e. to be constrained.

16.6 A Note on Chebyshev Actually Grübler was not the first to come up with this formula. More than ten years earlier in 1869 Chebyshev presented a paper with the title “On the parallelograms” to the Imperial Academy of Sciences of Saint Petersburg. See Chebyshev (1870). Chebyshev denotes an arbitrary straight line mechanism with the word ‘parallelogram’. Such a parallelogram is, he writes, a system of m straight lines moving in a plane and connected by means of n hinges, that each connect two straight lines. Moreover, the system is connected by means of hinges to v fixed points. In order to fix the position of one straight line three 3 variables are needed. The existence of a hinge corresponds to two equations. Chebyshev concludes that the number of independent variables equals 3 m − 2(n + v). So the condition that must be satisfied in order to have one degree of freedom is: 3m − 2(n + v) = 1. This is exactly Grübler’s result. In the literature the criteria to determine the number of degrees of freedom are sometimes called the Chebychev–Grübler–Kutzbach criteria. This includes analogous formula for spatial mechanisms. In 1929 Karl Kutzbach (1875–1942) gave a

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16 Ludwig Burmester, Kinematics as Part of Geometry

formula for the mobility of general spatial mechanisms. See Kutzbach (1929). For a more modern approach see Waldron (1966).

16.7 Grübler on Classifying Kinematical Chains Grübler immediately draws some conclusions. The condition implies, because 3n = 2g + 4 implies that n cannot be odd, that closed kinematical chains with cylindrical joints of freedom have an even number of links. Moreover, substitution  and one degree of i=2 n i = n and i=2 in i = 2g in Grübler’s formula yields. n2 = 4 +

 (i − 3)n i i=4

This implies that in closed kinematical chains with cylindrical joints and one degree of freedom the number of binary links is independent of the number of ternary links and is at least 4.   Grübler’s approach with the formulae 2g − 3n + 4 = 0, i=2 n i = n and i=2 in i = 2g yields in principle a way to classify kinematical chains with cylindrical joints and one degree of freedom. We know that n must be even. For n = 2 we get g = 1. This does not yield a closed chain. For n = 4 we get g = 4. The 4 joints must be binary. The chain is the four bar linkage. When n = 6 we get g = 7. From the equations n2 + n3 + n4 etc. = 6 and 2n2 + 3n3 + 4n4 + etc. = 14 we get n2 = 4 and n3 = 2. This gives us two closed chains, Watt’s chain and Stephenson’s chain. See Fig. 16.4. The names were coined by Burmester. Stephenson’s chain clearly has the abstract structure of Stephenson’s valve mechanism. Watt’s chain has the structure of a simplified version of one of Watt’s engines if we piston sliding inside the cylinder as a cylindrical joint. For n = 8 we get g = 10 and the equations n2 + n3 + n4 + etc. = 8 and 2n2 + 3n3 + 4n4 + etc. = 20 give us three groups of solutions. See Fig. 16.5.

Fig. 16.4 Watt’s chain and Stephenson’s chain. Source Grübler (1917), p. 17

16.8 The Burmester Theory and the Burmester Points

275

Fig. 16.5 Grübler’s 12 closed constrained 8-link kinematical chains with cylindrical joints. Source Grübler (1917), p. 17

Grübler’s Figs. 22a, b, c, d, e, f correspond to n2 = 4, n3 = 4 and n4 = 0. Grübler’s Figs. 23a, b, c, d correspond to n2 = 5, n3 = 2, n4 = 1 and his Figs. 24a and b correspond to n2 = 6, n3 = 0 and n4 = 2.

16.8 The Burmester Theory and the Burmester Points In Euclidean discrete position theory one studies two or more discrete positions of a moving Euclidean space in a fixed Euclidean space. Also in this area Burmester contributed considerably. As we have seen around 1830 Chasles studied two discrete positions in the plane and in space. Chasles generalized his results by allowing the moving space to undergo a similarity transformation.

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16 Ludwig Burmester, Kinematics as Part of Geometry

As far as I know the first treatment of three discrete positions 1, 2 and 3 in the plane was given by Auguste Grouard6 in 1870 in a session of the Société Philomatique in Paris. See (Grouard 1870). Grouard considered three discrete positions 1, 2 and 3 of a planar figure that is moved and at the same time subjected to a similarity transformation. His results were published without proofs. Three positions lead to three similarity poles P12 , P23 , P31 that form the pole triangle. Grouard discovered that the circumscribed circle of the pole triangle plays an important role. Consider a line l in the moving space. It coincides in the three positions with the lines l1 , l2 , l3 that are called homologous lines. Consider the triangle formed by three homologous lines and connect its vertices with the three corresponding similarity poles (the point of intersection Xij of lines i and j is connected with Pij ). This gives three concurrent lines that intersect in a point on the circumscribed circle. Grouard also gives the theorem that all lines containing three homologous points intersect in one point. In 1876 and 1877 Burmester published an important paper on discrete position theory. The paper consists of three parts. The paper is on the design of four-bar straight-line mechanisms by considering discrete positions of a moving plane. It all started when in 1876 Burmester read a paper by Kirsch on a new method to design a four bar straight line linkage. Kirsch considered a series of discrete positions of a given four bar mechanism and constructed for a large number of positions of the coupler plane the inflexion circle. He then chose in the moving plane a point P close to many points of intersection of the inflexion circles. Afterwards Kirsch analytically varied the dimensions of the mechanism in order to reduce the deviations of the curve described by P from a straight line as much as possible. This is an interesting method, although not very reliable. As Burmester remarked afterwards, the method gives no real control over the radius of curvature of the curve described by P. Yet the paper must have given Burmester the idea to consider more than two discrete positions of a moving plane. Chasles, who was primarily interested in instantaneous motion, had in that context answered most of the obvious questions concerning two discrete positions both in the plane as in space. Kirsch’ method, although inferior, suggested a new approach to straight-line mechanisms by considering more than two discrete positions. In Burmester (1876) Burmester, starting from a given four-bar mechanism, first considers three discrete positions S1 , S2 and S3 of a moving plane. And then he considers four positions. The idea is to find in the moving plane points that are in the three positions and subsequently in the four positions on a straight line. The curve described by such points in a sense approximates the straight line. We will not describe Burmester’s approach to the Euclidean three position theory. Instead we will sketch the rather elementary treatment that was given by Stephanos in Stephanos (1881–1882).

6

Auguste Antoine Grouard (1843–1929), pupil of the Ecole Polytechnique, became a colonel in the French army and a well-known expert on Napoleonic strategy.

16.8 The Burmester Theory and the Burmester Points

277

Fig. 16.6 The triangle of the three poles

When three positions are given there are in general three poles P12 , P23 , P31 . A rotation about Pij moves an object from position i to position j. It turns out that the triangle of the three poles has nice properties. See Fig. 16.6. All sets of homologous points H1 , H2 , H3 can be obtained by reflecting one particular point H in the three sides of the pole triangle. Clearly the angles of rotation are twice the angles of the pole triangle. This implies that when three homologous points are on a straight line also the footpoints of the perpendicular lines from H on the three sides will be on a straight line. The Simson-Wallace theorem says that this is the case if and only if the point H is on the circumcircle of the pole triangle. This answers Burmester’s first question: The points that are in three discrete positions on a straight line are on a circle in the moving plane. The positions in the fixed plane of the circle in the three positions can be obtained by reflecting the circumcircle of the pole triangle in the sides of that triangle. We can now add a fourth position of the moving plane. See Fig. 16.7. The points that are in positions 1, 2 and 3 on a straight line are in position 3 on the circle c3. The points that are in the positions 2, 3 and 4 on a straight line are in position 3 on the circle d3. These two circles intersect in two points: P23 and a second point B3 which is called the Ball point after R. S. Ball who discovered the point for the instantaneous case as we saw in Chap. 10 of this book. B3 is with the homologous points B1 and B2 on a straight line and at the same time with B2 and B4 on a straight line. Clearly B1 , B2 , B3 and B4 are on a straight line. In the second part of the paper Burmester attacked the more general problem of the loci of points that are in a number of discrete positions on a circle. First he considers three positions and proves that the locus of the centers of the circles that are determined by triples of homologous points on three homologous lines is a conic section through the three poles. This theorem immediately yields: There exists either one or three circles that go through four homologous points on four homologous lines. The circles correspond to real points of intersection of conic sections. By applying this theorem to the homologous lines of a pencil the locus of centers of quadruples

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Fig. 16.7 Four discrete positions

of four homologous points are the points of intersection of corresponding elements of two pencils of conic sections. Burmester proves analytically (the methods in the rest of the paper are synthetic) that this center point curve (‘Mittelpunktskurve’ in German), as he calls it, is a circular curve of third degree. He also shows that it is a so-called focal curve, which is the locus of the foci of all conic sections that touch four given straight lines.7 A focal curve is determined by four lines or four points; Burmester shows that two pairs of counterpoles, e.g. P12 P34 and P23 P41 , completely determine the center point curve (Pij Pkl is by definition a pair of counter poles, if i, j, k, l are all different). Burmester then shows that the locus of points that are in four positions on a circle is also a focal curve, which he calls the ‘Angriffskurve’ (‘angreifen’ means ‘to catch’ or ‘to take hold of’), because it is the locus of points where a bar can be attached to the moving system. From the text it is clear that Burmester’s ideas are still in development. Burmester identifies the fixed plane with position 1, which means that he does not seem to realize that by considering the situation from the point of view of the moving plane, we are also dealing with four positions of the fixed plane with respect to the moving one and it is because of that obvious that the ‘Angriffskurve’ must be of the same nature as the ‘center point curve. Finally Burmester considers five positions and he finds the points that are all five positions on a circle by intersecting two focal curves. Because from the nine 7 Burmester writes that the focal curve was investigated analytically by Salmon (in his Conic sections) and Eckart (Zeitschrift f. Math. u. Phys., Bd. 10, S. 321) and later synthetically by Schröter (Math. Ann. Bd. 5, S. 50 und Bd. 6, S. 85) and Durège (Math. Ann. Bd. 5, S. 83).

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points of intersection the circle points and three poles must be subtracted, he easily proves the existence of the four points that were later called the Burmester-points by R. Müller. In 1879 L. Geisenheimer published a paper in which he studied primarily equiform instantaneous and continuous planar kinematics. Yet, referring to Grouard and Burmester, on page 137 he proves that in discrete three position theory the poles P31 and P2 31 are symmetrical with respect to the segment P12 P23 . This implies that by reflecting the circumscribed circle of the triangle of the three poles in three sides of that triangle one obtains the ‘inflexion-circle’ in the three positions. It also implies that these inflexion-circles intersect in the orthocenter of the triangle. In his book Burmester’s treatment8 of planar discrete position theory is a slightly more elegant version of the treatment in his papers (1876 and 1877). He no longer uses the name ‘Angriffskurve’, but called it ‘circle point curve’. One of the admirable characteristics of Burmester’s work is the fact that he combined a great interest in theoretical results with an interest in applications. His 1877 paper on discrete position theory is no exception. The Burmester theory is immediately applied to Stephenson’s link mechanism for controlling the steam valve of a locomotive. The treatment of the discrete position theory given in the Lehrbuch der Kinematik is a somewhat more elegant summary of the results of the 1876 and 1877 papers (Fig. 16.8).

16.9 On the Reception of Burmester’s Work Burmester’s book is part of the development of scientific mechanical engineering in the nineteenth century. Before the nineteenth century machines had been studied traditionally in mechanics. When in the eighteenth century rational mechanics was created, the investigation of machines had become an application of rational mechanics. Because of the difficulties involved the results were limited. In the nineteenth century it became clear that the geometrical aspect of machines and mechanisms could be investigated very successfully. Yet the investigation of machines remained applied mathematics or applied mechanics. There was no independent science of machines with a status comparable to the status of the established disciplines. Moreover in the nineteenth century the theory of machines was taught on the European continent at polytechnical schools, which in general had a lower status than the universities. Scientific mechanical engineering was not born easily. For example, in 1877 J. Lüders wrote a booklet of 88 pages with the title Wider Herrn Reuleaux (Against Mr. Reulaux) in which he described Reuleaux‘ Theoretische Kinematik as a book without any value, not written for the specialist but meant to impress the public at large. Others criticized Reuleaux as well, because they failed to see the value of his work for mechanical engineering in practice. 8

Burmester (1888), pp. 602–623.

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Fig. 16.8 Figures accompanying the treatment of the Burmester theory in the 9th chapter of the Lehrbuch der Kinematik. The curve σ in Fig. 634 is the center point curve. In Fig. 638, σ consists of the line at infinity and a hyperbola

In the second half of the nineteenth century one can distinguish in Germany three different views of technology.9 The first view, represented by Franz Grashof (1826– 1893), for a long time chairman of the Association of German Engineers (Verein Deutscher Ingenieure), defines technology in the tradition of French positivism: 9

Hensel et. al. (1989), p. 167.

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technology is applied natural science and applied mathematics. Burmester must have shared this view. In the second view, represented by Reuleaux, the machine is in the development of mankind the essential element that determines man’s relation with nature. Reuleaux can indeed be seen as one of the first philosophers of technology, who attempted to characterize the general development of mankind on the basis of the fundamental notion of machine. As a result of this position he emphasized the need for an independent, unified theory of the machine. This theory would reserve a precise place for its application. The third view, represented by Alois Riedler (1850– 1936), is opposed to the dominance of theory in the first two views. For Riedler technology is more a socio-economic system of which theory is only one of the components. “Knowledge is a daughter of application, not the other way around”, Riedler wrote (Hensel et. al. (1989), p. 180). Words like ‘reality’, ‘organisation’ and ‘labour’ frequently occurred in his writings. He also opposed the specialization that resulted in curricula at technological universities that consisted of completely separate courses. What Grashof, Burmester and also Reuleaux, however, had in common, was the belief in the value of theoretical considerations and in deduction from the general to the particular. The difference was that Reuleaux defended a general theory of machines, independent of natural science and mathematics. Riedler was skeptical of all general theory. Another representative in Germany of the latter view was Th. Beck (1839–1917), factory owner and engineer from Darmstadt. Poppe had given the following definition “By machines, we mean those artificial arrangements by which motions may with advantage be produced, prevented, or transmitted in definite directions.”, and Reuleaux had commented: What has advantage to do with science? (Reuleaux, (1876), p. 586). Beck argued that the main task of mechanical engineering is to understand how to build a machine in such a way that it yields the greatest advantage (Hensel et. al. (1989), p. 199). When Reuleaux and Burmester wrote their books the gap between the theory of machines and the practice of building, using and maintaining machines was still considerable. This created tensions and Reuleaux felt under attack from Riedler for not being practical enough, while, on the other hand, he felt that his work was not taken seriously enough by people like Grashof and Burmester. In this particular position Reuleaux was unable to appreciate the value of Burmester’s book.

16.10 Reuleaux’ Criticism of Burmester In general Burmester’s book was well received. For example in 1886 Schumann wrote a very positive review of the first two installments in the Jahrbuch für die Fortschritte der Mathematik and in 1887 he repeated his positive opinion with respect to the third installment. There was, however, one remarkable exception. In 1889 Reuleaux wrote: His (Burmester’s—T. K.) book represents a phenomenon that probably has not yet occurred in our literature. As the main result of my investigation I must point out that the book does not contain one new thought, not even a small one, in the area covered by the title of the

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book. Yes, in order to honor the truth I must say that none of the laws of kinematics is treated completely correctly. Mr. Burmester turns out to be an amateur in kinematics. (Reuleaux (1894), p. LIII, italics are mine-T.K.).

This quotation is preceded by almost 20 (sic!) pages of severe criticism. Some of the criticism is undoubtedly justified. Wherever Burmester saw a possibility to apply his general theoretical considerations to a particular mechanism he did so, even if such an application seemed miles away from the practice of machine building. In this respect Burmester’s book was written clearly by a mathematician and not by an engineer. However, Reuleaux mainly criticized Burmester for using the word kinematics in an improper way. The above given quotation in fact says: the book does not contain any new contribution in the area of knowledge that I, Reuleaux, call kinematics; it may contain many new thoughts concerning other areas of knowledge. The whole criticism concerns the use of the word kinematics! Burmester defined kinematics in 1886 when he announced his book as follows: “Kinematics, which includes the geometrical theory of motion and its application to machines, was born from the connection of geometry with the notion of motion.”10 Reuleaux had defined kinematics as ‘Zwanglauflehre’, which can be translated with ‘the theory of constrained motion’. Kinematics was from Reuleaux‘ point of view identical with the theory of mechanisms. Reauleux wrote that kinematics is: “The science that deals with the question how a machine should be composed such that the movements, i.e. changes of position, of the parts with respect to each other are completely determined” (Reuleaux (1894) p. XVI). Reuleaux distinguished between theoretical and applied kinematics. Applied kinematics dealt with the design of machinery in practice on the basis of theoretical kinematics. It is clear that Reuleaux had defined kinematics as the core discipline in the new emerging science of machines that he envisaged. In Reuleaux‘ view of kinematics the geometry of motion was most useful but it played in kinematics only a supporting role. Kinematics included in this view much more than geometrical considerations. Reuleaux‘ goal in life was the creation of a science of machines with a status comparable to the other sciences. That is why he gave his Theoretische Kinematik a deductive structure; he had tried to make the science of machines with kinematics at its heart as rigid as Euclid’s Elements. In Reuleaux‘ perspective Burmester did not simply use the word kinematics in a different way, no, Reuleaux saw Burmester’s book as an attempt to reduce the science of machines to merely applied mathematics. In 1890 he wrote about Burmester and others: “At the same time geometry imagined, that is its self-appointed imperialist protagonists imagined, to easily unharness the numerous recently derived kinematical theorems with the right of the conqueror and drive them like patient lambs into their sheepcote.”11 Reuleaux felt he had to defend the science of machines against enemies like Burmester. It seems that Reuleaux sent his criticism to the Rector and the senate of the Technische Hochschule in Munich and later he even sent a copy of his 20 10 11

Quoted in Reuleaux (1894), p. XXXII. Reuleaux (1890), p. 248.

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page criticism of Burmester’s book to the Royal Bavarian Ministery of Education in Munich (Königliche Bayrische Kultusministerium) so that the ministery would see how incompetent a professor they had on their payroll. Burmester, well-positioned in Munich, reacted to this with a public defense, which obviously did not satisfy Reuleaux. Actually Burmester was not the real enemy. When he criticized Burmester Reuleaux was professor in Berlin at the Technische Hochschule and he was at the top of his power. In 1890–1891 he was even rector of the Hochschule. In 1888, however, Alois Riedler had become professor at the Hochschule. Riedler had as for mechanical engineering completely different views. He defended a curriculum with much less theory and more practical exercises. Riedler was a formidable opponent who clashed vehemently with Reuleaux. In the end Riedler won. He succeeded in drastically reducing the number of classes devoted to mathematics and other theoretical subjects. In 1896 Reuleaux retired and soon kinematics was no longer an obligatory subject in Berlin. H.-J. Braun characterized the situation in mechanical engineering in late nineteenth century Germany as a conflict about the correct method. That is what it was. Yet, one gets the impression that Reuleaux‘ self-image included infallibility which in combination with his overambitious theoretical program and his willingness to fight his opponents any time anywhere at great lengths, turned the opposition between points of view into real battles. In particular in Berlin the clash was very serious. Elsewhere things developed differently. It is interesting to compare the dramatic developments in Berlin with the situation in Dresden in the same period.12 In Dresden kinematics classes started in 1870. They were given by Ernst Hartig (1836–1900). Hartig was a specialist and pioneer with respect to measurement and test engineering. As we have seen Burmester became professor in Dresden for descriptive geometry in 1872. In 1874 Reuleaux‘ pupil Trajan Rittershaus became professor of kinematics and electro-technical machines in Dresden. In the same period Otto Mohr (1835–1918), specialist on graphical methods in statics and strength of materials, was professor of technical mechanics for the civil engineers. Moreover, in 1873 Gustav Zeuner (1828–1907), had become director of the Polytechicum in Dresden. Instead of fighting each other these men stimulated each other in different ways. Although in the course of time the subject played a lesser role in the curriculum, for example, because dynamical aspects of machines became more important, in Dresden kinematics continued to be an important discipline. In 1894 Mohr had succeeded Zeuner, who had returned to teaching, as professor of technical mechanics. In 1900 the successor of Otto Mohr was Martin Grübler (1851–1935). A younger representative of the kinematical tradition in Dresden is Hermann Alt (1889–1945) who worked on Burmester theory as well. Burmester’s work was generally well received. Lorenzo Alievi (1856–1941) wrote his treatise Cinematica della Biella Piana (Kinematics of the plane coupler) in Rome

12

Cf. Mauersberger (2001).

