Reflections on the Practice of Physics: James Clerk Maxwell’s Methodological Odyssey in Electromagnetism 9780367367282, 9780429351013

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Reflections on the Practice of Physics

This monograph examines James Clerk Maxwell’s contributions to electro­ magnetism to gain insight into the practice of science by focusing on scien­ tific methodology as applied by scientists. First and foremost, this study is concerned with practices that are reflected in scientific texts and the ways scientists frame their research. The book is therefore about means and not ends. Giora Hon is a Professor of the History and Philosophy of Science in the Department of Philosophy, University of Haifa. Bernard R. Goldstein is a Historian of Science and University Professor Emeritus at the University of Pittsburgh.

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History And Philosophy Of Technoscience Series Editor: Alfred Nordmann

TITLES IN THIS SERIES Environments of Intelligence: From Natural Information to Artificial Interaction Hajo Greif A History of Technoscience: Erasing the Boundaries between Science and Technology David F. Channell Visual Representations in Science: Concept and Epistemology Nicola Mößner From Models to Simulations Franck Varenne The Reform of the International System of Units (SI) Nadine de Courtenay, Olivier Darrigol, and Oliver Schlaudt (eds) The Past, Present, and Future of Integrated History of Philosophy of Science Emily Herring, Konstantin S. Kiprijanov, Kevin Jones and Laura M. Sel­ lers (eds) Nanotechnology and Its Governance Arie Rip Perspectives on Classification in Synthetic Sciences: Unnatural Kinds Edited by Julia Bursten Reflections on the Practice of Physics: James Clerk Maxwell’s Methodo­ logical Odyssey in Electromagnetism Giora Hon and Bernard Goldstein

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Reflections on the Practice of Physics James Clerk Maxwell’s Methodological Odyssey in Electromagnetism Giora Hon and Bernard R. Goldstein

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First published 2020 by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN and by Routledge 52 Vanderbilt Avenue, New York, NY 10017 Routledge is an imprint of the Taylor & Francis Group, an informa business © 2020 Giora Hon and Bernard R. Goldstein The right of Giora Hon and Bernard R. Goldstein to be identified as authors of this work has been asserted by them in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record has been requested for this book ISBN: 978-0-367-36728-2 (hbk) ISBN: 978-0-429-35101-3 (ebk) Typeset in Times New Roman by Swales & Willis, Exeter, Devon, UK

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For Naomi and Daniel

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Contents

1

Preface

x

Introduction

1

1.1 Methodology: framing scientific knowledge 1

1.2 An overview of Maxwell’s approach to methodology 5

1.3 Maxwell’s initial publication of 1856 (an abstract)—placing

methodology at the forefront 9

1.4 Methodology as an essential feature of scientific practice:

the case of Maxwell 12

1.5 The argument 14

2

Maxwell’s choice: Faraday vs. Ampère

26

2.1 Michael Faraday (1791–1867) and James Clerk Maxwell:

a unique relation 26

2.2 André-Marie Ampère (1775–1836): the contrast 41

3

Thomson, Stokes, Rankine, and Thomson and Tait 3.1 Introduction: methodology in electromagnetism 48

3.2 William Thomson (1824–1907): from analogy to

representation 48

3.3 George Stokes (1819–1903): idealization 57

3.4 William J. M. Rankine (1820–1872): energy—a novel

unifying concept 62

3.5 W. Thomson and Peter Tait (1831–1901): abstract

dynamics 65

3.6 Conclusion 70

48

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viii Contents 4

73

Station 1 (1856–1858): on Faraday’s lines of force 4.1 A novel methodology: modifying the methodology

of analogy 73

4.2 The structure of Maxwell’s argument 80

4.3 From the general to the specific 85 4.4 Confronting Ampère’s theory 87

4.5 Conclusion 88

4.6 Appendix: a bibliographical note on Maxwell 1858 93

5

98

Station 2 (1861–1862): on physical lines of force 5.1 5.2 5.3 5.4

Introduction 98

Preliminary: from instrumentalism to causality 98

The methodology: linking hypothesis to representation 100

Applying the methodology: assumptions and their

consequences 108

5.5 Part III: a landmark in the history of physics 110

5.6 Conclusion 121

6

Station 3 (1865): A dynamical theory of the electromagnetic field

127

6.1 Introduction 127

6.2 Part I: marking the goal—the construction of a formal

theory consisting of a set of general equations 128

6.3 Part II: the flywheel analogy 130

6.3.1 How does the analogy work? 130

6.3.2 Illustration vs. analogy 134

6.4 The methodology of reversing the argument 136

6.5 Intermediate summary 142

6.6 A physical theory in symbolic language 144

6.6.1 An example—the case of electric elasticity 148

6.7 Conclusion 150

7

Station 4 (1873): A treatise on electricity and magnetism

155

7.1 Introduction 155

7.2 Framework 157

7.3 Novel methodologies 158

7.3.1 Energy as a key concept in electrodynamics 159

7.3.2 Mathematical tools in the treatment of electrodynamics 165

7.3.3 Dimensionality of units 170

7.3.4 Analogies, illustrations, and working models as mechanical

representations 175

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Contents

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7.3.5 Mental imagery 183

7.4 The impact of the new methodologies on the construction

of the theory 185

7.4.1 The electric displacement and the displacement current 185

7.4.2 General equations of the electromagnetic field 190

7.4.3 From mathematics to physics: vortices and mechanism 195

7.5 Conclusion 199

8

Philosophical reflections on Maxwell’s methodological odyssey

209

8.1 Commitment 209

8.2 Modifications of methodologies 209

8.2.1 Station 1: analogy 210

8.2.2 Station 2: hypothesis 211

8.2.3 Station 3: textual description 213

8.2.4 Station 4: abstract dynamics 215

8.2.5 Transitions from one methodology to the next 217

8.3 Methodologies in Maxwell’s practice 218

8.3.1 The role of mathematics 219

8.3.2 Reciprocity of formulation: translation 220

8.3.3 Mental imagery 223

8.3.4 Analogy 227

8.3.5 Hypothesis: the micro-level and explanatory claims 229

8.3.6 Model and modeling 233

8.4 Concluding remarks 237

References Index

246

258

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Preface

There is no more powerful method for introducing knowledge into the mind than that of presenting it in as many different ways as we can. When the ideas, after entering through differ­ ent gateways, effect a junction in the citadel of the mind, the position they occupy becomes impregnable. James Clerk Maxwell (1871)1

The theme of this book is the practice of science as it is reflected in method­ ology. In our previous research endeavors we were concerned in general with concepts and their generation, specifically with the concept of sym­ metry (G. Hon and B. R. Goldstein. 2008. From summetria to symmetry: The making of a revolutionary scientific concept. Dordrecht: Springer). This intense study of one of the critical building blocks of scientific theory, made us realize that as much as concepts precede theories, so do methodologies. With the conclusion of this current project we are now persuaded that the study of concepts and methodologies ought to precede the study of theories. We believe that the widely held distinction between the context of discovery and the context of justification is the consequence of putting theory at the forefront as the principal theme of philosophical investigation in the phil­ osophy of science. But this is putting the cart before the horse. In effect, methodology combines seamlessly both the chosen research procedure and its rationale so that severing the two features does not properly reflect scien­ tific practice—the way scientists operate. In so far as the concern of the philosopher of science is the practice of science, the actual nuts and bolts of science, the philosopher quickly realizes that he or she has to adhere, in the first place, to the published text, namely, the historical record. The claim as it is made in the published text, indeed, the theory as it is constructed or derived, is then found to be the consequence of some strategy—a methodology, either implicit or explicit. James Clerk Maxwell’s electrodynamics allows us to explore the relationship of methodology with a commitment to a conceptual framework, a set of experimental results, and the construction of theory in the practice of a

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Preface xi major figure in the history of physics. Maxwell had a keen interest in meth­ odology and recognized it as a distinct domain worthy of discussion. In fact, at several critical junctures in his study of electromagnetism he pre­ sented his perceptive views on the methodology to be selected, given the goal he had in mind. The analysis of the different methodologies he applied in his study offers profound insights into the advancement of scientific knowledge. *** Several scholars graciously offered counsel and assistance, notably Oded Balaban, Jed Buchwald, Martin Carrier, Giovanni Galizia, John Norton, Jürgen Renn, Hans-Jörg Rheinberger, Friedrich Steinle, and Gereon Wol­ ters. The librarian, Matthias Schwerdt (Max Planck Institute for the History of Science), surpassed all reasonable expectation, extending first-rate service. The following institutions supported this project and provided favorable conditions for our collaboration: The University of Haifa, The University of Pittsburgh, Max Planck Institute for the History of Science (Berlin), Zukunftskolleg (Konstanz University), and the Alexander von Humboldt Foundation. We thank all these individuals and institutions.

Note 1 Maxwell, 1871a, 13.

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Introduction

1.1 Methodology: framing scientific knowledge This book is about means, not about ends. Our principal goal is to gain insight into the practice of science by focusing on scientific methodology as applied by scientists. We are concerned first and foremost with practices that are reflected in scientific texts; we are interested in the ways scientists frame their research. We take scientific methodology to convey modes of research, for it combines procedures of thought and action cast into argumentative structures intended to convince the reader of the validity of the resulting claims to knowledge. Such procedures play two distinct roles. In the first place, a procedure offers grounds for the knowledge that has been generated; indeed, it is designed to establish criteria which justify the resulting claim to knowledge. The second role which methodology plays is of no less importance, for it expresses heuristics, ways of advancing knowledge. Methodology thus involves both the formal context of justification and the informal context of discovery. We hold that for scientists methodology combines seamlessly both the selected research procedure and its rationale. Keen on analyzing claims to knowledge, philosophers, in contrast to scientists, have treated separately, at least in a certain tradition, the process of discovery and that of justification. It is our view, however, that scientific methodology—as this concept is applied in our study—need not be based on the distinction between discovery and justification.1 Our study may be considered, therefore, a contribution to what Andersen and Hepburn call the turn-to-practice, namely, “the recent movement in philosophy of science toward a greater attention to practice: to what scientists actually do.”2 In keeping with tradition, Andersen and Hepburn entitled their encyclopedic entry, “scientific method,” although what is meant in this entry is “scientific methodology.” In our study we examine the generation of scientific knowledge from the perspective of the scientist, not from that of the philosopher. Nevertheless, as historians and philosophers of science we still appeal to the analytical tools applied by philosophers of science with a view to uncovering the processes by which scientific knowledge is generated. This is why we draw a sharp distinction between “method” and “methodology.”

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Introduction

In order to highlight this distinction, consider the usage of “method” by Oliver Heaviside (1850–1925), a British technical engineer, mathematician, and physicist who contributed substantially to the domain of electrodynamics and helped recast Maxwell’s set of equations into modern notation, principally vectorial. In his Electrical papers (1894) he remarked: Owing to the extraordinary complexity of the investigation when writ­ ten out in Cartesian form (which I began doing, but gave up aghast), some abbreviated method of expression becomes desirable. I may also add, nearly indispensable, owing to the great difficulty in making out the meaning and mutual connections of very complex formulae. In fact the transition from the velocity-equation to the wave-surface by proper elimination would, I think, baffle any ordinary algebraist, unassisted by some higher method, or at any rate by some kind of shorthand algebra, I therefore adopt, with some simplification, the method of vectors, which seems indeed the only proper method.3 And for another instance by the same author: Professor Tait, in his “Quaternions,” gives two methods of finding the wave-surface; one from the velocity-equation, the other from the index equation.4 The context makes it clear that in these passages “method” stands for a procedure—a technique—and it can be either formal or physical. Contrast now this usage of “method” with the invocation of “methodology” by Ernst Mach (1838–1916), the celebrated Austrian physicist and philosopher. Mach opened his collection of essays Knowledge and error (1905) with these words: Without in the least being a philosopher, or even wanting to be called one, the scientist has a strong need to fathom the processes by means of which he obtains and extends his knowledge.5 Mach gave priority to science over philosophy, but he did recognize that the issue of obtaining and, indeed, extending scientific knowledge is philosophical. And he continued: The work of schematizing and ordering methodological knowledge [metho­ dologischen Kenntnisse], if adequately carried out at the proper stage of scientific development, must not be underrated. But one must emphasize that practice in inquiry, if it can be acquired at all, will be furthered much more through specific living examples, rather than through pallid abstract formulae that in any case need concrete examples to become intelligible . . . The work I have attempted in the interest of scientific

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Introduction

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methodology [naturwissenschaftlichen Methodologie] and the psychology of knowledge proceeds as follows . . . Perhaps even philosophers may one day recognize my enterprise as a philosophical clarification of scientific meth­ odology [naturwissenschaften Methodologie] and will meet me half-way. However, even if not, I still hope to have been useful to scientists.6 “Methodologie” is die Lehre von der Methode (the theory of method); its subject matter is the means by which knowledge, specifically scientific knowledge, is generated and extended. Methodology is precisely the term we need for describing the analysis we have undertaken. A method offers some technique for solving a specific problem which could be technical in form or content, e.g., a mathematical technique for solving an equation, the manipulation of some instrument, or an arrangement of a set of phenomena. A methodology is a much wider concept, it is the vehicle for generating knowledge. Scientific practice is a combination of methods, methodology, and content (theoretical and experimental). It is evident that methodology is applied in some context. Broadly speaking, one may discern different contexts in physics for methodology, e.g., discovery of phenomena, construction of theory, testing theory, and a few others, depending on the subject matter and, of course, on the choices the physicist makes. Our point of departure is the scientific text, namely, the historical data of scientific discourse rather than a philosophical doctrine. We seek to remain faithful to the data as we search for the critical features of scientific methodologies. This is, of course, a daunting task that needs to be limited in scope in order to make the analysis feasible. Selection has to be exercised, and the criteria we apply are insightfulness, ingenuity, as well as versatility in argumentation, effectiveness, and importance; in other words, we are looking for methodologies that succeeded in having an impact on the development of science. We search for case histories in which the scientist himself or herself expounded his or her own views on methodology: these are important discussions, albeit rare in scientific texts, that exhibit the value of exploring methodology in actual practice. The contrast of the scientist’s comments on scientific methodology with his or her philosophical analysis facilitates, we claim, a deeper understanding of the way science works. The investigation of electromagnetism by James Clerk Maxwell (1831–1879) over a period of a decade and a half is a treasure trove of methodologies. What is striking about Maxwell is that in pursuing his pathbreaking research in electromagnetism, he paused at times to reflect on his practice and to offer insights into his choice of methodology. In fact, when one looks at Maxwell’s overall contributions to electromagnetism, it is immediately apparent that he applied multiple methodologies. These methodologies (with an emphasis on the plural form) display Maxwell’s ingenuity and versatility but, more importantly, they throw light on the nature of science—its practice as well as the knowledge gained and extended. Our focus in the science of physics is thus on electromagnetism, and the scientist we have chosen is Maxwell.

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4

Introduction

The term “methodology” was available in the literature at the time Maxwell embarked on his fundamental study of electromagnetism.7 For usages in mid­ nineteenth century Britain of method in the sense of methodology, consider those by John Herschel (1792–1871) and William Whewell (1794–1866). In 1830 Herschel, one of the leading British scientists in his day, published a definitive account of ways of studying nature for his time. In A preliminary discourse on the study of natural philosophy Herschel reflected on the methodology of scientific inquiry in the spirit of Francis Bacon (1561–1626), relating observation to theory via inductive reasoning.8 In a letter to Herschel, Michael Faraday (1791–1867) remarked that the Preliminary discourse “made me a better reasoner & even experimenter and . . . a better philosopher.”9 And the British polymath Whewell, Master of Trinity College, Cambridge (1841–1866) where Maxwell was a fellow (1850–1856), described Herschel’s Preliminary discourse as one of the first considerable attempts to expound in any detail the rules and doctrines of that method of research to which modern science has owed its long-continued steady advance and present flourishing condition. Whewell continued and referred to Herschel as “a person who has not only verified his principles over a wide range of speculation, but used them with practical success.”10 Here we have a characterization of the work of a practicing scientist, not of a philosopher. We note further that Whewell used “method” here in the sense of methodology, despite the fact that he had used the term methodology elsewhere—as we have seen. This supports the claim that Maxwell was following a standard convention in his time for the usage of method in the sense of methodology. Throughout our study we maintain that Maxwell invoked the term method to mean “method” in some instances and “methodology” in others. He did not use the term “methodology” but, on several occasions, he did invoke the expression “method of investigation” which conveys exactly what is meant by “methodology.”11 We return to the context in which Maxwell operated. Maxwell acknowledged and accepted the phenomena discovered by André-Marie Ampère (1775–1836), Hans Christian Ørsted (1777–1851), Wilhelm Weber (1804–1891), William Thomson (1824–1907), and others but, principally, he depended on the conceptual framework and experimental achievements of Faraday. Maxwell began with a commitment to Faraday’s concept of lines of force and his goal was to represent electromagnetic phenomena mathematically. He set himself the goal of “translating” the results that Faraday had stated verbally into symbolic language. Throughout his research in electromagnetism, Maxwell maintained his commitment to the fundamental concept of lines of force, but the means for reaching his goal—the methodology for casting the phenomena symbolically—vary from one publication to another. In fact, this is the reason why we call Maxwell’s research in electromagnetism an odyssey—the commitment and the goal stayed the same, although the means changed from one publication to the next.

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Introduction

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The structure of the book reflects the trajectory we have discerned.12 We commence in Ch.1 with an illustration of what we mean by methodology in the hands of Maxwell as well as with an outline of the core argument of our book. We proceed—in Chs. 2 and 3—to throw light on Maxwell’s choice of commitment and to highlight a set of methodologies applied in electromagnetism before Maxwell embarked on his pathbreaking research. We then devote a chapter to each of Maxwell’s four principal publications in electromagnetism (Chs. 4–7). We conclude in Ch.8 with philosophical reflections on methodologies in electromagnetism as they were conceived by Maxwell.

1.2 An overview of Maxwell’s approach to methodology The figure of Maxwell looms large in the world of physics in the second half of the nineteenth century. He was educated at the Universities of Edinburgh and Cambridge; he then held a number of academic positions, notably the chair of Natural Philosophy at King’s College, London, and the first Cavendish Professor of Experimental Physics at Cambridge University.13 Maxwell’s achievement in electrodynamics is comparable to that of Isaac Newton (1643– 1727) in mechanics, in that both provided the fundamental theories that govern major domains in physics, and both were deeply concerned with methodology. Indeed, Newton and Maxwell were both innovators of scientific methodology. The making of a new methodology in the scientific realm is a rare event; it requires, at one and the same time, a leap of the imagination and an acuteness of mind to conceive a novel methodology and to put it to good use. Scientific theories may come and go, but a productive methodology is here to stay. Four extensive contributions comprise Maxwell’s corpus in electromagnetism. The first contribution, “On Faraday’s lines of force,” was published in 1858 but had its roots in work Maxwell had presented in 1856. The next paper, “On physical lines of force,” appeared in 1861–1862. The third contribution, the last one in the form of an article, came out in 1865, “A dynamical theory of the electromagnetic field.” Eight years then passed in which Maxwell fundamentally revised his approach to this physical domain. These endeavors culminated in 1873 in A treatise on electricity and magnetism. In this magnum opus Maxwell applied a variety of methodologies, treating the subject both didactically and as cutting-edge research. The importance of the Treatise cannot be exaggerated; its success marks a milestone in the treatment of a major domain of physics, both for the theory as well as for the methodologies Maxwell applied in the construction of electrodynamic theory. Notwithstanding the different methodology in each contribution, there is a coherent trajectory in this series of publications, reaching its pinnacle in the Treatise. As we noted, we call this trajectory Maxwell’s methodological odyssey. We therefore associate each publication with a station on this odyssey and treat the stations separately and sequentially. Maxwell’s study of electromagnetism consists therefore of four stations.

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Introduction

We restrict our study to Maxwell’s research in electromagnetism. To be sure, Maxwell contributed substantially to other domains of physics, notably the dynamical theory of gases (1860) and his later Theory of heat (1871).14 However, we choose to focus on electromagnetism for, in this domain, Maxwell exhibited versatility in pursuing different methodologies with the same goal always in mind. The single goal renders the transition from one methodology to the other an odyssey—a rare episode in the history of science, rich in philosophical insights concerning the generation of scientific knowledge. This unifying framework of the odyssey throws new light on Maxwell’s practice. The assessment of the electromagnetic corpus leads to the conclusion that Maxwell’s constant goal was to offer a theoretical framework for electromagnetism comparable to the theoretical framework that Newton created for mechanics. The stature and comprehensiveness of both theories, the sense in which they replaced previous work in their respective domains, and the way they served as a starting point for subsequent research, are all features that call for comparison. But here the similarity ends. While Newton’s Philosophiae naturalis principia mathematica (1687) exhibits a single systematic methodology which draws consequences from an axiomatic foundation, the theoretical work of Maxwell was exploratory without axiomatic foundations. Maxwell’s Treatise (1873) is, by all accounts, a work comparable to the Principia in importance and impact but not in its methodologies, for Maxwell, unlike Newton, applied a variety of methodologies which were essentially inductive. Our concern with methodology flows from Maxwell’s own discussions of methodology. His explicit interest in methodology was as a working scientist —an actor—not as an analyst. Maxwell’s many comments on methodology and, indeed, his research practices which were based on his ingenious methodologies, invite us to consider together the variety of methodologies he employed in his series of four contributions to electromagnetism. A central feature of Maxwell’s scientific writings is that he addressed almost at one and the same time both the student of physics and the physicist at the cutting edge of research. Put differently, he regarded both didactic considerations and fundamental research as complementary. Thus, Maxwell opened his review of the Elements (1867) by Thomson and Peter Tait (1831– 1901) on a seemingly pedagogical note, but underneath it lay a profound conception of science. Maxwell sketched two approaches to scientific research which we may call the mathematical and the experimental. The mathematical approach is characterized by casting the phenomenon at stake into mathematical equations: “the progress of science, according to this method, consists in bringing the different branches of science in succession under the power of the calculus.”15 By contrast, the experimental approach, “render(s) the senses familiar with the physical phenomena, and the ear with the language of science, till the student becomes at length able both to perform and to describe experiments of his own.”16 Maxwell regarded the two approaches as

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Introduction

7

legitimate, but neither of them, in his view, “can fully accomplish the still greater work of strengthening their reason and developing new powers of thought.”17 What are the drawbacks in each of the two approaches? In the formal approach one ends up with a symbolic language devoid of physical meaning, while the physical approach amounts to burying oneself under experimental details and the associated calculation thereby exhausting oneself to the point of losing sight of higher forms of thought. In his inimitable style, Maxwell commented on the disadvantages of both approaches: Both of [these approaches] are allowing themselves to acquire an unfruitful familiarity with the facts of nature, without taking advantage of the opportunity of awakening those powers of thought which each fresh revelation of nature is fitted to call forth.18 Clearly, Maxwell set the scene for the third method, which is supposed to harmonize thought with nature; philosophically, this is a loaded demand but a most effective one: Every science must have its fundamental ideas—modes of thought by which the process of our minds is brought into the most complete harmony with the process of nature—and these ideas have not attained their most perfect form as long as they are clothed with the imagery, not of the phenomena of the science itself, but of the machinery with which mathematicians have been accustomed to work problems about pure quantities.19 This is the lesson which Maxwell took from Louis Poinsot (1777–1859) and Carl Friedrich Gauss (1777–1855): the advantage of keeping before the mind the things themselves rather than ordinary symbols and forming a distinct idea of the result that is sought. This third way undoubtedly reflects Maxwell’s approach in his first contribution to electromagnetism in 1856 (published 1858)—physical analogy avoids the allurement of physical hypothesis and the sterility of mathematical symbolism. Maxwell located the Elements of Thomson and Tait (see ch. 3, § 3.5) against this background: The book before us shews that the Professors of Natural Philosophy at Glasgow [Thomson] and Edinburgh [Tait] have adopted this third method of diffusing physical science.20 In the passage that follows Maxwell reported that the text of the Elements had originated in 1863 when Thomson and Tait decided that in their introductory courses for students of physics they would only appeal to elementary mathematical techniques. Maxwell clearly praised Thomson and Tait, as well as Poinsot and Gauss, for their methodology, while at the same time “inventing” a genealogy for his own methodology.

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Introduction

Maxwell explicitly endorsed the methodology of reasoning physically by means of dynamical ideas, rather than by appealing to the analysis of pure quantity. For the student who progresses with his mathematical studies will then be “able to see in the mathematical equations the symbols of ideas which have been already presented to his mind in the more vivid colouring of dynamical phenomena.”21 The prime example is of course the differential calculus—“the method of reasoning applicable to quantities in a state of continuous change.”22 This explains (in part), as Maxwell observed, why Thomson and Tait had begun their textbook with the study of kinematics rather than statics, for kinematics is the science of mere motion considered apart from the nature of the moving body and the causes which produce its motion. This science dif­ fers from geometry only by the explicit introduction of the idea of time as a measurable quantity . . . Hence kinematics, as involving the smallest number of fundamental ideas, has a metaphysical precedence over stat­ ics, which involves the idea of force, which in its turn implies the idea of matter as well as that of motion.23 Maxwell expressed disappointment that Thomson and Tait missed the opportunity to introduce the student of physics to vector calculus as part of the dynamical perspective on physics, but he ended the review with much appreciation for the two authors for having emphasized the importance of both symbolic argumentation and the corresponding mental imagery. In the same vein, Maxwell challenged the mathematicians to translate their ideas, expressed in symbolic form, into “appropriate words without the aid of symbols,” thereby enlightening the laymen as well as clarifying to themselves the ideas that the symbols convey.24 This duality was of paramount importance for Maxwell. He considered Thomson and Tait’s approach to physics a model to emulate. In this perspective, methodology is critical for both cutting-edge science and pedagogical practice. Maxwell indicated that his intellectual debts were primarily methodological rather than for “facts” or, for that matter for “theory.” There is virtually nothing taken by Maxwell from Thomson and Tait’s book—except a methodology, for Thomson and Tait focused on mechanics, while Maxwell applied this methodology to electromagnetism. In a recent comparison of the methodologies of Newton and Maxwell, the philosopher of science, Peter Achinstein, was concerned with what he called, “scientific strategies.” He referred to “Newton’s methodological strategy,” and spoke of “Newton’s methodology.”25 When he turned to Maxwell, he referred to “a method of strategy,”26 and devoted a chapter to a discussion of three methodologies proposed by Maxwell that can be used in situations in which you don’t have evidence sufficient to establish a theory, or indeed, in some cases, any theory at all to establish. Philosophers and

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scientists concerned with general questions of methodology too often focus entirely on methods of proving or establishing theories. But, Max­ well emphasizes, there are more scientific activities involving theorizing than are dreamed of in these philosophies. I think he is right, and that the three different methodologies he presents are important and liberat­ ing ones for philosophers as well as scientists to consider.27 Achinstein’s remark is noteworthy in several respects. In the first place we see that in his view both philosophers and scientists consider methodology principally as justification procedures: “methods of proving or establishing theories.” As Achinstein suggested, this is far too limited a view of methodology. To be sure, methodology can provide the grounds for justification, but this does not exhaust what we call scientific methodology. Indeed, it is worth repeating what Achinstein noted, “there are more scientific activities involving theorizing than are dreamed of in these philosophies.” We take “scientific activities” to be guided by methodology. Thus, we understand this remark to convey precisely the richness of methodology. It is not merely about “proving or establishing theories.” To put the claim positively, methodology is about directives, heuristics, rules of research procedures. Finally, Achinstein invoked the term “liberation”; this is an apt description, for Maxwell is indeed an innovator of methodologies which show how the same problem can be approached from different points of departure in a concerted effort to enhance understanding of the problem, if not to solve it. Still, the title of Achinstein’s book, Evidence and method, indicates that he set himself the task of establishing, if not proving, a theory. Further, we note his interchangeable use of several terms such as “method” and “strategy” which results in some confusion. However, we are not concerned with assessing Achinstein’s analyses; rather, what is of interest to us here is the focus on methodology and the way Achinstein conceived it.28 To avoid confusion, we adhere to one term throughout, namely, methodology —the issue at the center of our discussion—and therefore it is important to clarify what exactly we mean by it. Given Achinstein’s remarks, methodology is not only a procedure to justify some theory, but the actual instructions that guide practice. In that latter sense methodology is also about discovery, heuristic rules on how to proceed in a scientific investigation. Is Maxwell’s conception of methodology captured by the context distinction, namely, justification and discovery? The findings of our study resoundingly show that the context distinction as well as Achinstein’s analysis of relating evidence to method does not cover the richness of Maxwell’s explicit discussions of methodology.

1.3 Maxwell’s initial publication of 1856 (an abstract)—placing methodology at the forefront We begin with Maxwell’s first publication on electromagnetism to illustrate the importance of methodology for him. At the meetings on 10 December 1855 and 11 February 1856 of the Cambridge Philosophical Society, Maxwell

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10 Introduction reported preliminary results he had obtained in his study of electromagnetism. An abstract of these presentations was published in 1856 in two parts in the Philosophical Magazine, under the title “Faraday’s lines of force.” This text deserves special attention since Maxwell spelled out his goal and the methodological means to achieve it. In fact, the abstract constitutes more a work in progress than a précis of the extended paper of 1858 (see ch. 4); still, it offers insight into the way Maxwell practiced physics. At the beginning of the abstract Maxwell indicated that the methodology he adopted was a modified version of the formal analogy invoked by Thomson: The method pursued in this paper [Maxwell, 1856] is a modification of that mode of viewing electrical phenomena in relation to the theory of the uniform conduction of heat, which was first pointed out by Profes­ sor W. Thomson, [1842] . . . Instead of using the analogy of heat, a fluid, the properties of which are entirely at our disposal [i.e., purely imaginary], is assumed as the vehicle of mathematical reasoning.29 Maxwell embarked on his study of electromagnetism with a bold statement concerning methodology. He stated at the very outset that in this research paper he would modify Thomson’s methodology of analogy that relates two distinct physical domains via the same mathematical structure. This is an important claim that deserves close attention. In the first place, Maxwell acknowledged the critical role which methodology plays as the engine for advancing research; second, he crucially proposed to modify Thomson’s analogy by appealing to imagination. It is noteworthy that Maxwell began the abstract by stating the method he intended to pursue, which he characterized as “mathematical analogy” and later a “method of investigation.”30 He then referred in this context to the “method of lines of force” that Faraday had proposed.31 Maxwell rendered Faraday’s concept of lines of force as “processes of reasoning” and considered it a method of investigation, a methodology in our terminology. Maxwell characterized the methodology as “mathematical reasoning”—a vehicle for furthering argumentation. He introduced an imaginary imponderable and incompressible fluid that permeates a medium whose resistance is directly proportional to the velocity of the fluid. The intention was to apply mathematical ideas obtained from the flow of the imaginary fluid to various parts of electrical science. For example, “the laws of electric conduction, as laid down by Ohm, are shown to agree with those of the imaginary fluid.”32 This fluid, together with its imagined physics, constitutes an imaginary physical domain which, as Maxwell put it, was entirely at his disposal. Thus, claiming an analogy between the imaginary domain and the physics of electromagnetism facilitated drawing consequences from the imagined domain to the target domain. Maxwell’s extraordinary move is discussed in detail in ch. 4: this usage of “imaginary” in physics was unprecedented for, although the term imaginary occurred elsewhere in the literature of physics, it did not mean the same thing.33

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Maxwell explained that in the (forthcoming) paper he will extend what he called Faraday’s “method of lines of force to the phenomena of electro­ magnetism, by means of a mathematical method founded on Faraday’s idea of an ‘electrotonic state.’” And he added, In order to obtain a clear view of the phenomena to be explained, we must begin with some general definitions of quantity and intensity as applicable to electric currents and to magnetic induction. It was shown . . . that electrical and magnetic phenomena present a mathemat­ ical analogy to the case of a fluid whose steady motion is affected by certain moving forces and resistances.34 He then remarked, within square brackets, that “[The purely imaginary nature of this fluid has been already insisted upon.].”35 We call attention to the expression “mathematical analogy.” Since the analogy is based on a certain mathematical structure, the modifier “mathematical” is undoubtedly apt. The use of this analogy is in one direction, namely, from an imaginary system to a real physical system, where the other direction is without interest. The contrast with Thomson is striking since Thomson only considered analogies between physically real systems. Thus, in Thomson’s analogy, the bidirectionality of the analogy is inherent to the methodology, while in Maxwell’s analogy directionality is optional—the researcher can choose to render the analogy unidirectional or bidirectional. The Physical Society of Berlin published a review of Maxwell’s abstract.36 The report began by briefly describing Thomson’s analogy between electrical phenomena and heat conduction. Notwithstanding the analogy, the German report claimed that Thomson did not intend to use Faraday’s concept of lines of force in the explanation of electrical phenomena. It then continued with an account of Maxwell’s contribution, noting that Maxwell replaced heat conduction with the flow of an imaginary imponderable and incompressible fluid. The reviewer added that, given the present state of knowledge, it cannot yet be decided if the analogy works or not. Evidently, Emil Jochmann, the author of this review, did not appreciate the innovative methodology which Maxwell introduced.37 Moreover, from an exchange of letters with Faraday in late 1857 it is apparent that Faraday too did not understand what Maxwell had done.38 Faraday expressed his discomfort with Maxwell’s mathematical argumentation since Faraday could not follow it; therefore, he asked Maxwell to translate his results into ordinary language. Maxwell’s starting point was Faraday’s description of the phenomena; hence, Faraday already had in hand the discussion in ordinary language. But, for Maxwell, the whole point of his research was to appeal to the power of mathematics in the search for new phenomena in electromagnetism. In other words, Maxwell was making a productive methodological claim which Faraday did not appreciate. The goal was to recast Faraday’s verbal description of the

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12 Introduction phenomena into mathematical symbolism in order to discover new phenomena. Put differently, for Maxwell a mathematical formulation of the phenomena could lead to new results unforeseen in the verbal descriptions. This was a kind of translation; Maxwell did not claim to be doing anything more than that.39

1.4 Methodology as an essential feature of scientific practice: the case of Maxwell Two fundamental features of scientific knowledge are based on theory, namely, prediction and explanation.40 The set of electromagnetic phenomena was no exception; early in the nineteenth century physicists, both on the continent and in Britain, were engaged in developing a theory, or theories, for predicting and explaining these phenomena. Indeed, when Maxwell came on the scene in the mid-nineteenth century there had already been several such theories of electromagnetic phenomena.41 These theories, which were based on the Newtonian conception of action at a distance, were not satisfactory to Maxwell: they did not account for the variety of electromagnetic phenomena in a unified way. This is how Maxwell viewed the state of the art in 1850s: The present state of electrical science seems peculiarly unfavourable to speculation. The laws of the distribution of electricity on the surface of conductors have been analytically deduced from experiment; some parts of the mathematical theory of magnetism are established, while in other parts the experimental data are wanting; the theory of the conduction of galvanism and that of the mutual attraction of conduct­ ors have been reduced to mathematical formulae, but have not fallen into relation with the other parts of the science. No electrical theory can now be put forth, unless it shews the connexion not only between electricity at rest and current electricity, but between the attractions and inductive effects of electricity in both states. Such a theory must accurately satisfy those laws, the mathematical form of which is known, and must afford the means of calculating the effects in the limiting cases where the known formulae are inapplicable.42 Maxwell clearly expressed dissatisfaction with the state of the art and listed some requirements for the theory he envisioned. Indeed, he was ready to take up the challenge of developing a new theory that met these demands. It is worth noting that Maxwell’s comment is not part of theory building; rather, it is an aside, sharing with the reader his position vis-à-vis the state of the art in the study of electromagnetism. Against this background, Maxwell sought a suitable methodology; this search for methodologies is the focus of our study. Maxwell was explicit in his scientific writings that an appropriate choice of methodology assists research and results in progress.

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We are persuaded that these reflections on heuristics and reasoning throw light on the way Maxwell practiced science, specifically in electromagnetism. For Maxwell the way forward was to have a mathematical formulation of the known phenomena and with this formulation to search for novel phenomena. It is seldom the case that a scientist of the stature of Maxwell shares with his reader reflections on scientific practice, that is, how to proceed with research. Typically, the scientist presents results without informing the reader of his motivation and form of reasoning. Evidently, Maxwell thought that such reflections are worthy of inclusion in his scientific works as they can assist the reader in several respects. First, motivation helps in identifying precisely the problem at issue. A clear formulation of the problem to be attacked is an important part of the research. Asking the right question can be most productive. In his autobiography, the philosopher Robin G. Collingwood (1889–1943) insightfully remarked: Here I was only rediscovering for myself, in the practice of historical research, principles which Bacon and Descartes had stated, three hundred years earlier, in connexion with the natural sciences. Each of them had said very plainly that knowledge comes only by answering questions, and that these questions must be the right questions and asked in the right order. And I had often read the works in which they said it; but I did not understand them until I had found the same thing out for myself.43 Second, knowledge of the reasoning underlying a certain practice clarifies the result and carries rhetorical force. And, third, exposing motivation helps the reader understand why—in this case—Maxwell departed from the consensus of contemporary physicists. There is, however, more to it than just informing the reader: in a sense, Maxwell was thinking out loud —indicating his concerns about methodology and how to approach the phenomena of electromagnetism. As we will see, the critical move is from Station 3 to Station 4, where the project was greatly expanded, surely under the influence of Thomson and Tait. Granted, there is nothing specific for which Maxwell was indebted to Thomson and Tait—his thinking was entirely original—but the Treatise on natural philosophy by Thomson and Tait (1867) increased Maxwell’s appreciation of methodological issues. In addition to recognizing that energy is a fundamental concept of physics, Maxwell was also stimulated to introduce in Station 4 a range of mathematical techniques as well as a new focus on pedagogy. Nevertheless, Maxwell did not explain why he moved from one methodology to the next. We focus on these changes and throw them into relief.44 Indeed, the bulk of our study is devoted to a description and analysis of Maxwell’s various methodologies in great detail; we also pay attention to Maxwell’s ongoing “conversation” with his sources and, more importantly, with himself, as it were.

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14 Introduction

1.5 The argument To the key question, what made Maxwell so successful in his treatment of electromagnetism, we respond: methodology. Indeed, Maxwell shared with his reader this insight and, at times, explicitly reflected in his scientific writings on the methodological steps that he had deliberately taken. Put differently, in the hands of Maxwell, methodology became the engine for scientific change. In the subsequent chapters of this book we offer a detailed account of the methodologies Maxwell developed and applied in the domain of electromagnetism. We show that Maxwell reflected deeply on methodological issues and came up with brilliant solutions that were not properly appreciated at the time by his fellow physicists. What was the state of the art in electromagnetism in the 1850s when Maxwell began his research? It was a divided scene: in Britain Faraday demonstrated experimentally novel electromagnetic phenomena and attempted to understand them by means of the unifying concept he introduced, namely, lines of force. Thomson showed with great ingenuity that electromagnetism has a mathematical structure analogous to heat conduction, and suggested that eventually all these phenomena would be understood mechanically. But it was not at all clear that Faraday’s concept of lines of force was the key for comprehending phenomena in this physical domain. Indeed, Thomson sided with prominent continental physicists such as Ampère, Gauss, and Weber, who held to the view inspired by Newtonian mechanics: action at a distance is the concept which unifies electromagnetism.45 How then to progress? This was probably the most urgent question which Maxwell faced when he came on the scene at the age of 24 with the intention of contributing to this domain of physics. In particular, he had to decide which conceptual framework to adopt. The concept of “lines of force” meant that the action of the force takes place along curved paths in contrast to the well-established “action at a distance” in which forces act in straight lines. Maxwell acknowledged that, at the time when he began his work on electromagnetism, he had hardly made a single experiment in this domain; hence, the intended contribution had to be abstract, that is, theoretical. His initial goal was to turn Faraday’s concept of lines of force into standard mathematical form. The expectation was that a mathematical formulation would suggest new physical connections unforeseen by Faraday and thus enrich the domain with new discoveries. The question persisted, how to proceed? The problem is the link between methodology and physical content: what methodology would advance the domain given the commitment to the concept of lines of force? Maxwell’s contribution to electromagnetism comprises four essays (three articles and a book) that we call stations. One of our concerns is the transition from one station to the next. Clearly, the transitions reflect Maxwell’s openness to new methodologies and his skill in adapting for his own purposes methodologies previously used by other physicists. Each transition is dramatic, but the most striking one, which most cogently

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demonstrates our claim that methodology is the engine for change, is the transition from Station 3 to Station 4. The latter station, the terminus of the odyssey, had an enormous impact on the development of modern physics. The success of Station 4 stands in contrast to the previous stations which had little impact prior to the publication of the Treatise in 1873. It is therefore of paramount importance to understand Maxwell’s change of methodology from Station 3 to Station 4. However, despite its lack of impact, the very first step Maxwell had taken in Station 1 was groundbreaking in the way he modified an existing methodology. But then, in Station 2, Maxwell abandoned the bold methodology of Station 1 and introduced a productive mechanical hypothesis. In short, each station and the transition from one to the next deserve close attention. The account is built chronologically along the transitions from one station to the next, analyzing each methodology and highlighting its features. In his first paper, Station 1, Maxwell pursued what he called “physical analogy.” On all accounts this was an original move, unheard of at the time, for Maxwell extended physics into the realm of imagination where he could control the physics as he wished with the goal of recasting the experimental discoveries of Faraday into mathematical formalism. Station 1 is not about a simplified version of reality (e.g., eliminating friction from consideration in a problem in mechanics). It is thus neither about idealization nor about abstraction; rather, Maxwell introduced an entirely imaginary system. Thomson had analogies between different physical systems based on their mathematical descriptions where the same set of equations applies to two different domains of physics. By changing the meaning of the variables and the constants in the equations in one system, one gets immediately the equations that apply in another system; hence, consequences of the equations apply to all systems described by the same set of equations, and can be checked experimentally. Maxwell ingeniously modified this scheme and introduced an analogy between a physical system and an imaginary system, where the same set of equations applies. He invented, or rather “imagined,” to use his word, a new kind of physics along the guidelines he discussed, and then used it in an analogy in order to develop a new formalism. The crucial move was to imagine the lines of force as tubes, fill them with incompressible fluid, complete with sources and sinks. Maxwell then developed an imaginary hydrodynamics to turn Faraday’s geometrical scheme into a “physical” one, which now included both the direction of the force and its intensity. This novel imaginary physics served as an analog which, in turn, facilitated the construction of a new formalism. Faraday’s geometrical scheme had the power to unify electromagnetic phenomena, but only geometrically; Maxwell rendered it physical with the introduction of the intensity of the force in addition to its direction. This was done by appealing, surprisingly, to imagination; indeed, that was the strength of the new methodology. We call this methodology “contrived analogy,” for Maxwell constructed an analogy which was only of interest in one direction: from his imaginary physical

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16 Introduction scheme to the physics of electromagnetism. Thomson invoked analogies that work in both directions, but Maxwell had analogies that work either in both directions or in one direction. This was an important modification which enriched substantially the methodology of analogy (ch. 4). Interestingly, there is no trace of the methodology of Station 1 in Station 2. Whereas in Station 1 Maxwell pioneered imaginary physics, in Station 2 he opted for a hypothesis which he then linked to a mechanical analogy for electromagnetism. He did not reveal his reasons for discarding the imaginary physics of Station 1, but in Station 2, as indicated in the title of the paper, he took a physical approach in stark contrast to the instrumental approach of Station 1. In the transition from Station 1 to Station 2, Maxwell turned to a conventional methodology, that of hypothesis at the micro-level, to explain phenomena at the macro-level. Maxwell called his proposed mechanical scheme, which was designed to depict the structure of matter at the micro-level, “hypothesis”; it was introduced to account for electromagnetic phenomena. This scheme is not imaginary in the sense of Station 1 where Maxwell had been explicit that his scheme was a contrived physical arrangement. The introduction of a hypothesis was—and still is—a common practice for physicists, whereas the methodology in Station 1 was radically different from any previous physical argument. Curiously, while Station 2 contains Maxwell’s two chief discoveries in electromagnetism, namely, the displacement current and the identification of light as an electromagnetic disturbance, he did not offer any explanation for the heuristic relation between the methodology he adopted in this station and the discoveries themselves. However, both discoveries are evidently consequences of Maxwell’s methodology, specifically his commitment to lines of forces and the way they are conceptualized in a medium (see ch. 5, § 5.5). Once these major discoveries were formulated, Maxwell felt sufficiently confident to reverse the argument and to consider the two discoveries assumptions in the foundation of the theory he constructed. In a relatively short time, in the subsequent station, Station 3, Maxwell cast Faraday’s empirical discoveries once again into mathematical equations. The methodology in Station 3 consists in translating the phenomena expressed verbally directly into mathematical equations; no hypothesis is involved, and Maxwell characterized the theory as dynamical. In Station 3 we witness once again the application of a different methodology from those applied in the previous studies. The various methodologies, coupled with the fundamental concept of lines of force, reflect Maxwell’s changing attitude. Originally, in Station 1, he introduced the concept of lines of force as an alternative to action at a distance; then, in Station 2, he considered lines of force a possible account at the molecular level in contrast to action at a distance which he took to be problematic. Subsequently, in Station 3, Maxwell no longer saw a need for an argument to justify the choice of lines of force over action at a distance; it was simply taken for granted as an assumption.

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While in Station 1 Maxwell avoided ontological commitments, in Station 2 he explicitly made physical—not merely conceptual—claims, notably that lines of force are physically real. Station 2 is often presented in the current literature as the prime example of early modeling in physics. Given the ontological assumption that lines of force are physically real, Maxwell sought to account for electromagnetic phenomena with a mechanical scheme—what he called, “molecular vortices” (an idea that Rankine had introduced: see ch. 3, § 3.4). However, we argue that the methodology Maxwell pursued in Station 2 is rather the traditional methodology of hypothesis. In fact, Maxwell referred to the mechanical scheme of molecular vortices as “hypothesis,” and the underlying logic of his methodology is the use of hypothesis as an assumption from which consequences were drawn. We elaborate the concepts of hypothesis and model and their use, and show that Maxwell’s methodology in Station 2 is not one of modeling. In fact, there is a twist to the argument, for Maxwell changed his perspective on the molecular vortices. We show that what he had considered systematically a hypothesis in Station 2, became a working model in Station 4. This significant change, which has not been noticed in the literature, originated —we argue—in a shift in Maxwell’s methodological perspective: the scientific details remain the same but the methodological framework changes and with it the status of the hypothesis. This is a key result—a consequence of treating Maxwell’s trajectory in electromagnetism in its entirety (see ch. 8, § 8.3.6). Beginning in Station 2, elasticity became increasingly a key property of the medium, which could help explain the discovery of the displacement current. This was another substantial change in the transition from Station 1 to Station 2. The discovery that light behaves like an electromagnetic disturbance came about when Maxwell took a quantity that had already been accurately measured and gave it physical meaning—consistent with the assumed physical ontology—with far reaching consequences (ch. 5). In Station 3 Maxwell began afresh, no longer pursuing the methodology of Station 2, returning to Faraday’s verbal descriptions of electromagnetic phenomena and translating them into mathematical symbolism expressed as a set of equations. Maxwell did not begin with a set of principles: the equations were not derived from principles and they are not presented in a deductive form. Therefore, the resulting set of equations does not exhibit an axiomatic structure. In Maxwell’s transformed version of Faraday’s methodology certain phenomena serve as assumptions of the theory. Once these assumptions are recast into symbolic language, formal inferences can be drawn with respect to physical properties of electrodynamics in quantitative terms. The translation from the verbal to the symbolic transformed Faraday’s methodology which is solely descriptive. As formulated, the mathematical equations facilitate quantitative calculation of Faraday’s qualitative descriptions (ch. 6). By paying special attention to methodology, Maxwell recognized that the methodology of abstract dynamics developed by Thomson and Tait in 1867 for mechanics could be adapted to electromagnetism, even though the content of these two domains is completely different. This insight grew out

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18 Introduction of shifting his attention from force to energy, and was based on the assertion that “energy is energy” whether it be mechanical, electromagnetic or anything else. It is noteworthy that in the transition from Station 3 to Station 4 Maxwell did not appeal to any new data (except for minor changes in the value for the speed of light) or to any new experiments. The changes are due to a shift in methodology, without which Maxwell would not have progressed from Station 3 to Station 4. The “new” concept was energy, and the “new” methodology was creatively adapted from Thomson and Tait’s “abstract dynamics”—which they had applied to mechanics—to electromagnetism. The appeal to abstract dynamics led Maxwell to introduce various mathematical concepts and methods that he had not invoked previously. In Station 3 Maxwell returned to a modified methodological version of Station 1 in which verbal descriptions of the phenomena were translated into symbolic language in formulas, but this time the equations had no “picture,” although an appeal was made to a mechanical illustration and analogy. It is in Station 3 that Maxwell introduced the concept of field as the bearer of electromagnetic phenomena (ch. 6). Our focus on methodology uncovers the argumentative structure of Station 3 which is in some sense the reversal of the argument of Station 2: the discoveries of Station 2 now serve as the assumptions of Station 3. While Maxwell did appeal to mechanical illustration and analogy to progress with the translation of the verbal descriptions of the phenomena to formulas, there is ultimately no specific mechanism to illustrate the elasticity of the medium, which is its sole physical property that accounts for the phenomena. In the final analysis, the dominant methodological strategy was to move from causal mechanical explanation to formal description of the phenomena. The goal from the beginning was, after all, the recasting of Faraday’s experimental discoveries into symbolic language. To be specific, Maxwell began to think seriously about energy in electromagnetism in Station 3, and then sought a way to treat it mathematically. This search was among the considerations that motivated him to appeal to Thomson and Tait’s methodology of abstract dynamics, where they discussed energy in mechanical systems. The appeal to Lagrange and Hamilton in electromagnetism was then open. Importantly, resolving the methodological issue preceded the mathematical treatment which complemented the transition from Station 3 to Station 4. Thus, when it came to the applications of mathematical techniques to physics, Maxwell’s moves were already embedded in a methodological framework. In fact, in these mathematical applications we see how Maxwell transformed methods into methodologies. In his hands mathematical methods, already available in the literature, became physical theories. He indicated whether, for example, a certain magnitude is scalar or vector, and whether the vector is defined with respect to translation or with respect to rotation, and then let the formalism exhibit the relations among the various magnitudes in physical realms where unforeseen relations could be explored. Maxwell introduced quaternions, line- and surface-integrals, the Lagrangian

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and the Hamiltonian, as well as topology, and put these mathematical techniques to use in a most productive way (ch. 7). Despite these dramatic changes which characterize the transition from Station 3 to Station 4, there are still elements of continuity: (1) an ongoing commitment to Faraday’s experimental data as well as to his concept of lines of forces; (2) translating Faraday’s verbal descriptions into equations, and (3) the claim that symbols and equations have to be translated back into meaningful physics expressed verbally. Other elements of continuity which are not due to Faraday, are (4) electromagnetic disturbances are propagated at the speed of light; (5) the medium for light is the same medium as that for electromagnetic disturbances (a discovery reported in Station 2), and (6) the displacement current (a discovery reported in Station 2). In Station 4 the didactic aspect flows from the emphasis on retranslation from the symbolic to the verbal—and the importance of giving physical meaning to symbols and equations. In fact, we show that the many didactic passages in Station 4 are not just addressed to a beginning student, for even advanced researchers need to pay attention to the physical meaning of symbols and equations. In Station 4 there are thus several methodologies which Maxwell pursued in parallel, but the coherence of the text has not generally been appreciated.46 One prominent aspect of consistency that arises in Station 4 is the status of analogy. Maxwell had begun in Station 1 with a concept of analogy which we characterize as “strong,” distinguishing it from “weak.” In a “strong” analogy inferences are drawn from one domain to another and vice versa, while in a “weak” analogy one’s intuition is enriched with respect to some phenomenon by referring informally to some other phenomena. In the latter case the analogy is didactic: making something plausible by appealing to the more familiar. The former case, strong analogy, has the advantage that inferences drawn in one domain are applied in another. A strong analogy therefore must maintain consistency; it is logically bound. A weak analogy, by contrast, need not be consistent—it is merely suggestive. In reviewing the entire trajectory of Maxwell’s methodological odyssey we are immediately struck by Maxwell’s versatility and originality: he demonstrated flexibility of mind and, at the same time, determination and conviction. Throughout his journey in electromagnetism Maxwell depended on Faraday both for his unifying concept of lines of force as well as for his verbal descriptions of phenomena. Evidently, Maxwell was not satisfied with his own treatment of these elements in Stations 1 to 3, despite their apparent success. In Station 4, then, Maxwell reviewed the domain of electromagnetism in an original way and revisited many of the themes and arguments he had put forward in the preceding three stations, appealing in this final station to several methodologies (ch. 7). From a philosophical perspective, we observe that the trend of the trajectory is toward greater abstraction while, at the same time, Maxwell insisted on mental images accessible without mathematical symbolism. As Maxwell put it,

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20 Introduction Scientific truth should be presented in different forms, and should be regarded as equally scientific, whether it appears in the robust form and the vivid colouring of a physical illustration, or in the tenuity and pale­ ness of a symbolical expression.47 This was not the view of many of Maxwell’s contemporaries, especially the German physicists cited by Maxwell at the outset, in the Preface, as well as at the end of Station 4.48 For the German physicists, parallel representation was not an issue. Indeed, Maxwell concluded the Treatise on this note: we ought to endeavour to construct a mental representation of all the details of its [the medium’s] action, and this has been my constant aim in this treatise.49 This concluding remark aptly characterizes the entire odyssey. In Station 1 there is the claim that turning the verbal into the symbolic may lead to new results by manipulating the symbols.50 In Station 4 Maxwell had translations in both directions, from the symbolic to the verbal and vice versa. The translation from the symbolic to the verbal is not for making new discoveries; rather, it is necessary for intelligibility (the symbols by themselves have no physical meaning). This suggests an “asymmetry” in the translation process. Maxwell did not attribute his two fundamental discoveries in Station 2 to the methodology of translation; rather, in some sense the discovery that electromagnetic disturbances move at the speed of light was a result of giving physical meaning to the symbols manipulated in the experiment by Rudolf Kohlrausch (1809–1858) and Wilhelm Weber (1804–1891), who failed to see that the quantity they found was physically real, not just close to the speed of light, as if by accident.51 For another case, Maxwell insisted on the physical reality of the formula for energy rather than mere symbols: the equation should be (mv)v, where both quantities, momentum and velocity, are vectors, in contrast to the common formulation of mv2, where the symbols and their manipulation have no physical sense.52 Heinrich Hertz (1857–1894) may well have been right in saying that Maxwell’s theory of electrodynamics is the set of Maxwell’s equations but, in contrast to Hertz, our interest is in Maxwell’s argument and the changes it underwent throughout the odyssey.53 While there were many other factors in developing the theory, we focus on the role of Maxwell’s insights into methodology that bore fruit throughout his study of electromagnetism. Our fundamental claim that methodology is an integral part of any scientific inquiry comes as no surprise, for it has been widely acknowledged by philosophers of science. However, given the turn-to-practice approach in current philosophy of science, we suggest following closely scientific practices in order to uncover in some detail the underlying methodology, whether it be explicit or implicit. Why, then, as a working scientist, was Maxwell so interested in methodology? Was Maxwell interested in

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methodology for philosophical reasons? This seems not to be the case, at least in conventional terms. For example, in a letter dated 23 December 1867, Maxwell wrote to Tait: I have read some metaphysics of various kinds and find it more or less ignorant discussion of mathematical and physical principles, jumbled with a little physiology of the senses. The value of the metaphysics is equal to the mathematical and physical knowledge of the author div­ ided by his confidence in reasoning from the names of things.54 A year later he wrote to Pattison, “You must see that my acquaintance with positivism whether as stated by Comte, Littré, or Mill is but slight . . .”55 In other words, Maxwell had little interest in philosophy of science per se and did not think much of what he had seen of it in recent publications—in so far as it related to physics. It is therefore plausible that his reflections on methodology are related to his transition from one methodology to another; above all, Maxwell’s concern was how to develop a new and successful methodology suitable to the goal he set himself, given the means available to him. This, however, is a philosophical issue. Indeed, Maxwell did address philosophical questions despite his apparent disdain for philosophy of science as it was practiced in his day.56 We argue that Maxwell made a conscious decision to begin with methodology. In fact, in Station 1, it was all a matter of methodology because Maxwell depended entirely on Faraday’s experimental findings and conceptual framework. At the outset Maxwell was committed to Faraday’s concept of lines of force rather than to the Newtonian action at a distance. This was probably a matter of deep intuition, and so the core of the problem was what methodology could make the concept of lines of force contend with the concept of action at a distance? What methodology should be applied then for progress to be made in electromagnetism? Maxwell’s contribution was methodological from the outset of his investigation of electromagnetism. Indeed, his response was to adopt a series of methodologies: in each of them he expressed, with increasing confidence, his commitment to the concept of lines of force. We claim that Maxwell’s productive search for new methodologies was a key to his success in turning the study of electromagnetism into a theory of electrodynamics. The analytical accounts we develop in the subsequent chapters present the evidence to support the overall thesis: Maxwell’s methodological “asides” were not just for the reader, but gave a (partial) perspective on the development of Maxwell’s own positions on methodology and how it is conducive to scientific progress. In sum, we explore the change of ideas which is exhibited by a sequence of texts. The rationale of this study is straightforward: we are interested in the way science has actually been practiced and in the way research has been carried out. Maxwell’s contributions to electromagnetism are very

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22 Introduction useful for us because he moved from one methodology to another while staying on course toward a single goal. Our study of this case in the history of physics reveals the pivotal role of methodology in advancing scientific research. As a case study, Maxwell’s methodological odyssey in electromagnetism throws light on the practice of physics. It shows that one and the same scientific problem can be approached from different perspectives, based on different metaphysical assumptions resulting in different procedures of research, that is, different methodologies. What makes the case of Maxwell intriguing is that this is a one-man show in which a variety of approaches were taken with a view to generating new physical knowledge.

Notes 1 For a fairly recent study of this traditional distinction, see Schickore and Steinle (eds.) 2006. For an overview, see the Introduction vii–xix. The contributions of Steinle (pp. 183–195) and Arabatzis (pp. 215–230) in this volume are particularly apt for our argument, as these essays indicate that the distinction needs to be assessed in specific historical contexts. 2 Andersen and Hepburn, 2016.

3 Heaviside, 1894, 2: 3.

4 Heaviside, 1894, 2: 12.

5 McCormack and Foulkes (trs.) 1976, xxxi, a translation of: “Ohne im geringsten

Philosoph zu sein oder auch nur heißen zu wollen, hat der Naturforscher ein starkes Bedürfnis, die Vorgänge zu durchschauen, durch welche er seine Kennt­ nisse erwirbt und erwitert” (Mach, [1905] 1906, v). 6 McCormack and Foulkes (trs.) 1976, xxxi–xxxiii, a translation of: “Die Arbeit der Schematisierung und Ordnung des methodologischen Kenntnisse, wenn sie im geeigneten Entwicklungsstadium des Wissens und in zureichender Weise ausgeführt wird, dürfen wir nicht unterschätzen. Es ist aber zu betonen, dass die Übung im For­ schen, sofern sie überhaupt erworben werden kann, viel mehr gefördert wird durch einzelne lebendige Beispiele, als durch abgeblasste abstrakte Formeln, welche doch wieder nur durch Beispiele konkreten, verständlichen Inhalt gewinnen. . .. Die Arbeit, welche ich im Interesse der naturwissenschaftlichen Methodologie und Erkenntnispsychologie auszuführen versucht habe, besteht in Folgendem. . .. Viel­ leicht erkennen sogar die Philosophen einmal in meinem Unternehmen eine Philo­ sophische Läuterung der naturwissenschaften Methodologie und kommen ihrerseits einen schritt entgegen” (Mach, [1905] 1906, v–ix, italics in the original). 7 See OED sub “methodology.” Whewell ([1837] 1858, 3: 327) invoked “method­ ology” in discussing Linnaeus’s Systema naturae. 8 It is of some interest that Herschel’s book was reprinted in 1851 (a “new edi­ tion”), closer to the time of Maxwell: see Herschel, [1830] 1851. For a modern edition, see Herschel, [1830] 1987. 9 James (ed.) 1993, 87 (Letter 623, Faraday to Herschel, 10 November 1832). 10 Whewell, 1831, 377, italics in the original. Cowles (2016, 726) paraphrased Whe­ well’s assessment of Herschel’s Preliminary discourse in the following way: “the first book to root scientific progress systematically in its peculiar method. . .. Herschel put methodology on the map.” 11 See n. 31, below. See also Maxwell, 1858, 27; 1873b, 50; 1873d, 1: 343, § 296; 1873d, 2: 197, § 571; and 1873d, 2: 249, § 638. Maxwell’s instructor in philosophy in Edinburgh, William Hamilton (1788–1856), invoked in his lec­ tures on logic the term “methodology” meaning “the Doctrine of Method”

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Introduction

12

13 14

15 16 17 18 19

20 21 22 23 24 25 26 27 28 29

23

(W. Hamilton, 1860, 2: 22 et passim). For the role of W. Hamilton in Max­ well’s education, see Marston, 2014, 261. Hunt (2015, 306 n. 4) commented on this trajectory and the transitions that took place: for discussion, see ch. 8, n. 21. One account which addresses Max­ well’s entire corpus on electromagnetism and makes claims about the transitions from one contribution to the next is Margaret Morrison’s brief and insightful essay, where the principal goal is “to introduce distinctions among the processes involved in constructing a fictional model, an idealization, and mathematical abstraction” (Morrison, 2009, 132). Morrison (2009, 120, 133 n. 8) relied heavily on Siegel, 1991. She recognized, as we do, that Maxwell’s work in electromagnet­ ism has to be considered in its entirety. However, since the objects of her study are fictional models, idealization, and abstraction, and the roles they play in sci­ entific research, rather than Maxwell’s research per se, her study of Maxwell’s contributions to electromagnetism is necessarily schematic. For Maxwell’s biography see, e.g., Campbell and Garnett, [1882] 1884. For over­ views of Maxwell’s life and works see, e.g., Goldman, 1983; Everitt, 2008; and Flood, McCartney, and Whitaker (eds.) 2014. See Maxwell, 1860, 1871d. One of the striking features of Maxwell’s Station 3 (1865) and Station 4 (1873) is the lack of commitment to the molecular hypothesis at the micro-level, in contrast to Station 2 (1861–1862)—we elaborate this point in chs. 5 and 6 (see, e.g., ch. 6, n. 66). In fact, the molecular hypothesis dominated Maxwell’s views on heat: see, for example, Maxwell, 1871d, Preface, vi and 286–287: “The last chapter is devoted to the explanation of various phenomena by means of the hypoth­ esis that bodies consist of molecules, the motion of which constitutes the heat of those bodies . . . All bodies consist of a finite number of small parts called molecules . . . The molecules of all bodies are in a state of continual agitation . . . But when we apply the methods of dynamics to the investigation of the properties of a system consisting of a great number of small bodies in motion the resemblance of such a system to a gaseous body becomes still more apparent.” This claim became the cele­ brated kinetic theory of gases. Evidently, Maxwell’s methodological approach to this physical domain was different from his approach to electromagnetism. In other words, Maxwell did not have a universal methodology for, even in the context of elec­ tromagnetism, he invoked multiple methodologies. Maxwell, 1873e, 399. Maxwell also reviewed the 2nd edition: see Maxwell, 1879. Maxwell, 1873e, 399. Maxwell, 1873e, 399. Maxwell, 1873e, 399. Maxwell, 1873e, 399. Cf. Hertz’s demand, namely,“. . . there must be a certain conformity between nature and our thought”[müssen gewisse Übereinstimmungen vorhanden sein zwischen der Natur und unserem Geiste] (Hertz [1894/1900] 1956, 1; Hertz, 1894, 1). Several important features in Hertz’s thinking can be traced back to Maxwell (see, e.g., n. 53, below). Maxwell, 1873e, 399. Maxwell, 1873e, 399. Maxwell, 1873e, 399. Maxwell, 1873e, 400. Maxwell, 1873e, 400. Achinstein, 2013, 83 and 91. Achinstein, 2013, 141. Achinstein, 2013, Preface xiv. For a balanced review of the book, see Kragh, 2013. Maxwell, 1856, 404. On p. 405 Maxwell added that this fluid is “imaginary.” For discussion of Thomson’s position, see ch. 3, § 3.2.