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in 1892. He published the book in Naples in 1895.13 It deals with the higher order properties of the curvature of the four bar coupler curve. Allievi was familiar with Burmester’s Lehrbuch der Kinematik and with Schoenflies‘ Geometrie der Bewegung of 1886. Schoenflies had pointed out that Burmester’s discrete position theory implied analogous elegant results in instantaneous kinematics, for example, points of stationary curvature are on a curve of the third degree, the instantaneous analogon of Burmester’s circle point curves. Moreover, in instantaneous kinematics the analytical methods of the differential calculus can be applied. Allievi drew the conclusion that the higher order properties of the curvature of the four bar coupler curve ought to be accessible to an analytical investigation. Allievi started at the second order level with the Euler-Savary equation and by differentiating he reached higher order results. This was a good idea. Half a century later A. E. Richard de Jonge rightly praised Allievi’s book. See Jonge (1943), p. 667.

16.11 Some Nineteenth Century Developments Elsewhere While in German Reuleaux and Burmester did their impressive work, considerable interest in kinematics existed also in France. We have seen that among the French mathematicians Michel Chasles (1793–1880) played a highly stimulating role in the development of the geometry of motion. Chasles was followed by Victor Mayer Amédée Mannheim (1831–1906). In 1880 Mannheim published in Paris his Cours de géométrie descriptive de l’École Polytechnique comprenant les éléments de la géométrie cinématique. The part on kinematics is on kinematic geometry. Mannheim wrote on kinematically generated curves and surfaces and their properties. Also Jean Gaston Darboux (1842–1917) worked in kinematics following Chasles. One of his fascinating results is the so called Darboux motion. This motion is a nontrivial spatial motion with only planar trajectories. Darboux motions can be obtained by composing a planar elliptic motion with a harmonic oscillation. The result is that all trajectories are ellipses. He described it as follows. A cylinder C rolls inside of a cylinder C’ with a radius twice as big while C slides up and down in such a way that points on C describe straight lines intersecting the axis of C’. Darboux proves that there are no other non-trivial spatial motions with only planar trajectories. His argument is roughly as follows. All points of a moving space describe planar curves and assume that for all planes in the fixed space there is a point of the moving space that describes a curve in it. We can then consider the inverse motion in which all planes pass through a fixed point. By considering two parallel planes it becomes obvious that all these planes passing through a fixed point envelope necessarily a cone of revolution. Darboux then applies the according to him obvious truth that the order of the trajectories of the points in a given movement is equal to the class of the 13

Alievi (1895).

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developable surface of the planes in the inverse movement. Because the developable surface is a circular cone and has class two, the trajectories of the points must be conics.14 Darboux published his result in 1890 and later in a note on algebraic movement added to a classic book on kinematics by Gabriel Xavier Paul Koenigs (1858– 1931).15 The book is Leçons de Cinématique, professées à la Sorbonne, Cinématique théorique, Paris, 1897. In a sense, in terms of kinematics, Koenig’s book closes the nineteenth century in France. For an abstract of the book I refer to (Lovett 1900). It is interesting that in 1926 another French kinematician, Raoul Bricard (1870–1943) showed that there is a spatial motion such that all point paths are spherical curves. The proof appeared in the first volume of his Leçons de cinématique.16

14

See also Jüttler (1993). Darboux (1890) and Koenigs (1897), pp. 352–390. 16 Bricard (1926), pp. 308–312. 15

Chapter 17

Albert Einstein, the Kinematics of Special Relativity

Abstract In classical kinematics the motion of an object is a function from a timeaxis to the set of possible positions of the object in space. Time is absolute in the sense that the time-axis is an omnipresent clock. When it is 12 o’clock in the origin, it is 12 o’clock in the entire space. In 1905 Einstein published a paper that shook the world. In this chapter we discuss the birth of the theory of special relativity. In order to explain the phenomena it turned out to be necessary to give up the absolute character of time and not only that. We also describe the twin-paradox.

17.1 Introduction It is important to distinguish between physical space, physical time, physical motion, on the one hand, and their mathematical models on the other hand. Until the twentieth century physical space was generally modelled as a three-dimensional Euclidean space. Time was modelled as a one-dimensional Euclidean space, the time-axis. A mathematical model of the motion of an object was a mapping of a time-interval on a set of positions of an object in space. In the course of time the context in which this was done changed considerably. For the astronomers in antiquity the physical universe had the shape of a huge sphere. Outside of this sphere there was nothing. In mathematics space was defined along the lines of Euclid’s Elements as a potentially infinite whole. Time was viewed as a one-dimensional quantity comparable to the length of a line segment. In the sixteenth and seventeenth centuries the Greek view of the universe was completely revised by Copernicus, Kepler, Galilei and Newton. Physical space was by Newton modelled by means of an actually infinite 3-dimensional Euclidean space and time by a one-dimensional space. In classical antiquity the axis of the sphere of the fixed stars and the earth did not move. They were absolutely at rest. A man sitting on the deck of a moving ship was at rest, but only relatively, not absolutely because he moved with respect to the earth. In the seventeenth century the Sun had become the center of the universe instead of the earth, but the view of the physical universe as an infinite Euclidean space made this © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Koetsier, A History of Kinematics from Zeno to Einstein, History of Mechanism and Machine Science 46, https://doi.org/10.1007/978-3-031-39872-8_17

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center look arbitrary. This change of perspective suggested for Newton and others the question whether absolute rest and motion exist or whether all motion and rest are relative. Newton had a very clear opinion on this matter. A Scholium in the Principia, positioned between the Definitions and the Laws of Motion, lays out his views on time, space, place, and motion. Space is absolute and exists independently of the existence of bodies. Time is also absolute and passes without relation to anything external. Absolute motion is a change in position in absolute space of an object. Absolute time and absolute space are separate aspects of objective reality. Others, like Leibniz had completely different ideas. For Leibniz space and time are not like containers in which things are, but systems of relations between things. Space is a way to order coexisting things and time is a relation of succession. Leibniz used the analogy with a genealogical tree which is an abstract system of relations between relatives. The bottom line is, however, that until the twentieth century space and time were viewed as separate aspects of reality. This changed in 1905 when Albert Einstein introduced the kinematics of special relativity. In this chapter we use the word ‘kinematics’ in the broad sense. A phenomenon is kinematical in the broad sense if it is independent of the specifics of the dynamics. It is kinematical in the narrow sense if it is an example of standard spatio-temporal behavior.1 Einstein also used the word ‘kinematics’ in the broad sense when he gave the title ‘Kinematical part’ to the section of his 1905 paper that we will be discussing. We saw above in Chap. 12 that one of the areas where the word kinematics started to play a role was theoretical mechanics. In textbooks it usually referred to the introductory sections on velocity and acceleration before masses and forces were introduced. That is still the case. Martínez has argued in 2009 that “the lost origins of Einstein’s relativity” lie in kinematics. What does that mean? In the nineteenth century some theorists, not only mathematicians but also physicists viewed theoretical mechanics as a mathematical theory. Martínez argues that in the course of the century physicists realize that theoretical mechanics is a physical theory meant to explain what we observe in physical reality. In the words of Martínez: “Belief in the abstract mathematical truth of kinematics was replaced by belief in the necessity of grounding concepts on perceptions or on feasible or idealized procedures of measurement.”2 For example, measuring velocity requires measuring time at different locations and this requires synchronous clocks at these locations. Einstein’s kinematics of special relativity emerged indeed from this background.

1

I owe this distinction to Janssen (2009), p. 28. For some authors this is a sensitive issue. They seem to find it misleading to use the word kinematics in the broad sense of the word. See also Janssen (2009). 2 Martínez (2009), p. 121.

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17.2 The Principle of Relativity In order to define the position of an object we need a frame of reference, for example a coordinate system, with respect to which we define the position. In classical mechanics not all reference systems are equivalent. We will call a reference system in which Newton’s Law of Inertia holds an inertial frame of reference. This law says that a body at rest remains at rest and a body in uniform motion remains in that motion unless acted on by force. Not all frames of reference are inertial. For example, a rotating frame of reference in which masses experience a centrifugal acceleration is not inertial. One of Einstein’s starting points was what I will call the classical principle of relativity which says that in all frames of reference moving uniformly with respect to an inertial frame of reference the laws of Newtonian mechanics hold. Years later Einstein illustrated it by imagining a physicist travelling with his laboratory in a completely closed railway wagon at a constant speed. By means of experiments based on Newtonian mechanics the physicist will not be able to determine whether the wagon moves or not because Newton’s laws are independent of the translation of the reference system. At the end of the nineteenth century roughly speaking physics consisted of two branches, mechanics and the theory of electromagnetism. Thermal phenomena could be understood by means of mechanics and the theory of electromagnetism encompassed optics, electricity and magnetism. Einstein was dissatisfied with the electro-magnetic foundation of physics as it was given, for example, by Lorentz, because it lacked simplicity.3 It did not satisfy certain aesthetic criteria. The problem, Einstein’s dissatisfaction, was to a large extent caused by methodological considerations. It was his opinion that if an asymmetry is inherent in the phenomena, we must accept it. However, asymmetries inherent in the theory but not in the phenomena are unacceptable. In his 1905 paper Einstein gives in the first sentences an example of a situation that did not satisfy him: “That Maxwell’s electrodynamics—the way, in which it is usually understood—when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena, is well known.”4 Einstein then gives an example of such an asymmetry. If we move a conducting ring in the neighborhood of a magnet at rest we generate a current in the ring. If we move the magnet instead of the ring in such a way that the relative motion of the two is the same as in the first case, we generate the same current in the ring. However, the electrodynamics of Einstein’s time gave in the two cases different explanations for the occurrence of the current. If we move the magnet, the changing magnetic field surrounding the magnet produces according to Maxwell’s theory an electric field, which in its turn produces in the conductor a current. If we do not move the magnet there is no changing magnetic field in the neighborhood of the magnet. However, now it is the electromotive force in the conductor that brings about the same current as in the other case. 3

Cf. John Stachel’s essay ‘What Song the Syrens Sang’: How Did Einstein Discover Special Relativity? in Stachel (2002). 4 Einstein (1905), p. 891.

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Moreover, the asymmetry is based on the essential difference between being at rest relative to the ether and moving relative to the ether. Yet this asymmetry cannot be detected in the phenomena. Einstein wrote: “Examples of this sort, together with the unsuccessful attempts to discover any motion of the earth relative to the ‘light medium’, lead to the conjecture that to the concept of absolute rest there correspond no properties of the phenomena, neither in mechanics, nor in electrodynamics […] (italics are mine—T. K.)”.5 The unsuccessful attempts that Einstein refers to very probably included the Michelson Morley experiment.6 This experiment was designed and sensitive enough to detect the orbital motion of the earth about the Sun. No such motion was detected. Einstein decided to generalize the classical principle of relativity so that it would also encompass electrodynamics. A physicist travelling with his laboratory in a completely closed railway wagon at a constant speed would not only be unable to determine whether the wagon moves or not by means of experiments based on Newtonian mechanics. Experiments based on electrodynamics would not be able to help the physicist find out whether the wagon moves either. The generalized principle became Postulate 1 in his 1905 paper: For every reference system in which the laws of mechanics are valid, the laws of electrodynamics and optics are also valid. Einstein called it the Relativity Principle. It was according to Einstein an inevitable conclusion if one looked at the available empirical data.

17.3 The Principle of the Constancy of Light and the Paradox Einstein added Postulate 2: The Principle of the Constancy of Light. It says: Light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body. This second postulate only states that light behaves like a wave and as such seems harmless. Yet the two postulates together lead to a paradox. The two postulates imply that light propagates in every reference system in which the laws of mechanics are valid (every inertial system) with the velocity c. With the usual Euclidean interpretation of velocity this implies c = c + v for any v. This is the problem that Einstein set out to solve in the years preceding the publication of the paper in 1905. His solution is a typical example of a major conceptual breakthrough. In his autobiography of 1946 Einstein seems to depict the breakthrough as the result of several factors: First there was a problem: the nature of light and in general the nature of electro-magnetic waves. Secondly a methodological view of what theoretical physics ought to be. It ought to be based on universal

5

Ibidem. It has been questioned whether the Michelson-Morley experiment did indeed influence Einstein in a direct way. It probably did. See Van Dongen (2009).

6

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formal principles. And then thirdly, his success in finding two seemingly incompatible fundamental principles: the principle of the constancy of the velocity of light and the principle of relativity (both based on experience according to Einstein). Then there is the nature of the breakthrough itself. Einstein wrote in 1946: “All attempts to clarify this paradox satisfactorily were condemned to failure as long as the axiom of the absolute character of time, viz., of simultaneity, unrecognizably was anchored in the unconscious. Clearly to recognize this axiom and its arbitrary character really implies already the solution of the problem. The type of critical reasoning which was required for this central point was decisively furthered, in my case, especially by reading of David Hume’s and Ernst Mach’s philosophical writings.”7 The crucial elements in the breakthrough were: The skeptical views of Hume and Mach with respect to the status of concepts. Both philosophers had warned that concepts only have meaning in so far as they are grounded in sense experience. Applied to space and time this led to the willingness to give up parts of classical Euclidean kinematics, in fact in particular the willingness to give up the axiom of the absoluteness of time. The solution to the paradox is found by Einstein in replacing Euclidean kinematics by a new un-orthodox kinematics. In Einstein’s paper the kinematical solution of the paradox immediately follows the introduction of the paper and subsequently the results are applied to electrodynamics: he applies it to radiation reflected from a moving mirror, and he briefly studies the consequences of his new kinematics for the dynamics of moving electrons.

17.4 The Willingness to Give Up the Axiom of the Absoluteness of Time Hume and Mach were skeptic philosophers in the sense that for them concepts are fictional in so far as they extend beyond sense experience. Einstein read Hume’s Treatise of Human Nature. Hume wrote about concepts and in particular of the limitations of human conceptualization. When writing about space and time Hume wrote, for example, “We may hence discover the error of the common opinion, that the capacity of the mind is limited on both sides, and that ‘t is impossible for the imagination to form an adequate idea, of what goes beyond a certain degree of minuteness as well as of greatness” (Book 1, Part 2, Section 1). An analysis of the crucial concept in Einstein’s paper, simultaneity, is completely absent in Hume. Yet, when one reads Hume, one can understand how the emphasis of the fact that concepts are based on our limited sense-experience and that it is dangerous to assume their validity beyond this experience, impressed Einstein.

7

Einstein’s autobiography in Schilpp (1951), p. 53.

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The same holds for Ernst Mach (1838–1916), who held similar skeptical views. In 1883 he published The science of mechanics, a critical and historical account of its development (Die Mechanik in ihrer Entwicklung historisch kritisch dargestellt) in which he argued that “A motion is termed uniform in which equal increments of space described correspond to equal increments of space described by some motion with which we form a comparison, as the rotation of the earth. A motion may with respect to another motion be uniform. But the question, whether a motion is in itself uniform is senseless.”8 Mach here rejected the notion of absolute space as senseless because it went beyond what sense experience tells us. It is clear how such ideas will have influenced Einstein. Mach had been pointing out the fictional character of Newtonian concepts without, however, demolishing them. He had been preparing the ground for Einstein. The two postulates created an interesting paradoxical situation: A Euclidean space k moves with velocity v with respect to another Euclidean space K with which it coincides. Something, one unique thing, an electromagnetic wave, light, moves with velocity c with respect to both spaces! It is not possible, unless we redefine one or more elements of the situation! From the point of view of Euclidean kinematics this paradox cannot be solved. The absolute character of time is so built-in, so fundamental in the Euclidean frame-work that the idea of the relativity of time will not pop up. Remarkably enough, as we will see, Hermann Minkowski (1864–1909) in 1908 suggested that the special theory of relativity could have been developed by a pure mathematician. Yet Einstein’s paradox was not a mathematical paradox, it was a physical paradox. Einstein viewed the situation in terms of measurements and not in terms of pure mathematics. The measurement of velocities involves the measurement of distances and time intervals. Imagine an experiment to check out the paradoxical situation. Imagine we actually measure the speed of a light wave in both systems and we measure the speed of the origin of System k with respect to system K. We execute these measurements very accurately. Is it possible that we discover that the speed of light is in both cases c and the origin of System k moves with a velocity v relative to System K? In our measurements time plays a central role. Einstein in his paper: “We have to take into account that all our judgments in which time plays a role are always judgments of simultaneous events. If, for instance, I say, “That train arrives here at 7 o’clock,” I mean something like this: the pointing of the small hand of my watch to 7 and the arrival of the train are simultaneous events.”9 The measurements of the velocity of light in System K and in System k and the measurement of the velocity of System k with respect to System K involve clocks. How do we determine whether two clocks, Clock 1 and Clock 2, positioned at different locations in some system, are synchronous? Einstein gives the following 8 9

Mach (1907), p. 224. Miller (1981), p. 393.

17.4 The Willingness to Give Up the Axiom of the Absoluteness of Time

293

definition of synchronicity in a system: The clocks are synchronous if always when a light signal sent at time τ 0 (Clock 1-time) from Clock 1 to Clock 2, reflected at time τ 1 (Clock 2-time) from Clock 2 back to Clock 1, returns at time τ 2 (Clock 1-time), we have τ I = 1/2(τ0 + τ2 )

(17.1)

This means the time of reflection τ 1 (on Clock 2) is always exactly in the middle of the time interval [τ 0 , τ 2 ] on Clock 1. This looks like a perfect definition if the velocity of light is an absolute constant as Postulate 2 guarantees. This definition enables one to establish synchronicity in System K and in System k. Joseph Sauter, a colleague of Einstein at the Patent Office in Zürich in 1905 later remembered that Einstein had told him: “Suppose one of the two clocks is on a tower at Bern and the other on a tower at Muri. At the instant when the clock of Bern marks noon exactly, let a luminous signal depart from Bern in the direction of Muri; it will arrive at Muri when the clock at Muri marks a time noon + t; at that moment, reflect the signal in the direction of Bern; if on the moment when it returns to Bern the clock in Bern marks noon + 2t, we will say that the two clocks are synchronized.”10 I will call such an experiment to check whether two clocks are synchronous a synchronicity experiment. We know that five weeks before Einstein finished the text of his 1905 paper he had a sudden and crucial insight: synchronicity can be established in System K and in System k, but it need not be the same synchronicity: observers in relative motion may disagree on which events are simultaneous. Einstein gave a lecture in Kyoto on December 14, 1922, in which he told that the paradox defined by his two postulates had given him almost a year of fruitless thoughts. He said: “But a friend of mine living in Bern helped me by chance. On beautiful day, I visited him and said to him: ‘I presently have a problem that I have been totally unable to solve. Today I have brought this ‘struggle’ with me.’ We then had extensive discussions, and suddenly I realized the solution. The very next day, I visited him again and immediately said to him: ‘Thanks to you, I have completely solved my problem.’ My solution actually concerned the concept of time. Namely, time cannot be absolutely defined by itself, and there is an unbreakable connection between time and signal velocity. Using this idea, I could now resolve the great difficulty that I previously felt. After I had this inspiration, it took only five weeks to complete what is known as the special theory of relativity.” Einstein’s friend was Michele Besso, the only individual mentioned by Einstein in the 1905 paper. In the nineteenth century in particular the railroads needed coordinated time. Also Germany was in 1891 still struggling with the problem. Clock coordination could be done simply by carrying around a clock bearing the correct time. In 1891, however, electrical systems were used as well. Peter Galison quoted from Count Helmuth von Moltke’s last address to the parliament on March 16, 1891: “That unity of time is indispensable for the satisfactory operation of railways is universally recognized, and is not disputed. But, meine Herren, we have in Germany five different units of 10

Josef Sauter, 50 Jahre Relativitätstheorie, Reprinted in Flükiger (1974), p. 156.

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time. In north Germany, including Saxony, we reckon by Berlin time; in Bavaria, by that of Munich; in Württemberg, by that of Stuttgart; in Baden by that of Carlsruhe; and on the Rhine Palatine by that of Ludwigshafen. […] This is, I may say, a ruin which has remained standing out of the once splintered condition of Germany, but which, since we have become an empire, it is proper should be done away with.”11 In Einstein’s 1905 paper on relativity the problem of the synchronization of clocks plays a central role. Galison argued that at the start of the twentieth century the practical problem of clock coordination was such an important problem that it is hard to believe that it had no effect on Einstein. Moreover, Einstein was at the time working at the Bern patent office where he entered a world in which patents concerning coordinated clocks and the networks of electrical chrono-coordination passed by.

17.5 Checking the Inspiration We do not know how Einstein spent the night between the two visits to Michele Besso. He must have checked where his sudden inspiration would lead to. Very probably he will have started with simple thought experiments of the kind that he describes in the second section of his 1905 paper. In that section Einstein first repeated the two principles: 1. The laws by which the states of physical systems undergo changes are independent of whether these changes of state are referred to one or the other of two coordinate systems moving relatively to each other in uniform translational motion. 2. Any ray of light moves in the ‘resting’ coordinate system with the definite velocity c, which is independent of whether the ray was emitted by a resting or by a moving body. Here is velocity = light path/time interval where time interval is to be understood in the sense of the definition of §1.” (In §1 Einstein had discussed the notion of synchronicity.). Einstein then imagines a rigid rod AB with finite length lying along the x-axis of a coordinate system. Let RAB be the length of the rod measured in the resting system. He imagines that the rod gets a constant velocity v in the direction of the positive x-axis. What then follows in the paper is in fact the description of a synchronicity experiment. We imagine two clocks at the ends A and B of the rod that synchronize with the clocks in the resting system. The clocks indicate at any instant the time in the resting system corresponding to the places where they happen to be. 11

Galison (2000), p. 365.

17.6 The Technical Development in the 1905 Paper

295

We now introduce two observers in the moving system, one in A and one in B. They check the synchronicity of the two clocks in A and in B. Let a ray of light depart from A, let it be reflected at B and let it reach A again, at respectively, time t A, time t B and time t' A . Because of the principle of the constancy of the velocity of light, although the light ray is released from a moving point, in the resting system it covers first the distance (t B − t A ) · c and then the distance (t A ' − t B ) · c. However, ) these distances ( are equal to, respectively, R AB + (t B − t A ) · v and R AB − t A' − t B · v. So we get tB − t A =

R AB R AB and t A' − t B = . c−v c+v

(17.2)

Clearly t B − t A /= t A' − t B . Conclusion: For observers in the resting system the two clocks are synchronous, because that is how they are set. Yet, observers in the positions A and B in the moving system traveling with the rod will conclude that the two clocks are not showing the same time. As long as velocity v is small the difference between the values of the two expressions will be very small and in practice unnoticeable. The next question that Einstein wants to answer is the following. What is in general terms the relation between the time in the moving system and the time in the resting system? We will sketch the solution given in the 1905 paper.

17.6 The Technical Development in the 1905 Paper Introduction In his 1905 paper Einstein considers two systems, System K, the resting system and System k, the moving system. In System K we have a Cartesian coordinate system OK , x, y, z and time t. In System k we have a Cartesian coordinate system Ok , ξ , η, ζ and time τ. System k moves with respect to System K with a constant velocity v (measured with respect to K) in the direction of the x-axis. The motion is such that at a certain moment the coordinate system Ok , ξ , η, ζ coincides with the OK , x, y, z coordinate system. During the motion an event occurs in Ok , with coordinates ξ , η, ζ at time τ. With respect to OK the same event has coordinates x, y, z at time t. Einstein’s goal is to determine the transformation rules: ξ = ξ(x, y, z, t) η = η(x, y, z, t) ζ = ζ (x, y, z, t) τ = τ (x, y, z, t)

(17.3)

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17 Albert Einstein, the Kinematics of Special Relativity

Einstein assumes that the equations must be linear because space and time are homogeneous. The homogeneity of space and time means that for the laws of physics all positions in space and all moments in time are equivalent. First Einstein concentrates on the time in the moving system: What is τ = τ (x, y, z, t)? We know that the clocks in the moving system are synchronous. This means that when we do a synchronicity experiment in the moving system for the clocks we have the relation τ1 = 1/2(τ0 + τ2 ). By looking at such a synchronicity experiment from the point of view of the resting system Einstein succeeds in turning this relation into a differential equation. The idea is that at time t in OK (corresponding to time τ0 in Ok ) a light signal leaves the origin of Ok in the direction of the positive ξ -axis to a point on that axis at a distance from Ok equal to Δx measured in OK . where there is a mirror. The mirror reflects at time τ 1 and the light returns in Ok at time τ2 . This would lead to the following expressions: τ0 = τ (vt, 0, 0, t), τ1 = τ (vt + vΔx/(c − v) + Δx, 0, 0, t + Δx/(c − v)) τ2 = τ (vt + vΔx/(c − v) + vΔx/(c + v), 0, 0, t + Δx/(c − v) + Δx/(c + v)) First, however, Einstein introduces a new variable x’ with x ' = x − v · t,

(17.4)

and he remarks that in this way a fixed point in the moving system, which is from the point (of view of ) the resting system moving with velocity v, corresponds to a fixed point x ' , y, z in the resting system. This is a clever move. In this way he gets rid of the time dependence in the first coordinate of the expressions for τ0, τ1, and τ2 . The new variable makes the argument simpler.12 In Einstein’s text the increment Δx is also called x ' . That is somewhat confusing because in this way in his text the variable x’ has a double meaning. On the one hand, x ' = x − vt. On the other hand, x ' is an increment Δx I will distinguish the two, but wherever I write Δx, Einstein writes x' . The signal starting at time τ0 from Ok is reflected at time τ1 and returns to Ok at time τ2 . We know for all three events the coordinates and time in the resting system K. Seen from system K the light ray moves towards a moving point that is at the beginning a distance Δx away. Because of the motion of the point this takes a time t = Δx/(c − v). After reflection we need t = Δx/(c + v) to get back to Ok . Moreover, we know that τ1 = 1/2(τ0 + τ2 ).

The introduction of the new variable x’ = x − v · t has led to discussions in the literature. See Miller (1981), pp. 208–209 and Martínez (2009), pp. 325–331. 12

17.7 Derivation of the Differential Equation for τ = τ (x ' , y, z, t)

297

This means that we have: τ0 = τ (0, 0, 0, t) τ1 = τ (Δx, 0, 0, t + Δx/(c − v)) τ2 = τ (0, 0, 0, t + Δx/(c − v) + Δx/(c + v)) τ1 = 1/2(τ0 + τ2 )

(17.5)

Nota bene, we are here considering the functions ( ) ( ) ( ) ( ) ξ = ξ x ' , y, z, t , η = η x' , y, z , ζ = ζ x' , y, z , τ = τ x' , y, z

(17.6)

The variables ξ , η, ζ , and τ are expressed in terms of x' , y, z, and t. Einstein realized that τ1 = 1/2(τ0 + τ2 ) gives us an equation which turns into a differential equation for τ when we let Δx vanish.

17.7 Derivation of the Differential Equation for τ = τ (x' , y, z, t) (17.5) and (17.1) yield [ ( { })] ( ) Δx Δx Δx 1 τ (0, 0, 0, t) + τ 0, 0, 0, t + + = τ Δx, 0, 0, t + 2 c−v c+v c−v (17.7) We can write: ] ( [ ( { }) ) Δx Δx Δx 1 τ 0, 0, 0, t + + − τ (0, 0, 0, t) =τ Δx, 0, 0, t + 2 c−v c+v c−v − τ (0, 0, 0, t) (17.8) This yields 1 ∂τ 2 ∂t

(

Δx Δx + c−v c+v

) =

∂τ Δx ∂τ + higher order terms in Δx. Δx + ∂x' c − v ∂t

After division by Δx we let Δx vanish and we get: ( ) 1 1 ∂τ ∂τ 1 1 ∂τ + = '+ , 2 c−v c + v ∂t ∂x c − v ∂t or in other words

(17.9)

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17 Albert Einstein, the Kinematics of Special Relativity

Fig. 17.1 Einstein’s second synchronicity experiment

∂τ ∂τ v = 0. + 2 ∂x' c − v 2 ∂t

(17.10)

) ( Strictly speaking the derivation concerns the point x ' , y, z, t = (0, 0, 0, t), but Einstein immediately assumes (that this holds for all values of x ' , y, z. ) ' In order to determine τ = τ x2 , y, z, t , we need two other differential equations. They are simple. We get: ∂τ ∂τ = = 0. ∂y ∂z

(17.11)

In order to prove this Einstein executes another synchronicity experiment (Fig. 17.1). At time t in the fixed system we send a signal from the origin of the moving system in the direction of the η-axis. It is reflected by a mirror on the η-axis at a distance Δy (measured in the fixed system) from the origin. The three events—the sending of the signal, the reflection and the arrival back in the origin—correspond in the moving system to moments τ0 , τ1 and τ2 . The light arrives at the position of the mirror after measured in the fixed system and the light is back in the a period tl = t + √cΔy 2 −v 2 origin of the moving system at t2 = t + This yields

√2Δy . c2 −v 2

)] ( ) ( Δy 2Δy = τ 0, Δy, 0, t + √ 1/2 τ (0, 0, 0, t) + τ 0, 0, 0, t + √ c2 − v 2 c2 − v 2 (17.12) [

This yields: Δy ∂τ Δy ∂τ ∂τ Δy + = + higher order terms inΔy. √ √ ∂t c2 − v 2 ∂y ∂t c2 − v 2

(17.13)

17.8 The Determination of ξ (x' , y, z, t), η(x' , y, z, t) and ζ (x' , y, z, t)

299

We divide by Δy and let Δy vanish and we get the equation ∂τ = 0. In an ∂y ∂τ analogous way we can deduce ∂ z = 0. ( ) can now solve( (17.10) and (17.11). Nota bene )that ξ = ξ x ' , y, z, t , η = ) ) ( ( We η x ' , y, z, t , ζ = ζ x ' , y,(z, t and )τ = τ x ' , y, z, t are linear functions. First, (17.11) implies that τ = τ x ' , y, z, t doesn’t contain y or z. We get from (17.10) that ) ( ( ) v ' (17.14) x τ x ' , y, z, t = a · t − 2 c − v2 a = ϕ(v) is a constant function depending only on v, says Einstein.

17.8 The Determination of ξ (x' , y, z, t), η(x' , y, z, t) and ζ (x' , y, z, t) A ray of light emitted at τ = 0 in the direction of the positive ξ -axis proceeds according to ( ξ = cτ or ξ = c · a · t −

v x' 2 c − v2

) (17.15)

( ) The event (ξ, 0, 0, τ ) in the moving system corresponds to x' , 0, 0, t in the fixed system. The principle of the constancy of the velocity of light in combination with the relativity principle implies that seen from System k the light ray covers a distance x’ in time t with a velocity c − v. So t=

x' c−V

(17.16)

Substitution in (17.15) yields the equation that we want: ( ξ =a·

c2 x' c2 − v 2

) (17.17)

This is an interesting argument. Equation (17.14) holds generally. It gives us the time τ for an arbitrary event (x, y, z, t) in the fixed system. The argument leading to (17.17) concerns only events that take place on the coinciding x-axis and ξ -axis. Does (17.17) hold generally? Do we need the homogeneous character of space—there is no preferred location for the coinciding axes—to make the argument conclusive? Similarly Einstein considers a ray moving along the η-axis. We have ( η=c· t−

) v ' . x c2 − v 2

(17.18)

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17 Albert Einstein, the Kinematics of Special Relativity

For this ray x' = 0

and

y /( ) = t, c2 − v 2

(17.19)

which yields c c η = a /( z. ) y and, analogously, ζ = a √ 2 c − v2 c2 − v 2

(17.20)

17.9 Towards the Formulae of the Lorentz Transformation Clearly now Einstein must get rid of the coordinate x' = x − vt. In order to simplify the equations somewhat further Einstein, moreover, substitutes / a(v) = ϕ(v) 1 −

v2 . c2

(17.21)

ϕ(v) is a for now unknown function of v. We get t − xv2 τ = ϕ(v) / c , 2 1 − vc2 x − vt ξ = ϕ(v) / , 2 1 − vc2

(17.22)

η = ϕ(v)y, ζ = ϕ(v)z. Einstein now checks the correctness of the equations as follows. He assumes that a light wave is transmitted from the two coinciding origins at time t = τ = 0. Assuming that the wave front is described in System K by the equation x2 + y2 + z2 = c2 t2 ,

(17.23)

what does it look like in system k? We can apply the equations. The answer, after some calculations is ξ 2 + η 2 + ζ 2 = c2 τ 2 .

(17.24)

17.10 The Twin Paradox

301

The last problem that Einstein must still solve is the determination of the function ϕ(v). In order to determine this function Einstein introduces a third System K' (with coordinates x ' , y ' , z ' , t ' ) that moves like K with respect to System k except for the fact that its velocity is −v. By applying (17.22) twice we can deduce that t ' = ϕ(v) · ϕ(−v) · t, x ' = ϕ(v) · ϕ(−v) · x, y ' = ϕ(v) · ϕ(−v) · y and z ' = ϕ(v) · ϕ(−v) · z Because the relations between x, y, z and x ' , y ' , z ' do not depend on time the systems K’ and K do not move with respect to each other. Obviously we have ϕ(v).ϕ(−v) = 1

(17.25)

Because η = ϕ(v)y the factor ϕ(v) represents the contraction or the extension of a unit length perpendicular to the direction of the motion. Because of symmetry considerations the degree of contraction cannot depend on the sign of the velocity and we have ϕ(v) = ϕ(−v)

(17.26)

From (17.25) and (17.26) Einstein deduces: ϕ(v) = 1. So finally we have the famous formulae: t − xv2 x − vt τ = / c ,ξ = / , η = y, ζ = z 2 2 1 − vc2 1 − vc2

(17.27)

17.10 The Twin Paradox In Section 4 of his paper Einstein considers a clock in the origin of the moving system. In the moving system it shows time τ. How fast does this clock run from the point of view of the resting system? In the resting system the coordinates of the clock are x = vt, y = 0, z = 0. When we substitute x = vt in τ=/

1 1−

( v ) t − x ( v )2 c2

(17.28)

c

we get / τ =t 1−

( v )2 c

(17.29)

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This means that from the point of view of the system at rest the moving clock stays behind. Every second the moving clock seems to stays behind 1/2(v/c)2 seconds (neglecting higher order terms). At this point Einstein wrote: “One sees immediately that this result still holds good when the clock moves along any polygonal line from A to B, even when the two points A and B coincide.”13 Einstein also assumed that this result could be extended to the case where the polygonal line is replaced by a continuous curve. These considerations seem to imply a paradox. Equations (17.27) imply: τ + ξ v2 ξ + vτ , t=/ c , x=/ 2 2 1 − vc2 1 − vc2

y = η, z = ζ

(17.30)

This means that also from the point of view of the moving system a clock in the origin of the resting system stays behind. So we seem to have two clocks of which each lags behind the other. This seems impossible. And indeed when two clocks are in the same frame of reference, it is impossible that they both lag behind the other. When two frames are moving with respect to each other, however, the two clocks in the origins of the frames are not in the same frame of reference. We have a similar situation in the case of the twin paradox. The twin paradox is a thought experiment in special relativity involving twins, one of whom makes a journey into space travelling at high speed. Because of the speed the clock of the travelling twin goes slower than the clock of the twin who stayed at home. Einstein’s remarks on a clock moving along a polygonal line imply that when the traveler returns home he finds that the twin who remained on Earth has aged more. This is paradoxical because kinematically there seems to be a perfect symmetry between the positions of the two twins. After all, if we do not consider masses and forces, during the first phase of the trip the twins are simply speeding away from each other and during the second phase of the trip they are approaching each other again. In reality forces are required and mass must be moved, but Einstein’s theory of special relativity is initially purely kinematical. So one could argue that on the bases of the symmetry each of the twins should paradoxically find the other to have aged less. We will return to the paradox in the next chapter.

13

Lorentz (1913), p. 36.

Chapter 18

Minkowski: The Universe Is a 4-Dimensional Manifold

Abstract In 1908 in a lecture Minkowski gave a brilliant reformulation of Einstein’s theory. He views the World as a 4-dimensional manifold that can be observed from reference systems with three space-axes and a time-axis. The transitions between the systems are determined by the Lorentz transformations. About space and time Minkowski said: “Space in itself and time in itself will be completely reduced to shadows and only a kind of union of the two will maintain independence”. In this chapter we sketch a part of Minkowski’s lecture. We conclude our discussion of the twin-paradox.