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24 Introduction 30 31 32 33 34 35 36 37 38

39 40 41 42 43 44 45 46 47

48 49 50 51 52 53

Maxwell, 1856, 404, 316. See also, Maxwell, 1858, 27. See ch. 4, n. 7. Maxwell, 1856, 316. See this usage in Maxwell, 1858, 29. Maxwell, 1856, 405. Consider, for example, the expression “imaginary magnetic matter” to which Thomson appealed in 1851 (see ch. 3, n. 30). Maxwell, 1856, 316, italics in the original. Maxwell, 1856, 316–317. Jochmann, 1859. Jochmann, 1859. On 7 November 1857 Faraday wrote to Maxwell acknowledging receipt of a few papers that Maxwell had sent him, and on 9 November Maxwell responded, explain­ ing some of his view on lines of force, especially as applied to gravitation. See James (ed.) 2008, 305–306, letter 3357. Cf. Campbell and Garnett, 1882, 288–290; Harman (ed.) 1990, 548–552. See ch. 4, n. 5. Hempel (1965) played a major role in setting up the philosophical discussion of these two fundamental features of scientific knowledge and their relations to sci­ entific theory: for details, see; Fetzer, 2017 and the rich bibliography therein. See, e.g., Darrigol, [2000] 2002. Maxwell, 1858, 27. Collingwood, [1939] 1978, 25. This question, namely, why did Maxwell change methodologies within the same physical domain, will be addressed in the next subsection of this chapter, § 1.5, where we present the argument of the book. See also ch. 8, § 8.3. For a survey of electromagnetism throughout the nineteenth century, see Darri­ gol, [2000] 2002. See, e.g., Duhem ([1906/1914] 1954/1974, 70 and 78–80); for details, see ch. 2, n. 24. Maxwell, 1870, 420. Compare this striking assertion with Hertz ([1893] 1962, 28): “Scientific accuracy requires of us that we should in no wise confuse the simple and homely figure, as it is presented to us by nature, with the gay gar­ ment which we use to clothe it. Of our own free will we can make no change whatever in the form of the one, but the cut and colour of the other we can choose as we please.” Evidently, Hertz agreed with Maxwell’s view. Maxwell, 1873d, 1: Preface xi–xiii and Maxwell, 1873d, 2: 437–438, § 866. Maxwell, 1873d, 2: 438, § 866. We return to the issue of mental representation in ch. 7, § 7.3.5; see, also, ch. 8, § 8.3.3. We may remark en passant that Maxwell did not make a discovery in this way, though others did as, for example, the discovery by Edwin Hall (1855–1938) of the effect named after him. See ch. 7, n. 1. See ch. 5, n. 64. See ch. 7, n. 36. Hertz, [1893] 1962, 20–21, 1892, 22–23: “Auf die Frage, ‘was ist die Maxwell’sche Theorie’ wüsste ich also keine kürzere und bestimmtere Antwort als diese: Die Maxwell’sche Theorie ist das System der Maxwell’schen Gleichungen.” Hertz and Heaviside independently developed mathematical formulations of electro­ dynamics, based on Maxwell’s theory, which eventually became the set of equations we call today, Maxwell’s equations. Both physicists sought to take into account the mutual relations between magnetism and electricity by giving Maxwell’s system a symmetrical structure. Both achieved this goal by formal means, that is, by an ingenious manipulation of the equations, and both were pleased to find clarity and completeness—as well as consistency—in this way. There is no doubt that Hertz and Heaviside had a major impact on physics in the twentieth century. For example, Einstein’s relativity theory of 1905 was in many respects a response to Hertz

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Introduction

25

(explicitly) and to Heaviside (implicitly). The idea that a physical theory may consist of a set of mathematical equations expressing formal relations without physical interpretation eventually became the rule in twentieth-century physics. Indeed, this is how quantum mechanics is generally regarded today. 54 Harman (ed.) 1995, 335. 55 Maxwell to Mark Pattison, 13 April 1868; in Harman (ed.) 1995, 362. 56 For a striking example, see ch. 5, n. 55.

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2

Maxwell’s choice Faraday vs. Ampère

2.1 Michael Faraday (1791–1867) and James Clerk Maxwell: a unique relation In 1881 Hermann von Helmholtz (1821–1894) delivered the Faraday Lecture before the Fellows of the Chemical Society in London. Helmholtz, a renowned German physicist who made significant contributions to several quite different domains of modern science, limited his address to that aspect of Faraday’s scientific work which, as Helmholtz put it, “I know the best from my own experiences and studies: I mean the theory of electricity.”1 Helmholtz was one of the leading continental physicists specializing in electromagnetism who adhered—as most of the continental physicists had done—to the Newtonian concept of action at a distance. It is therefore most striking to note Helmholtz’s praise for Faraday who did not accept this Newtonian principle in electromagnetism. Moreover, Helmholtz rightly linked his praise for Faraday with his appreciation of Maxwell’s accomplishments in electrodynamics.2 Helmholtz remarked: [Faraday’s] principal aim was to express in his new conceptions only facts, with the least possible use of hypothetical substances and forces. This was really a progress in general scientific method, destined to purify science from the last remnants of metaphysics. Now that the mathematical interpretations of Faraday’s conceptions regarding the nature of electric and magnetic force has been given by Clerk Maxwell, we see how great a degree of exactness and precision was really hidden behind his words, which to his contemporaries appeared so vague or obscure; and it is astonishing in the highest to see what a large number of general theories the methodical deduction of which requires the high­ est powers of mathematical analysis, he has found by a kind of intuition [Anschauung], with the security of instinct, without the help of a single mathematical formula.3 Two points are worthy of note. In the first place, Helmholtz insightfully referred to Faraday’s scientific methodology designed to be based only on

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Maxwell’s choice 27 facts, thus minimizing the appeal to hypothetical substances and forces.4 The aim then was to construct a physics without metaphysics. Second, Helmholtz expressed his admiration for Faraday who succeeded in developing the physics of electromagnetism in precise terms (to be sure, hidden from view) without the use of mathematics. Helmholtz rightly recognized that this exactness and precision was brought to light in Maxwell’s formal theory. Later in his address, Helmholtz explained Faraday’s extraordinary clear thinking: And now with quite a wonderful sagacity and intellectual precision Faraday performed in his brain the work of a great mathematician without using a single mathematical formula. He saw with his mind’s eye [wunderbar klaren und lebhaften Intuition] that by these systems of tensions and pressures produced by the dielectric and magnetic polarisation of space which surrounds electrified bodies, magnets or wires conducting electric currents, all the phenomena of electro­ static, magnetic, electro-magnetic attraction, repulsion, and induction could be explained, without referring at all to forces acting directly at a distance. This was the part of his path where so few could follow him; perhaps a Clerk Maxwell, a second man of the same power and independence of intellect, was necessary to reconstruct in the normal methods of science the great building; the plan of which Faraday had conceived in his mind and attempted to make visible to his contemporaries.5 In almost poetic terms, Helmholtz succeeded in capturing the unique relation between Faraday and Maxwell. Helmholtz’s generosity is worth emphasizing, for he praised Faraday’s success based on a concept that Helmholtz did not accept, namely, lines of force. It is the medium’s elasticity that accounts for electromagnetic phenomena. This idea was foreign to continental physicists including Helmholtz. And it was Maxwell who brought to light Faraday’s extraordinary ideas by casting them in formal language, that is, mathematics. In 1873 Maxwell paid a moving tribute to Faraday on the occasion of the journal Nature offering its subscribers the first in a long series of “Portraits of Eminent Men of Science.” This article was published in the same year that Maxwell brought Station 4 to completion. For some 15 years he had been under the spell of Faraday’s intellect—it was a formative experience which shaped his scientific approach to electromagnetism both for content and methodology. According to Maxwell, Faraday undertook no less a task than the investigation of the facts, the ideas, and the scientific terms of electro-magnetism, and the result was the remodelling of the whole according to an entirely new method.6

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28 Maxwell’s choice What was this “entirely new method,” which we take to be methodology? Maxwell continued his laudatory remarks by acknowledging that to some these praises would come as a surprise since Faraday was not a professional mathematician and in his writings one finds “none of those integrations of differential equations which are supposed to be of the very essence of an exact science.”7 Indeed, according to Maxwell, Faraday would not even understand the mathematical symbols which fill the pages of Poisson, Ampère, Weber, and Neumann. So Maxwell reiterated his rhetorical question: It is admitted that Faraday made some great discoveries, but if we put these aside, how can we rank his scientific method so high without dis­ paraging the mathematics of these eminent men?8 This remark brought Maxwell to the point: at stake was Faraday’s “scientific method.” What was so unique in Faraday’s methodology? Maxwell continued: The geometry of position is an example of a mathematical science established without the aid of a single calculation. Now Faraday’s lines of force occupy the same position in electro-magnetic science that pen­ cils of lines do in the geometry of position. They furnish a method of building up an exact mental image of the thing we are reasoning about. The way in which Faraday made use of his idea of lines of force in co­ ordinating the phenomena of magneto-electric induction shews him to have been in reality a mathematician of a very high order—one from whom the mathematicians of the future may derive valuable and fertile methods.9 Maxwell’s point is that some parts of mathematics, notably “geometry of position,” had not been cast in terms of equations.10 This is a general claim about mathematics which is not directly related to Faraday, and it goes to show that Maxwell was fully aware of mathematical concepts which could be well suitable to Faraday’s way of thinking. Mathematics, or more precisely, mathematical thinking, did not suffice for Faraday’s success. Maxwell stressed the introduction of “an entirely new method” and it is not clear, given our distinction between “method” and “methodology,” what exactly Maxwell had in mind in explaining Faraday’s novel physics with “new method”. Here is a trivial use of the term “method” by Faraday: The best method of removing the charge I have found to be, to cover the finger with a single fold of a silk handkerchief, and breathing on the stem, to wipe it immediately after with the finger, the ball B and its connected wire, &c. being at the same time uninsulated: the wiping

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Maxwell’s choice 29 place of the silk must not be changed; it then becomes sufficiently damp not to excite the stem, and is yet dry enough to leave it in a clean and excellent insulating condition.11 This is of course a routine usage of “method” in the context of performing some experiment (the details need not detain us). Maxwell, by contrast, took the term to indicate a pivotal practice, namely, the use of concept. The central idea is worth repeating, “[lines of force] furnish a method of building up an exact mental image of the thing we are reasoning about.” This does not refer to a practical solution to a preparation problem amidst some experimental procedure; rather, it refers to the application of an abstract concept—lines of force. The point we are making is that this too—in Maxwell’s view—is a “method.” The application of lines of force facilitates the creation of an exact mental image of electromagnetic phenomena and thus the arrangement of these phenomena. However, this is not yet the generation of scientific knowledge. In his compact language Maxwell argued that it is the reasoning about this mental image, about the arrangement of the phenomena, which results in knowledge. We call Maxwell’s usage of “method” which results in scientific knowledge, “methodology.” Thus, method—among other things, the use of a concept and the reasoning about phenomena based on the relevant concept—is methodology. We claim that this is a lesson Maxwell drew from Faraday’s practice. In Station 4 Maxwell declared that The ideas which I have attempted to follow out are those of action through a medium from one portion to the contiguous portion. These ideas were much employed by Faraday, and the development of them in a mathematical form, and the comparison of the results with known facts, have been my aim in several published papers. The comparison, from a philosophical point of view, of the results of two methods so completely opposed in their first principles must lead to valuable data for the study of the conditions of scientific speculation.12 The “two methods” to which Maxwell referred were based on “action at a distance” and “lines of force,” respectively. The former was promoted by, among others, Ampère, and the latter—as we have seen—by Faraday. And Maxwell then explained Faraday’s methodology: The whole history of this idea in the mind of Faraday, as shewn in his published researches, is well worthy of study. By a course of experi­ ments, guided by intense application of thought, but without the aid of mathematical calculations, he was led to recognize the existence of something which we now know to be a mathematical quantity, and which may even be called the fundamental quantity in the theory of electromagnetism. But as he was led up to this conception by a purely

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30 Maxwell’s choice experimental path, he ascribed to it a physical existence, and supposed it to be a peculiar condition of matter, though he was ready to abandon this theory as soon as he could explain the phenomena by any more familiar forms of thought.13 The central feature in Faraday’s methodology is the interaction of newly discovered experimental facts and the available theoretical framework, giving priority to facts over theory and opting for simple theory.14 Maxwell acknowledged that Faraday’s methodology resulted in verbal analysis of electromagnetic phenomena and thus informal, but he remarked approvingly: It was perhaps for the advantage of science that Faraday, though thor­ oughly conscious of the fundamental forms of space, time, and force, was not a professed mathematician. He was not tempted to enter into the many interesting researches in pure mathematics which his discover­ ies would have suggested if they had been exhibited in a mathematical form, and he did not feel called upon either to force his results into a shape acceptable to the mathematical taste of the time, or to express them in a form which mathematicians might attack. He was thus left at leisure to do his proper work, to coordinate his ideas with his facts, and to express them in natural, untechnical language.15 And Maxwell continued to specify his goal in the framework of Faraday’s methodology, It is mainly with the hope of making these ideas the basis of a mathem­ atical method that I have undertaken this treatise.16 This was Maxwell’s goal right from the beginning of his studies of electromagnetism, and it continued to be his prime motivation for the entire project. Indeed, this is one of the principal claims of our study, namely, methodologies may change, but the goal—this goal—remained the same throughout Maxwell’s endeavors in electromagnetism. Maxwell’s thorough knowledge of Faraday’s researches lies behind the claim that there are different legitimate methodologies for attaining scientific knowledge. Maxwell frequently referred to Faraday in laudatory terms and often quoted his master verbatim.17 For example, in his plan for the Treatise Maxwell referred to a passage in Faraday’s Experimental Researches (1838).18 Later in the book Maxwell quoted this passage at length: (§ 1297) The direct inductive force, which may be conceived to be exerted in lines between the two limiting and charged conducting sur­ faces, is accompanied by a lateral or transverse force equivalent to a dilatation or repulsion of these representative lines . . .; or the attractive

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Maxwell’s choice 31 force which exists amongst the particles of the dielectric in the direction of the induction is accompanied by a repulsive or a diverging force in the transverse direction.19 And Maxwell commented, “This is an exact account of the conclusions to which we have been conducted by our mathematical investigation.”20 Note that Faraday’s language is hypothetical (“may be conceived to be . . .”), and it does not contain any image or illustration. To be sure, Faraday invoked his concept of lines of force for understanding electromagnetic phenomena, but he did not appeal to other images or methods of analysis. Maxwell clearly recognized the implications of Faraday’s language. In his essay, “On action at a distance,” published in the same year as his Treatise, Maxwell remarked: By means of this new symbolism [the lines of force], Faraday defined with mathematical precision the whole theory of electro-magnetism, in language free from mathematical technicalities, and applicable to the most complicated as well as the simplest cases. And Maxwell continued by drawing attention to Faraday’s transition from geometry to physics: But Faraday . . . went on from the conception of geometrical lines of force to that of physical lines of force . . .21 However, this is definitely not Faraday’s mode of expression; rather, it is Maxwell’s.22 Maxwell’s independence of mind, that is, his move away from the methodology of Faraday as his own researches into electromagnetism developed, is clearly expressed in the structure of his magnum opus of Station 4. The Treatise does not begin with an account of the phenomena of electromagnetism; rather, its point of departure is a preliminary discussion, preceding Part I, on the measurement of quantities, in which several mathematical innovations are introduced and their application to physics is discussed.23 This way of thinking or “framework,” starting with mathematical operations, is different from Faraday’s methodology. Nevertheless, Maxwell did not proceed in deductive steps, as some continental physicists preferred. In fact, Maxwell’s procedure has often been seen as opaque, motivated by several methodologies which at times seem to contradict one another.24 But to return to the point of departure: Station 1, “On Faraday’s lines of force” (1858), calls attention right in its title to the motivating concept, Faraday’s lines of force. Since this is the fundamental concept of the entire project it is worth spelling it out. Maxwell referred to this issue as “a question which has been raised again and again ever since men began to think.” And he continued:

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32 Maxwell’s choice The question is that of the transmission of force. We see that two bodies at a distance from each other exert a mutual influence on each other’s motion. Does this mutual action depend on the existence of some third thing, some medium of communication, occupying the space between the bodies, or do the bodies act on each other immediately, without the intervention of anything else?25 The dependence on what Maxwell called “some third thing,” that is, a medium of communication, occupying space, is illustrated by a well-known simple experiment: if we strew iron filings on paper near a magnet, each filing will be mag­ netized by induction, and the consecutive filings will unite by their opposite poles, so as to form fibres, and these fibres will indicate the direction of the lines of force. The beautiful illustration of the presence of magnetic force afforded by this experiment, naturally tends to make us think of the lines of force as something real, and as indicating some­ thing more than the mere resultant of two forces, whose seat of action is at a distance, and which do not exist there at all until a magnet is placed in that part of the field.26 Against this background and in response to the leading question, namely, how force is transmitted, Maxwell remarked: The mode in which Faraday was accustomed to look at phenomena of this kind differs from that adopted by many other modern inquirers, and my special aim will be to enable you to place yourselves at Fara­ day’s point of view, and to point out the scientific value of that concep­ tion of lines of force which, in [Faraday’s] hands, became the key to the science of electricity.27 It is important to recognize that Maxwell contrasted the Newtonian conception of action at a distance with that of lines of force. The former not only lacks an intermediary medium, it cannot act along a curved line as indeed lines of force do. In the Preface to his Treatise, Maxwell stated: I have confined myself almost entirely to the mathematical treatment of the subject, but I would recommend the student, after he has learned, experimentally if possible, what are the phenomena to be observed, to read carefully Faraday’s Experimental Researches in Electricity. He will there find a strictly contemporary historical account of some of the greatest electrical discoveries and investiga­ tions, carried on in an order and succession which could hardly have been improved if the results had been known from the first, and

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Maxwell’s choice 33 expressed in the language of a man who devoted much of his atten­ tion to the methods of accurately describing scientific operations and their results.28 Maxwell suggested that a student of electromagnetism should begin by reading Faraday for Maxwell considered Faraday’s work to be didactic as well as cutting-edge research. In the previous literature of physics it was not at all common to combine pedagogical considerations and cutting-edge research, as Maxwell did in his Treatise. For example, it is not the case for Newton’s Principia. And here we see that Faraday served as a model to be emulated. At the time of publication of the Treatise Maxwell was also engaged in other projects; in an essay of 1873, Maxwell related Faraday’s fundamental concept of lines of force to a novel understanding of action at a distance. Maxwell considered the former concept a method of inquiry, “the independent method of investigation employed by Faraday in those researches in electricity and magnetism.”29 In this way Maxwell could compare directly the contributions of Faraday with those in the great tradition of the continental physicists. Maxwell first described the limitations of Faraday’s skills in traditional mathematics: Faraday, with his penetrating intellect, his devotion to science, and his opportunities for experiments, was debarred from following the course of thought which had led to the achievements of the French philo­ sophers, and was obliged to explain the phenomena to himself by means of a symbolism which he could understand, instead of adopting what had hitherto been the only tongue of the learned. This new symbolism consisted of those lines of force extending them­ selves in every direction from electrified and magnetic bodies, which Faraday in his mind’s eye saw as distinctly as the solid bodies from which they emanated.30 As we have seen, Faraday introduced the concept of lines of force in response to the question, How is force transmitted through space? Maxwell analyzed Faraday’s imaginative leap. The idea takes as its point of departure the observed distribution of iron filings subject to a magnetic field. This observation was not new but it was previously regarded, in Maxwell’s words, “as an interesting curiosity of science.” At this juncture Maxwell quoted directly from Faraday’s Experimental Researches: It would be a voluntary and unnecessary abandonment of most valuable aid if an experimentalist, who chooses to consider magnetic power as represented by lines of magnetic force, were to deny himself the use of iron filings. By their employment he may make many conditions of the power, even in complicated cases, visible to the eye at once, may trace

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34 Maxwell’s choice the varying direction of the lines of force and determine the relative polarity . . . By their use probable results may be seen at once, and many a valuable suggestion gained for future leading experiments.31 For the mathematicians the experiments with iron filings merely offered solutions for the direction of the resultant of two forces. Maxwell realized that for Faraday the lines of force, represented by the magnetized filings, form a system. The arrangement of the lines was not just a specific solution to some combination of forces, but a means for indicating the intensity of the force. The number of lines which pass through an area is a measure of the intensity of the force acting in this area. Moreover, each individual line has a continuous existence in space and time. Thus, every line of force preserves its identity during the whole course of its existence. According to Maxwell, this concept had far-reaching consequences; indeed, the whole theory of electromagnetism followed from it: By means of this new symbolism, Faraday defined with mathematical precision the whole theory of electro-magnetism, in language free from mathematical technicalities, and applicable to the most complicated as well as the simplest cases.32 Maxwell emphasized this point in the Treatise: A method which, in Faraday’s hands, was far more powerful is that in which he makes use of those lines of magnetic force which were always in his mind’s eye when contemplating his magnets or electric currents, and the delineation of which by means of iron filings he rightly regarded* as a most valuable aid to the experimentalist. And Maxwell explained, Faraday looked on these lines as expressing, not only by their direc­ tion that of the magnetic force, but by their number and concentra­ tion the intensity of that force, and in his later researches† he shews how to conceive of unit lines of force. I have explained in various parts of this treatise the relation between the properties which Fara­ day recognised in the lines of force and the mathematical conditions of electric and magnetic forces, and how Faraday’s notion of unit lines and of the number of lines within certain limits may be made mathematically precise.33 [Footnotes in Maxwell, 1873d, 2: 174, § 541] * Exp. Res., 3234 † Ib., 3122

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Maxwell’s choice 35 Indeed, Maxwell’s entire project may be understood as an attempt to make the concept of lines of force “mathematical precise” to serve as the unifying concept of the entire theory. Faraday, so Maxwell argued, had endowed the concept of lines of force with a wider physical meaning. According to Maxwell, Faraday perceived in the medium a state of stress, consisting of a tension, like that of a rope, in the direction of the lines of force, combined with a pressure in all directions at right angles to them. This is quite a new conception of action at a distance, reducing it to a phenomenon of the same kind as that action at a distance which is exerted by means of the tension of ropes and pressure of rods. When the muscles of our bodies are excited by that stimulus which we are able in some unknown way to apply them, the fibres tend to shorten themselves and at the same time to expand laterally. A state of stress is produced in the muscle, and the limb moves. This explanation of muscular action is by no means complete. It gives no account of the cause of the excitement of the state of stress, nor does it even investigate those forces of cohesion which enable the muscles to support this stress. Nevertheless, the simple fact, that it substitutes a kind of action which extends continuously along a material sub­ stance for one of which we know only a cause and an effect at a dis­ tance from each other, induces us to accept it as a real addition to our knowledge of animal mechanics. For similar reasons we may regard Faraday’s conception of a state of stress in the electro-magnetic field as a method of explaining action at a distance by means of the continuous transmission of force, even though we do not know how the state of stress is produced.34 Characteristically, Maxwell resorted to an analogy to clarify his claim that the system of lines of force, introduced by Faraday is, in fact, as Maxwell put it, “a method of explaining” how force is transmitted through the medium. Importantly, Maxwell acknowledged that neither Faraday nor he himself knew how the state of stress arises; both, however, were committed to the claim that lines of force do represent this state of stress of the medium as the force is transmitted through it. Philosophically, this is an interesting conundrum. On the one hand, Maxwell—following Faraday— claimed to have been able to explain “action at a distance,” but he was openly agnostic as to the mechanism which results in a state of stress. Maxwell compared the lines of force to ropes that transmit tension and pressure at right angles to them. He then developed the analogy of the mechanical action in animals which offers an explanation for the work that a living body can do, with no insight into the causes of muscular tensions. The analogies of ropes and muscles are clearly illustrative and not part of the argument. As we have noted, we call these analogies weak.

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36 Maxwell’s choice By introducing these weak analogies, Maxwell sought to make his theory plausible to the reader.35 Maxwell offered an extensive account of Faraday’s lines of force which Maxwell considered an insightful conceptual innovation. Although no text of Faraday is cited, it seems to us most likely that Maxwell had in mind the following passage in Faraday (1852), in recasting Faraday’s concept of lines of force in terms of a “state of stress”: To acknowledge the action in curved lines, seems to me to imply at once that the lines have a physical existence. It may be a vibration of the hypothetical aether, or a state or tension of that aether equivalent to either dynamic or a static condition; or it may be some other state, which though difficult to conceived, may be equally distant from the supposed non-existence of the line of gravitating force, and the inde­ pendent and separate existence of the line of radiant force (3251.). Still the existence of the state does not appear to me to be mere assumption or hypothesis, but to follow in some degree as a conse­ quence of the known condition of the force concerned, and the fact dependent on it.36 It is thus not just a matter of Maxwell’s translating Faraday’s words into symbols.37 Maxwell also translated Faraday’s words into his own terms (adding, e.g., the analogy of the rope, which is not in Faraday). Faraday did not invoke the term “stress,” let alone “state of stress,” although he did invoke “tension.” So Maxwell translated Faraday’s ideas into his own terminology. Put differently, Maxwell understood what Faraday perceived as a state of stress in the medium, even though this was not Faraday’s expression. In the introductory section of Station 1, that is, at the outset of his research in electromagnetism, Maxwell recast Faraday’s experimental results “in a convenient and manageable form.”38 The next step was a reference to Faraday’s Experimental researches as well as to his essay of 1852, “On the physical character of the lines of magnetic force.”39 What was there in Faraday’s paper that made Maxwell consider it foundational? Faraday began his essay of 1852 with a cautionary remark addressed to the reader: The following paper contains so much of a speculative and hypothetical nature, that I have thought it more fitted for the pages of the Philo­ sophical Magazine than those of the Philosophical Transactions. Still it is so connected with, and dependent upon former researches, that I have continued the system and series of paragraph numbers from them to it . . . The paper . . . depends much for its experimental support upon the more strict results and conclusions contained in [publications in the Philosophical Transactions].40

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Maxwell’s choice 37 Faraday announced his readiness to speculate, anchoring his speculations to the experimental researches that he had conducted. He considered it coherent to refer to his speculations with the same system of numbered paragraphs that he had used for his experimental researches. This move is striking since Faraday, already in his 60s, made it legitimate to speculate and to link these speculations to solid research; indeed, we will see that the freedom to speculate had a strong liberating effect on Maxwell.41 What was the nature of Faraday’s speculations? As the title of his essay indicates, it is the physical character of the lines of magnetic force. That is, the lines as “representants” are not disputed; what is at stake is the physical character of the lines. I have recently been engaged in describing and defining the lines of magnetic force (3070.), i.e., those lines which are indicated in a general manner by the disposition of iron filings or small magnetic needles, around or between magnets; and I have shown . . . how these lines may be taken as exact representants of the magnetic power, both as to dis­ position and amount . . . The question at present appears to be, whether the lines of magnetic force have or have not a physical existence; and if they have, whether such physical existence has a static or dynamic form.42 Faraday reminded the reader that the definition of lines of force he had given elsewhere “had no reference to the physical nature of the force at the place of action.” Indeed, he then stated that he desired to restrict the meaning of the term line of force, so that it shall imply no more than the condition of the force in any given place, as to strength and direction; and not to include (at present) any idea of the nature of the physical cause of the phenomena.43 But in this paper his approach was different: I am now about to leave the strict line of reasoning for a time, and enter upon a few speculations respecting the physical character of the lines of force, and the manner in which they may be supposed to be continued through space.44 Faraday praised the pursuit of speculation: It is not to be supposed for a moment that speculations . . . are useless, or necessarily hurtful, in natural philosophy. They should ever be held as doubtful, and liable to error and to change; but they are wonderful aids in the hands of the experimentalist and mathematician; for not only are they useful in rendering the vague idea more clear for the time,

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38 Maxwell’s choice giving it something like definite shape, that it may be submitted to experiment and calculation; but they lead on, by deduction and correc­ tion, to the discovery of new phenomena, and so cause an increase and advance of real physical truth, which, unlike the hypothesis that led to it, becomes fundamental knowledge not subject to change.45 And he added: though I value [speculations] highly when cautiously advanced, I con­ sider it as an essential character of a sound mind to hold them in doubt; scarcely giving them the character of opinions, but esteeming them merely as probabilities and possibilities, and making a very broad distinction between them and the facts and laws of nature.46 Faraday set the conditions that determine whether the lines are physical or not: If an action in curved lines or directions could be proved to exist in the case of the lines of magnetic force, it would also prove their physical existence external to the magnet on which they might depend; just as the same proof applies in the case of static electric induction.47 This stands in contrast to the transmission of force based on the assumption of action at a distance which must always take place in straight lines. Faraday acknowledged that the experimental data available at the time were not sufficient for a full comparison of the various “lines of power” (Faraday’s phrase). For example, they did not establish the claim that “the magnetic lines of force are analogous to those of gravitation, or direct actions at a distance; or whether, having a physical existence, they are more like in their nature to those of electric induction or the electric current.”48 Faraday revealed his train of thought as follows: If the northness and southness be considered so far independent of each other as to be compared to two fluids diffused over the two ends of the magnet (like the two electricities over a polarized conductor), then breaking the magnet in half ought to leave the two parts, one absolutely or differentially north in character, and the other south. Each should not be both north and south in equality of proportion, consider­ ing only the external force. But this never happens. If it be said that the new fracture renders manifest, externally, two new poles, opposite in kind but equal in force (which is the fact), because of the necessity of the case, then the same necessity exists also for the dependence and relation of the original poles of the original magnet, no matter what or where the first source of the power may be. But in that case the curved

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Maxwell’s choice 39 lines of force between the poles of the original magnet follow as a con­ sequence; and the curvature of these lines appears to me to indicate their physical existence.49 It is immediately evident that Faraday set his speculations in the context of experimental results of which the most fundamental is the fact that a monopole does not exist, for it is impossible to isolate a north or south magnetic pole. Any speculation on the nature of the magnet and its associated lines of force must be consistent with this well-established fact. Again and again Faraday speculated in one way or another, but he always returned to the fact that the magnet is bi-polar. The permanent co-existence of the two poles and the action they exert on each other which must be in curved lines, convinced Faraday that “the lines have a physical existence.”50 In fact, any suggestion regarding the nature of the magnet should offer “the necessary reason why no absolute charge of northness or southness is found in the two halves.”51 The nature of magnetic action is not clear; it was suggested that it might be a vibration of the hypothetical aether, a state or tension of the aether equivalent to either dynamic or a static conditions. This is indeed speculative but Faraday surmised that the state of the aether is real and not hypothetical, as it was the result of the applied force.52 Earlier in his researches Faraday considered the lines which are seen, e.g., by the disposition of iron filings, “representants of the magnetic power”; but now he proposed to go further. The lines are closed curves, passing in one part of their course through the magnet, and in the other part through the space around it. These lines are identical in their nature, qualities and amount, both within the magnet and without. If to these lines . . . we add the idea of physical existence . . . it will be seen at once that the probability of curved exter­ nal lines of force, and therefore of the physical existence of the lines, is as great, and even far greater, than before.53 Faraday adduced further support for the physical reality of the magnetic lines of force by appealing to the electric analog: I incline to the opinion that [the magnetic lines of force] have a physical existence correspondent to that of their analogue, the electric lines; and having that notion, am further carried on to consider whether they have a probable dynamic condition, analogous to that of the electric axis to which they are so closely and, perhaps, inevitably related, in which case the idea of magnetic currents would arise.54 Here we see that Faraday was ready to draw surprising consequences from his speculations. Indeed, the analogy he drew between magnetism and electricity went further, for “an absolute northness or southness, or an unrelated

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40 Maxwell’s choice northness or southness, [was] as impossible as an absolute or an unrelated state of positive or negative electricity.”55 This analysis led Faraday to the view that the medium or space around a magnet is as essential to magnetic action as the magnet itself, being part of the true and complete magnetic system. And Faraday resorted to a figurative expression to describe the nature of a magnet: “every part of the surface of the magnet, so to say, is pouring forth externally lines of magnetic force.”56 He went on to describe magnets “as the habitations of bundles of lines force.”57 This view is consistent with the claim that the magnetic lines of force are physical: they converge onto, or diverge from, the intervening material, depending on its magnetic quality. Now, if this external relation of the poles is denied, then—Faraday argued—“the internal relation must be denied also; and with it a vast number of old and new facts (3070. &c.) will be left without either theory, hypothesis, or even vague supposition to explain them.”58 This is a negative claim, but it shows the effectiveness of speculations in the context of a well-established set of experimental results. Faraday brought the paper to a conclusion by contrasting his earlier view based on experimentation and his current one which was speculative: Having applied the term line of magnetic force to an abstract idea, which I believe represents accurately the nature, condition, direction, and comparative amount of the magnetic forces, without reference to any physical condition of the force, I have now applied the term phys­ ical line of force to include the further idea of their physical nature. The first set of lines I affirm upon the evidence of strict experiment (3071, &c.). The second set of lines I advocate, chiefly with a view of stating the question of their existence; and though I should not have raised the argument unless I had thought it both important, and likely to be answered ultimately in the affirmative, I still hold the opinion with some hesitation, with as much, indeed, as accompanies any conclusion I endeavour to draw respecting points in the very depths of science . . ..59 This dense conclusion raised six issues for Maxwell: (1) two conceptions are elaborated, experimental and speculative; (2) the former is “affirmed,” the latter is “advocated”; (3) but on both accounts there is no commitment to the physical condition of the force; (4) thus, there is a clear understanding of what is real and what is speculative; (5) the lines of force can express the nature, condition, direction, and comparative amount of the magnetic forces; and, finally, (6) the commitment to the concept of “lines of force” is hesitant. As we will see, Maxwell responded fully to the first four issues, and he accepted Faraday’s claim that the lines express the direction and the “comparative amount” of magnetic forces, which Maxwell called “intensity.”60 Faraday promoted the concept of “lines of force” as a viable, physical concept for a unified understanding of electromagnetic phenomena. Nevertheless, he continued to entertain the concept of “action at a distance.”