18.1 Empiricists and Rationalists In seventeenth and eighteenth century philosophy we find a fundamental opposition between empiricists and rationalists. The rationalists held that our mind is a major source of true knowledge. According to them we can independently of sense experience acquire true knowledge about the external world. The empiricists held that, on the contrary, sense experience is the only source of knowledge about the external world. The Stanford Encyclopedia of Philosophy says: “The dispute between rationalism and empiricism concerns the extent to which we are dependent upon sense experience in our effort to gain knowledge. Rationalists claim that there are significant ways in which our concepts and knowledge are gained independently of sense experience. Empiricists claim that sense experience is the ultimate source of all our concepts and knowledge.” (Lemma on Rationalism versus Empiricism by Peter Markie). The opposition goes back to antiquity. Plato was a rationalist, while as for Aristotle we can say in the words of Dawes that “Insofar as it insists on both the reliability of the senses and their essential role in the gaining of knowledge, Aristotle’s view is broadly empiricist”.1 Well known seventeenth century rationalists are Descartes, Spinoza and Leibniz. Leibniz in his New Essays on Human Understanding argued, for example, as follows. 1

https://plato.stanford.edu/entries/empiricism-ancient-medieval/#ArisEmpi

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Koetsier, A History of Kinematics from Zeno to Einstein, History of Mechanism and Machine Science 46, https://doi.org/10.1007/978-3-031-39872-8_18

303

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18 Minkowski: The Universe Is a 4-Dimensional Manifold

The senses only give instances, they never give the whole. Yet all instances that confirm a general truth do not establish the universal necessity of it. So in order to arrive at necessary truths we must have a source of knowledge that differs from the senses. Leibniz admits that certain general truths would not have occurred to us without the senses. Yet, our insight in these truths is a priori, not based on sense experience. One element in Leibniz’ metaphysics is his belief in a pre-established harmony between body and soul and in general between all monads in the world. The monads are the simple substances, the basic units in Leibniz’ universe. Newton was inclined towards empiricism. He wrote: “We do not know the substances of things. We have no idea of them. We gather only their properties from the phenomena, and from the properties [we infer] what substances may be…And we ought not rashly to assert that which cannot be inferred from the phenomena.”2 Well known empiricists are Locke, Berkeley and Hume. Hume, for example, defines science as an empirical inquiry based on experience and observation. As for metaphysics or the possibility to acquire knowledge that goes beyond observation Hume is a skepticist. The science of nature is for Hume fallible; other observations may force us to adjust our theories. Rationalists are inclined to believe that absolute truth can be attained. Inevitably as for knowledge that goes beyond observation empiricists will be inclined to fallibilism. The opposition between empiricists and rationalists not only occurs in philosophy. The early Albert Einstein was an empiricist in the way he practiced physics.3 Einstein’s position with respect to mathematics changed in the course of time. Before 1912 Albert Einstein avoided sophisticated mathematics. After 1912 he realized that mathematics could be very useful to the physicist. His empiricism changed as well. After 1912 the idea of mathematical simplicity guided him in the development of the general theory of relativity. Moreover, he saw the quest as an attempt to understand reality in its depth.4 I will in this chapter describe Minkowski as a rationalist. Einstein’s 1905 paper on the special theory of relativity could not have been written if Einstein had not thought along the skeptical empiricist lines of Hume and Ernst Mach. On the other hand, as we will see Minkowski’s 1908 lecture in every respect breathes rationalism. Newton had defended the existence of absolute space and absolute time. Although Huygens and Leibniz had criticized these notions, they remained very influential. Only in the nineteenth century serious criticism was raised by Mach and others. The notion of the luminiferous ether as a substance that fills space is equivalent to a notion of absolute space. Einstein in his 1905 paper rejected both absolute space and absolute time. As we have seen his rejection of the dogma of absolute time is the core of his paper on the special theory of relativity. It is remarkable that Minkowski in fact restored the absolute character of reality by equating the 4-dimensional World that he defined with reality. The space-time frames of reference that observers use to observe reality are all relative, but what they 2

Gaukroger (2014), p. 15. Cf. Holton (1973), p. 224. 4 Einstein (1933), p. 14. 3

18.2 Developments in Geometry

305

observe, the World and its invariant properties, is absolute. This is the meaning of Minkowski’s opening sentence: “Space in itself and time in itself will be completely reduced to shadows and only a kind of union of the two will maintain independence.” The space and time that we observe are like the shadows that the prisoners on Plato’s cave are watching. They are a manifestation of true reality but should not be equated with it.

18.2 Developments in Geometry Hermann Minkowski had been one of Einstein’s teachers in Zürich. Actually Einstein took nine courses from Minkowski, more than from anyone else. He later admitted that he could have gotten a sound mathematical education in Zürich. However, he neglected mathematics. He had experienced the subject as split up in numerous time consuming specialties, without being able to distinguish the really valuable from the rest. Moreover, his interest in nature had been prevalent.5 Minkowski is reported to have described Einstein as a lazy dog. However Einstein surprised Minkowski by being able to formulate the special theory.6 And Minkowski surprised Einstein. In 1908 on September 21, Minkowski gave an influential lecture on the special theory of relativity. Before we can discuss it, we must make a few remarks on the development of pure mathematics in the nineteenth century and on Minkowski’s background. (i) (ii)

Geometries with dimension n > 3. The theory of invariants.

Several people were involved in the development of the concept of multidimensional geometries. In particular the Swiss mathematician Ludwig Schäfli (1814– 1895), Arthur Cayley and Bernard Riemann introduced the notion of spaces with more than three dimensions. In the period 1850–1852, Schäfli wrote his Theory of Multiple Continuity (Theorie der vielfachen Kontinuität) in which he developed the analytical linear geometry of ndimensional space. Linear equations with more than three variables were well-known. Yet, the geometrical interpretation failed. In his book Schäfli defined distances in ndimensional geometry, he discussed the n-dimensional sphere and he even calculated its volume. He gave much attention to polytopes, the higher dimensional analogon of polyhedra. Unfortunately, the publication of the book ran into difficulties, mainly because of its size. Arthur Cayley published English translations of some parts of Schäfli’s book in 1860. The complete text was only published posthumously in 1901. Cayley’s interest in higher dimensional geometries was stimulated by Hamilton’s invention of the quaternions. For example, Thomas Graves and Arthur Cayley independently discovered the octonians. The octonians are a non-associative extension 5 6

Einstein’s autobiography in Schilpp (1951), pp. 14–15. Pyenson (1977), p. 72.

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of the quaternions. The quaternions and the octonians suggested, respectively, 4- and 8-dimensional interpretations. As for the role of Bernard Riemann, he habilitated in 1854 in Göttingen with two papers. One of them, About the Hypotheses that form the ˝ Basis of Geometry (Uber die Hypothesen, welche der Geometrie zu Grunde liegen) contains a very general notion of n-dimensional geometries. He extended differential geometry to n-dimensions. In nineteenth century mathematics the notion of ‘invariant’ played an important role. Famous is Felix Kleins so-called Erlanger program published in 1872. See Klein (1872). Geometry had become an enormous field consisting of many, very different geometries. Klein succeeded in defining a point of view that restored the unity of the field. The basic idea was the following: given a manifold and a group of transformations, the features of the manifold that are not changed by the transformations of the group determine a particular geometry. A transformation is a one-to-one mapping of the manifold on itself. In other words: a particular geometry is determined by the invariants under a group of transformations. For example: distance is an invariant under Euclidean transformations, but not under projective transformations. In particular in the second half of the nineteenth century an algebraic theory developed which dealt with algebraic invariants. Cayley, Sylvester, Clebsch, Gordan and many others worked in this theory. For example the theory of invariants of binary forms is concerned with properties of homogeneous polynomials in two variables which are independent of the choice of coordinates, modulo a power of the determinant of the transformation corresponding to the change of coordinates. For example: if we subject the binary form f(x, y) = ax 2 + 2bxy + cy2 tot a linear transformation x = αx' + βy'; y = γ x' + δy' with r = αδ − βγ , we get f(x' , y' ) = a' x'2 + 2b' x' y' + c' y'2 . The expression D = b2 − ac is a simple invariant, because, as one can check easily D' = b'2 − a' c' = r 2 •D.

18.3 Hilbert’s Influence and Minkowski’s Rationalism In 1891 Hilbert attended a lecture in Halle by Hermann Wiener on the foundations and structure of geometry. Hilbert was impressed by Wiener’s abstract point of view in dealing with geometric entities. In the railway station in Berlin, on his way back to Königsberg, he is reported to have remarked to his companions: “One must be able to say at all times—instead of points, straight lines and planes—tables, chairs, and beer mugs.”7 This statement contains the modern axiomatic point of view of a mathematical theory in a nutshell: the axioms define the notions like point and line, and their names are strictly speaking irrelevant. In Grundlagen der Geometrie of 1899 Hilbert applied this point of view to geometry and in 1900 he applied the axiomatic method to arithmetic, i.e. the system of the real numbers, which he characterizes as a real Archimedean field that cannot be extended to a larger field of the same kind. Hilbert had become aware of the power of the axiomatic method. 7

Reid (1996), p. 57.

18.3 Hilbert’s Influence and Minkowski’s Rationalism

307

In 1886 Felix Klein had come to Göttingen which he, supported by Friedrich Althoff, the director of the Prussian university system, transformed into a major successful research center. It was, however, David Hilbert who, supported by Klein, turned Göttingen into the center of the mathematical world at the time. Hilbert came to Göttingen in 1895. In 1902 he was asked to come to Berlin, but he refused. No one had ever before rejected a call from Berlin, but Hilbert did after Althoff had promised to create a position for Hilbert’s friend Hermann Minkowski. After Minkowski’s appointment Hilbert allegedly said: “Now we are invincible!”.8 The start of the nineteenth century witnessed a revival of interest in Leibniz. In the United Kingdom, Bertrand Russell and in France, Louis Couturat wrote books on Leibniz. In Germany Ernst Cassirer published a book on Leibniz in 1902 in which he explained what Leibniz meant with his notion of pre-established harmony. He in fact argued that according to Leibniz reality follows from pure reason through the application of mathematics.9 Similar ideas can be found in Göttingen at the start of the century. Hilbert at the time developed a great interest in physics. Again the axiomatic method played a central role: his major objective was the axiomatization of physics. At the Second International Congress of Mathematics held in Paris in 1900 Hilbert defended the ‘pre-established harmony between nature and mathematics’. He meant that the physical sciences in which mathematics plays a central role ought to be treated axiomatically, as geometry. See Kragh (2015). In 1912 the Breslau philosopher Richard Hönigswald studied the problem of the pre-established harmony between mathematics and reality. He distinguished three possibilities: the harmony is rooted in psychological factors, it is rooted in experience or it is rooted in mathematics. Hönigswald opted for the third possibility. He agreed with the mathematician Adolph Kneser who had described the situation by means of the metaphor of the tree: The remarkable consequences of mathematics cannot be calculated in advance with certainty. However, they can be anticipated as the fruit of a tree in blossom.10 This is precisely the attitude that Minkowski exhibited in his 1908 lecture on relativity. This went very far, according to Minkowski in 1915: “To some extent, the physicist needs to invent these concepts [of the theory of relativity] from scratch and must laboriously carve a path through a primeval forest of obscurity; at the same time the mathematician travels nearby on an excellently designed road. […] It will become apparent, to the glory of the mathematicians and to the boundless astonishment of the rest of mankind, that the mathematicians have created purely within their imagination a grand domain that should have arrived at a real and most perfect existence, and this without any such intention on their part.”11 In this way fundamental physics seems to become a part of mathematics. Yet apparently Hilbert and Minkowski never left the idea that experiment must be the final arbiter of physical theory.12 8

Rowe (1989), p. 197. Pyenson (1977), p. 143. 10 Pyenson (1977), p. 148. 11 Quoted in Kragh (2015), pp. 218–219. 12 Kragh (2015), p. 520. 9

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18.4 Minkowski and Relativity Already in 1905, before the appearance of Einstein’s paper, Hilbert and Minkowski together with Herglotz and Wiechert organized a seminar on the theory of electrons. Yet only in October 1907 Minkowski wrote Einstein13 with the request to send him a copy of his 1905 paper which he wanted to use in a seminar organized by himself and Hilbert on partial differential equations in physics. In April 1908 Minkowski published a paper, The Basic equations for Electromagnetic Processes (Die Grundgleichungen für die electromagnetischen Vorgänge), in which he expressed Maxwell’s equations in four-dimensional form. It is remarkable that Einstein and Jakob Laub soon published an answer. They rejected the approach by means of a four-dimensional space as too complicated for the reader of Annalen der Physik. Yet in his lecture on September 21, 1908 in Cologne, Minkowski elaborated his four-dimensional apparatus. Einstein at first still remained skeptical, wondering what was really new. Yet Minkowski’s contribution was quite spectacular. As for the four-dimensional interpretation of the special theory of relativity Minkowski was preceded by Henri Poincaré (1854–1912), who published in 1906 a paper in which he discussed the possibility to interpret the world as a space with three spatial dimensions and one imaginary dimension of time. After having chosen the units of length and time in such a way that the velocity √ of light becomes 1 he could characterize a position by the coordinates: x, y, z, t − 1.14 Poincaré also showed that the Lorentztransformations form a continuous group with the invariant x 2 + y2 + z2 − t 2 . This was a crucial result and it certainly influenced Minkowski. Minkowski presented the theory in a very attractive form, adding some important new elements: the World-lines, the proper time, the Light Cone. Minkowski’s presentation of the theory was wonderfully in accordance with the view existing among many German scientists and mathematicians that there exists a pre-established harmony between pure mathematics and physics.15 When Minkowski at the end of his lecture said that certainly through experimental confirmation in the future those who do not like to give up old ideas would be reconciled by the thought of such a pre-established harmony, he referred to such feelings. One has to get used to it, he said in fact, but I offer you a new, even more beautiful pre-established harmony.

13

Walter (1996), p.8. Poincaré (1906). 15 Pyenson (1985), pp. 202–203. 14

18.5 A 4-Dimensional Interpretation of Newtonian Mechanics

309

18.5 A 4-Dimensional Interpretation of Newtonian Mechanics We have seen that Einstein’s 1905 paper is in every respect the paper of a physicist influenced by empiricist philosophers. Minkowski’s lecture is very much the lecture of a mathematician with a rationalist view of physics. Following Poincaré, Minkowski realized that Einstein’s new theory could only profit from a geometrical treatment in terms of 4-dimensional geometry and groups of transformations. First Minkowski interpreted Newtonian mechanics in 4-dimensional terms. He defined the World as the totality of all 4-tupels (x, y, z, t) of real numbers. The sub-space consisting of all points (x, y, z, 0), characterized by the equation t = 0 is a Euclidean space and the x-, y-, and z-axes are orthonormal. The t-axis plays a special role (See Fig. 18.1); in the first pages of his paper Minkowski is not very explicit about its status. In his drawings he draws it initially perpendicular to the other axes. Minkowski assumes that at every point in the World there is always something that can be observed. Thinking of matter or electricity he calls it ‘substance’. The initial coordinate system is used to define the World, but it is at the same time only one of many reference systems that we can use to describe the World. We can replace it with other reference systems. Let us play for a moment with the idea of Minkowski’s World. The 3-dimensional manifold characterized by t = 0 separates the upper part of the World with coordinates t > 0 and the lower part of the World with coordinates t < 0. See Fig. 18.1. Fig. 18.1 Minkowski’s World

t>0

z O y x

t , 2 2 2 c d x + dy + dz

(18.2)

which means that (

dx dt

)2

( +

dy dt

)2

( +

dz dt

)2 < c2 .

(18.3)

Every velocity is smaller than c! Here Minkowski admits that the true reason for the introduction of the group Gc is the actual behavior of light in physical space. He now proves that from the point of view of the two reference systems involved in Fig. 18.5 the Lorentz contraction will be observed. A rod e at rest in the x' Ot' system corresponds to the World lines that together make up the oblique column in Fig. 18.5. In the xOt system this rod is observed as e'. Similarly a rod d' at rest in the xOt system corresponds to the world lines that together make up the vertical column in Fig. 18.5. Nota bene, in the drawing the t-axis is vertical, perpendicular to the x-axis. This is not essential. In the x' Ot' system this rod is observed as rod d. Suppose now that d' at rest in the xOt system has in this system the unit length (OC in Fig. 18.4). Similarly suppose that rod e at rest in the x' Ot' system has in this system the unit length (OC' in Fig. 18.4). If for the direction of the t'-axis and / = v, one / the oblique column we have dx/dt

can prove rather easily that d' /e' = 1: 1 − vc2 and similarly e/d = 1: 1 − vc2 . The unit in the other system is observed as shortened in comparison with the unit in the 2

2

Fig. 18.5 The Lorentz contraction

t'

x' e

d

d'

e'

18.6 Special Relativity Deduced a Priori

315

own system: the Lorentz-contraction. As for the proof, see Fig. 18.4. Some analytical geometry in the xOt-plane yields: ) ) ) ( ( v 1 c+v c+v c v ' ' , ,B = , ,C = , and A = β β β cβ β cβ ) ( . √ β ' 2 2 , 0 with β = (c − v ) D = c '

(

At this point Minkowski refers to Einstein’s paper and admits that he was the first to bring down the notion of absolute time. However, he wrote: “Neither Einstein nor Lorentz shook the notion of space”. He added: “To go beyond the notion of space in an adequate way can only be assessed as the audacity of mathematical culture.”22 Yet, this is necessary for a true understanding of the Group Gc, says Minkowski. He finds here a reason to suggest a new name for Einstein’s postulate of relativity. Einstein’s postulate said that the laws of nature are the same in all reference systems moving with uniform velocity with respect to each other. From Minkowski’s point of view, however, the phenomena give us a 4-dimensional world, which is projected in space and time. With respect to the way in which this projection takes place we possess a certain liberty. Einstein’s relativity postulate says that the laws of nature are independent of the way in which this projection takes place. That is why Minkowski proposes the name Postulate of the absolute World, or short, World postulate. In the next section of his paper Minkowski treats the World as given. He chooses an arbitrary origin O and three space-axes and one time-axis. The cone c2 t 2 − x 2 − y2 − z2 = 0 with top O consists of two parts. The before-cone of O, corresponds to t < 0 and it consists of all points that can send light to O. It is also called the future light cone of O. The after-cone of O, corresponds to t > 0 and it consists of all points that can receive light from O. It is also called the past light cone (Fig. 18.6). By a suitable choice of the reference system any point in the area between the two cones can turn out to be simultaneous with O, later than O or before O. The points inside the before-cone are necessarily earlier and the points inside the after cone are necessarily after O. A very interesting application is the following. Minkowski considers a World line. The tangent to the World line can be chosen as t-axis. Let us call such an axis the τ-axis. We then have dτ > 0, dx = dy = dz = 0. Because of the invariance of c2 dt2 − dx2 − dy2 − dz2 under transformations for all reference frames we have for all Oxyzt reference frames: c2 dτ 2 = c2 dt 2 − d x 2 − dy 2 − dz 2 or . 1√ 2 2 dτ = (c dt − d x 2 − dy 2 − dz 2 ) c

(18.4)

Minkowski calls dτ the increment of the proper time corresponding to the World line. The proper time measured by a clock corresponding to the World-line is then 22

Lorentz (1913), p. 62.

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18 Minkowski: The Universe Is a 4-Dimensional Manifold

Fig. 18.6 Light cone in the two-dimensional case. Source https://en.wikipedia. org/wiki/Light_cone

∫ τ=

dτ.

(18.5)

In his 1905 article Einstein had mentioned the time dilation of a clock moving along a polygonal line from A to B and he had assumed that the result could be extended to the case where the polygonal line is replaced by a continuous curve. Minkowski nicely made this idea precise.

18.7 The Twin Paradox Let us consider the twin paradox from the point of view of Minkowski’s space-time World. We can observe the world from infinitely many reference frames moving with constant velocities with respect to each other. Such a set of reference frames is also called a set of inertial reference frames, Twin A, who stays at home, has his or her own reference frame, let us call it Frame I, in which he or she does not move. Let us assume that the other twin B, who makes a journey into space, travels first with a constant velocity V away from A. In order to start the trip B his or her personal reference frame which coincided originally with Frame I starts to coincide with Frame II. Let us assume that during the trip homewards B travels with a constant velocity −V. This means that in order to start the trip back home the personal reference frame of B must once more switch to a third reference frame, Frame III. And finally when he or she gets home B switches back to Frame I in order to reunite with twin A.

18.7 The Twin Paradox

317

Fig. 18.7 The trip of Twin B

Figure 18.7 illustrates the trip of Twin B. We assume that a minute on the clock of Twin B in Frames II and III corresponds to 1.5 min on the clock of Twin A in Frame I. Then when the Twins meet again the age of Twin A is 1.5 times the age of Twin B. The point of the twin paradox is based on the fact that kinematically the situation of the two twins seems to be symmetrical. From the point of view of each of them the other one is speeding away with a high velocity and then returns with that same velocity. And this is correct although it does not yield a contradiction. Relativity theory implies that things look different depending on the frame of reference. Time dilation and Lorentz contraction do illustrate this. When two frames of reference are moving with respect to each other from the point of view of each of the frames the clock in the other frame seems slower and distances seem to shrink. The twin paradox is essentially another illustration. Let us compare three situations: 1. We look at the world from the frame of reference of Twin A. Twin B leaves at a high speed V and returns at the same speed after having reached a point 1 lightyear away from Twin A in the reference system of Twin A. This is the situation we discussed above. 2. We look at the world from the frame of reference of Twin B. Twin A leaves at a high speed V and returns at the same speed after having reached a point 1 lightyear away from Twin B. 3. We now look at the world from a frame of reference which differs from the frames of reference of both Twin A and Twin B. With respect to this frame of reference the twins leave in opposite directions with speed V/2. When they are half a lightyear away from their point of departure they both return to that point with speed V/2. In situation 1 when Twin B meets Twin A again it turns out that Twin A is older than Twin B. In situation 2 Twin B turns out to be older than Twin A, while in situation 3 the Twins have the same age when they meet again although they would be both younger than individuals who stayed at the point they departed from and had the age of the twins when they departed. The twin paradox is a paradox in the sense that there seems to be a contradiction, although there is no inconsistency. The three situations are such that in each of them

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18 Minkowski: The Universe Is a 4-Dimensional Manifold

the two twins are speeding away from each other with speed V until the distance between them is one lightyear and then they approach each other again with speed V until they meet. The paradox is based on the assumption that this means that the three situations are equivalent. They are not. It all depends on the frame of reference from which things are observed. The literature on the twin paradox is sometimes confusing. There is general agreement about the fact that there is no symmetry between the twin who leaves and the twin who stays at home. There is also agreement about the fact that the difference in age is caused by the time dilation.

Chapter 19

Kinematics in the 20th Century

Abstract In this chapter we make some remarks about developments in the twentieth century. We saw in the preceding chapters that kinematics has its roots in geometry, mechanics and mechanical engineering. It is remarkable that in a sense it never released itself from its roots. It basically always remained a subordinate research area. It is striking that in this respect in the twentieth century the subject was mainly developed further as a research area in mechanical engineering. A number of mathematicians specialized in geometry played an important role in this.

19.1 The Twentieth Century The history of kinematics is a vast subject. In a very broad sense of the word every branch of physics has its own kinematics. There is a kinematics of the motion of rigid bodies, but there is, for example, also a kinematics for continuous deformations, one for wave motion and there is one to describe the motion of elementary particles. We have restricted ourselves to kinematics in geometry, in theoretical mechanics and in mechanical engineering. And even within this area there is a lot of work dealing with kinematics or related to kinematics that we have not touched upon. I will mention one example: Ball’s theory of screws. The idea of the instantaneous screw axis was a central element in Poinsot’s work in ‘geometrical mechanics’. In his Théorie nouvelle de la rotation des corps of 1834 he argued that one at the time had a very clear idea of the motion of a mass point in space. The trajectory is a curve. And although the analytical approach of Lagrange perfectly describes the movement of a rigid body in space, Poinsot felt he only had a very obscure idea of how a rigid body actually behaves in space. Because the center of gravity of a rigid body is a point and we have a clear idea of its trajectory, the problem left to be solved was to develop a clear picture of the motion of a rigid body about its center of gravity. Kinematics yielded the answer. At each moment there is an instantaneous axis of rotation. Poinsot concluded that “the rotation of a body about an axis which constantly varies in position around the same fixed point, is nothing other than the

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Koetsier, A History of Kinematics from Zeno to Einstein, History of Mechanism and Machine Science 46, https://doi.org/10.1007/978-3-031-39872-8_19

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19 Kinematics in the 20th Century

movement of a certain cone, the vertex of which is at this point, and which is currently rolling, without sliding, on the surface of another fixed cone with the same vertex.”1 Poinsot also showed that as many forces as we want applied in any way to a body, can always be reduced to a single force R and a single torque (S, −S) such that axis of the couple coincides with the line of action of R. He used this idea to treat the dynamics of a moving rigid body. The Irish astronomer Robert Stawell Ball (1840–1913) was very much impressed by Poinsot’s work. He realized the analogy between the instantaneous screw axis and the force and the torque in which the axis of the couple coincides with the line of action of the force. He started to work on that idea. In 1871 he read a paper to the Royal Irish Academy on his new theory of screws. See Ball (1875). In 1876 he published a book on the subject, Ball (1876). Its title is The Theory of Screws, A Study in the Dynamics of a Rigid Body. In the twentieth century Kenneth H. Hunt (1920–2002) in particular has applied screw theory in the kinematics of mechanisms.2 The present book sketches the history of kinematics up to and including the nineteenth century. What happened to kinematics in the twentieth century? As for classical theoretical mechanics, in the twentieth century it remained an important and widely applied theory but the kinematical part didn’t change much. In Chaps. 17 And 18 of this book we saw that with the appearance of relativity theory Newtonian kinematics underwent a major overhaul. Relativistic mechanics can also be divided into kinematics and dynamics. Kinematical considerations concerning positions, velocities and accelerations play in relativistic mechanics to a certain extent a role similar to the role they play in classical mechanical mechanics. On the other hand, as we saw in the case of special relativity there are clear differences. The kinematics of general relativity is beyond the scope of this book. In nineteenth century geometry was a fertile field where many new results turned out to be within reach. This generated interest in kinematics among mathematicians. Many of them contributed to the area. In the course of the twentieth century mathematicians moved their attention to more abstract subjects and most of them lost their interest in kinematics. In mechanical engineering the nineteenth century interest in kinematics did not disappear in the twentieth century. The first half of the twentieth century is precisely the period in which kinematics is applied in practice, and in particular in Germany. In the 1890s Riedler had been right: much of the kinematics that was taught at the German technological universities, the Technische Hochschulen, was not applied in practice. In Germany this changed drastically in the first decades of the twentieth century. The nineteenth century graphical methods to determine velocities, accelerations and radii of curvature were applied both in the analysis and the synthesis of mechanisms. Moreover, there were new developments. In 1931 the important German kinematician Rudolf Beyer (1892–1960) published his Technische Kinematik (Technical kinematics). Beyer starts his preface, next to a picture of Reuleaux, with the statement: “In the spirit of Franz Reuleaux, with Burmester in mind, in the 1 2

Poinsot (1834), pp. 11–12. Hunt (1978).

19.1 The Twentieth Century

321

footsteps of Ferdinand Wittenbauer.” The first 147 pages are devoted to Reuleaux’ theory of constraints in the spirit of the great man. The next 280 pages are written with Burmester in mind and the last 63 pages are devoted to dynamics in the footsteps of Wittenbauer. In the 1930s Rudolph Beyer was the spokesman of the German kinematicians and the situation is perfectly clear.3 The quarrels of the second half of the nineteenth century were forgotten and Reuleaux and Burmester were both celebrated as great kinematicians. Reuleaux’ lasting contribution turned out to have been his abstract approach of the machine: look at mechanisms and the problem of their classification in terms of kinematical pairs and kinematical chains. The bulk of the theory, however, at the time turned out to be, with Burmester in mind, the development of the general kinematical properties of motion in combination with their application to specific mechanisms. Worth mentioning here is the fact that Hermann Alt (1889–1954), professor in Dresden, further developed the Burmester theory.4 After the First World War, the United States emerged as the major economic power in the world. At the end of the twentieth century the United States still dominated, economically and scientifically. It is striking that the United States, despite its great economic successes, was a long time lagging behind as for kinematics. In 1942 and 1943 A. E. Richard de Jonge wrote two papers in which he introduced the kinematics that had been developed in Europe and in particular in Germany in the United States.5 Writing about the US De Jonge called ‘kinematic synthesis’, the least known but most interesting branch of kinematics. In De Jonge’s description the Burmester theory gets much attention. The author’s intention was to wake up the Americans by pointing out that in Europe, in particular in Germany, but elsewhere as well, important work in kinematics had been done. In America hardly any published work existed. De Jonge was right, but after World War II things changed. The man who introduced modern engineering kinematics in America was Ferdinand Freudenstein (1926–2006). When de Jonge published his papers on kinematics he was adjunct professor at the Polytechnic Institute of Brooklyn. In 1954 he worked for the Reeves Instrument Corporation in New York. He was enthusiastic when he read Freudenstein’s first two papers. De Jonge wrote: “The greatest benefit of the use of the author’s method is obtained when computing machines can be utilized.”6 Through Freudenstein and his pupils modern kinematics conquered America. As for engineering applications in the nineteenth century the emphasis in kinematics had been on planar mechanisms. In the twentieth century a considerable interest developed in spherical and spatial mechanisms. As for methodology until after World War II graphical methods prevailed. Only with the rise of the information age and the introduction of electronic computers graphical methods were replaced by analytical methods. Nineteenth century geometry in its analytical form in combination with the power of modern computers offered a wealth of new possibilities for engineers dealing with the design of mechanisms. 3

Kerle (2007), p. 184. Mauersberger (2006). 5 Jonge (1942) and Jonge (1943). 6 Roth (2007), p. 242. 4

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19 Kinematics in the 20th Century

The reader who wishes to get an idea of the state of affairs in kinematics at the end of the twentieth century, I refer to the classic handbook by Bottema and Roth (1979) for theoretical kinematics and to McCarthy (2000) for mechanism design. An excellent survey of the developments in the second half of the twentieth century can be found in a book edited in 1993 by Arthur G. Erdman, Modern Kinematics, Developments in the Last Forty Years.7 The book nicely illustrates Ferdinand Freudenstein’s influence. The book also shows that in mechanical engineering, kinematics is a very vast field that has many facets and cannot be easily summarized.

19.2 Institutionalization We saw that in the nineteenth century books entirely devoted to kinematics started to appear. At the time at universities chairs were introduced for ‘descriptive geometry and kinematics’. The first mathematical review journal, the Yearbook for the Advances in Mathematics (Jahrbuch für die Fortschritte der Mathematik), had during its entire existence (from 1868 to 1942) a special subsection, first of mechanics and later of geometry, was devoted to kinematics. The institutionalization of kinematics never went further than this. Yet in 1969 engineers from behind the Iron Curtain, the Western World and some crucial non-aligned countries, got together and founded The International Federation for the Theory of Machines and Mechanisms (IFToMM). It is remarkable that the foundation of IFToMM was prepared in the years of the Cold War when the rivalry and competition between the two superpowers, the United States and the Soviet Union, was huge. IFToMM was founded by scientists working in engineering. In the 1960s the Russian Academician Artobolevskii organized in the USSR a series of All-Union Conferences on contemporary problems in the theory of machines and mechanisms. In the same period the Americans in cooperation with the Europeans developed similar initiatives. In the Cold War engineers on both sides felt they could profit from each other. Two men played a crucial role in the foundation of IFToMM: Ivan I. Artobolevskii (1905–1977) and F. R. Erskine Crossley (1915–2017) from the United States. Artobolevskii represented the Russian theory of machines and mechanisms.8 He did important work in kinematics and he was very influential in Eastern Europe. In 1947–1952 he compiled a four volume work called Mechanisms (in Russian) published by the Academy of Sciences in which he gave drawings and descriptions of some 4000 mechanisms. In the 70 s a revised and extended version appeared.9 An English translation of the first two volumes appeared in 1976.10

7

Erdman (1993). Artobolevskii (2013). 9 Artobolevskii (1976–1980). 10 See also https://archive.org/details/ArtobolevskyMechanismsInModernEngineeringDesignVol1 and https://archive.org/details/ArtobolevskyMechanismsInModernEngineeringDesignVol21 8

19.3 Twentieth Century Mathematicians Working in Kinematics

323

When IFToMM was founded kinematics enjoyed considerable popularity among mechanical engineers. The interest in kinematics at the time is also confirmed by the fact that De Groot published a bibliography of kinematics in 1970 with roughly 7000 references.11 IFToMM published a journal, the Journal of Mechanisms, which was initially very much oriented towards kinematics. Later the journal was renamed Mechanism and Machine Theory. Right from the start IFToMM was in every respect an international organization. The USSR, Bulgaria, the German Democratic Republic, Hungary, Poland and Rumania represented Eastern Europe. The West was represented by the USA, Australia, the German Federal Republic, Italy and the United Kingdom. India and Yugoslavia were the non-aligned countries. On the other hand, big parts of the World were not represented. Right now IFToMM has 46 member organizations. The Ibero-American community is represented and so is Asia. The organization has grown but, inevitably, it has also changed its focus. In the 1960s, for example, classical kinematics of mechanisms remained a core discipline. However, in the past 50 years fast computing, sophisticated software and new applications have changed the theory of machines and mechanisms. Mechanical engineering has always been a multidisciplinary activity, but the number of disciplines involved has only grown. Kinematics, dynamics and gearing are classical subjects but the computer has changed them. We now have, for example, computational kinematics, multibody-dynamics and tribology. Really new subjects are robotics, mechatronics and micromachines.

19.3 Twentieth Century Mathematicians Working in Kinematics It is interesting that when in the course of the twentieth century most mathematicians lost their interest in classical geometry and also kinematics, a small group continued to work in kinematics.. Two of them were the Dutch mathematicians Oene Bottema (1901–1992) and Geert Remmert (Geert) Veldkamp (1907–1989). In particular Bottema and also the English mathematician E. J. F. (Eric) Primrose (1920–1998) worked in close cooperation with engineers in kinematics.12 Although he mainly worked alone also the Austrian mathematician Walter Wunderlich (1910– 1998) must also be mentioned in this context for his work in kinematics.13 Two other Austrian mathematicians working in kinematics in the twentieth century were Wilhelm Johann Eugen Blaschke (1885–1962) and Hans Robert Müller (1911–1999). Take the example of Oene Bottema. Bottema mastered the whole range of geometric theories that the nineteenth century had produced, combined with the

11

Groot (1970). For an obituary see https://iopscience.iop.org/article/10.1070/RM1998v053n04ABEH000066/ pdf 13 For more on Wunderlich’s work on kinematics see Husty (2007). 12

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19 Kinematics in the 20th Century

greater rigor that the twentieth century had brought. He was an ideal partner for the engineers. The geometrical methods of nineteenth century kinematicians often lacked rigor. An example of a rigorous approach to plane instantaneous kinematics is the approach based on instantaneous invariants, invented by Bottema.14 Bernard Roth wrote later about the moment he learned about this approach the following: “It was immediately clear to me that describing instantaneous properties by use of instantaneous invariants was a big advance. It provided a rigorous analytic method to derive planar curvature theory without the tedious, vague, semi-graphical, limit arguments then being used in textbooks and lectures. Moreover, it allowed for all the instantaneous geometric properties of planar motions to be described analytically under one theory.”15 The basic idea is extremely simple. I restrict myself to plane kinematics but in spherical and space kinematics the approach by means of instantaneous invariants has the same advantages. In the moving and in the fixed plane we introduce Cartesian systems of coordinates. We exclude instantaneous translation and we restrict ourselves to timeindependent properties. Because translation is excluded dϕ/dt /= 0 and we can then use the parameter ϕ, the angle between the x-axis and the X-axis, as the independent variable. Let us see how this works. If x, y; X, Y are the Cartesian coordinates of a point in the moving plane and the fixed plane respectively, we have X = x cos ϕ−y sin ϕ + a; Y = x sin ϕ + y cos ϕ + b,

(19.1)

a and b being functions of ϕ. We will denote differentiation with respect to ϕ with primes. We then have in general X ' = −x sin ϕ−y cos ϕ + a ' ; Y ' = x cos ϕ − y sin ϕ + b'

(19.2)

X '' = −x cos ϕ + y sin ϕ + a '' ; Y '' = − x sin ϕ − y cos ϕ + b''

(19.3)

X ''' = x sin ϕ + y cos ϕ + a ''' ; Y ''' = − x cos ϕ + y sin ϕ + b'''

(19.4)

For the instantaneous centre of rotation P we have X' = Y' = 0. This implies that in the moving system for the coordinates of P we have x P = a ' sin ϕ−b' cos ϕ; y P = a ' cos ϕ + b' sin ϕ,

(19.5)

and in the fixed system

14

Others have come up with other approaches. For example, in Blaschke and Müller (1956) the basic results in plane instantaneous kinematics are derived in a different way. 15 Roth (2015), p. 5.

19.3 Twentieth Century Mathematicians Working in Kinematics

325

X P = −b' + a; Y P = a ' + b

(19.6)

We will denote the values of the nth derivatives of the variables a and b for the position ϕ = 0 with the suffix n. This means that the zero-order properties (they concern merely position) in the position ϕ = 0 are determined by a0 and b0 . The first-order properties (they concern tangents) in the position ϕ = 0 are determined by a1 and b1 . And the second-order properties (they concern curvature) in the position ϕ = 0 are determined by a2 and b2 . Moreover, we choose the two systems of coordinates in a clever way: at the moment under consideration they coincide, while the common origin coincides with the instantaneous centre of rotation, and the (two coinciding) x-axes coincide with the common tangent to the two polodes. When at ϕ = 0 the two coordinate systems coincide we have in that position a0 = b0 = 0 and thus X = x and Y = y.

(19.7)

When the origin of the two systems coincides with the instantaneous centre of rotation at the moment ϕ = 0, we have a1 = b1 = 0 and thus from (19.2) we get X ' = −y and Y ' = x.

(19.8)

The y-coordinate of the velocity vector of the position of the pole in the fixed system is equal to a'' + b'. At the moment ϕ = 0 this equals a2 + b1 = a2, because b1 = 0. We let this vector coincide with the coinciding x-axes at this moment, so a2 = 0 as well. To avoid the case in which the velocity of the position of the pole in the fixed system is zero, we exclude the case that b2 = 0. We then get for the second-order properties X '' = −x and Y '' = −y + b2 .

(19.9)

Differentiation of (19.5) and (19.6) shows that at ϕ = 0 the y-coordinates of the velocity vectors of the pole on the two polodes are zero and that the x-coordinates are equal and both −b2 . Clearly the moving polode rolls on the fixed polode. Summarizing we have for these specially chosen reference systems the following expressions for the derivatives: X=x

X' = − y

X'' = x

X''' = y + a3

Y=y

Y' = x

Y'' = − y + b2

Y''' = − x + b3

We can finally choose the direction of the two coinciding x-axes at the moment ϕ = 0 opposite to the direction in which the pole is moving on the fixed polode. This means that b2 is positive. The constants an and bn with n > 1 are precisely defined and characteristic of the motion. Bottema called them instantaneous invariants. We

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19 Kinematics in the 20th Century

Fig. 19.1 Bobillier’s theorem

will illustrate the geometrical significance of b2 . The points in the moving plane that are in an inflexion point of their trajectory satisfy the relation X'':Y'' = X':Y', which gives the equation of the inflexion circle by means of (19.8) and (19.9): x2 + y2 −b2 y = 0. Clearly b2 is the diameter of the circle. We can now also easily derive the standard Euler-Savary formula. We need the well-known formula for the radius of curvature )3/2 X '2 + Y '2 . ρ = ' '' X Y − Y ' X '' (

In the position we are considering we have X = x, Y = y; X' = − y, Y' = x; X'' = − x, Y'' = − y + b2 and we get )3/2 x 2 + y2 . ρ= 2 x + y 2 − b2 y (

When the distance from the pole O to the point P is r and the distance from O to the center of curvature is r C = r − ρ, and we introduce polar coordinates x = rcosθ and y = rsinθ, with some algebra we find the standard Euler-Savary formula (Fig. 19.1). 1 1 1 − . = r rC b2 sin θ With the standard Euler-Savary formula we can prove Bobillier’s theorem. We discussed it in Chap. 10 of this book. The following proof is Veldkamp’s.16 The points α and β are the centers of curvature of the trajectories of, respectively, the points A and B at the moment ϕ = 0. O is the instantaneous center of rotation 16

Veldkamp (1963), pp. 13–14.