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Maxwell’s choice 41 He envisaged harmonizing the two different perspectives on the phenomena and took them to be on an equal footing with respect to the conservation of force.61 What then was Maxwell’s conception of Faraday’s methodology? Faraday was widely recognized as a brilliant experimentalist, but this did not entail taking his “speculations” seriously as Maxwell did. Moreover, considering Faraday to be a mathematician was a bold move, and no one else in the community (including Faraday himself!) had this understanding of what it meant to be mathematician (or, more precisely, a mathematical physicist). Maxwell had a unique view of mathematics, namely, it need not be expressed in equations or other formal structures. Faraday’s “speculations” were mathematical and only had to be put in terms of standard formalism for this to be obvious to others. The “geometry of position” offered Maxwell an example of a mathematical domain that did not involve calculations. Faraday’s lines of force occupy the same position in electromagnetic science as a pencil of lines do in the “geometry of position.” They furnish a method of constructing an exact mental image of the thing we are reasoning about. In short, Maxwell saw the advantages of Faraday’s methodology that remained hidden to others and then proceeded —as we will show—to develop successive theories of electromagnetism in which he sought to maintain the core concepts he inherited from Faraday.

2.2 André-Marie Ampère (1775–1836): the contrast In his essay of 1873, Maxwell reflected on the development of electromagnetism, given the point of departure of the Newtonian concept of action at a distance, and juxtaposing Ampère and Faraday.62 Maxwell had been consistent throughout the years in respecting the achievements of Ampère but, at the same time, Maxwell was critical of his approach. He remarked: Ampère, by a combination of mathematical skill with experimental ingenuity, first proved that two electric currents act on one another, and then analysed this action into the resultant of a system of push-and­ pull forces between the elementary parts of these currents. The formula of Ampère, however, is of extreme complexity, as com­ pared with Newton’s law of gravitation, and many attempts have been made to resolve it into something of greater apparent simplicity.63 Interestingly, Maxwell thought that the origin of this development was rooted in mathematics and astronomy from which Faraday departed: Whereas the general course of scientific method then consisted in the application of the ideas of mathematics and astronomy to each new investigation in turn, Faraday seems to have had no opportunity of

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42 Maxwell’s choice acquiring a technical knowledge of mathematics, and his knowledge of astronomy was mainly derived from books. Hence, though he had a profound respect for the great discovery of Newton, he regarded the attraction of gravitation as a sort of sacred mys­ tery, which, as he was not an astronomer, he had no right to gainsay or to doubt, his duty being to believe it in the exact form in which it was delivered to him. Such a dead faith was not likely to lead him to explain new phenomena by means of direct attractions.64 In any event, according to Maxwell, Faraday could not master the technical, mathematical details and therefore he had to conceive a new methodological approach which Maxwell was keen to follow. Ampère belonged then to a tradition in electromagnetism that followed Newton’s methodology; adherents of this tradition included Henry Cavendish (1731–1810), Charles-Augustin de Coulomb (1736–1806), and Siméon Denis Poisson (1781–1840) as well as Ørsted. But Faraday did not pursue this approach.65 In Maxwell’s first contribution to electromagnetism he contrasted Faraday with Ampère. As we have seen, Maxwell entitled the paper he read in December 1855 to the Cambridge Philosophical Society, “On Faraday’s lines of force.” In it he presented Faraday’s theory and proposed a methodology to translate a descriptive theory into a formal one. Within this analysis, Maxwell noted Ampère’s theory of the attraction of closed circuits. He considered the mutual action of magnets and electric currents in the framework of Ampère’s theory, but then proved two consequences of this theory by appealing to Faraday’s concept, namely, lines of force. In the years to come Maxwell continued to remind his readers of Ampère important achievements in electromagnetism, and then distanced himself from Ampère’s methodology by appealing to Faraday. Essentially, Ampère —in contrast to Faraday—accepted the application of action at a distance to electromagnetism. In Station 1 Maxwell dedicated a whole section to “the action of closed currents at a distance” whose “mathematical laws of the attractions and repulsions,” according to Maxwell, “have been most ably investigated by Ampère, and his results have stood the test of subsequent experiments.”66 As we will see, Maxwell—on the one hand—acknowledged the fundamental and solid contribution of Ampère and—on the other—expressed a critique of Ampère’s methodology for proceeding on the assumption of action at a distance and thus limiting the validity of the result. To be sure, Maxwell admitted that, “in the present state of science,” the Newtonian concept of action at a distance was still “warrantable,” but his deep intuition—under the direct influence of Faraday—led him to adopt Faraday’s concept of lines of force.67 Indeed, in the section, “Electro-magnetism,” Maxwell responded in the spirit of Faraday to Ampère’s theory:

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Maxwell’s choice 43 It is to be observed, that the currents with which Ampère worked were constant and therefore re-entering. All his results are therefore deduced from experiments on closed currents, and his expressions for the mutual action of the elements of a current involve the assumption that this action is exerted in the direction of the line joining those elements. This assumption is no doubt warranted by the universal consent of men of science in treating of attractive forces considered as due to the mutual action of particles; but at present we are proceeding on a different prin­ ciple, and searching for the explanation of the phenomena, not in the currents alone, but also in the surrounding medium.68 This is a revealing passage. Maxwell explicitly stated his position, namely, he was adopting Faraday’s methodological approach—in opposition to that of Ampère—of taking into consideration the surrounding medium as the bearer of the phenomena of electromagnetism: “at present we are proceeding on a different principle.” In Station 4 Maxwell once again contrasted Faraday with Ampère. Maxwell praised Ampère whose experimental investigation “established the laws of the mechanical action between electric currents, . . . one of the most brilliant achievements in science.” Indeed, Maxwell called Ampère the “Newton of electricity.” However, Maxwell remarked that Ampère’s method “does not allow us to trace the formulation of the ideas which guided it.” According to Maxwell, We are led to suspect . . . that [Ampère] discovered the law by some pro­ cess which he has not shewn us, and that when he had afterwards built up a perfect demonstration he removed all traces of the scaffolding by which he had raised it.69 This was not the case with Faraday who shared with his reader his unsuccessful, as well as his successful, experiments. The method which Faraday employed in his researches consisted in a constant appeal to experiment as a means of testing the truth of his ideas, and a constant cultivation of ideas under the direct influence of experiment.70 But it was not only the methodology that was unique: Faraday’s language, as Maxwell noted, was “somewhat alien from the style of physicists who have been accustomed to establish mathematical forms of thought.”71 Hence Maxwell offered this advice to the student of physics: Every student . . . should read Ampère’s research as a splendid example of scientific style in the statement of a discovery, but he should also study Faraday for the cultivation of a scientific spirit, by means of the

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44 Maxwell’s choice action and reaction which will take place between newly discovered facts and nascent ideas in his own mind.72 In Maxwell’s view, Faraday’s methodology exemplified the essence of the scientific spirit: Faraday sought to accommodate newly discovered facts within an existing theoretical framework although he was equally ready to abandon such frameworks when confronted with new facts. Nevertheless, Maxwell did acknowledge the fundamental achievements of continental physics to which Ampère contributed: The ideas which guided Ampère belong to the system which admits direct action at a distance, and we shall find that a remarkable course of speculation and investigation founded on these ideas has been carried on by Gauss, Weber, J. Neumann, Riemann, Betti, C. Neumann, Lorenz, and others, with very remarkable results both in the discovery of new facts and in the formation of a theory of electricity.73 Maxwell positioned himself against this continental tradition that he took to be based on the ideas of Ampère, siding completely with Faraday. Nevertheless, on several occasions he gave a detailed account of Ampère’s achievements, for example, in presenting Ampère’s experimental method to which Maxwell referred as “the null method of comparing forces”: Instead of measuring the force by the dynamical effect of communicat­ ing motion to a body, or the statical method of placing it in equilibrium with the weight of a body or the elasticity of a fibre, in the null method two forces, due to the same source, are made to act simultaneously on a body already in equilibrium, and no effect is produced, which shews that these forces are themselves in equilibrium.74 Maxwell conducted this analysis in the chapter headed “Ampère’s investigation of the mutual action of electric currents,” in which he gave an outline of what he termed as, “Ampère’s method,” under a running header, “Ampère’s scientific method.”75 Evidently, Maxwell considered it important to draw attention to Ampère’s methodology and to elaborate on it at some length. Maxwell appreciated the great accomplishments of Ampère both as an experimentalist and as a mathematical physicist. Still, he observed critically that “the currents used by Ampère, being produced by the voltaic battery, were of course in closed circuits.” And he continued to report that “no experiments on the mutual action of unclosed currents have been made.”76 This of course opened up the domain of electromagnetism to further research and called for a different approach in order to make progress. Indeed, a new mathematical method—the quaternions of Hamilton —was applied by Maxwell to address the widening aspects of electromagnetic phenomena for which the limited approach of Ampère could not account.77

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Maxwell’s choice 45

Notes 1 Helmholtz, 1881, 182. For the German original, see Helmholtz, [1881] 1896, 251. 2 In contrast to Helmholtz’s praise of Maxwell, H. F. Weber, Einstein’s physics teacher at the Polytechnic Institute in Zurich [ETH], simply ignored everything that came after Helmholtz. In fact, Einstein studied Maxwell on his own from Föppl (1894). For details, see Pais, 1982, 44. 3 Helmholtz, 1881, 182. For the German original, see Helmholtz, [1881] 1896, 252–253. 4 Helmholtz’s words are Principien wissenschaftlicher Methodik which we take to mean scientific methodology. 5 Helmholtz, 1881, 182. For the German original, see Helmholtz, [1881] 1896, 256–257. It is striking that the translator rendered wunderbar klaren und lebhaf­ ten Intuition as “the mind’s eye”—the very expression that Maxwell’s invoked in his reflections on Faraday: see nn. 30 and 33, below. 6 Maxwell, 1873c, 398. 7 Maxwell, 1873c, 398. 8 Maxwell, 1873c, 398. 9 Maxwell, 1873c, 398–399. On the geometry of position, see ch. 7, nn. 192, 193, and 196. 10 Maxwell depended on the work of J. B. Listing (1808–1882), citing his “Census räumlicher Complexe” (1861). See Maxwell, 1873d, Preliminary, 1: 16, § 18. In 1847 Listing coined the term Topologie in German for this new mathematical domain, but it was not immediately adopted in English: see Listing, [1847] 1848, 6. Maxwell considered using the expression (in Latin) geometria situs as well as the expression (in French) géométrie de position, but decided against them; instead, he opted for a literal translation of the French, namely, geometry of position. Typically, topo­ logical concepts and results belong to pure mathematics but, as Epple argued, “Maxwell viewed topology as linked to natural philosophy for conceptual reasons.” On this view, “topology was the science investigating the properties of physical con­ tinuity in actual space.” Epple, 1998, 381, italics in the original. 11 Faraday, 1838, 13, § 1203. 12 Maxwell, 1873d, 2: 146, § 502. 13 Maxwell, 1873d, 2: 173–174, § 540. 14 For Faraday’s methodology of exploratory experimentation, see Steinle, [2005] 2016. 15 Maxwell, 1873d, 2: 163, § 528. 16 Maxwell, 1873d, 2: 163, § 528. 17 Tait (1883a, 647): “[Maxwell] began by reading with the most profound admir­ ation and attention the whole of Faraday’s extraordinary self-revelations, and proceeded to translate the ideas of that master into the succinct and expressive notation of the mathematicians.” Indeed, as Maxwell himself reported in the Preface to the Treatise (1873d, 1: ix): “before I began the study of electricity I resolved to read no mathematics on the subject till I had first read through Fara­ day’s Experimental Researches on Electricity.” 18 Maxwell, 1873d, 1: 59, § 59. 19 Maxwell, 1873d, 1: 131, § 109; Faraday, 1838, 37, § 1297; reprinted in Faraday, [1839–1855] 1965, 1: 409. 20 Maxwell, 1873d, 1: 131, § 109. 21 Maxwell, 1873b, 52. 22 For the explicit transition from geometry to physics, see Maxwell, 1858, 29–30; and ch. 4, nn. 30–32. This is an example of what Helmholtz considered the col­ laboration of Faraday and Maxwell.

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46 Maxwell’s choice 23 See ch. 7, n. 7 and §§ 7.2 and 7.3.2. 24 See, notably, Duhem ([1906/1914] 1954/1974, 70 and 78–80), who sharply criticized Maxwell for inconsistency; Duhem’s views in this matter have been very influential. See also Ariew and Barker, 1986. Poincaré too was critical of Maxwell’s Treatise. Generally, they insisted on algebraic formulations of the laws of electrodynamics, whereas Maxwell thought this was only one of two ways to express these laws (see ch. 8, nn. 35–39). 25 Maxwell, 1873b, 44. 26 Maxwell, 1861–1862, 161–162. See also n. 31, below. 27 Maxwell, 1873b, 44, italics in the original. 28 Maxwell, 1873d, 1: Preface xiii. 29 Maxwell, 1873b, 50. 30 Ibid. This description of the “mind’s eye” in connection with Faraday also occurs in the Preface to the Treatise: see n. 33, below. 31 Maxwell, 1873b, 50–51: in Maxwell’s footnote the reference is to Faraday § 3284 although it should be to § 3234. Cf. Faraday, 1838; [1839–1855] 1965, 2: 397, § 3234. 32 Maxwell, 1873b, 52. 33 Maxwell, 1873d, 2: 174, § 541, where Maxwell included the following cross-refer­ ences: §§ 82, 104, 490. 34 Maxwell, 1873b, 52. 35 Maxwell, 1873b, 54. See also ch. 1, § 1.5, and ch. 4, n. 10. 36 Faraday, 1852a, 410, § 3243, italics in the original. 37 For the discussion of Maxwell’s view of translation, see ch. 8, § 8.3.2. 38 Maxwell, 1858, 29. 39 Faraday, 1852a. 40 Faraday, 1852a, 401. 41 See, e.g., ch. 4, nn. 19 and 20. 42 Faraday, 1852a, 405, § 3251. Cf. ibid., 401–402, § 3243. 43 Faraday, 1852b, 26, § 3075, italics in the original. 44 Faraday, 1852a, 402, § 3243. 45 Faraday, 1852a, 402. 46 Faraday, 1852a, 402, § 3244. 47 Faraday, 1852a, 406, § 3254, italics in the original. 48 Faraday, 1852a, 407, § 3256. 49 Faraday, 1852a, 409, § 3261, italics in the original. 50 Faraday, 1852a, 410, § 3263. 51 Faraday, 1852a, 411, § 3264. 52 See n. 50, above. 53 Faraday, 1852a, 410, § 3264. 54 Faraday, 1852a, 413, § 3269. 55 Faraday, 1852a, 417, § 3277. 56 Faraday, 1852a, 423, § 3289. 57 Faraday, 1852a, 426, § 3295. 58 Faraday, 1852a, 427, § 3297. 59 Faraday, 1852a, 427–428, § 3299, italics in the original. 60 See, e.g., Maxwell, 1858, 30. For discussion of the directionality of the force (geometry) and its intensity (physics), see ch. 4, n. 31 and the associated text. 61 See Faraday, 1852b, 26–27, § 3075; and Faraday, [1857/1858] 1859, 455–459. 62 Maxwell, 1873b; see nn. 21, 25, and 27, above. For Ampère’s theory, see Ampère, [1826] 1883. 63 Maxwell, 1873b, 49–50. 64 Maxwell, 1873b, 50.

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Maxwell’s choice 47 65 66 67 68 69 70 71 72 73 74 75 76 77

See Maxwell, 1873c, 398. Maxwell, 1858, 48. Maxwell, 1858, 48. Maxwell, 1858, 55. Maxwell, 1873d, 2: 162–163, § 528. Maxwell, 1873d, 2: 162, § 528. Maxwell, 1873d, 2: 162, § 528. Maxwell, 1873d, 2: 163, § 528. For other advice to the student, see ch. 7, n. 62. Maxwell, 1873d, 2: 146, § 502. Maxwell, 1873d, 2: 147, § 504. Maxwell, 1873d, 2: 146–147, §§ 502–504. Maxwell, 1873d, 2: 151, § 509. Maxwell, 1873d, 2: 158–159, § 522. For further discussion of the introduction of novel mathematical techniques into electrodynamics, see ch. 7. It should be noted that in Station 4 Maxwell offered an analysis of a few other continental physicists such as Helmholtz and Weber.

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3

Thomson, Stokes, Rankine, and Thomson and Tait

3.1 Introduction: methodology in electromagnetism Maxwell viewed his contributions to electromagnetism in the context of the work of his immediate predecessors and was often explicit about his sources of inspiration.1 We therefore turn our attention to a few physicists who explicitly made methodological claims to which Maxwell responded. We analyze the methodological contributions of four prominent physicists which serve as background to Maxwell’s study of electromagnetism: (1) Thomson’s formal analogy between physical systems; (2) Stokes’s concept of jelly which offers an analog to the ether with jelly as an idealization for facilitating mathematical treatment; (3) Rankine’s energeticist program; and, finally, (4) Thomson and Tait’s novel approach to Newtonian mechanics which served as the immediate background to Maxwell’s Treatise of electricity and magnetism (1873). To be sure, Maxwell was well aware of the contributions of other physicists, but he did not respond to their methodologies. The philosophical position we take emphasizes critical texts with which Maxwell was “in conversation.” Our interest is thus limited to the actual practice of physics, and not to general philosophical debates which took place at the time, for example, between the philosophers William Whewell (1794–1866) and John Stuart Mill (1806–1873).2

3.2 William Thomson (1824–1907): from analogy to representation Thomson was one of the central figures of Victorian science who helped shape physics throughout the nineteenth century; indeed, he contributed to virtually all domains in the physical sciences, including the mathematical analysis of electromagnetism, the formulation of the second law of thermodynamics, and the conceptual foundation for the temperature of absolute zero, to name but a few of his groundbreaking efforts. In 1866 Queen Victoria conferred the honor of knighthood on Thomson in appreciation of his “services in connection with the Atlantic Telegraph, and in recognition of [his] high position in science,” and in 1892 he was elevated to the peerage with the title Baron Kelvin of Largs.3 Early on in his long career, under the influence of Faraday, Thomson set out to contribute to the study of electromagnetism. Within a span of five

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Sources of inspiration 49 years he published three seminal papers, the first of which was, “On the uniform motion of heat in homogeneous solid bodies, and its connexion with the mathematical theory of electricity” (1842). In it he pointed to a connection between disparate phenomena, namely, heat and electricity. In his “On the mathematical theory of electricity in equilibrium” (1846), Thomson made a much stronger claim which regarded these two phenomena as posing an identical mathematical problem, thus rendering them formally analogical. In the third contribution, “On a mechanical representation of electric, magnetic, and galvanic forces” (1847), he abandoned both “connection” and “analogy” and opted for “representation.” For the first time, laws governing different phenomena, in this case, electricity and magnetism, were represented by the mechanical laws of equilibrium of elastic bodies. Thomson mathematically proved the existence of three different states of equilibrium by means of the strain propagated in an elastic solid, each representing (his word) a different physical phenomenon.4 He believed he was on the way to formulating a mechanical theory of electric and magnetic forces in terms of the equations of equilibrium of an elastic solid. Thomson’s first publication at the age of 18 is most impressive. The paper appeared anonymously in the Cambridge Mathematical Journal in February 1842. In 1854, when Thomson republished the paper, he added the following note: The general conclusions established in it show that the laws of distribu­ tion of electric or magnetic force in any case whatever must be identical with the laws of distribution of the lines of motion of heat in certain perfectly defined circumstances . . . they constitute a full theory of the characteristics of lines of force, which have been so admirably investi­ gated experimentally by Faraday, and complete the analogy with the theory of the conduction of heat . . .5 At the end of the introductory note to this reprint, Thomson remarked that “the analogy with the conduction of heat on which these views are founded, has not, so far as the author is aware, been noticed by any other writer.”6 However, in the original paper Thomson did not consider the phenomena of heat and electricity analogous; rather, he suggested a possible connection. In 1845 Thomson characterized the fact he had discovered about heat and electricity analogy. Instead of a mere connection, for Thomson problems in electricity and heat were now identical; he recognized the generality of the claim: Corresponding to every problem relative to the distribution of electricity on conductors, or to forces of attraction and repulsion exercised by electrified bodies, there is a problem in the uniform motion of heat which presents the same analytical conditions, and which, therefore, considered mathematically, is the same problem.7

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50 Sources of inspiration This is a classical formulation of inference by analogy: given a formal correspondence between two distinct physical domains, “considered mathematically,” the problems are interchangeable, and so are the solutions. Thomson was confident that, in view of their identical analytical features, one theory could offer insights into the other; indeed, features of one can be replaced by features of the other: The problem of distributing sources of heat, according to these conditions, is mathematically identical with the problem of distributing electricity in equilibrium . . . In the case of heat, the permanent temperature at any point replaces the potential at the corresponding point in the electrical system, and consequently the resultant flux of heat replaces the resultant attraction of the electrified bodies, in direction and magnitude. The problem in each case is determinate, and we may therefore employ the elementary principles of one theory, as theorems, relative to the other . . .8 Here we have a striking case of an argument by analogy, which we call “strong analogy.”9 Thomson claimed that two distinct disparate physical phenomena could be described by the same mathematical formulation, and he marked the corresponding interchangeable elements, e.g., temperature in the case of heat and potential in the case of electricity. Thomson succeeded in transforming his argument regarding the phenomena of heat and electricity from mere connection to a much more specific relation. In 1873, in his review of the reprint of this paper, Maxwell summarized Thomson’s achievement: Thomson . . . points out that these two problems [heat and electricity], so different both in their elementary ideas and their analytical methods, are mathematically identical, and that, by a proper substitution of electrical and thermal terms in the original statement, any of Fourier’s wonderful methods of solution may be applied to electrical problems. The electrician has only to substitute an electrified surface for the surface through which heat is supplied, and to translate temperature into electric potential, and he may at once take possession of all of Fourier’s solutions of the problem of the uniform flow of heat. Maxwell was very much impressed, and continued: “To render the results obtained in the prosecution of one branch of inquiry available to the students of another is an important service done to science . . .”10 This analogical relation between two distinct phenomena, based on identical mathematical formulations, facilitated the development of a physical theory without any commitment to a physical hypothesis; that is, the approach was completely phenomenalistic in that only macroscopic quantities such as temperature, quantity of heat, thermal conductivity, and electric force played a role.11 Although the general principle of analogical

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Sources of inspiration 51 reasoning was not new, Thomson recognized the power of this methodology for clarifying the complex phenomena of electricity and magnetism, without a commitment to the hypothetical claims of any physical theory. As Smith and Wise rightly emphasized, Thomson’s commitment to this methodology was deeply held, and he never discarded it.12 Thomson used this methodology in his proof of the analogy between heat and electricity. In this respect, he considered himself a disciple of Joseph Fourier (1768–1830): Now the laws of motion for heat which Fourier lays down in his Théorie Analytique de la Chaleur, are of that simple elementary kind which consti­ tute a mathematical theory properly so called; and therefore, when we find corresponding laws to be true for the phenomena presented by electrified bodies, we may make them the foundation of the mathematical theory of electricity: and this may be done if we consider them merely as actual truths, without adopting any physical hypothesis, although the idea they naturally suggest is that of the propagation of some effect by means of the mutual action of contiguous particles; just as Coulomb, although his laws naturally suggest the idea of material particles attracting or repelling one another at a distance, most carefully avoids making this a physical hypoth­ esis, and confines himself to the considerations of the mechanical effects which he observes and their necessary consequences.13 Thomson concluded his paper by explicitly calling the relation he established between heat and electricity analogy and suggesting, cautiously, that specific physical explanations may eventually be discovered for the various physical phenomena. It is, no doubt, possible that such forces at a distance may be dis­ covered to be produced entirely by the action of contiguous particles of some intervening medium, and we have an analogy for this in the case of heat, where certain effects which follow the same laws are undoubt­ edly propagated from particle to particle. It might also be found that magnetic forces are propagated by means of a second medium, and the force of gravitation by means of a third.14 Interestingly enough, Thomson envisaged similarity among distinct physical phenomena such as magnetism and gravitation since—so he thought—all required a medium. Still, he ended his paper on an agnostic note: We know nothing, however, of the molecular action by which such effects could be produced, and in the present state of physical science it is necessary to admit the known facts in each theory as the foun­ dation of the ultimate laws of action at a distance. (22 November 1845)15

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52 Sources of inspiration We see that in this early stage of Thomson’s long career he was engaged in arguing for the analogy between heat and electricity. But, more importantly for us, the analogy did not lead Thomson to decide between the dominant concept of action at a distance and the concept of lines of force which Faraday was in the process of developing. Rather, Thomson accepted the received view that action at a distance offers a proper conceptual foundation for the various theories. Clearly, in contrast to Faraday,16 he was not committed to the concept of lines of forces. The successful analogy between the two distinct phenomena of heat and electricity set Thomson on a search for further analogies. In his diary he wrote: Glasgow, 31 October 1846, 11.45 P.M.—I have this evening . . . after thinking on Faraday’s discovery of the effect of magnetism on transpar­ ent bodies and polarized light, been recurring to my idea (which occurred to me in the May term) which I had to give up, about magnet­ ism and electricity being capable of representation by the straining of an elastic solid constituted in a peculiar way.17 And a month later, on 28 November, he recorded, “I have at last succeeded in working out the mechanico-cinematical (!) representation of electric, magnetic, and galvanic f ces.”18 Thomson did not, however, invoke “analogy”; rather, he was thinking in terms of “representation,” the idea being that one theory, namely, a mechanical theory, could formally exhibit features of another theory, that of electromagnetic phenomena. Still, in the following year, on 11 June 1847, Thomson wrote to Faraday: I enclose the paper which I mentioned to you as giving an analogy for the electric and magnetic forces by means of the strain, propagated through an elastic solid. What I have written is merely a sketch of the mathematical analogy. I did not venture even to hint at the possibility of making it the foundation of a physical theory of the propagation of electric and magnetic forces, which, if established at all, would express as a necessary result the connection between electrical and magnetic forces, and would show how the purely statical phenomena of magnet­ ism may originate either from electricity in motion, or from an inert mass such as a magnet. If such a theory could be discovered, it would also, when taken in connection with the undulatory theory of light, in all probability explain the effect of magnetism on polarized light.19 Thomson believed he was on track to formulating a mechanical theory of electric and magnetic forces in terms of the equations of equilibrium of an elastic solid in which magnetic forces would be represented by angular displacements. He could not restrain his enthusiasm in imagining the consequences of such a physical theory were it to be successful. The paper to which Thomson was referring is remarkable, short, and insightful ([1847]

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Sources of inspiration 53 1882). Thomson mathematically proved the existence of three different states of equilibrium of a strained solid, each representing—as he used the word—a different physical phenomenon. Thomson did not invoke analogy, or analogies; rather, he considered his proofs to demonstrate three different “representations.” The appeal to “representation” in the title of the paper, as well as in the body of the paper, is worthy of comment. The verb “represent” is paraphrased in one instance (in the first paragraph of the paper) as “give a state of the solid.” It appears that Thomson regarded the mathematical equations as expressing some physical state, and the state—in turn—represents some physical phenomenon. “Representation” is not a technical term here. An equation represents some physical law which is put in mathematical terms. This does not seem to be an innovation; rather, it is similar to the usage of a graph as representing a function (in analytic geometry). Indeed, the expression “graphical representation” was very common in this period.20 For the first time, laws governing several different phenomena were represented by the mechanical laws of equilibrium of elastic bodies. We see here a transition from analogy to representation. Thomson opened the paper by taking his cue from Faraday: Mr. Faraday, in the eleventh series of his Experimental Researches in Electricity [Faraday, 1838], has set forth a theory of Electrostatical Induction, which suggests the idea that there may be a problem in the theory of elastic solids corresponding to every problem connected with the distribution of electricity on conductors, or with the forces of attrac­ tion and repulsion exercised by electrified bodies. The clue to a similar representation of magnetic and galvanic forces is afforded by Mr Fara­ day’s recent discovery of the affection with reference to polarized light, of transparent solids subjected to magnetic or electromagnetic forces. I have thus been led to find three distinct particular solutions of the equations of equilibrium of an elastic solid, of which one expresses a state of distortion such that the absolute displacement of a particle, in any part of the solid, represents the resultant attraction at this point, produced by an electrified body; another gives a state of the solid in which each element has a certain resultant angular displacement, repre­ senting in magnitude and direction the force at this point, produced by a magnetic body; and the third represents in a similar manner the force produced by any portion of a galvanic wire; the directions of the forces in the latter cases being given by the axes of the resultant rotations impressed upon the elements of the solid.21 The point of departure was the experimental findings of Faraday which Thomson linked to the mathematical theory of an elastic solid; Thomson then proceeded to develop his ingenious approach: The general equations of equilibrium of an elastic solid have been inves­ tigated by Mr Stokes*, without the assumption of any relation between

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54 Sources of inspiration the “cubical compressibility” and the elasticity, with reference to vari­ ations of form which are not accompanied by change of volume.22 [Footnote in Thomson, [1847] 1882, 77] * In a paper “On the Friction of Fluids in Motion, and of the Equilibrium and Motion of Elastic Solids,” read at the Cambridge Philosophical Society, 14 April 1845. See Trans., Vol. viii. Part 3. Thomson’s idea was to exploit the result which Stokes had obtained. As Thomson indicated, Stokes’s equations express the conditions of the interior equilibrium of an incompressible elastic solid. And Thomson added: These equations are to be employed for the representation of the forces in the several physical problems considered in this paper.23 Thomson sent Stokes a draft of this paper, requesting comments. Stokes responded: You asked me to look at the proof sheet of your article in the [Cam­ bridge and Dublin] Mathl Journal in which you referred to my paper. I had no alteration to make in it. You seem to be engaged in important speculations. Perhaps the jelly-like fluid that we once spoke of may be made in your hands to explain the law of the mutual action of electric currents and the phenomena of the induction of these currents.24 Many years passed before Thomson referred in print to this jelly-like fluid, while the jelly idea continued to be very much on Stokes’s mind.25 Be that as it may, we note that Stokes fully approved of Thomson’s approach. Once again Thomson concluded his paper on an agnostic note: I should exceed my present limits were I to enter into a special examin­ ation of the states of a solid body representing various problems in elec­ tricity, magnetism, and galvanism, which must therefore be reserved for a future paper. (Glasgow College, 28 November 1846)26 As the biographer of Thomson, Silvanus P. Thompson (1851–1916), noted, “it remains to add that the ‘future occasion’ did not occur till 1889.”27 Indeed, after more than 40 years, Thomson believed he reached the goal he had set for himself in his youthful work (Thomson, 1847). In that paper, Thomson avoided conjectures concerning the underlying physical states and was content to represent three different phenomena with the same formalism; he did not have a “mechanical” representation despite the paper’s title. In 1889, after “considering the subject for forty-two years,” Thomson was ready, as he phrased it, to go “below the surface” and determine the internal relations, namely, the mechanical relations, among

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Sources of inspiration 55 ether, electricity, and ponderable matter. This commitment to a mechanical world picture led to his search for mechanical models.28 A few months after the publication of this seminal paper on the mechanical representation of electric forces, Thomson reflected on the solution for the equations of equilibrium of an elastic solid: The application of the solution . . . in the mathematical theory of elastic solids, is analogous in some degree to a method of treating certain questions in the theory of heat, which was indicated in a paper “On the Uniform Motion of Heat in Solid Bodies, and its Connexion with the Mathematical Theory of Electricity.” [Thomson, 1842] Instead of the three components of the flux of heat at any point, we have to consider the three components of the rotation of an element of distorted solid; instead of a source of heat, we have a source of strain round the point of application of a force. If the solid were incompressible, there would be as close a connexion with the mathematical theory of electro-magnetism, as was shewn to subsist between the theories of heat and electricity: this follows at once from the theorem given in the former paper, with reference to the “mech­ anical representation” of the electro-magnetic force due to an elem­ ent of galvanic arc.29 Thomson thus noted a linkage between the two papers, for both the theory of heat and the theory of electricity, or at least aspects of these theories, are analogous to the mathematical theory of elastic bodies. An important feature of Thomson’s usages of the term “analogy” is that it relates to physical domains, despite its being formal or mathematical. For Thomson the physical phenomena of electromagnetism, galvanism, heat, and elasticity, share the same mathematical formalism; hence, they can be treated analogically. Analogy, on this view, is a relation between two physical domains, whereas when Thomson treated an imaginary physical situation, he used the expression “as if”: The action of a magnetic solenoid is the same as if a quantity of posi­ tive or northern imaginary magnetic matter numerically equal to its magnetic strength were placed at one end, and an equal absolute quan­ tity of negative or southern matter at the other end.30 We are not concerned with the technical aspects of a magnetic solenoid, which Thomson defined as an infinitely thin bar of any form, longitudinally magnetized, with an intensity varying inversely as the area of the normal section in different parts.31 Rather, what interests us is the fact that unlike Maxwell, Thomson did not set up analogies between imaginary and physical domains.32 Still, Thomson suggested an analogy (his term) for the distribution of magnetism in a solenoid:

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56 Sources of inspiration When an incompressible fluid flows through a tube of variable infinitely small section, the velocity (or in reality the mean velocity) in any part is inversely proportional to the area of the section. Hence the intensity and direction of magnetization, in a solenoid, according to the definition, are subject to the same law as the mean fluid velocity in a tube with an incom­ pressible fluid flowing through it. Again, if any finite portion of a mass of incompressible fluid in motion be at any instant divided into an infinite number of solenoids (that is, tube-like parts), by following the lines of motion the velocity in any one of these parts will at different points of it be inversely proportional to the area of its section. Hence the intensity and dir­ ection of magnetization in a solenoidal distribution of magnetism, according to the definition, are subject to the same condition as the fluid-velocity and its direction, at any point in an incompressible fluid in motion.33 While Thomson called this relation between an incompressible fluid flowing through a tube and the intensity and direction of magnetization in a solenoid an analogy, he did not pursue it, for he considered it “so obvious that it is scarcely necessary to point it out.”34 As we will see, this is remarkably close to the setup of the imaginary fluid which Maxwell introduced in Station 1—at the outset of his studies of electromagnetism—in his quest to recast Faraday’s lines of force into mathematical formalism. Indeed, in this footnote, Thomson described an incompressible fluid flowing in tubes in such an extraordinarily precise way that one might imagine that Maxwell was the author! In 1856 Thomson reported to the Royal Society that a suggestion by Rankine had caught his attention: the concept of “molecular vortices” could be the key for comprehending a wide variety of phenomena mechanically.35 However, it was not at all clear to Thomson how this might work: The explanation of all phenomena of electro-magnetic attraction or repulsion, and of electro-magnetic induction, is to be looked for simply in the inertia and pressure of the matter of which the motions constitute heat. Whether this matter is or is not electricity, or is itself molecularly grouped; or whether all matter is continuous, and molecular heteroge­ neousness consists in finite vortical or other relative motions of contigu­ ous parts of a body; it is impossible to decide, and perhaps in vain to speculate, in the present state of science.36 In 1873 Maxwell referred to his own theory of molecular vortices as a development of this idea of Thomson which “though rough and clumsy compared with the realities of nature, may have served its turn as a preliminary hypothesis.”37 According to Buchwald, in the 1840s, Thomson was not particularly interested in “electrostatics but introduced it only as an example of an attraction theory that can be used to solve problems in heat theory.”38 In this view, Thomson was primarily concerned with the relation between the

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Sources of inspiration 57 mathematics of Laplace’s theory of attraction and Fourier’s theory of heat conduction. Electrostatics was a vehicle for understanding the transmission of heat in terms of attractive forces, that is, on the basis of action at a distance.39 Thomson recognized the value of this methodology and its techniques but, typically, once he set up an analogy, he did not draw consequences from it. Thomson had an ambivalent attitude to Faraday’s conceptual framework, for he was not convinced that Faraday’s concept of lines of force was better than the concept of action at a distance. Smith and Wise aptly characterized the difference between Thomson and Maxwell with respect to their appeals to analogy: [Thomson] regularly restricted one side of an analogy physically so as to produce mathematical identity, but never appealed to analogy to enlarge the physical scope of a theory. That is, he never employed math­ ematical analogy to justify introducing new physical terms. James Clerk Maxwell, by contrast, would develop speculative analogies with great effect, thereby expanding Thomson’s limited analogies into his own quite different field theory. Thomson’s stricture indicates once again the intensity of his commitment to (what he regarded as) non-hypothetical theory. Recognizing that analogy alone could lead to uninterpretable phantasies, he required full empirical reference on both sides of an ana­ logy. Without it, the analogy could not support a mathematical theory “properly so called.”40 This is an important juncture in our argument for, in contrast to Thomson’s practice of discarding possible analogies between a physical domain and an imaginary one, Maxwell, as we will see, set up a contrived, mathematical analogy. In this case we discern the extent to which Maxwell distanced himself from Thomson. And Maxwell did not share Thomson’s ambivalence toward Faraday’s lines of force even though, in 1855, when Maxwell read his first paper on electromagnetism, he was only 24 years old and very much under the influence of Thomson. It is abundantly clear that Maxwell developed his study of electromagnetism against the background of Thomson’s productive application of powerful methodologies.