19.3 Twentieth Century Mathematicians Working in Kinematics

327

at this moment. The horizontal line is the common tangent to the polodes in O. The points A and B are not collinear and the line through A and B does not go through O. The polar coordinates for the points A, α, B and β are, respectively, (r 1 ,θ 1 ), (ρ 1 ,θ 1 ), (r 2 ,θ 2 ) and (ρ 2 ,θ 2 ). With Euler-Savary we have 1 1 1 1 1 1 − = and − = . ρ1 r1 b2 sin θ1 ρ2 r2 b2 sin θ2 We now use the lines OA and OB as the u- and w-axis of an oblique coordinate system. The equation of the pole tangent p is u · sin θ1 + w · sin θ2 = 0. The equations of the lines AB and αβ u w + =1 r1 r2

u w + = 1. ρ1 ρ2

These last two equations give us the equation of the line q connecting O and the point of intersection of AB and αβ: (

) ( ) 1 1 1 1 − − u+ w = 0. ρ1 r1 ρ2 r2

With the two applications of the Euler–Savary formula this last equation can be written as u · sin θ2 + w · sin θ1 = 0. Comparing the equations of the lines p and q shows that they are symmetrical with respect to the bisector of the angle between OA and OB. This proves Bobillier’s theorem. When the derivative of the expression for the radius of curvature is zero the curvature of a trajectory is stationary. With respect to our special systems of reference we find then the following equation for the curve of stationary curvature: ( 2 ) ( ) x + y2 (a3 x + b3 y) + 3b2 x x2 + y2 − b2 y = 0 In Chap. 10 of this book we discussed Robert Ball’s attempt to show the existence of points that have a third-order contact with the tangent to their trajectory. Such points are nowadays called Ball points. The approach by means of instantaneous invariants gives an elegant way to find them.17 They obviously are points of intersection of the 17

See for example Bottema and Roth (1979), p. 280.

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19 Kinematics in the 20th Century

inflexion circle and the curve of stationary curvature. The equation of the inflexion circle is x2 + y2 − b2 y = 0. Of the six points of intersection three coincide with the pole. The isotropic points are also points of intersection. When a3 /= 0 there is only one Ball point and its coordinates are x =−

b2 b3 a3 a32 + b32

y=

b2 a32 . a32 + b32

The origin, that is the pole, is in general not a Ball point. At the moment under consideration it is in general in a cusp of the curve it describes.

Bibliography

Alievi L (1895) Cinematica della Biella Piana, Naples Allis V, Koetsier T (1991) On some paradoxes of the Infinite. Br J Philos Sci 42:187–194 Allis V, Koetsier T (1995) On some paradoxes of the infinite II. Br J Philos Sci 46:235–247 Almagest (1984) translated and annotated by Toomer GJ, Duckworth, The Old Piano Factory, London Alberto A (2004) Martínez, Kinematic subtleties in Einstein’s first derivation of the Lorentz transformations. Am J Phys 72:790–798 Ampère A-M (1829) Solution d’un problème de dynamique, suivie de considérations sur le problème général des forces centrales. Annales de mathématique pures et appliquées. Tome XX, pp 6–58 Ampère A-M (1834) Essai sur la philosophie des sciences. Paris Arago F (1854) Notices biographiques. Tome I, Paris Aristotle (1936) Mechanica, sometimes attributed to Aristotle. Loeb Classical Library, Cambridge (Mass.) Aristotle (1929) The Physics, with an English translation by Philip H. Wicksteed and Francis M Cornford, The Loeb Classical Library, London (William Heinemann Ld), Cambridge, Massachusetts (Harvard University Press) Aronhold SH (1872) Grundzüge der kinematischen Geometrie. Verhandlungen Der Verein Zur Beförderung Des Gewerbefleiβes in Preuβen 51:129–155 Artobolevskii II (1976) Past, present and future of the theory of machines and mechanisms. Mech Mach Theory 11:353–361 Artobolevskii II (1976–1980) Mechanisms in modern engineering design, a handbook for engineers, designers and inventors, vol 5. MIR Publishers, Moscow Artobolevskii II (2013) Mechanisms for the generation of plane curves. Elsevier Auger L (1962) Un savant méconnu: Gilles Personne de Roberval (1602–1675). Librairie Scientifique A. Blanchard, Paris Augustine (1955) Confessions and Enchiridion, newly translated and edited by Albert C. Outler. Westminster Press, Philadelphia Bernays P (1965) Hilberts Untersuchungen über die Grundlagen der Arithmetik. In: Hilbert D (ed) Gesammelte Abhandlungen, Dritter Band, [Repr.]. Chelsea Publ. Corp., New York, pp 196–216 Baez J, Octonions T (2002) Bull Amer Math Soc 39:145–205 Ball RS (1870–1874) Notes on applied mechanics. I. Parallel Motion. II. The contact of cams. In: Proc Roy Irish Acad Sci 1:243–245 Ball RS (1875) The theory of screws, a geometrical study of the kinematics, equilibrium and small oscillations of a rigid body. Trans Roy Irish Acad Sci 25:157–218 Ball RS (1876) The theory of screws, a study in the dynamics of a rigid body. Dublin © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Koetsier, A History of Kinematics from Zeno to Einstein, History of Mechanism and Machine Science 46, https://doi.org/10.1007/978-3-031-39872-8

329

330

Bibliography

Bell ET (1973) Men of mathematics. Simon and Schuster, New York Belhoste B, Belhoste J-F (1990) La théorie des machines et les roues hydrauliques. Cahiers D’histoire Et De Philosophie Des Sciences 29:1–32 Bernoulli J (1692) Linae cycloidales, evolutae, etc. Acta Eruditorum, pp 207–213, pp 291–296 (Tables III and IV) Bernoulli J (1742) De centro spontaneo rotationis. In: Bernoulli J (ed) Opera Omnia. Tom. IV, Lausanne, pp 265–273 Domenico Bertolini Meli (1990) The relativization of centrifugal force in the eighteenth century. Isis 81:23–43 Biener Z, Schliesser E (eds) Newton and empiricism. Oxford Academic Bion N (1709) Traité de la Construction et des Principaux Usages des Instruments de Mathématique. Paris Blaschke W, Müller HR (1956) Ebene Kinematik. R. Oldenbourg, München Blåsjö V (2016) Transcendental curves in the Leibnizian Calculus, Ph. D. thesis, University of Utrecht Bobillier E (1837) Géométrie, 3rd edn. Paris Bobillier E (1880) Cours de Géométrie, 15th edn. Paris Bos HJM (1996) Johann Bernoulli on exponential curves, ca. 1695: innovation and habituation in the transition from explicit constructions to implicit functions. Nieuw Archief voor Wiskunde (IV) 14:1–19 Bos HJM (1988) Tractional motion and the legitimation of transcendental curves. Centaurus 31:9–62 Bos HJM (2001) Redefining geometrical exactness, Descartes’ transformation of the early modern concept of construction. Springer Bottema O (1961) Some remarks on theoretical kinematics 1. On instantaneous invariants, 2. On the application of elliptic functions in kinematics. In: Proceedings of the international conference for teachers of mechanisms, Yale University, New Haven, pp 157–164 Bottema O, Roth B (1979) Theoretical kinematics. Amsterdam Boyer CB (1945) Historical stages in the definition of curves. National Math Mag 19:294–310 Boyer CB (1947–1948) Note on epicycles & the ellipse from Copernicus to Lahire. Isis 38:54–56 Boyer CB (1956) History of analytic geometry. New York Bradley RE, Sandifer E (eds) (2007) Leonhard Euler, life, work and legacy (Studies in the History and Philosophy of Mathematics, Volume 5). Elsevier Science Publishers Bresse J (1853) Mémoire sur un théorème nouveau concernant les mouvements plans et sur l’application de la cinématique a la détermination des rayons de courbure. Journal de l’École Polytechnique 35:89–115 Bricard R (1926) Leçons de cinématique, Tome I, Cinématique théorique. Paris Bruin F, Vondjijdis A (1971) (translators and editors). The books of Autolykos: on the moving sphere and on risings and settings. American University of Beirut, Beirut Bulmer-Thomas I (1939a) Selections illustrating the history of Greek mathematics, vol I, from Thales to Euclid. London William Heinemann Ltd & Harvard University Press Bulmer-Thomas I (1939b) Selections illustrating the history of Greek mathematics, vol II, from Aristarchus to Pappus. London William Heinemann Ltd & Harvard University Press Bulmer-Thomas I (1984) Plato’s astronomy. Class Q New Ser 34:107–112 Blanc C, de Haller P (eds) Leonhardi Euleri Opera Omnia, Series 2, Vol. 17, Leonhardi Euleri, Commentationes Mechanicae ad theoriam machinarum pertinentes, vol 3. Bern Burmester L (1876) Über die Geradführung durch das Kurbelgetriebe. Der Civilingenieur 22:597– 606 (plus Table 30) Burmester L (1877) Über die Geradführung durch das Kurbelgetriebe. Der Civilingenieur 23:227– 250, 319–342 (plus Tables 10, 11, 14, 15, 16) Burmester L (1880) Ueber die momentane Bewegung ebener kinematischer Ketten. Der Civilingenieur 26:248–285 (plus Tables 14–18) Burmester L (1888a) Lehrbuch der Kinematik, vol 1. Die ebene Bewegung, Leipzig Burmester L (1888b) Atlas zum Lehrbuch der Kinematik, vol 1. Die ebene Bewegung, Leipzig

Bibliography

331

Camus CÉL (1766) Cours de mathématique, troisième partie, Élémens de mécanique statique. Tome second, Paris Camus CÉL (1837) A treatise on the teeth of wheels: demonstrating the best forms which can be given to them for the purposes of machinery. London Cannon JT, Dostrovsky S (2012) The evolution of dynamics: vibration theory from 1687 to 1742:. Springer Cantor M (1900) Vorlesungen ¨Uber Geschichte der Mathematik, Band II, 2nd edn. Stuttgart Caparrini S (2002) The discovery of the vector representation of moments and angular velocity. Arch Hist Exact Sci 56:151–181 Capecchi D (2012) History of virtual work laws, a history of mechanics prospective. BirkhauserSpringer ˇ Capek M (1976) The concepts of space and time, their structure and development. Reidel, Dordrecht Carnot L (1797) Essai sur les machines en général. Defay, Dijon (Also published in Oeuvres mathématiques du citoyen Carnot. Bâle, J. Decker, 1797, pp. v–xvi+1–124) Giulio Carlo Ignazio (1854) Elementi di cinematica applicata alle arti. Torino Carnot L (1803) Principes fondamentaux de l’équilibre et du movement. Paris Cauchy A-L (1899) Sur les mouvements que peut prendre un système invariable, libre, ou assujetti à certaines conditions. Exercices de mathématiques, Chez De Bure Frères, Paris 70–90 (Also in Oeuvres, IIe Série, Tome VII, Paris 1899, pp 94–120) Cayley A (1869) On the mechanical description of a nodal bicircular quartic. Proc Lond Math Soc 1–2:100–106 Cayley A (1876a) On three-bar motion. Proc Lond Math Soc 7:136–166 Cayley A (1876b) On the bicursal sextic. Proc Lond Math Soc 7:166–172 Ceccarelli M (2000) Screw axis defined by Giulio Mozzi in 1763 and early studies on helicoidal motion. Mech Mach Theory 35:761–770 Ceccarelli M (ed) (2007) Distinguished figures in mechanism and machine science, their contributions and legacies, part 1. Springer Ceccarelli M (ed) (2010) Distinguished figures in mechanism and machine science, their contributions and legacies, part 2. Springer Ceccarelli M (ed) (2014) Distinguished figures in mechanism and machine science, their contributions and legacies, part 3. Springer Ceccarelli M (ed) (2020) Distinguished figures in mechanism and machine science, their contributions and legacies, part 4. Springer Ceccarelli M, López-García R (eds) (2022) In: 7th International symposium on history of machines and mechanisms (HMM), explorations in the history and heritage of machines and mechanisms. Springer Ceccarelli M (2022) Paolo Frisi with contributions to modern Mechanics. In: IX Convegno di Storia dell’Ingegneria - Napoli 2022, 5th International Conference on History of Engineering - Naples - Italy – 2022, 14 p Chasles M (1830) Note sur les propriétés générales du système de deux corps semblables entre eux, placés d’une manière quelconque dans l’èspace; et sur le déplacement fini, ou infiniment petit d’un corps solide. Bulletin des sciences mathématiques, physiques et chimiques, Tome XIV, pp 321–336 Chasles M (1843) Propriétés géométriques relatives au mouvement infiniment petit d’un corps solide libre dans l’espace. Comptes rendus des séances de l’Académie des Sciences, Tome XVI:1420–1442 Chasles M (1845) Constructions des rayons de courbure des courbes décrites dans le mouvement d’une figure plane qui glisse sur son plan. Journal de mathématiques pures et appliquées 10:204– 208 Chasles M (1860a) Propriétés relatives au déplacement fini quelconque, dans l’espace, d‘une figure de forme invariable. Comptes rendus hebdomadaires des séances de l’Académie des Sciences, Tome 51:855–863

332

Bibliography

Chasles M (1860b) Suite du Mémoire sur le déplacement fini quelconque, dans l’espace, d‘une figure de forme invariable. Comptes rendus hebdomadaires des séances de l’Académie des Sciences, Tome 51:905–914 Chasles M (1861a) Sur le déplacement d’une figure de forme invariable dans l’espace (suite). Comptes rendus hebdomadaires des séances de l’Académie des Sciences, Tome 52:77–85 Chasles M (1861b) Sur le déplacement dune figure de forme invariable dans l’espace (suite) Comptes rendus hebdomadaires des séances de l’Académie des sciences, Tome 52:189–197 Chasles M (1861c) Sur le déplacement dune figure de forme invariable dans l’espace (suite). Comptes rendus hebdomadaires des séances de l’Académie des sciences, Tome 52:487–501 Chasles M (1877–1878) Mémoire de géométrie sur la construction des normales à plusieurs courbes mécaniques. Bulletin de la Société Mathématique de France 6:1877–78, 208–231 Chebyshev PL (1962a) Sur les parallélogrammes. In: Chebyshev (1962b), pp 85–106 Chebyshev PL (1962b) Oeuvres de P. L. Tchebycheff. Tome II, Chelsea, New York Graney ChM (2011) Coriolis effect, two centuries before Coriolis. Phys Today 64:8–9 Clagett M (1971) Greek science in antiquity. New York. Reprinted in 1971 by arrangement with Abelard-Schumann, Ltd. Clagett M (1956) The Liber de motu of Gerard of Brussels and the Origins of Kinematics in the West. Osiris 12 Clagett M (1959) The science of mechanics in the Middle Ages. University of Wisconsin Press Clagett M (1984) Archimedes in the Middle Ages, volume Five, parts I–V. Phila-delphia Coriolis G-G (1829) Du calcul de l’effet des machines. Paris Coriolis G-G (1832) Mémoire Sur le principe des forces vives dans les mouvemens relatifs des Machines. Journal de l’École Polytechnique, Tome XIII (XXIe Cahier), pp 268–302 (References are to the reprint in Coriolis (1990]) Coriolis G-G (1835) Mémoire Sur les équations du mouvement relatif des systèmes de corps. Journal de l’École Polytechnique, Tome XV(XXIVe Cahier), pp 142–154 (References are to the reprint in Coriolis (1990]) Coriolis G-G (1829) Traité de la mécanique des corps solides et du calcul de l’effet des machines. Seconde édition, Paris (The last part of this book is a reprint of Coriolis (1829) Coriolis G-G (1990) Théorie mathématique des effets du jeu de billard suivi des deux célèbres mémoires publiés en 1832 et 1835 dans le Journal de l’École Polytechnique. Éditions Jacques Gabay, Sceaux ˇ Cornford FM (1976) The invention of space. In: Capek (1976), pp 3–17 Corry L, Minkowski H (2010) Relativity and the axiomatic approach to physics. In: Petkow (2010), pp 3–42 Coxon AH (1968) The Fragments of Parmenides, A critical text with introduction, translation, the ancient testimonia and a commentary. Van Gorcum, Assen/Maastricht, The Netherlands, Wolfeboro, New Hampshire, USA Crowe MJ (1967) A history of vector analysis. University of Notre Dame Press. Notre Dame Czwalina A (ed) Archimedes Werke. Darmstadt Darboux G (1890) Sur le déplacement d’une figure invariable. Annales scientifiques de l’École Normale Supérieure, Serie 3, vol 7, pp 323–326 Descartes (1649) Geometria (translated by Frans van Schooten, annot. F. Debeaune). Leiden Descartes R (1659–1661) Geometria. Second enlarged edition by F. van Schooten of the first 1649 edition, includes treatises by Van Schooten, Hudde, Heuraet, Debeaune and De Witt, 2 vols. Amsterdam Descartes R (1954) The Geometry of Rene Descartes with a facsimile of the first edition (translation and edition by Smith DE, Latham ML). Dover Publications, Inc. New York, (Originally published in 1925 by the Open Court Publishing Co.) Kirsch (1876) Zur Theorie der Geradführungen. Der Civilingenieur 22:321–334 (Mit Tafel 17) Groot J de (compiler) (1970) Bibliography on kinematics. Two volumes. Eindhoven University of Technology, Eindhoven, Holland

Bibliography

333

Hire P de la (1694) Mémoires de mathematique et de physique contenant un traité des épicycloïdes. Paris Hire P de la (1706) Traité des roulettes. Histoire de l’Académie royale des sciences, Année 1706, Paris, 1731, pp 340–389 Jonge AER de (1942) What is wrong with ‘Kinematics’ and ‘Mechanisms’? Mechanical Engineering, April 1942, pp 273–278, and the author’s closure, Mechanical Engineering, pp 747–751 Jonge AER de (1943) A brief account of modern kinematics. Trans ASME, pp 663–683 Lanz JM de, Bétancourt y Molina A de (1808) Essai sur la composition des machines: Programme du cours élémentaire des machines pour l’an 1808 par M. Hachette. Paris Lanz JM de, Bétancourt y Molina A de (1817) Analytical essay on the construction of machines: translated from the French of M.M. Lanz and Bétancourt. London Ocagne M d’ (1905) Le calcul simplifié par les procédés mécaniques et graphiques. Paris, 2ième Edn Prony GCFMR de (1790) Architecture hydraulique, vol I. Paris Prony GCFMR de (1796) Architecture hydraulique, vol II. Paris Prony GCFMR de (1826) Sur le parallélogramme du balancier de la machine à feu. Annales des Mines XII, pp 81–91 Rowe DE (1989), Klein, Hilbert, and the Gottingen Mathematical Tradition. Osiris 5:186–213 Saint-Venant AJCB de (1851) Principes de Mécanique fondés sur la Cinématique. Paris Alembert JLR d’ (1749) Recherches sur la Précession des Équinoxes et sur la Nutation de l’Axe de la Terre. Paris Aubuisson de Voisins J-F d’ (1834) Traité d’Hydraulique à l’ Usage des Ingénieurs. Chez F. G. Levrault, Paris Dickinson HW, Jenkins R (1927) James Watt and the Steam Engine, Ox-ford Dicks DR (1970) Early Greek Astronomy to Aristotle. Thames and Hudson Drabkin IE (1950) Aristotle’s wheel: notes on the history of a paradox. Osiris 9 Drachmann AG (1963) The mechanical technology of Greek and Roman Antiquity. Munksgaard, Copenhagen Dugas R (1988) A History of Mechanics. Dover Publications, New York Duhamel JMC (1838) Note sur la Méthode des Tangentes de Roberval. Mémoires présentés par divers savants à l’Académie royale des Sciences de l’Institut de France, Sciences Mathématiques et Physiques. Tome V, Paris, pp 257–266 Dupont P (1962–1963) Nuove dimostrazioni della formula di Eulero-Savary e suo studio storico. Atti della Reale Accademia delle scienze di Torino 97:927–962 Dupont P (1963–1964a) Il centro istantaneo di rotazione nei moti rigidi piani, dal punto di vista storico, critico, applicativo. Atti della Reale Accademia delle scienze di Torino 98:331–354 Dupont P (1963–1964b) Il teorema di Bobillier, il problema della retta tangente alle polari e sua aplicazione per la determinazione del centro delle accelerazioni. Atti della Reale Accademia delle scienze di Torino 98:442–471 Dupont P (1963–1964c) Determinazione grafica del centro di curvatura della polare fissa nei moti rigidi piani. Atti della Reale Accademia delle scienze di Torino 98:489–513 Dijksterhuis EJ (1950) De Mechanisering van het Wereldbeeld. Amsterdam Dijksterhuis EJ (1956) Archimedes. Ejnar Munksgaard, Copenhagen Einstein A (1905) Zur Elektrodynamik bewegter Körper. Ann Phys 17:891–921 Einstein A (1933) On the methods of theoretical physics, Herbert Spencer Lecture. Clarendon Press, Oxford Erdman AG (1993) Modern kinematics, developments in the last forty years. Wiley Euler L (1752) E177 Découverte d’un nouveau principe de Mécanique. Mémoires de l’académie des sciences de Berlin 6:185–217 (Also E177. Opera Omnia: Ser 2, vol 5, pp 81–108) Euler L (1754) De aptissima figura rotarum dentibus tribuenda. Novi commentarii academiæ scientiarium Petropolitanæ 5(1754/5):1760, 299–316 (Also E249, Opera Omnia: Series 2, Volume 17, pp. 119–135)

334

Bibliography

Euler L (1767) Supplementum de figura dentium rotarum. Novi Commentarii aca-demiae scientiarum Petropolitanae 11:207–231 (Also E330, Opera Omnia: series 2, 17, pp. 196–219) Farey F (1827) A treatise on the Steam Engine, vol 1. London Farrington B (1944) Greek science, its meaning for us. Penguin Books Ferguson ES (1962) Kinematics of mechanisms from the time of Watt. Contributions from the Museum of History and Technology, United States National Museum. Bulletin 228, Washington D.C., Paper 27, pp 185–230 Flükiger M (1974) Albert Einstein in Bern: Das Ringen um ein neues Weltbild; Eine Dokumentarische Darstellung über den Aufstieg eines Genies. Paul Haupt Berne, Berne Fowler DH (1994) Continued fractions. In: Grattan-Guinness I (ed) Companion encyclopedia of the history and philosophy of the mathematical sciences, vol I, London and NewYork, pp 730–740 Frajese A (1951) Sur la signification des postulats euclidiens. Archives Internationales d’histoire des Sciences 30:383–392 Franke R (1979) Zur Geschichte des Cardan-Gelenks. Technikgeschichte 46:3–19 Galison P (2000) Einstein’s clocks: the place of time. Crit Inq 26:355–389 Gaukroger S (1995) Descartes, an intellectual biography. Clarendon Press, Oxford Gaukroger S (2014) Empiricism as a Development of Experimental Natural Philosophy. In: [Biener&Schliesser,2014], pp 15–38 Geikie A (1923) Obituary notice Sir Alfred Bray Kempe 1849–1922. In: Proceedings of the Royal Society of London, Obituary notices of fellows deceased, pp. i–x Descartes R (1683) Geometria a Renato Des Cartes, una cum notis Florimundi de Beaune, opera atque Studio Francisci à Schooten (third edition). Amsterdam Gillespie CC (1971) Lazare Carnot Savant. Princeton University Press, Princeton Goodstein JR (1984) The rise and fall of Vito Volterra’s world. J Hist Ideas 45:607–617 Grant E (1966) Nicole Oresme, De proportionibus proportionum and Ad pauca respicientes (Edited, translated and annotated by Edward Grant). The University of Wisconsin Press, Madison Grant E (1974) A Source Book in Medieval Science. Harvard Grootendorst AW (1999) De kegelsneden bij Jan de Witt. Nieuw Archief voor Wiskunde, 4e serie, deel 17:409–425 Grouard A (1870) Recherches sur les figures planes semblables. Bulletin de la Société Philomatique de Paris. Séance du 22 janvier 1870, pp 4–6 Grübler M (1883) Allgemeine Eigenschaften der zwanglaufigen ¨ ebenen kinematischen Ketten. Der Civilingenieur 29:168–200 Gr¨Ubler M (1889) Wandlungen der Kinematik in der Gegenwart. Der Civilingenieur, pp 221–236 Grübler M (1917) Getriebelehre, Eine Theorie des Zwanglaufes und der ebenen Mechanismen. Berlin Hachette JNP (1828) Traité élémentaire des machines, quatrième édition. Corby, Paris Hachette JNP (1830) Histoire des Machines à Vapeur depuis leur Origine jusqu’à nos Jours. Paris Halsted GP (1895) Biography: Pafnutij Lvovitsch Tchebychev. The American Mathematical Monthly 2(3):61–63 Harari O (2003) The concept of existence and the role of constructions in Euclid’s elements. Arch Hist Exact Sci 57:1–23 Harman PM (1982) Energy, Force, and Matter, The Conceptual Development of Nineteenth-Century Physics. Cambridge University Press, Cambridge Hart H (1875) On certain conversions of motion. The Messenger of Mathematics IV, pp 82–87 and pp 116–120 Hartenberg RS, Denavit J (1964) Kinematic synthesis of linkages. McGraw-Hill, New York Heath TL (1921a) A History of Greek mathematics, vol. I, from Thales to Euclid. Oxford (Reprint as a Dover Edition in 1981) Heath TL (1921b) A history of Greek mathematics, vol. II, from Aristarchus to Diophantus. Oxford (Reprint as a Dover edition in 1981) Heath TL (1932) Greek astronomy, edited by Ernest Baker, London, J.M. Dent & Sons LTD

Bibliography

335

Heiberg JL (ed) (1910–1915) Archimedes Opera, 2nd edn, vol 3. Teubner, Leipzig (Reprint in Stuttgart, 1972) Hellige HD (1991) Leitbilder und historisch-gesellschaftlicher Kontext der frühen wissenschaftlichen Konstruktionsmethodik, artec-Paper Nr. 8, Januar 1991. See https:// www.uni-bremen.de/fileadmin/user_upload/sites/artec/Publikationen/artec_Paper/008_paper. pdf Hensel S, Ihmig KN, Otte M (1889) Mathematik und Technik im 19. Jahrhundert in Deutschland. Soziale Auseinandersetzung und philosophische Problematik. Göttingen, Vandenhoeck& Ruprecht Hermann J (1716) Phoronomia, sive de viribus & motibus corporum solidorum & fluidorum. Amsterdam Holditch H (1842) On rolling curves [Read December 10, 1838]. Cambridge Philosophical Transactions 7:61–86 Holditch H (1858) Geometrical theorem. The Quarterly Journal of Pure and Applied Mathematics 2:38 Holton G (1973) Thematic origins of scientific thought. Harvard University Press Hunt KH (1978) Kinematic geometry of mechanisms. Clarendon Press, Oxford Husty M (2007) Walter Wunderlich (1910–1998). In: Ceccarelli (2007), pp 371–392 Huygens C (1703) Descriptio automati planetarii. In: Christiani Hugenii Zelemii, dum viveret, toparchae. Opuscula postuma Huygens C (1929) Oeuvres complètes. Tome XVI, Percussion, Den Haag Huygens C (1937) Oeuvres complètes. Tome XIX, Mécanique théorique et physique 1666–1695, Den Haag Huygens C (1944) Oeuvres complètes, Tome XXI, Cosmologie. Den Haag Itard J (1970) Etienne Bobillier. Dictionary of Scientific Biography, vol. II. New York Janssen M (2009) Drawing the line between kinematics and dynamics in special relativity. Stud Hist Philos Mod Phys 40:26–52 Jan de Witt Elementa Curvarum Linearum, Liber Primus: Tekst, vertaling, inleiding en commentaar door A.W. Grootendorst, Amsterdam (1997) Joella GY (1988) Unrolling time, Christiaan Huygens and the mathematization of nature. Cambridge University Press, Cambridge John D (2004) Norton, Einstein’s Investigations of Galilean Covariant Electrodynamics prior to 1905. Arch Hist Exact Sci 59:45–105 Jüttler B (1993) Über zwangläufige rationale Bewegungsvorgänge. In: Sitzungsberichte der Österreichischen Akademie der Wissenschaften, Abt. II, Bd. 202,1993, pp 117–132 Kant I (1786) Metaphysische Anfangsgründe der Naturwissenschaft, Riga Kapovitch M, Millson JJ (2002) Universality theorems for configuration spaces of planar linkages. Topology 41:1051–1107 Kempe AB (1873) A general method of solving equations of the nth degree by mechanical means. Messenger Math II: 51–52 Kempe AB (1875) On some new linkages. Messenger Math IV: 121–124 Kempe AB (1876) On a general method of describing plane curves of the nth degree by linkwork. In: Proceedings of the London Mathematical Society, volumes 1–7, issue 1, November 1875, pp 213–216 Kempe AB (1877) How to draw a straight line, a lecture on linkages. London, 1877 (References are to the reprint in 1977 by the National Council of Teachers of Mathematics, U.S.A.) Kennedy ABW (1887) Mechanics of machinery. London Kennedy ES (1966) Late medieval planetary theory. Isis 57:365–378 Kenneth KJ (1966) The constraint analysis of mechanisms. J Mech 1:101–114 Kerle H (2007) Kurt Hain (1908–1995). In: Ceccarelli (2007), pp 183–215 Klein F (1872) Vergleichende Betrachtungen über neuere geometrische Forschungen. Erlangen Klein F (2016) Elementary mathematics from a higher standpoint: volume II: geometry. Springer

336

Bibliography

Knorr WR (1975) The evolution of the euclidean elements, a study of the theory of incommensurable magnitudes and its significance for early Greek geometry. D Reidel Publishing Company, Dordrecht Knorr WR (1978) Archimedes and the spirals: the heuristic background. Hist Math 5:43–75 Knorr WR (1983) Construction as existence proof in ancient geometry. Anc Philos 3:125–148 Knorr W (1986) The ancient tradition of geometric problems. Birkhäuser, Boston Koenigs GXP (1897) Leçons de Cinématiques professées à la Sorbonne, Cinématique théorique. Paris Koetsier T (1983) A contribution to the history of kinematics, part I and part II. Mech Mach Theory 18:37–42, 43–48 Koetsier T (1986) From kinematically generated curves to instantaneous invariants: episodes in the history of instantaneous planar kinematics. Mech Mach Theory 21:489–498 Koetsier T, Allis V (1997) Assaying supertasks. Logique & Analyse 40:291–313 Koetsier T (2004) A Note on Gaspard-Gustave de Coriolis (1792–1843). In: Huang T (ed) Proceedings of the eleventh world congress in mechanism and machine science, vol 2, China Machine Press, Tianjin, pp 916–920 Koetsier T, Bergmans L (2005) Mathematics and the divine, a historical study. Elsevier, Amsterdam Koetsier T (2007) Euler and kinematics. In: [Bradley&Sandifer,2007], pp 167–194 Koetsier T, Burmester L (2010) (1840–1927). In: [Ceccarelli,2010], pp 43–64 Koetsier T (2019) The ascent of GIM, the global intelligent machine, a history of production an information machines. Springer Koetsier T (2022) Kinematic modelling in greek astronomy: the design of heavenly mechanisms. In: [Ceccarelli&López-García,2022], pp 279–291 Krafft F (1970) Dynamische und Statische Betrachtungsweise in der Antiken Mechanik. Franz Steiner Verlag GMBH, Wiesbaden Kragh H (2015) Mathematics and physics: the idea of a pre-established harmony. Sci Educ 24:515– 527 Kurrer K-E (2002) Geschichte der Baustatik, Auf der Suche nach dem Gleichgewicht. Ernst & Sohn, Wiley Company Kusukawa S (1996) Bacon’s classification of knowledge. In: Peltonen M (ed) The Cambridge Companion to Francis Bacon, Cambridge, pp 47–74 Kutzbach K (1929) Mechanische leitungsverzweigung, ihre gesetze und anwendungen. Maschinenbau 8(21):710–716 Lamar Crosby H Jr (1955) Thomas of Bradwardine His Tractatus de Proportionibus, Its significance for the development of mathematical physics, Edited and translated by Lamar Crosby H Jr. The University of Wisconsin Press, Madison Lagrange J-L (1788) Mécanique analytique. Paris Le Ru V (1994) La force accélératrice: un exemple de définition contextuelle dans le Traité de Dynamique de d’Alembert. Revue d’histoire des sciences 47:475–494 Liguine V (1875) Nouvelles Annales de Mathématiques, 2ieme Serie 14, pp 529–560 Lindberg DC (ed) (1978) Science in the middle ages. University of Chicago Press, Chicago Lipkin L (1870) Ueber eine genaue Gelenk-Geradfuhrung. Bulletin De L’académie Impériale Des Sciences De Saint-Pétersbourg 16:57–60 Littlewood JE (1953) A mathematician’s miscellany. Methuen, London Lorentz HA, Einstein A, Minkowski H (1913) Das Relativitätsprinzip, eine Sammlung von Abhandlungen, mit Anmerkungen von A. Sommerfeld und Vorwort von O. Blumenthal. Leipzig, Berlin Loria G (1911) Spezielle algebraische und transzendente ebene Kurven. Theorie und Geschichte, Leipzig Loria G, Vassura G (1919) Opere di Evangelista Torricelli, vol I, Parte II, Faenza Lovett EO (1900) Koenigs’ lectures on kinematics. Bull Amer Math Soc 6(7):299–304 McCarthy JM (2000) Geometric design of linkages. Springer, New York

Bibliography

337

Mach E (1907) The science of mechanics, a critical and historical account of its development. Chicago Michael Sean Mahoney (1973) The Mathematical Career of Pierre de Fermat (1601–1665). Princeton University Press Mahoney MS (1978) Mathematics (in the Middle Ages). In: [Lindberg,1978], pp 145–178 Mallik AK (1994) Kinematic analysis and synthesis of mechanisms. CRC Press Mannheim A (1889) Étude d’un déplacement particulier d’une figure de forme invariable par des procédés élémentaires et purement Géométriques. Rend Circ Mat Palermo 3:131–144 Marchisotto EA (1992) Lines without order. Am Math Mon 99:738–745 Marsden W (1969) Greek and Roman Artillery. Historical Development, Oxford Martínez AA (2009) Kinematics, the lost origins of Einstein’s relativity. Baltimore Mauersberger K (2001) Zur Geschichte der Getriebetechnik an der TH Dresden vor Lichtenheld. Wissenschaftliche Zeitschrift Der TH Dresden 50:9–18 Mauersberger K (2006) Theory of mechanisms and machines at Dresden University of Technology: a historical review. Dresden Mendell H (2007) Two traces of two-step Eudoxan proportion theory in Aristotle: a tale of definitions in Aristotle with a moral. Arch Hist Exact Sci 61:32–37 Mersenne M (1963) Correspondence du P. M. Mersenne. Tome VIII, Paris Roverval GP de (1699) Mémoires de l’Académie Royale des Sciences contenant les ouvrages adoptez par cette académie avant Renouvèlement en 1699. Tome III, Ouvrages de M. Roberval, pp 1–67 Miller AI (1981) Albert Einstein’s special theory of relativity. Addison-Wesley Mohr KF (1841) Uber die Gorgonmachine. Dingler’s polytechnisches Journal 81:16–18 (Figs. 34– 37) Mueller I (1981) Philosophy of mathematics and deductive structure in Euclid’s elements. The MIT Press, Cambridge Muirhead JP (1854) The origin and progress of the mechanical inventions of James Watt, Vols I, II and III, London Mulder HM (1990) Pure, mixed and applied mathematics the Changing perception of mathematics through history. Nieuw Archief Voor Wiskunde 4(8):27–41 Murdoch JE, Sylla ED (1978) The science of motion (in the Middle Ages). In: Lindberg (1978), pp 206–264 Moon FC (2007) The Machines of Leonardo da Vinci and Franz Reuleaux, Kinematics of Machines from the Renaissance to the 20th Century. Springer Morin et al. (1875) Prix Montyon, Mécanique, Comptes rendus hebdomadaire des séances de l’Académie des Sciences LXXX, 1875, pp 1469–1470 Morrow GR (1970) (Translation and Introduction), Proclus: a commentary on the first book of Euclid’s elements. Princeton University Press, Princeton Mozzi G (1763) Discorso matematico sopra il rotamento momentaneo dei corpi. Stamperia di Donato Campo, Napoli Neugebauer O (1953) On the “hippopede” of Eudoxus. Scripta Mathematica 19:225–229 Neugebauer O (1975) A history of ancient mathematical astronomy. Springer Neugebauer O (1983) Astronomy and history, selected essays. Springer, New York Newton I (1745) Sir Isaac Newton’s two treatises of the quadrature of curves and analysis by equations of an infinite number of terms explained etc., Translation by John Stewart, London Newton I (1967) The mathematical papers of Isaac Newton (Edited by Whiteside DT), vol 1, Cambridge Nicole F (1707) Méthode générale pour déterminer la nature des courbes formées par le roulement de toutes sortes de courbes sur une autre courbe quelconque. Mémoires de l’Académie des sciences de l’Institut de France, vol 20, pp 81–97 Nikulin D (2002) Matter, imagination and geometry. Ontology, natural philosophy and mathematics in Plotinus, Proclus and Descartes, Aldershot

338

Bibliography

Nolle H (1974a) Linkage coupler curve synthesis, a historical review I, developments up to 1875. Mech Mach Theory 9:147–168 Nolle H (1974b) Linkage coupler curve synthesis, a historical review II, developments after 1875. Mech Mach Theory 9:325–348 Nolle H (1975) Linkage coupler curve synthesis, a historical review III, spatial synthesis and optimization. Mech Mach Theory 10:41–55 O’Meara DJ (2005) Geometry and the divine in Proclus. In: Koetsier&Bergmans (2005), pp 133–145 Palagia O, Choremi-Spetsieri A (eds) (2007) The panathenaic games. Oxford Books, Oxford Palter R (1972) Kant’s formulation of the laws of motion. Synthese 24:96–116 PAM (1914) Obituary notice Samuel Roberts, 1827–1913. In: Proceedings of the Royal Society of London, Series A, LXXXIX, pp xx–xxi Pappus (1986) Pappus of Alexandria, book 7 of the collection, part 1. Introduction, text, and translation, Edited with translation and commentary by Alexander Jones. Springer, New York Peaucellier CN (1864) Lettre de M. Peaucellier, capitaine du Génie (à Nice). Nouvelles Annales de Mathématiques. Journal des Candidats aux Écoles Polytechnique et Normale (2me série) 3:414–415 Peaucellier CN (1872–1873) Note sur un balancier articulé à mouvement rectiligne. Journal de Physique: 388–390 Peaucellier CN (1873) Note sur une question de géométrie de compas. Nouvelles Annales de Mathématiques (2me série) 12:71–78 Peaucellier CN (1876) Note sur l’emploi des systèmes articulés à liaison complète, en géométrie, en mécanique et dans les sciences appliquées. Mémorial de l’Officier du Génie 25:369–389 Persson A (1998) How do we understand the Coriolis force? Bull Amer Meteor Soc 79:1373–1385 Petersen O (1974) A survey of the almagest. Odense University Press Petkow V (ed) (2010) Minkowski spacetime: a hundred years later. Springer Plato (1973) The collected dialogues including the letters (Cornford’s translation), Edited by Hamilton E, Cairns H with and introduction and prefatory notes, Bollingen series LXXI, Princeton Plotinus (1917–1930) Enneads 4, Tractate 8, The Soul’s Descent into the Body, Translated by Stephen MacKenna and B. S. Page. PL Warner, Publisher to the Medici Society, London Poincaré H (1954) Sur la dynamique de l’électron, Rendiconti del Circolo Matematico di Palermo 21, 1906, pp 129–176. Reprinted in Œuvres de Henri Poincaré, Tome IX, Paris, 1954, pp 494–586 Poinsot L (1834) Théorie nouvelle de la rotation des corps. Paris Polster H (1912) Kinematik. Berlin und Leipzig Poncelet J-V (1839) Mécanique Industrielle, exposant les principes de statique et de dynamique, les organes mécaniques et les moteurs. Meline, Caus et compagnie, Bruxelles Poncelet J-V (1848) Lehrbuch der Anwendung der Mechanik auf Maschinen, Band II, Darmstadt (Deutsch herausgegeben von C. H. Schnuse) Proclus (1970) A commentary on the first book of Euclid’s elements. Translated with Introduction and Notes by Morrow GR, Princeton University Press, Princeton Pyenson L (1977) Hermann Minkowski and Einstein’s special relativity theory. Arch Hist Exact Sci 17:71–95 Pyenson L (1985) The Young Einstein, the Advent of relativity. Bristol, Boston, Adam Hilger Rankine WJM (1861) Manual of applied mechanics: with numerous diagrams. London Ravetz J (1961) The representation of physical quantities in eighteenth century physics. Isis 52:7–20 Reid C (1996) Hilbert. Springer Résal H (1862) Traité de cinématique pure. Mallet-Bachelier, Paris Reuleaux F (1875) Theoretische Kinematik. Berlin Reuleaux F (1876) The kinematics of machinery, outlines of a theory of machines, Translated by Alex B. W. Kennedy, London Reuleaux F (1890) Ueber das Verhältnis zwischen Geometrie, Mechanik und Kinematik. Zeitung des Vereins Deutscher Ingenieure 34:217–225, 243–248 Reuleaux F (1900) Lehrbuch der Kinematik, Band 2. Braunschweig