3.3 George Stokes (1819–1903): idealization In Station 1, his first fundamental contribution to electromagnetism, “On Faraday’s lines of force” (1858), Maxwell referred to two papers by Stokes, “On the conduction of heat in crystals” and “On the dynamical theory of diffraction.”41 There is no need to analyze these papers for the purposes of our argument; rather, we focus on Stokes’s position regarding the ether. Here we note a striking difference between the mature scientist, Stokes, and the young Maxwell. Again, our concern is methodological, not theoretical.

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58 Sources of inspiration The theme of relative motion plays an important role in the phenomenon of aberration. In his paper of 1845 on the aberration of light Stokes began with the well-known difficulty that the phenomenon is easily explained by the emission theory of light but was problematic in the undulatory theory, which “explains so simply and so beautifully the most complicated phenomena, that we are naturally led to regard aberration as a phenomenon unexplained by it, but not incompatible with it.”42 In order to address this difficulty, the properties of the ether had to be re-examined; hence the study of the constitution of the luminiferous ether. In 1848 Stokes referred to his earlier papers of 1845 and 1846, and reported that he had proved mathematically that the phenomenon of aberration may indeed be reconciled with the undulatory theory of light, on the assumption that the ether has some distinctive properties. He then added: It becomes an interesting question to inquire on what physical proper­ ties of the ether this sort of motion can be explained. Is it sufficient to consider the ether as an ordinary fluid, or must we have recourse to some property which does not exist in ordinary fluids, or, to speak more correctly, the existence of which has not been made manifest in such fluids by any phenomenon hitherto observed?43 Stokes sought to prove that a normal fluid, say, water, would be unstable if it were to exhibit the properties required for the ether. This paper, we emphasize, is about “the constitution of the ether.” Stokes attempted to facilitate the conception of the strange properties of this curious substance: The following illustration is advanced, not so much as explaining the real nature of the ether, as for the sake of offering a plausible mode of conceiving how the apparently opposite properties of solidity and fluid­ ity which we must attribute to the ether may be reconciled.44 We can surmise from this suggestion that, in the first place, Stokes considered the ether a phenomenon that is difficult to comprehend; as we will see, for Stokes the ether was not an imaginary scheme. The fact that he referred to “the real nature of the ether” indicates that, having presupposed the ether, its physical existence was taken for granted. But then its properties were not comprehensible: was it a fluid, or was it an elastic solid? Was it a little of this and a little of that? In other words, was it some kind of “in between” matter? The “illustration” which follows was intended to help the reader gain an intuitive grasp of the properties of the ether. Stokes continued: Suppose a small quantity of glue dissolved in a little water, so as to form a stiff jelly. This jelly forms in fact an elastic solid: it may be con­ strained, and it will resist constraint, and return to its original form

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Sources of inspiration 59 when the constraining force is removed, by virtue of its elasticity; but if we constrain it too far it will break. Suppose now the quantity of water in which the glue is dissolved to be doubled, trebled, and so on, till at last we have a pint or quart of glue water. The jelly will thus become thinner and thinner, and the amount of constraining force which it can bear without being dislocated will become less and less. At last it will become so far fluid as to mend itself again as soon as it is dislocated. Yet there seems hardly sufficient reason for supposing that at a certain stage of the dilution the tangential force whereby it resists constraint ceases all of a sudden. In order that the medium should not be dis­ located, and therefore should have to be treated as an elastic solid, it is only necessary that the amount of constraint should be very small. The medium would however be what we should call a fluid, as regards the motion of solid bodies through it. The velocity of propagation of normal vibrations in our medium would be nearly the same as that of sound in water; the velocity of propagation of transversal vibrations, depending as it does on the tangential elasticity, would become very small.45 Stokes proceeded by supposing the successive dilution of a substance in water; this is, of course, the well-known process of abstraction, which begins with something that occurs in nature, and ends with an idealized substance. He put forward an example in which the properties of a solid and a fluid may be reconciled and coexist in a continuous mode with no contradiction. Although Stokes did not say that this is the way he had been thinking about the curious constitution of the ether, it stands to reason that it is related to his own intuition. Notice that the fluid as well as the elastic solid are physical entities, and Stokes was intent on manipulating their known properties. He then concluded: Conceive now a medium having similar properties, but incomparably rarer than air, we have a medium such as we may conceive the ether to be, a fluid as regards the motion of the earth and planets through it, an elastic solid as regards the small vibrations which constitute light. The idea is somehow to retain the (different) physical properties of a fluid and an elastic solid at one and the same time. He then proposed an analogy (albeit without the term) in order to understand the ether better: Perhaps we should get nearer to the true nature of the ether by conceiv­ ing a medium bearing the same relation to air that thin jelly or glue water bears to pure water. The sluggish transversal vibrations of our thin jelly are, in the case of the ether, replaced by vibrations propagated with a velocity nearly 200,000 miles in a second: we should expect, à priori, the velocity of propagation of normal vibrations to be incompar­ ably greater. This is just the conclusion to which we are led quite

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60 Sources of inspiration independently, from dynamical principles of the greatest generality, combined with the observed phenomena of optics.*46 [Footnote in Stokes, [1848] 1883, 13] * See the introduction to an admirable memoir by Green, “On the laws of the Reflexion and Refraction of Light at the common surface of two noncrystallized media.” Cambridge Philosophical Transactions, Vol. vii, p. 1. Stokes summed up this creative exercise: the ether is a medium rarer than air, a fluid with respect to the motion of objects through it, but a solid with respect to vibrations. To be sure, the jelly is an artificial device, but Stokes invoked it for grasping “the true nature of the ether.” Stokes introduced an analogical relation with respect to the “jelly”: ether is to air as thin jelly is to pure water. The reference to George Green (1793–1841) is indicative of the methodology Stokes adopted.47 In the introduction to the paper which Stokes cited, Green acknowledged that we are so perfectly ignorant of the mode of action of the elements of the luminiferous ether on each other, that it would seem a safer method to take some general physical principle as the basis of our reasoning, rather than assume certain modes of action, which, after all, may be widely different from the mechanism employed by nature . . .48 Stokes followed Green’s advice and constructed the “jelly” on known principles such that it offers an analogy for the way the ether could function. This is a plausibility argument, based on analogy. The jelly functions as an analog to the ether and facilitates the comprehension of this curious substance. We see Stokes engaged in combining several distinct phenomena, upon which he imposed some specific restrictions so that the behavior of these phenomena “illustrates” by analogy the mysterious nature of the ether. Stokes’s jelly is an ideal substance, a perfect fluid, “abstracted” from what is observed in nature: A perfect fluid is an ideal abstraction, representing something which does not exist in nature. All actual fluids are more or less viscous, and we arrive at the conception of a perfect fluid by starting with fluids such as we find them, and then in imagination making abstraction of viscosity.49 This is a revealing comment. Stokes added this note in the reprint edition of his papers; here we have his own explanation of what he wrote in his original publication. Stokes’s appeal to an ideal fluid has to do with the treatment of a “perfect” fluid; this means that some properties are “idealized” (e.g., ignoring friction) or eliminated altogether. The perfect fluid is something that only exists

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Sources of inspiration 61 in the mind of the physicist. Still, it is important to note that the point of departure for arriving at this abstraction is the real stuff, a fluid that possesses a measure of viscosity. The purpose of this methodology of abstraction from a physical case is to enhance intuition for comprehending a mysterious substance. Hence, it is clear that Stokes’s jelly is not imaginary in the sense that Maxwell used the term.50 To understand better the difference between Maxwell and Stokes in this regard, we turn to another paper of Stokes in which he appealed both to an imaginary entity and to analogy. In his paper of 1849 on the internal friction of fluids in motion Stokes provided a framework for many of his research projects and drew consequences with respect to several phenomena. His point of departure is physical: a fluid in contact with a solid. He then developed the following argument: According to the hypotheses adopted, if there was a very large relative motion of the fluid particles immediately about any imaginary surface divid­ ing the fluid, the tangential forces called into action would be very large, so that the amount of relative motion would be rapidly diminished. Passing to the limit, we might suppose that if at any instant the velocities altered dis­ continuously in passing across any imaginary surface, the tangential force called into action would immediately destroy the finite relative motion of particles indefinitely close to each other, so as to render the motion continu­ ous; and from analogy the same might be supposed to be true for the sur­ face of junction of a fluid and solid.51 As before, we focus on the methodology which Stokes followed; we are not concerned with the relevant physical theory. He introduced an imaginary (mathematical) surface into the fluid, and then argued that there is an analogy between the relative motion in the imaginary case and a physical fluid flowing past a (real) surface immersed in fluid. Thus, the mathematically accessible analysis of the imaginary case corresponds to that of the physical setup. In other words, the use of an imaginary element in this context is a tool for analyzing a physical situation. The analogy then is between a junction of fluid with an imaginary surface and a junction of fluid with a solid surface. This use of “imaginary” does not correspond to the “purely imaginary nature” of the fluid which Maxwell exploited in the 1850s. To be sure, Stokes’s system is idealized in that there is no claim that it exists in nature, but its relations to physical phenomena are categorically different from Maxwell’s considerations.52 For Stokes “imaginary” is a mathematical concept: the surface is a boundary between regions of a substance, and this boundary is imaginary because there is nothing physical about it. Here we have a typical example of the use of “imaginary” in mathematical physics at the time Maxwell composed his paper on “Faraday’s lines of force.” In the next chapter—Station 1—we will see that Maxwell’s usage of “imaginary” is quite different from that of Stokes.

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3.4 William J. M. Rankine (1820–1872): energy—a novel unifying concept Rankine is another prominent physicist to whom Maxwell referred on a few occasions.53 In his “Outlines of the Science of Energetics” (1855), Rankine explained that Energy, or the capacity to effect changes, is the common characteristic of the various states of matter to which the several branches of physics relate; if, then, there be general laws respecting energy, such laws must be applicable, mutatis mutandis, to every branch of physics, and must express a body of principles as to physical phenomena in general.54 It can immediately be seen that Rankine sought to unify the various branches of physics by developing a body of principles applicable to physical phenomena in general. In Rankine’s view, energy was the overarching concept for this purpose. Rankine then reported: In a paper read to the Philosophical Society of Glasgow on the 5th of January 1853, a first attempt was made to investigate such principles, by defining actual energy and potential energy, and by demonstrating a general law of the mutual transformations of those kinds of energy, of which one particular case is a previously known law of the mechanical action of heat in elastic bodies, and another, a subsequently demon­ strated law which forms the basis of Professor William Thomson’s Theory of thermo-electricity.55 He then continued to explain that The object of the present paper [Rankine, 1855] is, to present in a more systematic form, both these and some other principles, forming part of a science whose subjects are, material bodies and physical phenomena in general, and which it is proposed to call the SCIENCE OF ENERGETICS.56 Rankine characterized this science of energetics as a physical theory arrived at by (what he called) the “abstractive method” which he distinguished from the “hypothetical method”: According to the ABSTRACTIVE method, a class of objects or phenom­ ena is defined by describing, or otherwise making to be understood, and assigning a name or symbol to, that assemblage of properties which is common to all the objects or phenomena composing the class, as perceived by the senses, without introducing anything hypo­ thetical. According to the HYPOTHETICAL method, a class of objects or phenomena is defined according to a conjectural conception of their

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Sources of inspiration 63 nature, as being constituted in a manner not apparent to the senses, by a modification of some other class of objects or phenomena whose laws are already known. Should the consequences of such a hypothet­ ical definition be found to be in accordance with the results of obser­ vation and experiment, it serves as the means of deducing the laws of one class of objects or phenomena from those of another.57 The distinction is insightful. Rankine characterized two different methodologies. To be sure, Rankine invoked the term “method” but, in the context of our discussion, he was describing two distinct methodologies. According to Rankine, the abstractive methodology consists of an empirical generalization devoid of hypothetical elements, while the hypothetical methodology is based on reducing the phenomena to some conjectural elements, not accessible empirically, but governed by a known set of laws. A prime example of the abstractive methodology is mechanics. The principles of the science of mechanics, the only example yet existing of a complete physical theory, are altogether formed from the data of experience by the abstractive method . . . The laws of the composition and resolution of motions, and of the composition and resolution of forces, are expressed by propositions which are the consequences of the definitions of motion and force respectively. The laws of the relations between motion and force are the consequences of certain axioms, being the most simple and general expressions for all that has been ascertained by experience respecting those relations.58 For the hypothetical methodology, Rankine offered the following examples: The fact that the theory of motions and motive forces is the only com­ plete physical theory, has naturally led to the adoption of mechanical hypotheses in the theories of other branches of physics; that is to say, hypothetical definitions, in which classes of phenomena are defined con­ jecturally as being constituted by some kind of motion or motive force not obvious to the senses (called molecular motion or force) as when light and radiant heat as defined as consisting in molecular vibrations, thermometric heat in molecular vortices, and the rigidity of solids in molecular attractions and repulsions. The hypothetical motions and forces are sometimes ascribed to hypo­ thetical bodies, such as the luminiferous aether; sometimes to hypothetical parts, whereof tangible bodies are conjecturally defined to consist, such as atoms, atomic nuclei with elastic atmospheres, and the like.59 Perceptively, Rankine warned against the misuse of the hypothetical methodology:

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64 Sources of inspiration The neglect of the caution already referred to, however, has caused some hypotheses to assume, in the minds of the public generally, as well as in those of many scientific men, that authority which belongs to facts alone, and a tendency has consequently often evinced itself to explain away, or set aside, facts inconsistent with these hypotheses, which facts, rightly appreciated, would have formed the basis of true theories; thus the fact of the production of heat by friction, the basis of the true theory of heat, was long neglected, because inconsistent with the hypothesis of caloric; and the fact of the production of cold by electric currents, at certain metallic junc­ tions, the key (as Professor William Thomson recently showed) to the true theory of the phenomena of thermo-electricity, was, from inconsistency with prevalent assumptions respecting the so-called “electric fluid,” by some regarded as a thing to be explained away, and by others as a delu­ sion. Such are the evils which arise from the misuse of hypothesis.60 Maxwell echoed this warning against the inappropriate use of hypothesis in his first contribution to electromagnetism. The physicist should be on guard against being lured by a “favourite hypothesis.”61 As we will show in the next chapter, Rankine’s cautionary remark assumed fundamental importance for Maxwell when he embarked on his study of electromagnetism. Rankine further explained that “axiom differs from . . . hypothesis in this, that the axiom is simply the generalized allegation of the facts proved by experience, while the hypothesis involves conjectures as to objects and phenomena which never can be subjected to observation.”62 It is noteworthy that Rankine contrasted the two methodologies which, in fact, form a sequence, namely, the hypothetical ultimately leads to the abstractive.63 Consider mechanical hypotheses: their tendency, according to Rankine, is to combine all branches of physics into one system, by making the axioms of mechanics the first principles of the laws of all phenomena; an object for the attainment of which an earnest wish was expressed by Newton. In the mechanical theories of elasticity, light, heat, and electri­ city, considerable progress has been made towards that end.64 The various mechanical hypotheses will be combined ultimately in an axiomatic theory via the abstractive methodology. We now reach the critical claim that most likely triggered Maxwell’s methodological decision to use “energy” as the overarching concept in electromagnetism, beginning in Station 3 and then developed into a fundamental concept in Station 4: Instead of supposing the various classes of physical phenomena to be consti­ tuted in an occult way of modifications of motion and force, let us distin­ guish the properties which those classes possess in common with each other, and so define more extensive classes denoted by suitable terms. For axioms,

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Sources of inspiration 65 to express the laws of those more extensive classes of phenomena, let us frame propositions comprehending as particular cases, the laws of the par­ ticular classes of phenomena comprehended under the more extensive clas­ ses. So shall we arrive at a body of principles, applicable to physical phenomena in general, and which being framed by induction from facts alone, will be free from the uncertainty which must always attach even to those mechanical hypotheses whose consequences are most fully confirmed by experiment. This extension of the abstractive process is not proposed in order to supersede the hypothetical method of theorizing; for in almost every branch of molecular physics it may be held, that a hypothetical theory is necessary as a preliminary step to reduce the expression of the phenomena to simplicity and order, before it is possible to make any progress in framing an abstractive theory.65 Rankine advised the scientist to go beyond the hypothetical methodology. He admitted that the hypothetical stage is usually necessary to simplify the phenomena, but progress can only be made when the second, abstractive, step is taken. The preliminary hypothetical step is always prone to uncertainty because the hypothetical remains inaccessible. To move forward, one has to capitalize on the simplicity obtained by the hypothetical methodology and then generalize by induction from the phenomena to a body of principles. The use of such principles avoids dependence on the occult features of the hypothetical, of conjectured entities that can easily mislead the researcher. The overarching nature of energy allowed precisely this—a general analysis with no hypothetical commitment: The peculiar terms which will be used in treating of the Science of Energetics are purely abstract; that is to say, they are not the names of any particular object, nor of any particular phenomena, nor of any par­ ticular notions of the mind, but are names of very comprehensive clas­ ses of objects and phenomena. About such classes it is impossible to think or to reason, except by the aid of examples or of symbols. Gen­ eral terms are symbols employed for this purpose.66 In 1867 Thomson and Tait pursued this suggestion in their classic Treatise on natural philosophy. As Moyer remarked, they produced “the first comprehensive work to consistently and fully incorporate the laws of energy . . . Thomson and Tait significantly increased the clarity, power, and range of abstract dynamics.”67 We now turn to Thomson and Tait.

3.5 W. Thomson and Peter Tait (1831–1901): abstract dynamics In the spirit of Rankine, Thomson and Tait set themselves the task of rewriting mechanics as “the dynamics of energy.”68 Similarly, it would be correct to say that in Station 4 Maxwell rewrote electromagnetism “as the

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66 Sources of inspiration electrodynamics of energy.” This move was most ingenious, consolidating as it did Maxwell’s achievements in this domain. In the Preface to their Treatise on natural philosophy (1867), Thomson and Tait declared that One object which we have constantly kept in view is the grand principle of the Conservation of Energy. According to modern experimental results, especially those of Joule, Energy is as real and as indestructible as Matter. It is satisfactory to find that Newton anticipated, so far as the state of experimental science in his time permitted him, this magnifi­ cent modern generalization.69 Thomson and Tait then remarked: In the second chapter we give Newton’s Laws of Motion in his own words, and with some of his own commentaries—every attempt that has yet been made to supersede them having ended in utter failure. Perhaps nothing so simple, and at the same so comprehensive, has ever been given as the foundation of a system in any of the sciences. And they added what seems to be an afterthought: The dynamical use of the Generalized Cöordinates of Lagrange, and the Varying Action of Hamilton, with kindred matter, complete the chapter.70 They interpreted Newton’s third law as the law of conservation of energy and “would reverse Lagrange while restoring Newton.”71 Nevertheless, the works of Lagrange and Hamilton played a key role for Thomson and Tait in recasting mechanics in terms of abstract dynamics. The ultimate goal of Thomson and Tait’s Treatise was to develop abstract dynamics applying the law of conservation of energy to any (limited) material system: The law of energy may then, in abstract dynamics, be expressed as follows:– The whole work done in any time, on any limited material system, by applied forces, is equal to the whole effect in the forms of potential and kinetic energy produced in the system, together with the work lost in friction. This principle may be regarded as comprehending the whole of abstract dynamics, because, as we now proceed to show, the conditions of equi­ librium and of motion, in every possible case, may be immediately derived from it.72

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Sources of inspiration 67 In this context Thomson and Tait introduced Lagrange’s analysis: A material system, whose relative motions are unresisted by friction, is in equilibrium in any particular configuration if, and is not in equilib­ rium unless, the work done by applied forces is equal to the potential energy gained, in any possible infinitely small displacement from that configuration. This is the celebrated principle of virtual velocities which Lagrange made the basis of his Mécanique Analytique.73 This is a critical juncture in the argument developed by Thomson and Tait.74 As Smith pointed out, Thomson and Tait made conservation of energy the principal concept with the consequence that force becomes a derivative concept.75 In fact, Thomson and Tait’s appeal to abstract dynamics is based on this argument: The problem of finding the motion of a system subject to any unvarying kinematical conditions whatever, under the action of any given forces, is thus reduced to a question of pure analysis.76 Thomson and Tait contrasted their account of kinematics with that of dynamics and in so doing called attention to some of the problems which Maxwell later faced in electrodynamics: Ch. I, Kinematics There are many properties of motion, displacement, and deformation, which may be considered altogether independently of such physical ideas as force, mass, elasticity, temperature, magnetism, electricity. The prelim­ inary consideration of such properties in the abstract is of very great use for Natural Philosophy, and we devote to it, accordingly, the whole of this our first chapter; which will form, as it were, the Geometry of our subject, embracing what can be observed or concluded with regard to actual motions, as long as the cause is not sought.77 They stressed the fact that in kinematics the cause is not sought, but this is not the case in dynamics: Ch. II, Dynamical laws and principles We now come to the consideration, not of how we might consider such motion, etc., to be produced, but of the actual causes which in the mater­ ial world do produce them. The results of the present chapter must there­ fore be considered to be due to actual experience, in the shape either of observation or experiment.78 Evidently, the goal of dynamical analysis is to uncover the causes of the phenomena. Thomson and Tait suggested that their general approach to

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68 Sources of inspiration mechanics can be applied to other branches of physics, but the details are sketchy: Every material system subject to no other forces than actions and reactions between its parts, is a dynamically conservative system, as defined above, § 271.79 But it is only when the inscrutably minute motions among small parts, possibly the ultimate molecules of matter, which constitute light, heat, and magnetism; and the intermolecular forces of chemical affinity; are taken into account, along with the palpable motions and measurable forces of which we become cognizant by direct observations, that we can recognise the universally conservative character of all natural dynamic action, and per­ ceive the bearing of the principle of reversibility on the whole class of nat­ ural actions involving resistance, which seem to violate it.80 It is noteworthy that in delineating the physical features of a conservative material system and the role which resistance plays in undermining the idealized conditions, Thomson and Tait included magnetism. This shift in the conceptual foundation of mechanics (and by extension of all physics) amounted to replacing force, the (traditional) causal agent in physical science, with “extremum principles,” e.g., “least action.” Thomson and Tait then appealed to Hamilton: Maupertuis’ celebrated principle of Least Action has been, even up to the present time, regarded rather as a curious and somewhat perplexing prop­ erty of motion, than as a useful guide in kinetic investigations. We are strongly impressed with the conviction that a much more profound import­ ance will be attached to it, not only in abstract dynamics, but in the theory of the several branches of physical sciences now beginning to receive dynamic explanations. As an extension of it, Sir W. R. Hamilton [Phil. Trans. 1834, 1835] has evolved his method of Varying Action, which undoubtedly must become a most valuable aid in future generalisations.81 Indeed, abstract dynamics required generalization, but of course the idea was that “pure analysis” would be applicable to various domains of physics. Thomson and Tait emphasized the applicability of their approach to a variety of physical domains that were beginning to receive dynamical explanations. Clearly, electromagnetism was one of these new branches of physics. Thomson and Tait then presented Hamilton’s form of Lagrange’s equations of motion in terms of generalized co-ordinates: the “canonical form” of the equations of motion of a system.82 When they came to electromagnetism, Thomson and Tait referred to the theories of Ampère and Weber:83 Weber[’s theory], which professes to supply a physical basis for Ampère’s Theory of Electro-dynamics . . . assumes that an electric current consists in

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Sources of inspiration 69 the motion of particles of two kinds of electricity moving in opposite direc­ tions through the conducting wire; and that these particles exert forces on other such particles of electricity, when in relative motion, different from those they would exert if at relative rest. In the present state of science this is wholly unwarrantable, because it is impossible to conceive that the hypoth­ esis of two electric fluids can be true, and besides, because the conclusions are inconsistent with the Conservation of Energy, which we have numberless experimental reasons for receiving as a general principle of nature. It only adds to the danger of such theories when they happen to explain further phenomena, as those of induced currents are explained by that of Weber.84 While Thomson and Tait highlighted the misleading character of Weber’s theory, they still held to the Newtonian unifying concept of action at a distance: Every particle of matter in the universe attracts every other particle with a force, whose direction is that of the line joining the two, and whose magnitude is directly as the product of their masses, and inversely as the square of their distance from each other. Experiment shows . . . that the same law holds for electric and magnetic attraction; and it is probable that it is the fundamental law of all natural action, at least when the acting bodies are not in actual contact.85 The critical point is, of course, the claim that the law is probably fundamental in all cases of attraction and, as such, applicable to electromagnetism as well as to gravitation. Thomson and Tait recognized the advantage of appealing to Lagrange and Hamilton: [A]lthough this method of dealing with a system of connected particles is very simple, so far as the law of energy merely concerned, Lagrange’s methods, whether that of “equations of condition,” or, what for our present purposes is much more convenient, his “generalized co-ordin­ ates,” relieve us from very troublesome interpretations when we have to consider the displacements of particles due to arbitrary variations in the configuration of a system.86 No “very troublesome interpretations” will burden the analysis and one can proceed with abstract dynamics without any interpretative commitment, thus circumventing one of the fundamental problems in dynamics—to which Thomson and Tait called attention earlier in their account—namely, establishing the causes of the phenomena.87 It appears that Thomson and Tait were very much in favor of applying the concept of energy to all domains of physics, including electrodynamics, but they offered no specific guidelines to reach this goal. This was left to a later time and perhaps to other physicists.88

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3.6 Conclusion We have examined four different methodologies: (1) formal analogy developed by Thomson who restricted it to physical systems; (2) idealization applied by Stokes where jelly serves as an analog to the ether to facilitate mathematical treatment, thereby “extending” reality, pushing it to some limiting case; (3) energy introduced by Rankine as a unifying concept; and, (4) “abstract dynamics,” a completely novel approach to Newtonian mechanics developed by Thomson and Tait, based on the principle of conservation of energy. As we shall see, Maxwell responded to all these methodologies. Maxwell did not restrict himself to Thomson’s conservative practice, for he moved to a mathematical analogy in which one domain was imaginary; he thus modified Thomson’s conception of analogy; at the same time, Maxwell distanced himself from Stokes’s usage of idealization, a limiting argument, which was too restrictive for Maxwell’s purposes. Maxwell adopted Rankine’s conception of energy within the framework of electrodynamics, and he applied Thomson and Tait’s methodology of “abstract dynamics” to electromagnetic phenomena. Maxwell responded creatively to his sources, transforming their methodologies to suit his commitments and goals. In subsequent chapters we describe and analyze in detail, station by station, how Maxwell’s new methodologies were built on the success of his predecessors. Foremost among them was Faraday, but others—as we have noted—also had a role to play.

Notes 1 See, e.g., Maxwell, 1858, 67.

2 See, e.g., Losee, 1983; Cobb, 2011.

3 Thompson, 1910, 1: 499, and 2: 907, 913–915; Smith and Wise, 1989, ch. 23.

The scale for measuring temperature with respect to absolute zero is called Kelvin.

4 See Thomson to Faraday, 11 June 1847, in Thompson, 1910, 1: 203–204.

5 Thomson, 1872, 1 n; cf., 1842; 1854a.

6 Thomson, 1872, 2 n; cf., 1842; 1854a.

7 Thomson, 1872, 27; cf., 1845; 1846, 84, [1854b] 1872.

8 Thomson, 1872, 28–29, 1846, 85, italics in the original.

9 For the distinction we draw between weak and strong analogy, see ch. 1, § 1.5.

10 Maxwell, 1873a, 219; reprinted in, [1890] 1965, 2: 302. 11 Cf. Knudsen, 1985, 156–157; Smith and Wise, 1989, 211–212. This methodo­ logical approach did not escape the attention of the young Maxwell, who later implemented it with great success in developing physical ideas without adopting a physical theory (see ch. 4, n. 5). 12 Smith and Wise, 1989, 229.

13 Thomson, [1854b] 1872, 29, 1846, 86, italics in the original.

14 Thomson, [1854b] 1872, 37, 1846, 93.

15 Thomson, 1846, 93.

16 Faraday, 1852a.

17 Thompson, 1910, 1: 197.

18 Ibid., exclamation point in the original.

19 Ibid., 203–204, italics in the original.

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Sources of inspiration 71 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53

54 55 56 57 58 59

See, e.g., Weisbach, [1845] 1848, 34; Price, 1852, 266. Thomson, [1847] 1882, 76. Ibid., 77. Cf. Stokes, [1849b] 1880; reprinted in Stokes, 1880, 75–129. Thomson, [1847] 1882, 77. Stokes to Thomson, 13 March 1847, in Wilson 1990, 1: 6 (letter no. 6). See ch. 3, § 3.3. Thomson, [1847] 1882, 80. Thompson, 1910, 1: 198. See Thomson, 1890a, 501–502; cf., 1889, 50. See also, 1890b, 442, 450, 462: §§ 14, 28, 44. Thomson, 1848, 87–89; reprinted in, 1882, 98–99. Thomson, 1851, 271, § 71, italics in the original. Thomson, 1851, 269, § 66. For Maxwell’s use of analogy between imaginary and physical domains in Sta­ tion 1, see ch. 4. Thomson, 1851, 273 n, § 74. Ibid., 273, § 74. On Rankine see, e.g., ch. 3, § 3.4. Thomson, 1856–1857, 152. Maxwell, 1873a, 220. Buchwald, 1977, 105. Buchwald, 1977, 113. Smith and Wise, 1989, 212. Maxwell, 1858, 40 and 67. See Stokes, [1851a] 1901, [1851b] 1883. Stokes, [1845] 1880, 134. Stokes, [1848] 1883, 8. Stokes, [1848] 1883, 12. Stokes, [1848] 1883, 12 Stokes, [1848] 1883, 12–13. Maxwell too refers to Green; see Maxwell, 1858, 57. Green, [1842] 1871, 245. Stokes, [1849a] 1880, 311. For Maxwell’s appeal to imagination, see ch. 1, n. 33, and the associated text. See also ch. 4, § 4.1. Stokes, [1849b] 1880, 96. Within the context of his philosophy of “As If” [Die Philosophie des Als Ob], Hans Vaihinger ([1911] 1968, 222–225) discussed, but did not fully appreciate, Maxwell’s new methodology. Harman (ed.) 1995, 2: 417 (a letter from Maxwell to Tait, dated 3 August 1868): “Rankine, 1867 in a very short statement in the Phil Mag on Conservation has expressed several things very well about energy, force and effect.” In the Treatise (Maxwell, 1873d, 2:416, § 831) Maxwell referred to Rankine’s “molecular vor­ tices,” without citing any specific work. In fact, Rankine described his hypothesis of molecular vortices in his paper of 1855. So, despite the absence of references in the Treatise to Rankine, 1855, it is clear that Maxwell was well aware of Ran­ kine’s contributions to “energetics.” Duhem ([1906/1914] 1954/1974, 52–55) referred extensively to Rankine. Cf. Moyer, 1977, 252–255. Rankine, 1855, 385–386 (section vii). Rankine, 1855, 386 (section vii), italics in the original. Rankine, 1855, 386 (section vii), small caps in the original. Rankine, 1855, 382 (section ii), small caps in the original. Rankine, 1855, 383 (section iii). Rankine, 1855, 383–384 (section iv), italics in the original.

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72 Sources of inspiration 60 Rankine, 1855, 384–385 (section v). 61 Maxwell, 1858, 27. See ch. 4, n. 7, and the associated text. As Thomson and Tait indicated (1867, Preface viii), the allure of hypothesis is not the only prob­ lem here, for “analytical subtleties” harbor their own difficulties. 62 Rankine, 1855, 390 (section ix). Thomson and Tait characterized “axiom” in the following way (1867, 178, italics in the original): “[§] 243. An Axiom is a propos­ ition, the truth of which must be admitted as soon as the terms in which it is expressed are clearly understood. . .. as the properties of matter might have been such as to render a totally different set of laws axiomatic, there laws must be con­ sidered as resting on convictions drawn from observation and experiment, not on intuitive perception.” The similarity to Rankine’s characterization is evident. 63 This characterization of two contrasting methodologies corresponds to Einstein’s distinction between “constructive theories” and what he called “principle theor­ ies.” Einstein considered these two kinds of theories independent of one another. See Einstein, [1919/1954] 1981, 223. 64 Rankine, 1855, 384 (section v). 65 Rankine, 1855, 385 (section vi). 66 Rankine, 1855, 386 (section viii), italics in the original. 67 Moyer, 1977, 255. 68 Wise, 2005, 528. 69 Thomson and Tait, 1867, Preface vi, italics in the original. As Smith (1998, 197) put it, “the authors of the Treatise clothed their radical reforms of natural phil­ osophy in a hitherto unrecognized interpretation of Newton’s Principia.” Cf. Thomson and Tait, 1867, 161, § 206. For historical perspective on the structure and content of this textbook, see Smith, 1998 (ch. 10) and Wise, 2005. Smith (ibid.) also called attention to Thomson and Tait’s search for a British pedigree for their innovations, avoiding any hint of dependence on recent continental sources. Wise (2005, 529) concurred: Thomson and Tait sought authority beyond Lagrange in “the unrivaled hero of British science, Newton.” 70 Thomson and Tait, 1867, Preface vii, italics in the original. 71 Wise, 2005, 529. See also Smith, 1998, 203, 207. 72 Thomson and Tait, 1867, 200, §§ 287, 288. For the theory of energy in physical science, see pp. 188–189, §§ 271–272. 73 Thomson and Tait, 1867, 200, § 289. 74 See Wise, 2005, 530–531. 75 Smith, 1998, 207. 76 Thomson and Tait, 1867, 206, § 293, italics in the original. 77 Thomson and Tait, 1867, 1, § 1, italics in the original. 78 Thomson and Tait, 1867, 161, § 205, italics in the original. 79 See n. 72, above. 80 Thomson and Tait, 1867, 195, § 278. 81 Thomson and Tait, 1867, 231, § 318, italics in the original. 82 Thomson and Tait, 1867, 254, § 330. 83 In 1867 Thomson and Tait did not cite Maxwell’s publications of stations 1 to 3. 84 Thomson and Tait, 1867, 311–312, § 385. 85 Thomson and Tait, 1867, 345, § 458, italics in the original. 86 Thomson and Tait, 1867, 236, § 322. 87 See nn. 77 and 78, above. 88 On being asked why there were no subsequent volumes to Thomson and Tait (1867), which they called “volume 1,” Thomson replied that the works of Rayleigh on sound, Maxwell on electrodynamics and Lamb on hydrodynamics had largely covered the subject. See Thompson, 1910, 1: 476–477. Cf. Moyer, 1977, 256.

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Station 1 (1856–1858) On Faraday’s lines of force

4.1 A novel methodology: modifying the methodology of analogy The paper, “On Faraday’s lines of force” ([1856] abstract, 1858), was based on a new approach to electromagnetism, in contrast to the dominant continental theories which were based on the traditional Newtonian concept of action at a distance.1 Station 1 calls attention right in its title to the motivating concept, Faraday’s lines of force. In some respects, Maxwell’s first contribution to the study of electromagnetism is the most revolutionary, although each of the subsequent three contributions is innovative in its own way. There is no discernible difference in the methodology outlined in the abstract of 1856 from the methodology in the paper of 1858; naturally, many more details are provided in the paper than in the abstract.2 As in the abstract, Maxwell began with a discussion of scientific methodology, which is quite rare in an essay on a technical subject. He thought deeply about the epistemic means by which scientific knowledge, specifically in physics, is generated. In this discussion he emphasized that scientific methodology can work productively in the service of scientific research. We focus our attention on Part I of the paper where Maxwell set up the analogy of an incompressible fluid with electricity and magnetism and examined its consequences. We are not concerned with the theoretical claims of Maxwell’s paper since we are principally interested in scientific methodology. Our goal, then, is to determine the precise nature of the methodology which Maxwell adopted in this paper. Maxwell stated his goal of casting Faraday’s experimental researches in electromagnetism into mathematical form in the following way: In this outline of Faraday’s electrical theories, as they appear from a mathematical point of view, I can do no more than simply state the mathematical methods by which I believe that electrical phenomena can be best comprehended and reduced to calculation, and my aim has been to present the mathematical ideas to the mind in an embodied form, as systems of lines or surfaces, and not as mere symbols, which

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74 Station 1 (1856–1858) neither convey the same ideas, nor readily adapt themselves to the phe­ nomena to be explained.3 Maxwell eloquently formulated his goal: Faraday’s experimental findings should be presented to the mind as mathematical ideas in an embodied form. What is this embodied form? A powerful methodology was required to direct this research in order to coherently represent Faraday’s experimental results in a mathematical form that was physically meaningful. In addition to providing Maxwell with a great variety of experimental results, Faraday put forward the concept of lines of force, a systematic conceptual framework which unified the rich array of his experimental findings.4 Maxwell sought to establish novel mathematical connections based on Faraday’s concept of lines of force in the expectation that these connections would be physically meaningful in ways unforeseen by Faraday. Put differently, Maxwell hazarded a guess: Faraday’s conceptual speculations, which proved successful in the formulation of a general law, may prove even more successful in reaching greater generality, when cast into conventional mathematical form. The idea is that mathematical connections will lead experimentalists to find unexpected physical connections. In Maxwell’s view, the success of this research should be measured by the richness of the physical connections that can be established mathematically: By the method which I adopt, I hope to render it evident that I am not attempting to establish any physical theory of a science in which I have hardly made a single experiment, and that the limit of my design is to shew how, by a strict application of the ideas and methods of Faraday, the connexion of the very different orders of phenomena which he has discovered may be clearly placed before the mathematical mind.5 Maxwell stated what he was not aiming to achieve, namely, establishing a physical theory in a scientific domain in which he had hardly done any experiments; rather, he was interested in exploring the possibility that Faraday’s concept of lines of force was a viable option in electromagnetism. Maxwell does not elaborate his usage of theory here, but his point of departure is the set of qualitative laws in electromagnetism established by Faraday (to be recast in mathematical terms), and not a set of abstract postulates about the nature of electromagnetism. Indeed, Maxwell never presented a set of postulates and avoided appealing to a deductive system throughout his study of electromagnetism. Following the opening methodological section, Maxwell continued with a general comment on the state of the art: electrical science is well founded; the results have been obtained analytically from experiments, and speculation has no useful role to play. What is required to be shown is the connection between the states of static and current electricity as well as that

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between attractions and inductive effects of both states. But Maxwell complained that the mathematical apparatus applied to these issues thus far had been too intricate and might hinder progress. For this reason he sought simplification and reduction of the results of previous investigations to a manageable form. Maxwell referred to the student of this physical domain and the difficulties he may encounter: In order . . . to appreciate the requirements of the science, the student must make himself familiar with a considerable body of most intricate mathematics, the mere retention of which in the memory materially interferes with further progress. The first process therefore in the effec­ tual study of the science, must be one of simplification and reduction of the results of previous investigation to a form in which the mind can grasp them.6 It is evident that Maxwell had both the student and the advanced researcher in mind, for the difficulty—as Maxwell conceived of it—applied to both of them. Maxwell then advised: We must therefore discover some method of investigation which allows the mind at every step to lay hold of a clear physical conception, with­ out being committed to any theory founded on the physical science from which that conception is borrowed, so that it is neither drawn aside from the subject in pursuit of analytical subtleties, nor carried beyond the truth by a favourite hypothesis.7 What is striking in these opening remarks is that Maxwell appealed to methodology to ease the difficulties associated with this physical domain. For now, it is sufficient to note that methodology in Maxwell’s hands offers not only formal justification and informal heuristic procedures for discovery, but didactic principles as well. Maxwell introduced his analysis by expressing concern for the precise roles of mathematical formulas and physical hypotheses. In the former case, if one depends exclusively on mathematical formulation, one may lose sight of the phenomena under consideration, whereas in the latter case one sees the phenomena through a mediator, the hypothesis, and this may bias one’s approach to the phenomena. For Maxwell a hypothesis is a claim or assumption about the nature of the micro-level underlying the phenomena from which the phenomena can be deduced. He cautioned researchers not to be lured by a “favorite hypothesis” that can lead astray, nor should they be seduced by “analytical subtleties.” He thus insisted on avoiding, on the one hand, any “favorite hypothesis” about the micro-level from which the macro-phenomena could be deduced and, on the other, any purely formal expression of “analytical

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76 Station 1 (1856–1858) subtleties” without consideration of its physical meaning. Maxwell was not against physical hypothesis nor was he against formalism; as we will see, he appealed to both in the course of his methodological odyssey. What he objected to was a one-sided approach. According to Maxwell, the proper way to proceed in the absence of a theory is to apply the methodology of physical analogy. Maxwell then addressed the role of analogy: In order to obtain physical ideas without adopting a physical theory we must make ourselves familiar with the existence of physical analogies. By a physical analogy I mean that partial similarity between the laws of one science and those of another which makes each of them illustrate the other.8 Maxwell spelled out exactly what he meant by analogy, and a physical one at that:9 the similarity in the laws of the two physical systems is partial and the analogy works both ways. For the analogy to work formally both ways it must be “strong,” which means inferences can be drawn from one domain of phenomena to the other and vice versa. As we claimed, this stands in contrast to “weak” analogy which is merely suggestive by referring informally to some other known set of phenomena. In a weak analogy inferences are not drawn and the analogy is primarily didactic, intended to persuade the reader of the plausibility of the claim. In Station 1 Maxwell made the analogy part of the argument, thus rendering it strong. However, as his studies of electromagnetism grew, he began introducing weak analogies that were not part of the argument.10 Maxwell’s profound remarks, albeit brief, on the methodology of analogy were most likely put forward in anticipation of a discussion of the accepted Newtonian theory of attraction at a distance: a theory of gravitation which had been extended by continental physicists to other physical phenomena such as electromagnetism. Maxwell praised the Newtonian theory: its mathematical concepts are well known; the formulas have mathematical significance and are amenable to reasoning; and their results have been experimentally confirmed. Indeed, no formula in the physical sciences has been as well confirmed as that of the action of bodies on one another at a distance. Maxwell then illustrated the strength of the formula, using as his example the conduction of heat in uniform media which, on the face of it, is strikingly different from the physical relations of attraction. In this physical context the principal quantities are temperature, flow of heat, and conductivity, while the concept of force seems entirely alien to this physical domain. And yet, Maxwell noted, the mathematical laws of the motion of heat in homogeneous media are identical in form with those of attractions varying inversely as the square of the distance. We have only to substitute source of heat for

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center of attractions, flow of heat for accelerating effect of attraction at any point, and temperature for potential, and the solution of a problem in attractions is transformed into that of a problem in heat.11 This is very reminiscent—indeed, almost word for word—of Thomson’s conception of analogy, though the analogous domains are not the same.12 The search for strong analogies was thus immediately successful, for Maxwell was able to cite a relevant work by Thomson (1842): “This analogy between the formulae of heat and attraction was, I believe, first pointed out by Professor William Thomson in the Camb. Math. Journal, Vol. III.”13 The principle of physical analogy between theories in two distinct physical domains came directly from Thomson. Maxwell accepted Thomson’s demonstration of the analogy between heat conduction and the theory of electricity at an early stage in his research. This is exactly what we call strong analogy. The stage was now set for the first move. Maxwell reported that he had sought “to bring before the mind, in a convenient and manageable form, those mathematical ideas which are necessary to the study of the phenomena of electricity.”14 Maxwell stated that he had adopted the “process of reasoning” applied by Faraday in his researches. He admitted that this process had been interpreted mathematically by other scholars, notably by Thomson,15 but since the process is of “an indefinite and unmathematical character” there is room for recasting it into a form familiar to professional mathematicians.16 For Maxwell the way in which Faraday made use of his idea of lines of force in co­ ordinating the phenomena of magnetoelectric induction shews him to have been in reality a mathematician of a very high order—one from whom the mathematicians of the future may derive valuable and fertile methods.17 Maxwell’s claim surely came as a surprise to his contemporaries (and it is still shocking to some modern readers), for Faraday did not provide any equations or quantitative laws. Rather, his approach is qualitative, and not what would usually be described as mathematical. Yet Maxwell was insistent on this point and probably considered himself this “mathematician of the future”; indeed, he was the mathematician who recast Faraday’s experimental results mathematically. The method of analogy which Maxwell adopted is purely instrumental and limited: by strictly applying the ideas and “processes of reasoning” of Faraday, Maxwell intended to display before the mathematical mind the connection between very different orders of phenomena that Faraday had discovered. Maxwell’s response to Faraday’s speculations was very different from his response to those of Thomson (see chs. 2 and 3). Faraday’s advocacy of the

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78 Station 1 (1856–1858) physical reality of lines of force was completely accepted by Maxwell who had, however, no supporting evidence beyond what Faraday had already presented. For Maxwell, Faraday’s lines of force were most likely to be physical, not just “representants” (as Faraday called them). In Maxwell’s view: We have not as yet come to any facts which would lead us to choose any one out of these three theories, that of lines of force, that of imagin­ ary magnetic matter, and that of induced polarity. As the theory of lines of force admits of the most precise, and at the same time least theoretic statement, we shall allow it to stand for the present.18 It then made sense to recast Faraday’s claims in mathematical form. As we have seen (ch. 1), to accomplish this task Maxwell introduced a contrived analog to electromagnetic phenomena. Maxwell could proceed with his speculations, for Faraday had provided a precedent for the value of this kind of activity. As Maxwell remarked in concluding the preamble of his paper: If the results of mere speculation which I have collected are found to be of any use to experimental philosophers, in arranging and interpreting their results, they will have served their purpose, and a mature theory, in which physical facts will be physically explained, will be formed by those who by interrogating Nature herself can obtain the only true solu­ tion of the questions which the mathematical theory suggests.19 And, again, when outlining his plan of research in the opening sections of Part II of the paper, Maxwell reflected on what he had accomplished in Part I: In the following investigation . . . the laws established by Faraday will be assumed as true, and it will be shewn that by following out his speculations other and more general laws can be deduced from them. If it should then appear that these laws, originally devised to include one set of phenomena, may be generalized so as to extend to phenomena of a different class, these mathematical connexions may suggest to physi­ cists the means of establishing physical connexions; and thus mere speculation may be turned to account in experimental science.20 Maxwell accepted Faraday’s assessment of speculation, and was convinced that the concept of curved lines of force offered a viable alternative to the dominant theories of electromagnetism, which were based on action at a distance that takes place in straight lines. But, above all, the ultimate goal was to seek unforeseen “physical connexions” that flow from “mathematical connexions” and could thus enrich the relevant experimental science. Maxwell was convinced that Faraday was right about lines of force; they are surely significant. Indeed, he was committed to this idea. Faraday was right because of his methodology, that is, by positing lines of force Faraday

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was able to discover experimentally a variety of phenomena that could then be connected to one another by a unifying concept of some generality. But Faraday’s arguments were not mathematical and had not convinced anyone to abandon “action at a distance,” which was well established and successful in a variety of domains.21 For Maxwell the problem was how to recast Faraday’s insight so that new physical connections become apparent and thus turn this approach into a competitor to “action at a distance.” To do so he needed a mathematical technique that was powerful and persuasive. He turned to Thomson’s “new” methodology of analogy which had already proven to be successful. But it was not powerful enough to suit Maxwell’s needs. To get this to work, Maxwell appealed to an analogy with an imaginary incompressible fluid (again, without inventing the concept). The methodology of analogy was a tool, not a goal. And Maxwell used it in ways that had not been explored by Thomson;22 “mathematically identical” systems need not be physically real: in a pair of such systems one of them could be imaginary and the other physically real. So the fundamental methodology was Faraday’s and the overall methodology was Thomson’s, but the insight in combining these two elements in an unforeseen way was Maxwell’s. Here we witness a radical innovation in the paper. Maxwell added “imagination” to Thomson’s methodology. His terminology reflects this modification, for Maxwell transformed Thomson’s methodology of “physical analogy” into “mathematical analogy,” based on a scheme of an imaginary incompressible fluid.23 “Imagination” is not at all a vague term in this rich methodological context; on the contrary, we take it to be a technical term. It is permissible to invent an artificial scheme whose properties are entirely at one’s disposal to serve as a vehicle for mathematical reasoning in the search for possible physical connections. The domain then could be imaginary, but the analogy—to function successfully —must be strong. It may then be put forward as a challenge to the experimentalist to test the existence of these predicted connections. While we see similarities of Maxwell’s methodology with the methodologies of Thomson and Stokes, we also explore the differences among them which are significant (see ch. 3, §§ 3.2 and 3.3). Maxwell was a great scientist as well as a great innovator of methodology;24 and these innovations set him apart from Faraday, Thomson, and Stokes. Still, he summed up the methodological preamble with no hint of his intellectual virtuosity: In treating the simpler parts of the subject I shall use Faraday’s mathematical methods as well as his ideas. When the complexity of the subject requires it, I shall use analytical notation, still confining myself to the development of ideas originated by the same philosopher.25 As we have argued, these methods are in fact methodologies. Faraday was best known at the time for his qualitative experimental findings, and so it

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80 Station 1 (1856–1858) was extraordinary for Maxwell to refer to Faraday’s methodologies as mathematical. Indeed, no one else (including Faraday himself) considered him to be a mathematician.26

4.2 The structure of Maxwell’s argument Following the methodological preamble, Maxwell formally opened the paper: “I have in the first place, to explain and illustrate the idea of ‘lines of force.’”27 Maxwell began by addressing a fundamental phenomenon in electricity and magnetism. A small positively charged body placed in the vicinity of a body electrified in any manner experiences a force urging it in some direction. When the small body is charged negatively it will experience the same force but in the opposite direction. This holds too for magnetism, where the positive and the negative charges are replaced by the north and south poles. One can find a line passing through any point of space, expressing directions for the motion of a test particle subject to electric and magnetic forces. The curve thus obtained expresses the direction of the resultant force at each point, hence the concept of “line of force.” This is Maxwell’s mathematical definition of the concept of lines of force, which contrasts with Faraday’s initial attempts to characterize this unifying concept as well as his later operational description of it.28 Of course, many other lines may be drawn, expressing the direction of the force exerted at each point in space. This is what Maxwell called “a geometrical model of the physical phenomena.”29 Maxwell’s “geometrical model” only represents the direction of the force. To make it physical a method has to be designed to indicate, in addition to direction, the intensity of the force at any point; after all, the two principal features of any physical force are its direction and intensity. Maxwell indeed italicizes both terms, namely, direction and intensity.30 In this account Maxwell followed Faraday closely.31 The geometrical model for the concept of lines of force had therefore to be modified to capture direction and intensity at the same time. We note that Thomson had already remarked that, for a scheme to be physical, it must express the intensity as well as the directionality of the force.32 Maxwell then took a step beyond Faraday’s “process of reasoning” and rethought the concept of lines of force: If we consider these curves not as mere lines, but as fine tubes of vari­ able section carrying an incompressible fluid, then, since the velocity of the fluid is inversely as the section of the tube, we may make the vel­ ocity vary according to any given law, by regulating the section of the tube, and in this way we might represent the intensity of the force as well as its direction by the motion of the fluid in these tubes. This method of representing the intensity of a force by the velocity of an imaginary fluid in a tube is applicable to any conceivable system of

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forces, but it is capable of great simplification in the case in which the forces are such as can be explained by the hypothesis of attractions varying inversely as the square of the distance, such as those observed in electrical and magnetic phenomena.33 This is an extraordinarily bold step that requires unpacking. In fact, this creative act transformed a method into a methodology. As we have seen, according to Maxwell, Faraday’s application of the concept of lines of force constituted “a method of building up an exact mental image of the thing [one is] reasoning about”;34 to be specific, in this case a magnetic phenomenon, that is, the arrangement of iron fillings, is treated spatially. But this is a method of geometry, not of physics. The “method of representing the intensity of a force,” which is physics, facilitates an effective mathematical representation of lines of force and it, in turn, functions as the vehicle of reasoning for setting up a contrived analogy. The idea of turning the lines into tubes of different cross sections, filling them with an incompressible fluid, and letting this fluid flow in the tubes at varying velocities in order to represent both the direction and intensity of a force, form the necessary requirements, according to Maxwell, for setting up the analogy with electromagnetic phenomena. By introducing this contrived analogy between imaginary physics and the physics of electromagnetism, Maxwell showed how a method could be transformed into a methodology. This is a major finding of our study. Three points regarding this bold step are worthy of consideration. First, to the best of our knowledge, after this single instance in his paper of 1858 Maxwell did not call a scheme of this kind a “model”; rather, he considered it a “method of representing” physical quantities such as the intensity of a force. Indeed, he seldom used the term “model” and never invoked the expression “geometrical model” again. Second, while the concept of an incompressible fluid was not new,35 the idea of transforming the lines of force into tubes filled with this fluid which flows at varying velocities, depending on the cross sections of the tubes, goes far beyond Faraday’s concept of lines of force. We recall that a system of tubes was put forward in Thomson’s paper of 1851 on a mathematical theory of magnetism;36 but Thomson belittled the idea, relegating it to a footnote, and ultimately discarding it.37 Maxwell, however, was of a different opinion. His objective was clear: to extend Faraday’s concept to include the intensity of the force in addition to its direction. In effect, Maxwell made two claims: (1) direction and intensity need to be considered in a physical arrangement, and (2) a specific physical construction of tubes filled with an incompressible fluid meets the requirement of illustrating at one and the same time both intensity and direction. Third, Maxwell stressed the generality of the methodology he devised, for it can represent any kind of force. In particular, it does not exclude the attracting force of action at a distance which varies inversely as the square

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82 Station 1 (1856–1858) of the distance, as observed in electric and magnetic phenomena. This is a significant remark which is probably intended to undermine possible objections that, in principle, the methodology excludes the dominant theory based on action at a distance. No, it can represent that theory too, but Maxwell was not interested in pursuing this option. Rather, he stressed the characteristics of the forces related to those of electricity and magnetism: in the general case, there are interstices between the tubes; but in the case of electric and magnetic forces the tubes may be so arranged that there will be no interstices. Under these circumstances the space will be filled with the tubes and they will turn into surfaces, “directing,” as Maxwell put it, “the motion of a fluid filling up the whole space.”38 Given that the forces are electric and magnetic, one typically regards them as attractive and repulsive between certain points. But Maxwell simplified the mathematical analysis without loss of generality by representing them, both in magnitude and direction, “by the uniform motion of an incompressible fluid.”39 Maxwell then listed the steps to come: I propose . . . first to describe a method by which the motions of such a fluid can be clearly conceived; secondly to trace the consequences of assuming certain conditions of motion, and to point out the application of the method to some of the less complicated phenomena of electricity, magnetism, and galvanism; and lastly to shew how by an extension of these methods, and the introduction of another idea due to Faraday, the laws of attractions and inductive actions of magnets and currents may be clearly conceived, without making any assumptions as to the physical nature of electricity, or adding anything to that which has been already proved by experiment.40 The goals are (1) to offer a mathematical method for describing the motion of the fluid in question; (2) to draw consequences from some specific conditions imposed on the motion of the fluid and then, by analogy, apply these results to the less complicated phenomena of electricity, magnetism, and galvanism; and (3) to extend these methods to the laws of attraction and induction without, most importantly, assuming anything about the nature of electricity, or adding anything hypothetical which had not already been established by experiment. The key move of this three-stage methodological sequence is a mathematical application based on strong analogy without—and this is the critical point—any ontological commitment to the nature of electricity. Indeed, later on in the paper he formulated this position explicitly: In these six laws I have endeavoured to express the idea which I believe to be the mathematical foundation of the modes of thought indicated in the Experimental Researches. I do not think that it contains even the

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shadow of a true physical theory; in fact, its chief merit as a temporary instrument of research is that it does not, even in appearance, account for anything.41 The goal, then, was not to account for the phenomena; rather, it was to cast them mathematically. Maxwell included important cautionary remarks. He did not wish to devise a theory designed to explain the cause of the phenomena, for it would be premature to do so. At the same time, he sought generality and precision in the description of the phenomena. And he concluded: By referring everything to the purely geometrical idea of the motion of an imaginary fluid, I hope to attain generality and precision, and to avoid the dangers arising from a premature theory professing to explain the cause of the phenomena.42 Maxwell’s choice was clear: he sought a mathematical (geometrical) theory of generality and precision without a commitment to any causal explanation. He ended with a remark similar to Faraday’s in 1852, namely, speculation is useful both for experimentalists and for mathematicians. In this view, speculation can give definite shape to vague ideas and ultimately lead to physical truths.43 Maxwell now got to the heart of the matter: the body of the paper consists of two parts: (I) “Theory of the motion of an incompressible fluid,” and (II) “On Faraday’s ‘Electro-tonic state.’” But, as indicated above, we focus mainly on Part I, which concerns the methodology and its application. Maxwell proceeded to stipulate the properties of the imaginary fluid: The substance here treated of must not be assumed to possess any of the properties of ordinary fluids except those of freedom of motion and resistance to compression. It is not even a hypothetical fluid which is introduced to explain actual phenomena. It is merely a collection of imaginary properties which may be employed for establishing certain theorems in pure mathematics in a way more intelligible to many minds and more applicable to physical problems than that in which algebraic symbols alone are used. The use of the word “Fluid” will not lead us into error, if we remember that it denoted a purely imaginary substance . . .44 This is a rich statement: in the first place Maxwell explicitly denied that the proposed fluid is hypothetical. In his view hypotheses are put forward for the purpose of explaining phenomena, and this is not the objective of Maxwell in this paper. Maxwell therefore stressed that the “fluid” is imaginary with selected attributes. The sole reason for assigning imaginary

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84 Station 1 (1856–1858) properties is to render the mathematical treatment intuitively physical; or, in Maxwell’s striking phrase, to make the mathematics “more intelligible” than mere algebraic symbols in mathematical equations. The use of imagination will not mislead the researcher, Maxwell argued, if it is remembered that the “Fluid” denotes a purely imaginary substance with the following property: The portion of fluid which at any instant occupied a given volume, will at any succeeding instant occupy an equal volume.45 This is the property of incompressibility which furnishes, according to Maxwell, the following law: “the unit of quantity of the fluid will . . . be the unit of volume.”46 Maxwell then defined “lines of fluid motion” as those lines drawn in such a way that their direction always indicates the direction of the fluid motion. Now, if the direction and velocity of the fluid motion at any fixed point in space is independent of time, then the motion will be steady; otherwise, it will be variable. And Maxwell stipulated that his analysis presupposes steady motion. Next he defined a “tube of fluid motion” generated by lines of motion that form a tubular surface. No part of the fluid can flow across the tube, for no line of motion crosses the tubular surface which Maxwell characterized as “imaginary.” Since the fluid is incompressible and no quantity of the fluid can escape through the side of the tube, Maxwell could deduce a conservation law: the same quantity which crosses a section of the tube in a unit of time will cross a subsequent section. Again, due to the property of incompressibility, a unit volume which passes through a section of the tube in a unit of time, can be equated with a “unit tube of fluid motion.” Maxwell proceeded to define the motion of the whole fluid by means of a system of unit tubes. The system thus obtained is a “geometrical construction which completely defines the motion of the fluid.”47 Maxwell described these unit tubes as follows: A unit tube may either return into itself, or may begin and end at dif­ ferent points, and these may be either in the boundary of the space in which we investigate the motion, or within that space. In the first case there is a continual circulation of fluid in the tube, in the second the fluid enters at one end and flows out at the other. If the extremities of the tube are in the bounding surface, the fluid may be supposed to be continually supplied from without from an unknown source, and to flow out at the other into an unknown reservoir; but if the origin of the tube or its termination be within the space under consideration, then we must conceive the fluid to be supplied by a source within that space, capable of creating and emitting unity of fluid in unity of time, and to be afterwards swallowed up by a sink capable of receiving and destroy­ ing the same amount continually.48

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Maxwell saw no contradiction in the characterization of these two modes: The properties of the fluid are at our disposal, we have made it incom­ pressible, and now we suppose it produced from nothing at certain points and reduced to nothing at others.49 In fact, Maxwell characterized a sink as a source with a negative sign. The phrase “at our disposal” is the key to the way Maxwell modified Thomson’s methodology of analogy.50 This move is extraordinary. Maxwell created his own artificial physics to suit the purpose he had set himself. Indeed, later in the paper he made the following revealing remark: We may conceive of the electro-tonic state at any point of space as a quantity determinate in magnitude and direction, and we may represent the electro-tonic condition of a portion of space by any mechanical system which has at every point some quantity, which may be a velocity, a displacement, or a force, whose direction and magnitude correspond to those of the supposed electro-tonic state. This representation involves no physical theory, it is only a kind of artificial notation.51 Note that the mechanical system representing the electromagnetic phenomenon is not unique; it has only to fulfill certain physico-spatial requirements—no theory is involved. It is striking that the whole methodological move is based on “artificial notation.” Evidently, the analogy is no longer between two well defined physical domains as was the case with Thomson; rather, Maxwell constructed an analogy (he did not find it; he invented it) between a rich, physical domain of experimental physics and a scheme of an imaginary fluid whose properties are determined at the will of the physicist. It is worth noting that Maxwell did not idealize the physics at stake as Stokes did;52 instead, he stipulated it.53

4.3 From the general to the specific With the theory of the motion of an incompressible fluid in place, Maxwell proceeded to Section 2 of Part I, where he developed what he called a theory of the uniform motion of an imponderable incompressible fluid through a resisting medium.54 The move is from the most general case to a specific one, where resistance is involved. The fluid is supposed to have no inertia and its motion is opposed by a force due to the resistance of the medium through which the fluid flows. Maxwell restricted the case to a uniform medium whose resistance is the same in all directions. The stipulated conditions make it possible to formulate the following law: Any portion of the fluid moving through the resisting medium is directly opposed by a retarding force proportional to its velocity.55