Bibliography

339

Reuleaux F (1894) Der Konstrukteur, Ein Handbuch zum Gebrauch beim Maschinen-Entwerfen, Friedrich Vieweg und Sohn, Braunschweig, 4e umgearbeitete und erweiterte Auflage Reuleaux F (1963) The kinematics of machinery, outlines of a theory of machines by Franz Reuleaux, with a new introduction by Eugene S. Ferguson, Dover Publications, Inc., New York Riccioli GB (1651) Almagestum Novum, vol 2. Bologna, Italy Rittershaus T (1875a) Zur heutigen Schule der Kinematik. Der Civilingenieur 21:425–438 Rittershaus T (1875b) Zur Theorie der Quintenz-Waage. Der Civilingenieur 21:45–50 Rittershaus T (1884) Kinematik, Algemeine Encyklopedie der Wissenschaften und Künste. Ersch JS, Gruber JG (eds) vol. 36, pp 87–95 Roberts S (1866) On the mechanical description of some species of circular curves of the third and fourth degrees. Proc Lond Math Soc 1–2:125–136 Roberts S (1869a) On the pedals of conic sections. Proc Lond Math Soc 1–3:88–99 Roberts S (1869b) On the motion of a plane under certain conditions. Proc Lond Math Soc 1– 3:286–319 Roberts S (1876) On three bar motion. Proc Lond Math Soc 7:216–255 Robinson PJ (1994) Evangelista Torricelli. Math Gaz 78:37–47 Robison J (1822) A system of mechanical philosophy, vol 2. London Rockett AM, Szüsz P (1992) Continued fractions. World Scientific, Singapore Roegel D (2012) Napier’s ideal construction of the logarithms, 12 September 2012 (last version: 1 April 2021), LOCOMAT project, the LORIA Collection of Mathematical Tables: http://loc omat.loria.fr Rosenauer N, Willis AH (1967) Kinematics of mechanisms. New York Roth B (2007) In memoriam of Professor Freudenstein (1926–2006), Editorial in Transactions of the ASME. J Mech Des 129:241–242 Roth B (2015) On the advantages of instantaneous invariants and geometric kinematics. Mech Mach Theory 89:5–13 Saliba G (1979) The first non-ptolemaic astronomy at the Maraghag school. Isis 70:571–576 Salmon WC (ed) (1970) Zeno’s paradoxes. Indianapolis Savary F (1835–1836) Leçons et cours autographiés, Notes sur les machines, École Polytechnique, Unpublished lecture notes, available at the Bibliothèque Nationale in Paris Schadwill S (1876) Das Gliedervierseit als Grundlage der Kinematik. Verhandlungen des Vereins Zur Forderung ¨ des Gewerbfleisses in Preussen 55:378–447 Schilpp PA (ed) (1951) Albert Einstein: philosopher-scientist. Tudor Publishing Company, New York Schoenflies A (1886) Geometrie der Bewegung in synthetischer Darstellung. Leipzig Sidoli N, Sait K (2009) The Role of Geometrical Construction in Theodosius’s Spherics. Arch Hist Exact Sci 63:581–609 Smith C, Norton Wise M (1989) Energy and Empire, a biographical study of Lord Kelvin. Cambridge Stachel JJ (2002) Einstein from “B” to “Z”, Birkhäuser Stan M (2022) Phoronomy: space, construction, and mathematizing motion. In: McNulty MB (ed) Kant’s Metaphysical foundations of natural science: a critical guide. Cambridge, pp 80–97 Stephanos C (1881–1882) Sur quelques propriétés du système de trois figures égales situées dans un même plan. Bulletin de la Société Philomatique 7ième série 6:12–19 Sutherland D (2014) Kant on the construction and composition of motion in the phoronomy. Can J Philos 44(5/6):686–718 (Special issue: Mathematics in Kant’s Critical Philosophy) Sylla ED (1973) Medieval concepts of the latitude of forms: the Oxford calculators. In: Archives d’Histoire Doctrinale et Littéraire du Moyen Age, Tome XL, pp 223–283 Sylvester JJ (1875) History of the plagiograph. Nature 12:168 and 214–216 Sylvester JJ (1909) On recent discoveries in mechanical conversion of motion. In: The collected mathematical papers of James Joseph Sylvester, vol. 3. Cambridge, pp 7–25 Taton R (1975) Jean Victor Poncelet. Dictionary of scientific biography, vol 11. New York

340

Bibliography

Plutarch (1961) The Loeb Classical Library, Plutarch’s Moralia in fifteen volumes, volume IX, 697c–771e with an English translation by Edwin L. Minar Jr., F. H. Sandbach, W. C. Helmbold. William Heinemann Ltd., London, Harvard University Press, Cambridge Massachusetts Newton I. (1969) The correspondence of Isaac Newton: volume 1, 1661–1675 1st edn (Edited by H. W. Turnbull). Cambridge Thomson J (1970) Tasks and super-tasks, Analysis XV, 1954–55, pp 1–13, Reprinted in Salmon (1970), pp 89–102 Transon A (1845) Méthode géométrique pour les rayons de courbure d’une certaine classe de courbe. Journal de Mathématiques Pures et Appliquées 10:148–156 Toomer G (1976) Diocles: on burning mirrors. Springer Van Dongen J (2009) On the role of the Michelson-Morley experiment: Einstein in Chicago. Arch Hist Exact Sci 63:655–663 Varignon P (1700) Manière générale de déterminer les Forces, les Vitesses, les Espaces, & les Temps, une seule de ces quatre choses étant données […], Mémoires de l’Académie Royale, 1700, pp 22–27 Veldkamp GR (1963) Curvature theory in plane kinematics, Ph. D. thesis. Delft University of Technology Veldkamp GR (1970) Kinematica. Scheltema & Holkema, Amsterdam Vincent AJH (1836) Essai d’une théorie du parallélogramme de Watt, Mémoires de la Société royale des sciences de l’agriculture et des arts de Lille, 1836/37, pp 5–30 et Fig. 1–16 Vincent AJH (1848) Note sur la théorie dus parallélogramme de Watt, Nouvelles Annales de. Mathématiques 7:64–68 Vitrac B (2005) Quelques remarques sur l’usage du mouvement en géométrie dans la tradition euclidienne: de Platon et Aristote à ‘Umar Khayyam, Fahrang. Q J Hum Cult Stud 18:1–56 van der Waerden BL (1961) Science awakening. New York Pappus (1933) Pappus d’Alexandrie: La Collection Mathématique avec une Introduction et des Notes, Édition et traduction Paul Ver Eecke, 2 vols. Albert Blanchard, Paris et Bruges Walter SA (1996) Hermann Minkowski et la mathématisation de la théorie de la relativité restreinte 1905–1915. These Université de Paris VII Westfall RS (1980) Never at rest, a biography of Isaac Newton. Cambridge Whewell W (1840) The Philosophy of the Inductive Sciences, founded upon their history, vol 1. London and Cambridge Whiteside DT (1967) The mathematical papers of Isaac Newton, vol 1, Cambridge, pp 1664–1666 Widenmann G (1844) Ampère, André-Marie, Natürliches System aller Naturwissenschaften: eine Begegnung deutscher und französischer Speculation. Stuttgart Wilson L (1959) Scott, the significance of ‘hard bodies’ in the history of scientific thought. Isis 50:199–210 Willis R (1841) Principles of mechanism. London and Cambridge Winter TN (2007) The mechanical problems in the corpus of Aristotle, Faculty Publications, Classics and Religious Studies Department, University of Nebraska, Lincoln Wolfson PR (2001) The crooked made straight: Roberval and Newton on tangents. Am Math Mon 108:206–216 Lakatos I (1978) Imre Lakatos, Philosophical Papers, Volume 2, Mathematics, Science and Epistemology, Worrall J, Currie G (eds). London Yavetz I (1998) On the homocentric spheres of Eudoxus. Arch Hist Exact Sci 52:221–278 Zeuthen HG (1896) Die geometrische Construktion als “Existenzbeweis” in der Antiken Geometrie. Math Ann 47:222–228 Capellen WM zur (1966) Kinematics—a survey in retrospect and prospect. J Mech 1:211–228

Index

A Acceleration pole, 188 Achilles Zeno’s paradox, 4 Alievi Lorenzo, 283 Althoff Friedrich, 307 Ampère André-Marie, 186, 191, 192, 206, 211, 213–216, 220, 223, 253 Analysis versus synthesis, 260 Anaxagoras of Clazomenae, 24 Apollonius of Perga, 21, 33, 34, 40, 42, 46, 72–77, 83, 127, 136, 139, 142 Arago François, 231 Archimedes, 11, 21, 24–26, 34, 39, 40, 45, 47, 48, 53, 57, 58, 65, 89, 90, 103, 222 Archytas, 24–29, 33, 34, 37, 51, 69, 124, 125 Aristotle, 3, 4, 8, 11, 12, 45, 50, 51, 63, 72, 83, 89, 93, 147, 303 Aronhold Siegfried, 267 Aronhold-Kennedy theorem, 167, 168, 173 Artobolevskii Ivan I., 322 Augustine St., 45, 49, 50 Autolycus of Pitane, 65–69, 88

B Ball Robert S., 175, 176, 189, 190, 277, 319, 320, 327, 328 Berkeley Bishop George, 304 Besso Michele, 293, 294 Bétancourt Agustín de, 199–201 Beyer Rudolf, 320 Blaschke Wilhelm Johann Eugen, 323 Bobillier Etienne, 175, 176, 183–185, 187, 233, 326, 327 Bos Henk J. M., 49, 125, 127–130 Bottema Oene, 323–325 Boulton Matthew, 226, 231 Bradwardine, 88, 89, 92–94 Bresse Jacques Antoine Charles, 175, 176, 186–189, 221 Bricard Raoul, 285 Burmester Ludwig, 253, 265–268, 271, 274–284, 320, 321

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Koetsier, A History of Kinematics from Zeno to Einstein, History of Mechanism and Machine Science 46, https://doi.org/10.1007/978-3-031-39872-8

341

342 C Camus Charles-Étienne-Louis, 164, 165, 170, 200 Carnot Lazare, 191–193, 202 Cartesian oval, 132 Casali Giovanni, 87, 89, 95–97 Cauchy Augustin Louis, 156, 172, 174–180 Cayley Arthur, 241, 243–245, 249–252, 305, 306 Cayley diagram, 249, 250 Chasles Michel, 175, 176, 179–184, 186, 233, 275, 276, 284 Chebyshev Pavnuty, 239–241, 245, 273 Cissoid of Diocles, 42, 43, 103 Clavius, 48 Cleostratus of Tenedos, 66 Compound planar mechanisms, 269 Conchoid, 40, 41, 130, 230 Coriolis Gaspard Gustave de, 176, 191, 193, 201–203, 205–210, 223 Crossley F. R. Erskine, 322

D D’Alembert Jean le Rond, 98, 108, 109, 172–174, 211, 212 Darboux Jean Gaston, 284, 285 Delaunay Charles, 223 Descartes René, 103, 113, 114, 118, 119, 121–123, 125–130, 132–136, 142–144, 152, 153, 155–158, 303 Dichotomy Zeno’s paradox, 4 Diderot Denis, 211, 212 Dijksterhuis Eduard Jan, 31, 34 Diocles, 42, 43

Index Doubling of the cube, 21, 23, 25, 28, 32, 37 Duangle, 257 Duhamel Jean Marie Constant, 223 Dumbleton John, 94, 95

E Eccentric model, 76, 78 Elliptic motion, 84, 139, 163, 166, 284 Epicycle model, 72, 73, 78, 80–82 Epicycloidal gearing, 164 Eratosthenes, 25, 29, 34, 37–40 Erdman Arthur G., 322 Euclid’s Elements, 8, 10, 16, 21–23, 53, 92, 150, 163, 287 Eudoxus, 21, 25, 29, 31, 33, 34, 37, 47, 56, 57, 59, 69–72, 77, 83, 97, 127 Euler-Savary equation, 170, 173, 187 Euler-Savary formula, 155, 166–171, 186, 200, 326 Eutocius, 25, 26, 29, 30, 33, 39

F Fermat Pierre, 113, 114, 118–121, 133 Fluxions Newton’s version of the calculus, 146, 147 Frajese Attilio, 19, 20 Frisi Paolo, 173 Fundamental theorem of the calculus, 143–146

G Galilei Galileo, 53, 96, 97, 103, 106, 150, 158, 192, 287 Gear trains Huygens on, 135, 151 Gerard of Brussels, 89, 92, 94 Grootendorst Albert, 119 Grübler Martin, 268, 271–275 Grübler’s formula, 273

Index H Hachette Jean Nicolas Pierre, 198–200, 219, 233 Hart Harry, 245 Hart’s inversor, 246, 248 Heath Thomas, 8, 24, 29–31, 34, 64 Heron of Alexandria, 34–36 Heytesbury William, 94, 95 Hipparchus, 76–80, 83 Hippias of Elis, 46, 47 Hippopede, 70, 71 Hire Philippe de La, 155, 157–164, 166 Holditch Hamnett, 220 Hönigswald Richard, 307 Hume David, 291, 304 Huygens Christiaan, 117, 135, 143, 147–152, 154, 158, 159, 220, 304

I Immanuel Kant, 108, 109 Inflexion circle, 155, 157–159, 162, 166, 169, 187, 190, 276, 326, 328 Instantaneous center of rotation, 155, 157, 175–178, 180, 184, 188–190, 267, 269, 326 International Federation for the Theory of Machines and Mechanisms (IFToMM), 322, 323 Isaac Newton on tangents, 104

J John Napier on logarithms, 114 John of Holland, 88 Jonge A. E. Richard de, 284, 321

K Kempe Alfred Bray, 239–241, 244–252, 268

343 Kennedy Alexander Blackie William, 167 Kinematical chain, 255 Kinematical pairs higher, 257 Klein Felix, 306 Knorr Wilbur Richard, 19, 29, 31, 33, 34, 42, 47, 55 Koenigs Gabriel Xavier Paul, 285 Krafft Fritz, 51 Kutzbach Karl, 273 L Lakatos Imre, 23 Lanz José María de, 199–201, 217, 219 Leibniz Gottfried Wilhelm, 104, 110, 117–119, 135, 143, 146, 152–154, 158, 288, 303, 304, 307 Lipkin Lipman, 236 Littlewood’s paradox, 13–15 Locke John, 304 Lorentz transformation, 300 M Mach Ernst, 291, 292, 304 Mahoney Michael Sean, 120, 121 Mannheim Victor Mayer Amédée, 187, 236, 284 Martínez Alberto A., 288 Mechanical Problems Aristotelian corpus, 51, 52, 147 Menaechmus, 28, 29, 31–34, 37, 126 Menon Plato’s dialogue, 7 Merton College group, 89, 92, 97 Merton-theorem, 87, 95 Minkowski Hermann, 292, 304–316 Mohr

344

Index

Karl Friedrich, 233 Otto, 283 Monge Gaspard, 191–193, 198–202, 217 Mozzi Giulio, 173, 174, 179 Muirhead James P., 231 Müller Hans Robert, 323 Myth of Er, 64

Prony Gaspard Clair Francois Marie Riche, baron de, 231, 232, 239 Ptolemy, 37, 72, 73, 75, 76, 78–85 Pythagoras, 2, 5–7, 10, 21, 23, 42, 53, 64, 154

N Neusis-construction, 36, 39, 40 Nicomedes, 37, 39–41, 47

R Ramelli Agostino, 254 Redtenbacher Ferdinand, 254 Résal Henri, 211, 216, 220, 224 Reuleaux Franz, 110, 224, 247, 253–260, 263, 265–267, 279, 281–284, 320, 321 Reye Theodore, 266 Riedler Alois, 281, 283, 320 Riemann Bernard, 305, 306 Rittershaus Trajan, 266, 271, 283 Roberts Samuel, 234, 239, 241 Roberts’ configuration, 248 Roberval Gilles Personne de, 101–104, 156, 158, 221 Rosenauer Nicolai, 169 Russell Bertrand, 12

O Omar Khayam, 18 Oresme Nicole, 87, 89, 95–97

P Pantograph, 225–227, 247, 268 Pappus of Alexandria, 22, 40, 41, 45–49, 57–61, 114, 119, 127, 128, 130–134 Parmenides, 1–6, 64 Peaucellier Charles Nicolas, 225, 234–236, 240–243, 245 Phillips Edouard, 269 Phoronomy, 110, 260 Plagiograph, 247–249 Plato, 1–3, 6–8, 11, 15, 19, 21, 24–26, 28, 29, 31–33, 37, 42, 63–69, 88, 303, 305 Plotinus, 2, 15, 16 Plücker Julius, 241 Plutarch, 32 Poincaré Henri, 308, 309, 312 Poinsot Louis, 156, 319, 320 Polode, 156, 166, 167, 178, 180, 184, 260, 325 Poncelet Jean Victor, 186, 187, 192, 201, 202, 204, 207, 221, 241 Proclus, 1, 2, 15–18, 21, 41, 46, 133, 163

Q Quadratrix, 45–49, 60, 103, 105, 106, 132, 134, 152

S Saint-Venant Adhémar Jean Claude Barré de, 224 Sauter Joseph, 293 Schadwill C. L., 271 Scheiner Christoph, 225 Schiaparelli Giovanni Virginio, 69

Index Schoenflies Arthur, 175, 284 Schooten Frans van, 130, 135–144 Socrates, 1, 3, 4, 7, 24, 64 Solla Price Derek de, 213 Sporus of Nicaea, 48, 49 Squaring of the circle, 21, 23, 24 Stephenson’s motion, 269 Supertask, 13 Swineshead Richard, 94, 95 Sylvester James Joseph, 239, 241–249, 251, 252, 306 T Timaeus Plato’s dialogue, 6 Torricelli Evangelista, 97, 101, 102, 114, 117 Tractrix, 153, 154 Transon Abel, 186 Trisection of any angle, 21, 23–25, 31, 39, 45, 47, 58, 122, 124 Tusi-couple, 84, 85 Twin paradox, 301, 302, 316–318 V Varignon

345 Pierre, 98 Veldkamp Geert Remmert, 167, 323, 326 Vincent Alexandre Joseph Hidulphe, 234

W Watt James, 226 Whewell William, 216 Widenmann Gustav, 215 Wiener Hermann, 306 Willis Albert Henry, 169 Robert, 211, 216, 220, 227, 239 Winter Thomas Nelson, 51 Witt Jan de, 135–137 Wunderlich Walter, 323

Y Yoder Joella, 158

Z Zeno, 1–8, 11–15 Zeuthen’s thesis, 19