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86 Station 1 (1856–1858) Maxwell noted that the properties of this fluid are only those which are formally assigned to it: it has neither inertia nor mass. Thus resistance and pressure are the only features to be considered.56 To make the physical conditions more realistic, Maxwell considered the property of resistance anisotropy and formulated a system of linear equations, that is, the “equations of conduction.”57 The incompressible fluid, which has no inertia and flows through tubes with varying resistance depending on the direction of the flow, is now ready to be applied formally in an analogy to lines of force. I have now to shew how the idea of fluid motion . . . may be modified so as to be applicable to the sciences of statical electricity, permanent mag­ netism, magnetism of induction, and uniform galvanic currents . . .58 Evidently, the scheme of an imaginary fluid in motion through tubes of various cross sections with differing resistance has to be manipulated to suit the phenomena of electromagnetism. Put differently, the active modification of the scheme amounts to building, or rather contriving, the analogy: the result is an appropriate correspondence between the scheme and the phenomena under consideration. Indeed, Maxwell explicitly referred to the analogy between electrical problems and the fluid motion, and concluded that “the lines of forces are the unit tubes of fluid motion, and they may be estimated numerically by those tubes.”59 After showing that the analogy works for statical electricity, Maxwell turned to the theory of dielectrics, which is based on another concept in Faraday’s arsenal of new ideas. As Maxwell remarked, “Faraday expresses this by the conception of one substance having a greater inductive capacity, or conducting the lines of inductive action more freely than another.”60 By modifying the resistance in different media, one can get an analog to a dielectric, rendering the conductivity stronger or weaker depending on the case. The next theme is the theory of permanent magnets. Here the idea is to rely on the fact that the laws for electricity and magnetism are “mathematically identical.” A magnet is conceived as being composed of “elementary magnetized particles” which are ordered linearly with alternating poles, north and south. The linkage may be considered a line of force and the analogy of the fluid may be applied to “illustrate” the various phenomena. The scheme of tubes consisting of cells where the sources of the fluid and its sinks alternate from cell to cell, just as north and south poles alternate in what Maxwell called “imaginary magnetic matter.”61 Maxwell then considered the theory of paramagnetic and diamagnetic induction. Here he began with Faraday’s conception of “the magnetic field” in which the phenomena of paramagnetic and diamagnetic induction may be explained by the efficiency of conduction of the lines of force in comparison with the medium.62 Maxwell developed a qualitative theory of repulsion and attraction for magnetic induction, relying on the behavior of

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the fluid in the tubes. He considered what he called three theories, namely, (i) lines of force, (ii) imaginary magnetic matter, and (iii) induced polarity, and explicitly chose the first alternative, for it is the most precise and the least “theoretic.”63 He remarked that “the theory of lines of force is capable of the most perfect adaptation to this class of phenomena.” And he added that in this case the principles of induced polarity and (Thomson’s) imaginary magnetic matter are “of little use.”64 The next topic was “the theory of the conduction of current electricity.”65 Maxwell pointed out that in calculating the laws of constant electric currents the fluid motion theory works best. He also referred to the theories of Georg Ohm (1789–1854), Gustav Kirchhoff (1824–1887), and Georg H. Quincke (1834–1924).66 Maxwell asserted that, according to the received opinions, it is assumed that a current of fluid moving uniformly in conducting circuits must overcome an opposing resistance by exploiting an electro-motive force that has to be provided at some part of the circuit. Given the current, the pressure has to vary through the circuit. This pressure—electrical tension—is in fact physically identical with the potential in statical electricity; hence, it offers a means of connecting the two sets of phenomena, namely, electric current and electro-motive force. Thus, Maxwell’s confidence in the scheme of imaginary fluid increased; as he noted, “the analogy between statical electricity and fluid motion turns out more perfect than we might have supposed.”67

4.4 Confronting Ampère’s theory Maxwell arrived at a critical juncture, the action of closed currents at a distance. He remarked that the mathematical description of attractions and repulsions of conductors was the subject of successful studies by Ampère, and the laws he stated have been confirmed experimentally (see ch. 2, § 2.2). The challenge was, therefore, how to account for this success in terms of Maxwell’s novel approach? Maxwell proceeded to explain Ampère’s approach, praising it while at the same time revealing its limitations and pointing toward the general case. According to Maxwell, a single assumption underlies the theory, namely, the action of an element of one current upon the element of another current is an attractive or repulsive force, acting in the direction of the line joining the two elements. This well-formulated claim, established experimentally, has, however—so Maxwell argued—only been demonstrated for closed currents, either in conductors or in fluids. Since Ampère’s law may not cover the general case, Maxwell wished to reconsider the issue without the limiting conditions: “it will be more conducive to freedom of investigation if we endeavour to do without it [Ampère’s assumption].”68 Maxwell elaborated the various aspects of the law and highlights the mutual relation between electricity and magnetism: If we examine the lines of magnetic force produced by a closed current, we shall find that they form closed curves passing round the current and

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88 Station 1 (1856–1858) embracing it, and that the total intensity of the magnetized force all along the closed line of force depends on the quantity of the electric current only.69 Given the phenomena, the conceptual analysis, and the contrived analogy, Maxwell could now offer a technical recasting of Ampère’s law: The number of unit lines* of magnetic force due to a closed current depends on the form as well as the quantity of the current, but the number of unit cells† in each complete line of force is measured simply by the number of unit currents which embrace it. The unit cells in this case are portions of space in which unit of magnetic quantity is pro­ duced by unity of magnetizing force. The length of a cell is therefore inversely as the intensity of the magnetizing force, and its section inversely as the quantity of magnetic induction at the point.70 [Footnotes in Maxwell, 1858, 49] * [Faraday,] Exp. Res. (3122). See Art. (6) of this paper. † Art. (13) Evidently, the concept of lines of force in the analogical guise of tubes and fluid flowing in them offered Maxwell the imagery with which he could approach Ampère’s law in a novel way. And he formulated the following rule: “the whole number of cells due to a given current is therefore proportional to the strength of the current multiplied by the number of lines of force which pass through it.”71 This principle, according to Maxwell, lies at the root of Ampère’s law, especially in the case of the action between two currents. Maxwell then concluded: “All the actions of closed conductors on each other may be deduced from this principle.”72 Appealing now to his scheme of imaginary tubes filled with an incompressible fluid, Maxwell was able to dress up Ampère’s law in new clothes and obtain an insight which the continental physicists had not reached, for they had not examined the general case. In other words, Maxwell implied that there is no need to invoke action at a distance, and that the concept of lines of force is sufficient in this context. To be sure, this is not a critique of action at a distance, although Maxwell claimed that Ampère’s law does not cover all cases. This is a strong, positive claim for lines of force: using the concept of lines of force Ampère’s law becomes an instance of a general law. In sum, Maxwell’s innovative methodology worked: he was able to confirm previous results and then to extend them.

4.5 Conclusion In concluding the first part of Station 1, Maxwell explicitly declared that his intention was to “discover a method”—and we take “method” here to mean

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“methodology.”73 Maxwell clearly indicated that he sought a methodology based on a thorough analysis of the mechanical phenomena of elasticity in solids and viscous fluids. In this methodology the physical (expressed verbally) precedes the mathematical (expressed in symbols). This is a critical element of our argument and worthy of amplification. Maxwell “hope[d] to discover” a methodology with which he could take advantage of the work he had already done on elasticity of solids and viscous fluids in 1853.74 However, in 1858, in Station 1, he did not find such a methodology, even though he was fully aware of its presumed significance for the description of the luminiferous medium. Maxwell’s “hope” came to naught, for apparently he did not yet see a connection between elasticity and electromagnetic phenomena. Still, he was on the lookout. Evidently, Maxwell considered methodology an engine of research, a conception of scientific practice which was vindicated, we will see, when Maxwell moved on to Station 2. Maxwell was concerned that the symbolic language he had introduced might be a hindrance to the reader’s understanding of the phenomena. In Part II of the paper, having established analytically a set of general theorems in preparation for calculating electric and magnetic states, he cautiously remarked: The discussion of these functions would involve us in mathematical for­ mulae, of which this paper is already too full. It is only on account of their physical importance as the mathematical expression of one of Faraday’s conjectures that I have been induced to exhibit them at all in their present form. By a more patient consideration of their relations, and with the help of those who are engaged in physical inquiries both in this subject and in others not obviously connected with it, I hope to exhibit the theory of the electro-tonic state in a form in which all its relations may be distinctly conceived without reference to analytical calculations.75 This is clearly an indication that Maxwell sought a dual approach, namely, mathematical formalism on the one hand, and physical descriptions of the phenomena of electromagnetism on the other, a goal that Maxwell pursued throughout his odyssey. Given that Faraday’s concept of electro-tonic state is not sufficiently clear, whereas the continental theories of Weber, Ampère, and Neumann had been well established, Maxwell posed the following rhetorical question: What is the use then of imagining an electro-tonic state of which we have no distinctly physical conception, instead of a formula of attrac­ tion which we can readily understand?76 “A formula of attraction” is a clear reference to the Newtonian feature of action at a distance which all the continental theories had presupposed, and

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90 Station 1 (1856–1858) indeed is “readily” understood. Why then seek another, completely different, approach, which lacked a distinctly physical conception and, moreover, was dependent on one’s imagination? And Maxwell responded: I would answer, that it is a good thing to have two ways of looking at a subject, and to admit that there are two ways of looking at it. Besides, I do not think that we have any right at present to understand the action of electricity, and I hold that the chief merit of a temporary theory is, that it shall guide experiment, without impeding the progress of the true theory when it appears. There are also objections to making any ultimate forces in nature depend on the velocity of the bodies between which they act. If the forces in nature are to be reduced to forces acting between particles, the principle of the Conservation of Force requires that these forces should be in the line joining the particles and functions of the distance only. The experiments of M. Weber on the reverse polar­ ity of diamagnetics, which have been recently repeated by Professor Tyndall, establish a fact which is equally a consequence of M. Weber’s theory of electricity and of the theory of lines of force.77 Maxwell’s reply to his rhetorical question is methodological as well as epistemological. Scientific knowledge may be formulated in different ways; in fact, Maxwell held that when an idea is presented to the mind in multiple ways, it may quickly solidify and become “impregnable.”78 Moreover, knowledge of electromagnetism was not well understood at the time and objections could be made against the dominant view that was based on the concept of action at a distance. There were then advantages in developing a theory based on the concept of lines of force as long as it led to productive experiments and did not impede the development of what Maxwell called “true theory.” All in all, we observe in Station 1 a process of construction: unlike Faraday, Maxwell had full confidence in the concept of lines of force as the appropriate concept for grasping electromagnetic phenomena; and, unlike Thomson, Maxwell described an artifact—an imaginary scheme—which he set into an analogical relation with newly discovered electromagnetic phenomena. It is important then to note a shift away from the approaches of Faraday and Thomson. At the outset of Station 1 Maxwell reflected on the difficulty in approaching the subject afresh: the domain is “peculiarly unfavourable to speculation.” He then proceeded to describe the state of “electrical science” (as he termed it): despite many partial successes, the subject as a whole was not coherent and it was difficult to comprehend “a considerable body of most intricate mathematics, the mere retention of which in the memory materially interferes with further progress.” Maxwell called for “simplification and the reduction of the results of previous investigation to a form in which the mind can grasp them.”79 This meant that for Maxwell

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a new methodology was required to achieve his goal; he already had it in mind when he reported his preliminary results but, this time, in the published paper, it was fleshed out. Maxwell drew on the works of two principal predecessors: (1) Faraday’s Experimental Researches and his essay of 1852 on the physical reality of magnetic lines of force;80 and (2) the early papers of Thomson on electromagnetism which depended on the methodology of formal analogy and the concept of “imaginary magnetic matter.”81 Maxwell adapted mathematical formalism cautiously to serve the purpose of recasting Faraday’s results into symbolic form. For methodology Maxwell built on Thomson’s construction of formal and, what we call “strong,” analogies in which the mathematical identity of the laws, or equations, in two (or more) different physical domains are recognized, and inferences in one domain can be turned into inferences in another domain.82 Maxwell, however, applied Thomson’s methodology of formal analogy with a twist. This is a critical step in our argument. Maxwell, we argue, appealed to imaginary physics: the analog to electromagnetism is a system of an imaginary incompressible fluid, where “imaginary” is not to be conflated with “ideal” (see ch. 3, § 3.3). Maxwell constructed an imaginary physical system, contrived solely for the purpose of developing a mathematical scheme applicable to a specific physical domain. He could then draw consequences from this imaginary system to the physical domain of electromagnetism that was rich in experimental results. This strong analogy comes complete with a new feature. As Maxwell put it, “the mathematical ideas obtained from the fluid are . . . applied to various parts of electrical science.”83 This is a clear statement of unidirectionality—a novel aspect in the methodology of strong analogy, the result of modifying Thomson's two-way strong analogy. For Maxwell’s predecessors the role of an analogy was to go from the known (or better known) to the unknown (or partially known). But this was not Maxwell’s goal, for what is known about the imaginary system is not known from phenomena; rather, it is contrived. The “known” in an analogy (in physics) refers to what is known about some physical system. Maxwell did not suggest that his imaginary system will be found in nature at some time, for it was contrived for a specific purpose and was not intended to imitate nature. In a “normal” analogy the two systems have the same ontological status, but for Maxwell the ontological status of the imaginary system is not the same as the physical system. In sum, Maxwell set up a unidirectional relation from an imaginary system of tubes containing incompressible fluid to the physical system on which Faraday had experimented; this methodology was entirely new. We take the outline of Maxwell’s methodology in Station 1 to be the following: 1

The goal: Show that two competing conceptual frameworks of physical domain A are equally successful.

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92 Station 1 (1856–1858) (1.1) Specifically, show that Faraday’s concept of lines of force, recast into mathematical formalism, is a competitor to the concept of action at a distance. 2

The means: Modify the existing methodology of analogy to suit the goal. (2.1) Specifically, instead of using the analogy with some physical domain B, appeal to mathematical analogy with a completely imaginary physical situation: turn the lines of force into tubes “capable” of carrying fluid. (2.2) Draw consequences from some specific conditions imposed on the imaginary substance as it flows in the tubes, and apply, by analogy, the results to domain A without assuming anything about its nature.

Maxwell extended physics into an imaginary realm where he could control the physics as he wished with the goal of recasting Faraday’s experimental discoveries into mathematical formalism. Moving into the imaginary realm meant that Maxwell ruled out explaining the phenomena. He was fully aware of the limitations of his ingenious methodology and stated that he did not seek explanation of the phenomena. Maxwell’s innovation was to appeal to the power of imagination, and its range was not limited to known physical systems. We argue, then, that Maxwell harnessed a carefully constructed imaginary system to facilitate the recasting Faraday’s discoveries in electromagnetism in standard mathematical language. This new methodology is one of the tools that gave Maxwell the ability to go beyond his predecessors, notably Thomson and Faraday. Maxwell’s contribution was a novel methodological development with farreaching consequences. The unidirectional relation from an imaginary system to a physical one contrasts with Thomson’s methodology of strong analogy. The essence of analogy is proportionality and, by its nature, it is bidirectional. However, Maxwell did something else and it was unprecedented. To be sure, he built on Thomson’s work, but what he did was not at all what Thomson had done. Again, what Thomson accomplished is incontrovertible for, once he pointed to the mathematical analogy of the equations in distinct physical domains, there is no denying it. But the use of an imaginary system is entirely different: there are no underlying “facts” in this system to be taken into account; rather, everything would seem to depend on the effectiveness of the “analogy” either for accounting for data already known, or for correctly predicting new phenomena.84 The imaginary system is “instrumental”; it has no intrinsic interest. This was not the way physics had been practiced before Maxwell. Rather than introducing a new term for his novel methodology, Maxwell expanded the meaning of “analogy” to include this unidirectional relation. He thus made the direction of the strong analogy dependent on the interest of the researcher. Maxwell conjured up an analog for electromagnetic phenomena in “a very contrived

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way,” as North put it,85 and could thereby apply the mathematics he had developed for Faraday’s experimental findings. Maxwell succeeded in achieving his goal, namely, accounting for known electromagnetic phenomena by a unifying concept—lines of force. This had not been done before. But without his new methodology, Maxwell’s paper of 1858 falls apart: there is no content independent of this methodology. The data at Maxwell’s disposal were available to every other physicist at the time, but appealing to an imaginary system was not the way physicists proceeded. Maxwell gave the impression that he was simply applying Thomson’s methodology—he did not stress the novelty of his methodology. We however identify it as a new methodology even though Maxwell did not make this claim. Maxwell’s success in this paper surely gave him confidence but, as far as we can tell, Maxwell’s innovative approach was not appreciated by other physicists at the time.86 Maxwell raised a question that was his alone; it was not a burning issue for the “community,” namely, Can one represent electromagnetic phenomena mathematically based on Faraday’s concept of lines of force? Disregarding the means for answering the question (i.e., the methodology), Maxwell’s answer is “yes”; in fact, the result is as good as that reached by means of action a distance. But this was certainly not a reason to abandon action at a distance since the new alternative was no better (or so it seemed). Moreover, Maxwell did not introduce any new discoveries, and it was not at all clear how one could build on Maxwell’s work in Station 1 to find new phenomena in electromagnetism or how to apply the new methodology in other domains. One of the key questions at the time was, What is electricity? But Maxwell avoided this issue entirely in Station 1; indeed, he strongly objected to any explanatory move concerning the phenomena. Station 1 was critical for demonstrating to himself that his approach, based on Faraday’s concept of lines of force, is viable. But this was of no great interest to his contemporaries who were ardent supporters of the competing concept, action at a distance. In the next chapter we turn to Station 2 where Maxwell discarded the appeal to formal analogy between physical systems. Indeed, he set aside his physical analogy, and opted for another methodology altogether. However, despite the change in methodology, he maintained his commitment to the concept of lines of force.

4.6 Appendix: a bibliographical note on Maxwell 1858 Maxwell’s paper at the center of our discussion in this chapter, “On Faraday’s lines of force” (Maxwell 1858), was read at meetings of the Cambridge Philosophical Society on 10 December 1855 and 11 February 1856, and these are often the only dates given in citations of it: shortly after it was read, an abstract of it appeared, which we designate “Maxwell, 1856.” The paper was first published in 1858 in Transactions

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94 Station 1 (1856–1858) of the Cambridge Philosophical Society 10, Part I, pp. 27–83, and then bound with vol. 10, Part II, which appeared in 1864, with an added title page for the entire volume. However, many historians and philosophers of science refer to Maxwell, 1858 as Maxwell, 1856. We have found internal evidence, previously unnoticed, that Maxwell modified his paper after it was read in February 1856. Maxwell, 1858, 46, alludes to a paper by Quincke, only citing Verdet’s French summary that appeared in June 1856. Maxwell offered an English version of the French title, without mentioning the date of Quincke’s German original that appeared in March 1856 (according to Verdet, 1856, 203 n. 1). Maxwell may have made other modifications as well. Maxwell’s correspondence allows us to follow the progress of this essay between 1855 and the official publication date of 1858. In a letter to his father, dated 24 September 1855, Maxwell reported “I am getting my electrical mathematics in shape, and I see through some parts which were rather hazy before; but I do not find very much time for it at present . . .” And on 3 December 1855 he added “I have also to get ready a paper on Faraday’s Lines of Force for next Monday.” Then, on 12 March 1856, Maxwell reported to his father: I have just written out an abstract of the second part of my paper on Faraday’s Lines of Force. I hope soon to write properly the paper of which it is an abstract. It is four weeks since I read it. I have done nothing in that way this term, but I am just beginning to feel the elec­ trical state come on again, and I hope to work it up well next term.87 On 3 December 1856 Maxwell wrote to Tait, “Faraday’s lines of force is in a state of proof. I shall send you a copy”; and on 15 February 1857 Maxwell wrote again to Tait “The analogical argument on Faradays Lines of Force is in the Press & will shortly be published.”88 On 7 November 1857 Faraday wrote to Maxwell acknowledging receipt of a few papers that Maxwell had sent him, and on 9 November Maxwell responded, explaining his view on lines of force, especially as applied to gravitation.89 On 14 November Maxwell wrote to Thomson “I have just had correspondence with Faraday on the ‘Lines of Force’ as applied to gravitation. What a painful amount of modesty he has when he talks about things which may possibly be of a mathematical cast.”90 In sum, it seems that Maxwell only completed the abstract of this paper in early March 1856 at which time the full paper had not yet been written, and that he had proofs of the full paper in December 1856. While it is true that many papers in the nineteenth century were distributed as preprints to the author’s close friends and associates, in the case of Maxwell, 1858 the evidence suggests that preprints only began to circulate toward the end of 1857.

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Notes 1 See, e.g., Darrigol, [2000] 2002 and, for action at a distance in electrodynamics, see Woodruff 1962. For the publication history of Maxwell’s first paper in elec­ tromagnetism, see ch. 4, § 4.6. 2 For an analysis of the abstract (Maxwell, 1856), see ch. 1, § 1.3.

3 Maxwell, 1858, 51.

4 Faraday, 1852a; for Faraday’s view and his influence on Maxwell, see ch. 2,

§ 2.1. 5 Maxwell, 1858, 29. 6 Maxwell, 1858, 27. 7 Maxwell, 1858, 27. 8 Maxwell, 1858, 28. See also Turner, 1955. 9 Interestingly, Maxwell’s expression “physical analogy” in 1858 contrasts with his earlier expression “mathematical analogy” used in the abstract of 1856 for

describing his methodology: see ch. 1, nn. 31 and 35.

10 For the distinction strong vs. weak analogy, see ch. 1, § 1.5; for examples of

weak analogies in Station 4, see, e.g., ch. 7, nn. 98, 99 and 105.

11 Maxwell, 1858, 28, italics in the original.

12 See ch. 3, § 3.2 and especially n. 8.

13 Maxwell, 1858, 28. Cf. Thomson, 1842. See also ch. 3, § 3.2.

14 Maxwell, 1858, 29.

15 Cf., e.g., Buchwald, 1977. 16 Maxwell’s distinction between the “unmathematical character” of Faraday’s pro­ cesses of reasoning and those of the “professed mathematicians” echoes the dis­ tinction which Faraday himself drew between the experimentalist and the mathematician, although he advised both of them not to shy away from specula­ tion. See ch. 2, § 2.1, n. 45, and Faraday, [1857/1858] 1859, 458–459. 17 Maxwell, 1873c, 398–399. 18 Maxwell, 1858, 45. 19 Maxwell, 1858, 30. 20 Maxwell, 1858, 52. 21 See, e.g., Thomson, [1854b] 1872, 37; cf. Smith and Wise, 1989, 218. 22 Thomson, [1854b] 1872, 28. 23 See Maxwell, 1856, 316. 24 See, e.g., Achinstein, 1964, 1987, and 2010; Hesse, 1973. 25 Maxwell, 1858, 29. 26 For Maxwell’s use of the term method, see ch. 1, § 1.1. For discussion of Fara­ day as a mathematician, see ch. 2, § 2.1. 27 Maxwell, 1858, 29. 28 See, e.g., Faraday, 1838, 20, § 1231; and Faraday, 1852b, 26–27, §§ 3075, 3076. 29 Maxwell, 1858, 30. For a detailed discussion of model and modeling, see ch. 8, § 8.3.6.

30 Maxwell, 1858, 30. Maxwell has “intensity” where the sense is the magnitude of

the force.

31 E.g., Faraday, 1832b, espec. pp. 177ff.

32 Thomson, 1851, 273 n, § 74. See also ch. 3, § 3.2, n. 33.

33 Maxwell, 1858, 30.

34 Maxwell, 1873c, 398. See also ch. 2, n. 9.

35 A fluid which does not reduce its volume with increased pressure is incompress­ ible. See the discussion of “incompressible elastic solids” in Maxwell, 1853. The idea was already in the physics literature in the first half of the nineteenth century. 36 Thomson, 1851, 273 n, § 74. See ch. 3, nn. 33 and 34.

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96 Station 1 (1856–1858) 37 For discussion of Thomson, 1851, see ch. 3, § 3.2, especially nn. 32 and 33. Maxwell (1858, 44 n) cites this paper of Thomson. 38 Maxwell, 1858, 30. 39 Maxwell, 1858, 30. 40 Maxwell, 1858, 30. 41 Maxwell, 1858, 66, italics in the original; see also n. 58, below. 42 Maxwell, 1858, 30. 43 See ch. 2, § 2.1. 44 Maxwell, 1858, 30–31. 45 Maxwell, 1858, 31, italics in the original. 46 Maxwell, 1858, 31. 47 Maxwell, 1858, 32. 48 Maxwell, 1858, 32, italics in the original. 49 Maxwell, 1858, 32. 50 For Thomson’s methodology, see ch. 3, § 3.2. 51 Maxwell, 1858, 65. 52 For discussion of Stokes’s idealization, see ch. 3, § 3.3. 53 In his paper on similarity and analogy Mach (1902, 14) claimed that Maxwell had idealized the properties of the incompressible fluid. Mach thus missed Max­ well’s innovative move. On the methodology of idealization, see the discussion of Stokes in ch. 3, § 3.3. 54 Maxwell, 1858, 33, italics in the original. It is noteworthy that Maxwell modified the “incompressible fluid” with the term “imponderable.” This adjective is typic­ ally associated with an all-pervading fluid such as the ether whose properties diverge from the conventional properties of fluid—a feature that leaves much room for the imagination. This is the only instance of this term in Station 1. 55 Maxwell, 1858, 34, italics in the original. 56 Maxwell, 1858, 34–35. 57 Maxwell, 1858, 39, § 28. 58 Maxwell, 1858, 42. 59 Maxwell, 1858, 43. 60 Maxwell, 1858, 43–44, italics in the original. 61 Maxwell (1858, 44 n) acknowledged that the expression, “imaginary magnetic matter,” comes from Thomson, 1851. For discussion of this concept, see ch. 3, § 3.2, n. 30. 62 Maxwell, 1858, 44–45. 63 Maxwell, 1858, 45. 64 Maxwell, 1858, 46. 65 Maxwell, 1858, 46. 66 Maxwell, 1858, 46. Maxwell does not specify any particular work by Ohm, but he may have had in mind Ohm’s principal contribution to electromagnetic theory, Ohm, 1827. On the reception of Ohm in England, see Heidelberger, 2010, 257 n. 4. Maxwell cited “Kirchhoff, Ann de Chim. xli. 496” which, in fact, refers to Verdet, 1854, a summary in French of an article by Kirchhoff in 1849. Maxwell also cited “Quincke, [Ann. de Chim.] xlvii. 203,” which is a reference to Verdet, 1856, a summary in French of an article by Quincke that appeared earl­ ier in 1856 (for the significance of Verdet’s paper in dating Maxwell’s work, see ch. 4, § 4.6). 67 Maxwell, 1858, 46. 68 Maxwell, 1858, 48. 69 Maxwell, 1858, 49, italics in the original; cf. Wise, 1979, 1314. 70 Maxwell, 1858, 49. 71 Maxwell, 1858, 49.

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72 Maxwell, 1858, 50. 73 Maxwell, 1858, 51. For discussion of the distinction between method and meth­ odology, see ch. 1, § 1.1. 74 In his “On the equilibrium of elastic solids” Maxwell developed his own approach to elasticity based principally on Stokes’s analysis. In Note B of the paper; Maxwell (1853) made it clear that he did not deduce the fundamental equations of his paper from the laws of molecular action at the micro-level; rather, the laws were deduced from experiments involving phenomena at the macro-level. This methodology was employed, as we will see, in electromagnet­ ism where Maxwell sought a “mechanical conception” of the relevant phenom­ ena. In Station 2, Part III, Maxwell reported that finally he had “discover[ed] a method of forming a mechanical conception” of electromagnetic phenomena, based on his insight on the elasticity of the medium: see ch. 5, n. 60. 75 Maxwell, 1858, 65. 76 Maxwell, 1858, 67. Maxwell revisited the concept of electrotonic state in the Treatise (1873), where he praised Faraday for the acuteness of “his mind’s eye” (Maxwell, 1873d, 2: 174, §§ 540–541). 77 Maxwell, 1858, 67, italics in the original. 78 Maxwell, 1871a, 13. See the motto of this book. 79 Maxwell, 1858, 27. 80 Faraday, 1852a. See also ch. 2, § 2.1. 81 For a few selected references, see Maxwell, 1856, and 1858, 29, 44, 51, and 67. See also ch. 3, § 3.2. 82 For “strong” analogy, see ch. 1, § 1.5. 83 Maxwell, 1856, 404. 84 See n. 20, above. 85 North, 1981, 129. 86 For a German response, see ch. 1, nn. 37 and 38. We discuss possible reasons for the shift in methodology from Station 1 to Station 2 in ch. 8, § 8.2.5. 87 Campbell and Garnett, 1882, 216, 221, 252–253. 88 Harman (ed.) 1990, 481, 495. 89 Campbell and Garnett, 1882, 288; Harman (ed.) 1990, 548–552. Also, on 7 November 1857 Maxwell received a letter from Tyndall, Professor of Physics at the Royal Institution, London, in which Tyndall acknowledged receipt of some of Maxwell’s papers, including the paper on Faraday’s lines of force (Campbell and Garnett, 1882, 288). 90 Harman (ed.) 1990, 556.

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Station 2 (1861–1862) On physical lines of force

5.1 Introduction It is evident that in 1861–1862, as in 1856–1858, Maxwell made a conscious decision to begin with a methodological discussion. What methodology should be applied for progress to be made in recasting electromagnetism into symbolic language and, at the same time, maintaining a firm grasp of the relevant phenomena? Maxwell was committed to Faraday’s concept of lines of force rather than to the Newtonian action at a distance: he noted, as we have seen, the “beautiful illustration” afforded by iron filings strewn on paper in the presence of magnetic force which “tends to make one think of the lines of force as something real.”1 Maxwell’s commitment to the concept of lines of force was probably a matter of deep intuition, and so the core problem was what methodology could make the concept of lines of force contend successfully with the concept of action at a distance? Moreover, this methodology had to facilitate analysis in symbolic terms, that is, a formal representation. Maxwell’s solution in Station 2 was a hypothesis at the micro-level. To be sure, there were other hypotheses that offered insights into the experimental laws of electromagnetism, but what is new in Station 2 is the introduction of a hypothesis that links the relevant phenomena and its experimental laws to the action of a medium via a set of mechanical assumptions. No previous analysis invoked a medium to explain electromagnetic phenomena. In fact, in Station 2 of Maxwell’s methodological odyssey consequences drawn from mechanical assumptions at the micro-level offer a causal explanation of electromagnetic phenomena at the macro-level. The key term then in Station 2 is “hypothesis,” a methodology that serves to link electromagnetic phenomena to a medium.

5.2 Preliminary: from instrumentalism to causality Maxwell’s paper, “On physical lines of force,” consists of four parts: Part I was published in March 1861; Part II in April and May 1861; and Parts III and IV in January 1862. There was a delay in publication between Part II and Part III and, as we will see, Maxwell changed his mind in this interval:

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indeed, he made major discoveries at this time, which led to new perspectives on the nature of electromagnetism. In Station 2 Maxwell adopted the expression “mechanical illustration” which is reminiscent of Stokes’s appeal to illustration.2 Maxwell invoked “mechanical illustrations” replacing the expression “geometrical model” which he had used in Station 1.3 Whereas the contrived analogy which Maxwell had developed in Station 1 was merely instrumental in showing how the theory of electromagnetism could be based on the concept of lines of force, in Station 2 the goal was to account for the phenomena; Maxwell now sought a mechanical hypothesis to explain the phenomena by deducing them from the hypothesis. The change in the underlying metaphysics is dramatic: Maxwell discarded the instrumental approach; instead, he appealed to a hypothetical ontology that supports causal argumentation with the goal of explaining the phenomena. Maxwell reiterated his dissatisfaction with “the explanation founded on the hypothesis of attractive and repellent forces.” And he continued: My object in this paper is to clear the way for speculation in this direction [relating phenomena to the medium], by investigating the mechanical results of certain states of tension and motion in a medium, and comparing these with the observed phenomena of mag­ netism and electricity. By pointing out the mechanical consequences of a new hypothesis, Maxwell wished to be of some use to those who consider the phenomena as due to the action of a medium, but are in doubt as to the relation of this hypothesis to the experimental laws already established, which have generally been expressed in the language of other hypotheses.4 By “other hypotheses” Maxwell was undoubtedly referring to action at a distance which he in fact rejected. Maxwell’s goal in introducing his hypothesis is stated succinctly: to offer an alternative to the Newtonian methodology. Maxwell now distinguished Station 2 from Station 1. I have in a former paper [(1858), Station 1] endeavoured to lay before the mind of the geometer a clear conception of the relation of the lines of force to the space in which they are traced. By making use of the conception of currents in a fluid, I shewed how to draw lines of force . . . In the same paper I have found the geometrical significance of the “Electrotonic State” and have shewn how to deduce the mathemat­ ical relations between the electrotonic state, magnetism, electric cur­ rents, and the electromotive force, using mechanical illustrations to assist the imagination, but not to account for the phenomena.5

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100 Station 2 (1861–1862) Maxwell informed the reader that in Station 1 he had not sought to account for the phenomena, for he only formulated mathematical relations among various aspects of electromagnetism. Mechanical illustrations were introduced therefore merely to assist the imagination. He then remarked in Station 2: I propose now to examine magnetic phenomena from a mechanical point of view, and to determine what tensions in, or motions of, a medium are capable of producing the mechanical phenomena observed.6 Note the striking shift from Station 1. The term “producing” indicates that the argument concerns causes and is no longer based solely on analogy. Undoubtedly, the shift from analogy to causes is significant. While in Station 1 the illustrations had been mechanical, albeit imaginary, in Station 2 Maxwell took a “mechanical point of view” for “mechanical phenomena.” This means that in Station 2 Maxwell sought a way to account for, e.g., magnetic phenomena, mechanically. The mechanism was not intended merely as an illustration; rather, it was assumed to be ontologically viable. In other words, in Station 2 Maxwell offered a plausibility argument. Previously, in Station 1, Maxwell had modified the methodology of analogy by an appeal to imaginary physics which was entirely at his disposal to facilitate recasting Faraday’s experimental findings into symbolic form; here, however, in Station 2, he sought an explanation of the phenomena by introducing a hypothesis at the micro-level that has a deductive, causal relation to the macro-level of the phenomena. One indication of this significant shift in methodological perspective is that in Station 2 Maxwell posed the ontological questions: What is the nature of electricity? Is it one substance, two substances, or not a substance at all? And, then, in what way does electricity differ from matter, and how is it connected to it?7 These fundamental questions had been avoided in Station 1 and, in response to them, Maxwell applied a different methodology from the one he had followed previously in Station 1.8

5.3 The methodology: linking hypothesis to representation At this stage of his odyssey, Maxwell proposed a remarkable hypothesis concerning the physical origin of the magnetic field: molecular vortices whose axes coincide with the lines of force.9 As Maxwell put it, at stake was the physical connection of vortices with electric currents.10 The scheme consisted of cells within which vortices rotate that were supposed to be separated by layers of particles serving the double purpose of transmitting motion from one cell to another and by their own motions constituting an electric current.11 According to Maxwell, “the phenomena of induced currents are part of the process of communicating the rotatory velocity of the vortices from one part of the field to another.”12 This is clearly a physical explanation, not a mathematical description.

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Maxwell indicated that he wished to go beyond Thomson’s “mechanical representation”: The author [Thomson] of this method of representation does not attempt to explain the origin of the observed forces by the effects due to these strains in the elastic solid, but makes use of the mathematical analogies of the two problems to assist the imagination in the study of both.13 Our discussion of Thomson’s methodology in his early papers bears out Maxwell’s view; undoubtedly, Maxwell was a master of methodological issues. Maxwell appealed to “representation,” qualified as mechanical. When he referred to the figure with the mechanical scheme of the molecular vortices,14 he used various grammatical forms of “represent.”15 In fact, he called this arrangement, a “hypothesis.” He concluded the second part of the paper by indicating explicitly that the hypothesis of molecular vortices is an imaginary device designed to imitate electromagnetic phenomena: We have now shewn in what way electro-magnetic phenomena may be imitated by an imaginary system of molecular vortices. Those who have been already inclined to adopt an hypothesis of this kind, will find here the conditions which must be fulfilled in order to give it mathematical coherence, and a comparison, so far satisfactory, between its necessary results and known facts.16 It is worth noting that Maxwell didn’t just refer to a single electromagnetic phenomenon; rather, he referred to phenomena (in the plural). In other words, the comparison between the consequences drawn from the hypothesis and the known facts is universal. The imitation offered by the hypothesis concerns “an imaginary system,” so the imitation is not (what we call) a weak analogy for didactic purposes which, as we will see, ropes and screws suggest. The term “system” stands out; it makes clear that the hypothesis is not, in fact, an analogy at all.17 The imitation is at the level of phenomena and not with respect to the laws governing the phenomena. Thus, no similarity in the mathematical structure of the laws is addressed, in contrast to what Maxwell attributed to Thomson in Station 1.18 In short, in Station 2 Maxwell moved on and opted for another methodology altogether. We note that this “imaginary system of molecular vortices” depended on mechanical devices. The system was meant to be a description of a possible mechanism, albeit with difficulties; still, it was plausible, even though no proof of its existence was available. In fact, no such proof was anticipated, but in principle the system could have functioned as described. This was not the “imagination” of 1858, where “the purely imaginary nature of the fluid” indicated that it was contrived with no claim for plausibility, let alone physical reality. Just as in 1858 Maxwell expanded the concept of analogy, in 1861 he expanded the concept of hypothesis. Put differently, the methodology

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102 Station 2 (1861–1862) of 1861 is a modified version of the traditional methodology of hypothesis. Here too Maxwell was inventive and innovative on methodological issues. In Station 1 the mechanical system is imaginary: it is contrived with no claim to reality. By contrast, in Station 2 the system is regarded as physical, although its status is somewhat ambiguous with respect to reality, for the system of molecular vortices constitutes a possible physical explanation. To be sure, the explanation is plausible, but there is no claim that any experimental evidence is available to support it. The vortices are assumed at the micro-level, and intended to be a means of explaining macro-level phenomena. Clearly, the scheme in Station 2 of molecular vortices is epistemologically and ontologically different from the scheme in Station 1. Maxwell explicitly discussed the mechanical difficulties posed by the hypothesis of molecular vortices. The difficulties had to do with representing electricity as a translational phenomenon and magnetism as a rotational one, both occurring simultaneously. These mechanical characteristics of electricity and magnetism were due, as Maxwell noted, to one of Faraday’s discoveries.19 The problem was to find a way to link a motion of translation with one of rotation. It appears from all these instances that the connexion between magnet­ ism and electricity has the same mechanical form as that between cer­ tain pairs of phenomena, of which one has a linear and the other a rotatory character . . . In this paper I have regarded magnetism as a phe­ nomenon of rotation, and electric currents as consisting of the actual translation of particles, thus assuming the inverse of the relation between the two sets of phenomena.20 And Maxwell added, “all the direct effects of any cause which is itself of longitudinal character, must be themselves longitudinal, and that the direct effects of a rotatory cause must be themselves rotatory.” Hence the problem: A motion of translation along an axis cannot produce a rotation about that axis unless it meets with some special mechanism, like that of a screw, which connects a motion in a given direction along the axis with a rotation in a given direction round it; and a motion of rotation, though it may produce tension along the axis, cannot of itself produce a current in one direction along the axis rather than the other.21 The difficulty, however, had already been addressed by the continental theories which were based on the unifying concept of action at a distance. Moreover, Thomson also held to this view, as Maxwell remarked: The theory that electric currents are linear, and magnetic forces rotatory phenomena, agrees so far with that of Ampère and Weber; and the hypothesis that the magnetic rotations exist wherever magnetic force

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extends, that the centrifugal force of these rotations accounts for magnetic attractions, and that the inertia of the vortices accounts for induced cur­ rents, is supported by the opinion of Professor W. Thomson.22 Apparently, Maxwell was strongly motivated to find an account of the translational and rotational motions on the assumption of lines of force. His goal, once again, was to show that the concept of lines of force succeeded as well as, or better than, the concept of action at a distance. As he intimated: In fact the whole theory of molecular vortices, developed in this paper has been suggested to me by observing the direction in which those investigators who study the action of media are looking for the explan­ ation of electro-magnetic phenomena.23 Maxwell explained how his hypothesis of molecular vortices works mechanically, meeting the challenge of combining two contrasting motions. For this purpose, he introduced an analogy, namely, the two simultaneous mechanical motions of a screw, translation and rotation, at the macro-level, are analogous to the function of the molecular vortices at the micro-level which, in turn, underlie the phenomena of electricity and magnetism, respectively. The analogy is only suggestive and therefore it is a weak analogy; consistency here is not required. Still, the use of hypothesis in Station 2 required consistency since with this methodology conclusions are deduced in a logical structure. Again, as we have argued, this methodological approach contrasts with the methodology of Station 1, as Maxwell explained: In that paper [Station 1] I have stated the mathematical relations between this electrotonic state and the lines of magnetic force . . . and also between the electrotonic state and electromotive force . . . We must now [in Station 2] endeavour to interpret them from a mechanical point of view in connexion with our hypothesis.24 We have then an explicit statement that in Station 2 Maxwell set himself a different goal from the one he had grappled with in Station 1. By “our hypothesis” Maxwell meant a mechanical scheme that could be visualized. Indeed, for the benefit of the reader Maxwell drew a figure depicting this scheme; hence the appeal to mechanical illustrations. While in the first paper he took a formal-analogical approach with no didactic images, in the second, as the title indicates, he took a physical approach, appealing explicitly to a mechanical illustration. Most commentators take this illustration—a drawing of a mechanical contraption—to be a model but, in fact, Maxwell called it “hypothesis” and treated it as such.25 In that sense the methodology of Station 2 is fundamentally traditional.

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104 Station 2 (1861–1862) The question however persists, is “hypothesis” here a description of reality or merely a way of representing reality by means of something simpler or better known? Further evidence is required in order to consolidate the claim that indeed there occurred a fundamental change in methodology. Maxwell assumed, without argument, that electromagnetism is subject to a mechanical explanation. With this assumption, “hypothesis” offers a “physical” (i.e., mechanical) description of the nature of electromagnetism; it is a hypothesis in the sense that experimental evidence to support it was lacking at the time. But Maxwell seems confident that it will be forthcoming. The contrast with the previous paper of 1858 is striking, for experiments are not relevant to the “physics” of the imaginary system with the incompressible fluid, simply because there is no claim that this system is real. Therefore, Maxwell made his claim in Station 2 dependent on experimental evidence: If, by the same hypothesis, we can connect the phenomena of magnetic attraction with electromagnetic phenomena and with those of induced currents, we shall have found a theory which, if not true, can only be proved to be erroneous by experiments which will greatly enlarge our knowledge of this part of physics.26 Maxwell sought consequences of the mechanical hypothesis in the hope that experimental evidence will settle the issue. For example, he claimed that “the stress in the axis of a line of magnetic force is a tension, like that of a rope.”27 This weak analogy to a rope is explanatory in nature by offering a well-known illustration. But it is noteworthy that Maxwell claimed that the stress is a tension, for tension is “physical,” not an analogy. In any event, in contrast to Station 1, Maxwell sought in Station 2 to explain the phenomena: The next question is, what mechanical explanation can we give of this inequality of pressures in a fluid or mobile medium? The explanation which most readily occurs to the mind is that the excess of pressure in the equatorial direction arises from the centrifugal force of vortices or eddies in the medium having their axes in directions parallel to the lines of force. This explanation of the cause of the inequality of pressures at once suggests the means of representing the dipolar character of the line of force.28 Evidently, the hypothetical vortices are introduced for explanatory purposes. This is not the case with the imaginary fluid in Station 1. Maxwell commented that the imaginary fluid was not hypothetical for it was not introduced to explain actual phenomena. The fluid’s set of imaginary properties was employed for establishing certain mathematical theorems by making the argument more intelligible where physical issues are at stake rather than only the usage of algebraic symbols.29

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In 1858 Maxwell had made up his own rules for an imaginary system; in 1861 the rules are “standard” (i.e., the properties of the vortices in this hypothesis are those that apply to all vortices, nothing about them is “contrived”) to be applied consistently. In sum, the term “hypothesis” is a clue that the scheme has to be logically consistent and physically explanatory. It is noteworthy that when Maxwell turned his attention from magnetic to electrical phenomena, he expanded the same hypothesis of vortices, adding “a layer of particles, revolving each on its own axis in the opposite direction to that of the vortices.”30 Maxwell then explained that this mechanism means that the layer of particles acts as idle wheels.31 These wheels were placed in a gear mechanism where one wheel drives another in the same direction: The hypothesis about the vortices . . . is that a layer of particles, acting as idle wheels, is interposed between each vortex and the next, so that each vortex has a tendency to make the neighbouring vortices revolve in the same direction with itself.32 In discussing the effect of idle wheels, Maxwell referred to “contrivances, as, for instance, in Siemens’s governor for steam-engines.”33 clearly invoking an analogy with a physical mechanism. The point is that Maxwell did not introduce a new hypothesis to account for electrical phenomena; he was committed to a molecular system that exhibits simultaneously translational and rotational motions. By contrast, as we will see (ch. 7), a characteristic of the illustrations in Maxwell’s Treatise—Station 4—is that they are didactic, inconsistent, and avoid claims to be physical descriptions. Maxwell shared with the reader his great difficulty in conceptualizing the existence of vortices in a medium, “side by side, revolving in the same direction about parallel axes.” But once he overcame these difficulties, he turned to the examination of “the relations which must subsist between the motions of the vortices and those of the layer of particles interposed as idle wheels between them.”34 The study had commenced with the investigation of the statical forces of the system; it then evolved into an examination of the dynamics of a mechanism, specifically, “to determine the forces necessary to produce given changes in the motions of the different parts.”35 Maxwell proceeded to prove a series of propositions, all of which presuppose the mechanical system of molecular vortices. For example, he argued, “it appears . . . that, according to our hypothesis, an electric current is represented by the transference of the moveable particles interposed between the neighbouring vortices.”36 After his success in proving several mechanical propositions, Maxwell stated with confidence: We have thus obtained a point of view from which we may regard the relation of an electric current to its lines of force as analogous to the relation of a toothed wheel or rack to wheels which it drives.37

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106 Station 2 (1861–1862) We have here the following relation: a is to b as c is to d, where a and b are electric current and lines of force, respectively, vs. c and d which are a toothed wheel (or rack) and the wheels which it drives, respectively. The comparison is between electric current and a toothed wheel which is mechanical at the macro-level, not at the micro-level like vortices. However, the relation c to d at the macro-level is in fact a representation of the hypothetical claim at the micro-level: particles placed between contiguous vortices will be pushed in one direction, and if the process were to be reversed, namely, if the particles are forced in one direction, “by any cause” are Maxwell’s words, then they would make the vortices revolve in that direction. In other words, the vortices and the particles maintain a mutual mechanical relation, each one can move the other in the same direction. This is the principle of a rack and pinion in which translation and rotational motions are coupled in such a way that one may activate the other.38 In this way a mechanical “point of view” is obtained in which electricity and magnetism are linked together.39 Maxwell continued his determination of the forces which produce changes in the system by appealing to a driving wheel when it is put into motion violently. At stake is the impulse acting on the axle of a wheel in a machine when the actual velocity is suddenly given to the driving wheel, the machine being previously at rest. Maxwell then considered the opposite case: If the machine were suddenly stopped by stopping the driving wheel, each wheel would receive an impulse equal and opposite to that which it received when the machine was set in motion. This impulse may be calculated for any part of a system of mechanism, and may be called the reduced momentum of the machine for that point. In the varied motion of the machine, the actual force on any part arising from the variation of motion may be found by differentiating the reduced momentum with respect to the time, just as we have found that the elec­ tromotive force may be deduced from the electrotonic state by the same process.40 The expression, “just as,” is of course the key logical move of this analogy. Two systems are compared, one mechanical and the other electromagnetic. It appears that in both domains differentiating some specific phenomenon with respect to time results in the force. These are two independent analyses, one of electricity and the other of a mechanical system, and they both display the same process. In fact, they are linked by the hypothesis. What does “the same process” mean? In Prop. VIII Maxwell turned his attention to finding the relation between the alterations of motion of the vortices and the forces which they exert on the layer of particles between them.41 Based on his calculations he determined that the relation between the alterations of motion . . . and the forces exerted on the layers of particles between the vortices, or, in the language of

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our hypothesis, the relation between changes in the state of the mag­ netic field and the electromotive forces thereby brought into play.42 By “the same process” Maxwell meant that mechanically the micro-level functions in the same way as the macro-level and this is obtained “in the language of our hypothesis.” As Maxwell put it, “we must . . . endeavour to interpret [the mathematical relation between the electrotonic state and electromotive force] from a mechanical point of view in connexion with our hypothesis.”43 Maxwell’s claim is that (1) the system of molecular vortices is a viable mechanical system; (2) the system imitates electromagnetic phenomena; and (3) the system is represented by rack and pinion since the system of molecular vortices was constructed to transmit motions in both translation and rotation. What we have here are, respectively, three well recognized and distinct physical domains, namely, (i) a mechanical arrangement of transmission of motion both translational and rotational at the micro-level —the system of molecular vortices; (ii) electromagnetic phenomena; and (iii) a mechanical device consisting of rack and pinion. Note that the latter two domains are at the macro-level. It is worth emphasizing that the arrangement at the micro-level, namely, the system of molecular vortices, is mechanically conceivable, although it is provisional and temporary.44 Given these three distinct physical domains, how does the methodological argument work? Let us look closely at the three different domains and the relations among them: at the micro-level we have the system of molecular vortices while the phenomena of electromagnetism and the mechanical device of rack and pinion lie at the macro-level. The arrangement of rack and pinion represents the mechanical functioning of the system of molecular vortices, while the latter offers at the micro-level a mechanical imitation of electromagnetic phenomena at the macro-level. The mechanical arrangement of rack and pinion offers then an external perspective from which the manipulation of symbols (mathematical deduction) and physical intuition (causal explanation) can be controlled. To be sure, the interest is in one direction, from mechanical transmission to electromagnetism, and Maxwell pointed to a direct relation between the two domains, namely, the system of molecular vortices and the phenomena of electromagnetism.45 There is no distinction in their ontological status; Maxwell seemed to place the two domains, namely, electromagnetism and mechanics, on an equal ontological footing, while pursuing a causal explanation that is mechanical in nature in both cases. How does the methodology of hypothesis work in Maxwell’s hands? In Station 1 information flowed in one direction: from the imaginary system of tubes filled with an incompressible fluid to the physics of electromagnetism. In Station 2 information also flows unidirectionally, but in this case the route is complex. Information first flows from a well-understood mechanical device of rack and pinion at the macro-level to a feasible hypothetical

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108 Station 2 (1861–1862) system of molecular vortices at the micro-level. In Maxwell’s terms, rack and pinion represent at the macro-level the mechanical arrangement of a vortical system at the micro-level, combining translational (shaft) and rotational (toothed wheel) motions simultaneously. The mechanical representation at the macro-level throws light on the vortical system at the micro-level and helps control it. But this is not the end of it for, in its turn, the vortical system at the micro-level imitates the phenomena of electromagnetism at the macro-level. This is the hypothesis. Of course, Maxwell’s goal was not to explain a real mechanical system on the basis of a hypothetical vortical system. After all, Maxwell had proposed the vortical hypothesis for the purpose of explaining electromagnetic phenomena. The properties of these vortices are those of vortices in general,46 but their arrangement is controlled by Maxwell; for example, he introduced idle wheels in the shape of particles into the vortical system. In this way, Maxwell was able to draw important consequences from the molecularvortex hypothesis for explaining electromagnetic phenomena.

5.4 Applying the methodology: assumptions and their consequences To get a good grasp of Maxwell’s methodology in 1861, it may be beneficial to analyze at length the assumptions Maxwell made and the results he obtained as he set them down in concluding Part II of the paper. We focus principally on issues relevant to methodology and consistency, and we follow Maxwell’s summary point by point. We begin with point (1): (1) Magneto-electric phenomena are due to the existence of matter under certain conditions of motion or of pressure in every part of the magnetic field, and not to direct action at a distance between the mag­ nets or currents. The substance producing these effects may be a certain part of ordinary matter, or it may be an aether associated with matter.47 Maxwell began by asserting that electromagnetic phenomena are not “due to” action at a distance. In contrast to Maxwell’s view in Station 1, action at a distance is no longer considered equivalent to lines of force: “due to” is the language of causation, not the terminology of analogy. Maxwell’s confidence in the unifying concept of lines of force had grown substantially. But, interestingly, here he associated the lines with matter, be it ordinary or aetherial. This belief, that matter is the carrier of electromagnetic phenomena, is not even considered in Station 1. Clearly, at the outset of this summary Maxwell is unequivocal in seeking a physical account of the phenomena. Concepts are neither right nor wrong (neither true nor false); they are to be judged by their usefulness in some context. It follows that the concept of action at a distance is either helpful or not in accounting for electromagnetic phenomena. We note that Maxwell asserted that action at a distance has no physical reality; by contrast, lines of force are physical.

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What is the ontological status of the molecular vortices whose interactions represent these lines of force? Maxwell is unambiguous: lines of force are physical. For Maxwell, the explanandum is “lines of force,” while the explanans is “molecular vortices.” Maxwell was willing to consider alternatives to these vortices, but not to the lines of force—the object of his inquiry. (2) The condition of any part of the field, through which lines of mag­ netic force pass, is one of unequal pressure in different directions, the direction of the lines of force being that of least pressure, so that the lines of force may be considered lines of tension.48 The lines of force are in fact lines of tension; this makes them physical. (3) This inequality of pressure is produced by the existence in the medium of vortices or eddies, having their axes in the direction of the lines of force, and having their direction of rotation determined by that of the lines of force. The expression “produced by the existence” is physical talk: Maxwell did not hesitate to assert that vortices exist in the medium. The remaining issue, as far as Maxwell is concerned, is the size of the vortices: “it is probably very small as compared with that of a complete molecule of ordinary matter.”49 This claim makes no sense if the vortices are not real. That is, the claim regarding size presupposes that the vortices are real. (4) The vortices are separated from each other by a single layer of round particles, so that a system of cells is formed, the partitions being these layers of particles, and the substance of each cell being capable of rotating as vortex.50 In this passage Maxwell referred to substance: the rotating vortex is material and has a clearly defined boundary to form a distinct rotating cell. In sum, we have here a physical system that obeys the laws of mechanics. In (5) Maxwell began by describing the motion of the particles which form the layer that separates the vortices from each other (see point (4)). The irregular motions of the particles result in heat, and in effect “these particles . . . play the part of electricity.”51 According to this hypothesis the motion of translation constitutes an electric current, and the tangential pressures created by the irregular motion of the particles, constitutes electromotive force. Clearly, Maxwell’s analysis is physical, not formal. Maxwell confessed that the analysis is “awkward”: The conception of a particle having its motion connected with that of a vortex by perfect rolling contact may appear somewhat awkward. I do not bring it forward as a mode of connexion existing in nature, or even

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110 Station 2 (1861–1862) as that which I would willingly assent to as an electrical hypothesis. It is, however, a mode of connexion which is mechanically conceivable, and easily investigated, and it serves to bring out the actual mechanical con­ nexions between the known electromagnetic phenomena; so that I ven­ ture to say that any one who understands the provisional and temporary character of this hypothesis, will find himself rather helped than hindered by it in his search after the true interpretation of the phenomena.52 This is a revealing passage. Maxwell began on a negative note, namely, he stated that he did not claim that this “mode of connexion” exists in nature; indeed, he did not even propose it as “an electrical hypothesis.” The hypothesis of molecular vortices was intended to do more epistemic work than just serve as an illustration; it was supposed to give an idea of the way the phenomena might be produced in nature—it was clearly explanatory, not illustrative. But Maxwell never meant to suggest that this is the true description of the phenomena. He considered the scheme of molecular vortices a “provisional” hypothesis that succeeded in many ways in accounting for the phenomena. But Maxwell had an epistemic problem here: he embarked on a search for an explanation, something which he had not done in Station 1. He now stated that the goal was to find “the true interpretation” of the phenomena, assuming it would be mechanical. Maxwell insisted on the provisional status of the hypothesis he put forward; nevertheless, in his view it is likely to help in the search for the true interpretation. Point (6) elaborates on the mechanical relation between electric current and the vortices: the discussion offers a physical account of electromotive forces based on this mechanical relation. In the remaining two points another physical account is offered, namely, the results of relative motions. In points (7) and (8) Maxwell drew consequences regarding the forces which arise from the relative motions of electromagnetic elements, be they rotational or translational.53 Overall, the concluding pages of Part II of the paper on physical lines of force is a tour de force where Maxwell took advantage of a mechanical illustration to draw consequences for electromagnetic phenomena. It is noteworthy that, when Maxwell recapitulated the assumptions he had made and the results he had obtained, he offered verbal descriptions rather than algebraic equations. In Part I and Part II of Station 2 Maxwell displayed a consistent approach which he distinguished from the methodology he had applied in Station 1. There he took a formal-analogical approach while here, in Station 2, as the title of the paper indicates, he took a physical approach, proposing a hypothesis based on a mechanical illustration.54

5.5 Part III: a landmark in the history of physics Maxwell concluded the first two parts of Station 2 with a methodological remark of considerable interest:

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The facts of electro-magnetism are so complicated and various, that the explanation of any number of them by several different hypotheses must be interesting, not only to physicists, but to all who desire to under­ stand how much evidence the explanation of phenomena lends to the credibility of a theory, or how far we ought to regard a coincidence in the mathematical expression of two sets of phenomena as an indication that these phenomena are of the same kind. We know that partial coin­ cidences of this kind have been discovered; and the fact that they are only partial is proved by the divergence of the laws of the two sets of phenomena in other respects. We may chance to find, in the higher parts of physics, instances of more complete coincidence, which may require much investigation to detect their ultimate divergence.55 Maxwell raised a fundamental question in philosophy of science, namely, can explanation of phenomena function as evidence for the credibility of a theory? No wonder he posed this question “not only to physicists.” Maxwell did not doubt that explanation can indeed offer credibility; rather, he was wondering how much credibility can be obtained from an explanation? It was then a question of degree and it reflected the nature of the problem with which Maxwell was grappling. At stake was the issue of coincidence: “How far we ought to regard a coincidence in the mathematical expression of two sets of phenomena as an indication that these phenomena are of the same kind.” This was, of course, Maxwell's experience with the analogy which Thomson had discovered between electricity and heat, and the problem continued to haunt Maxwell with the relation between electromagnetism and mechanics. However, a most productive challenge awaited Maxwell in Part III of Station 2 where the key to success was the decision to interpret physically what appeared on the face of it a mere coincidence. Put concisely, given his commitment to the unifying concept of lines of force, Maxwell’s confidence in the significance of a numerical coincidence increased, and this, in turn, led him to endow it with physical meaning. This was the background for his fundamental discoveries. Part III of Station 2 is a landmark in the history of physics; it is extraordinary in terms of both methodology and physical content.56 Maxwell presented two major discoveries in electromagnetic theory which are interrelated: the phenomenon of the displacement current and the correlation of light with electromagnetic phenomena. At this stage Maxwell considered the medium for light and the medium for electromagnetic phenomena as possibly distinct, sharing, however, some similar properties. While adhering to Faraday’s unifying concept of lines of force and exploiting his experimental findings, Maxwell went well beyond Faraday. This was a decisive moment in Maxwell’s scientific quest in electromagnetism. His explicit reliance on methodology as the engine of discovery proved productive. Part III begins with a summary of the preceding two parts. In Part I of Station 2 it was shown how magnetic induction “may be accounted for” on the hypothesis of the molecular vortices. This language is hypothetical and

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112 Station 2 (1861–1862) explanatory. In Part II a mechanism was described by which the rotations of the vortices could produce the magnetic lines of force. This talk is physical. Maxwell then offered some details of his conception of this mechanism: I conceived the rotating matter to be the substance of certain cells, div­ ided from each other by cell-walls composed of particles which are very small compared with the cells, and that it is by the motions of these particles, and their tangential action on the substance in the cells, that the rotation is communicated from one cell to another.57 Maxwell argued that, to transmit rotation from the exterior to the interior parts of each cell, the substance in the cells should “possess elasticity of figure, similar in kind, though different in degree, to that observed in solid bodies.”58 This is an important claim: the similarity “in kind,” albeit not “in degree,” suggests a possible linkage between the luminiferous medium and the medium of electromagnetic phenomena. Based on the results up to this point, Maxwell offered an insightful assertion on the nature of the media for light and electromagnetic phenomena: The undulatory theory of light requires us to admit this kind of elasti­ city in the luminiferous medium, in order to account for transverse vibrations. We need not then be surprised if the magneto-electric medium possesses the same property.59 This is a significant remark, coming as it did at the time when no relation had been conceived between the media for optical and electromagnetic phenomena. Indeed, Maxwell expressed no surprise when he reported later in this part of Station 2 that he found such a relation. The vortices form a medium for the transmission of electromagnetic disturbances, and this medium seemed to share properties with the luminiferous medium, both of which are distinct from ordinary matter. The question, however, remains: are these two media one and the same? This is the problem of coincidence. Just as the luminiferous medium (i.e., the ether) is taken to be real, so the electromagnetic medium may be taken as real. Maxwell endowed the medium for electromagnetic phenomena with the mechanical property of elasticity, for he had grounds to believe that it is the same as the luminiferous medium. It is this feature that eventually liberated him from any commitment to a specific mechanical illustration. Moreover, what Maxwell failed to achieve in Station 1, he accomplished in Station 2: the discovery of “a method of forming a mechanical conception of this electro-tonic state adapted to general reasoning.”60 This “mechanical conception” is the elasticity of the medium which, for a solid under stress, results in displacement—a well-known phenomenon at the time which the young Maxwell had also studied.61 It is striking that in Station 1, at the outset of his studies of the physical domain of electromagnetism, Maxwell

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considered the elastic vibrations of the luminiferous medium a key for unlocking the physics of this domain. In the opening section of Station 1, Maxwell remarked: The . . . analogy, between light and the vibrations of an elastic medium, extends much farther, but, though its importance and fruitfulness cannot be over-estimated, we must recollect that it is founded only on a resemblance in form between the laws of light and those of vibrations. By stripping it of its physical dress and reducing it to a theory of “transverse alternations,” we might obtain a system of truth strictly founded on observation, but probably deficient both in the vividness of its conceptions and the fertility of its method.62 Indeed, as it is evident in Station 2, Maxwell made elasticity the principal property of the medium so that there was more than “a resemblance in form” to vibrations of a solid body. Maxwell was not certain that his hunch would be productive, but in Station 2 the issue was resolved, for elasticity became one of the key concepts that facilitated progress in electromagnetism. We pause to take stock of the situation, for our focus on methodology has yielded new results. This is one of the critical moments in Maxwell’s methodological odyssey, where we see Maxwell’s methodology at work: the physics (expressed verbally) precedes the mathematics (expressed symbolically). The first section of Part III was designed to present Maxwell’s physical argumentation. Maxwell abandoned the methodologies he employed in Parts I and II of Station 2, not to mention the methodology of Station 1, and embarked on a novel argument. Based on his commitment to the concept of lines of force, Maxwell argued that electromagnetic phenomena are propagated in a medium. This is a qualitative argument. In addition, there already was a medium for the transmission of light. Hence, if the medium for electromagnetic phenomena were the same as the luminiferous medium, that is, if one took the coincidence seriously, it would follow that the properties of the latter apply to the former. In particular, this means that the elasticity of the medium, required by the undulatory theory of light, is a property of the medium for electromagnetic phenomena. Once this is established, the relation between elasticity and displacement phenomena in the electromagnetic domain followed immediately according to the principles of elasticity in mechanics. All that was needed to validate this novel argument was empirical evidence that the two media are in fact the same. Maxwell found this evidence in the precise measurements by Kohlrausch and Weber of the ratio of electrodynamic measure of the integral current to electrostatic units. The German physicists were aware that their measurements resulted in a number which is surprisingly close to the value for the velocity of light but the coincidence had no consequences for them.63 Kohlrausch and Weber were not committed to a medium for the propagation of electromagnetic disturbances, and so it made no sense for

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114 Station 2 (1861–1862) them to ask if the medium for light (which was widely accepted at the time) was the same as the medium for electromagnetic disturbances. Given their commitment to “action at a distance” for electromagnetic phenomena, the numerical agreement was simply a puzzle for which they had no explanation. Maxwell, by contrast, took this coincidence seriously, clinching his argument.64 For Maxwell this was an argument from a phenomenon, not from any hypothesis or analogy. But, as Maxwell realized, it was compatible with the vortex hypothesis. The two discoveries, the displacement current and that light is an electromagnetic phenomenon, were in fact conceived together as consequences of Maxwell’s interpretation of the results of Kohlrausch and Weber’s measurement. The key concept was elasticity which explains why Maxwell considered the displacement current a phenomenon.65 Here we see that Maxwell realized that the concept of lines of force was clearly superior to action at a distance. What was puzzling to Kohlrausch and Weber was in fact the “answer” for Maxwell, and one of his major discoveries. The claim that the medium for electromagnetic phenomena was the same as the luminiferous medium had two related consequences: (1) electromagnetic phenomena are propagated at the speed of light, and (2) that there is a displacement current. In the secondary literature these two discoveries have been treated separately, with many conjectures on the origin of the displacement current. Indeed, Maxwell did not reveal the steps leading to this discovery. Siegel depicted the situation well: The question of the origin of the displacement current has been, and continues to be, the object of much interest and concern . . . The cen­ trality of this episode in the history of physics, its paradigmatic status as an example of theoretically motivated innovation, and its prominence in the pedagogy of physics have all contributed to making it a topic of prime concern for historians of physics. Unfortunately . . . the matter remains . . . obscure.66 Mindful of this cautious remark and faithful to our inquiry into Maxwell’s methodological trajectory, we restrict our attention to the evidence at hand, namely, Maxwell’s own text. We now turn to an account of the two discoveries and the methodology that led to them. Maxwell’s language at the beginning of Part III throws light on his attitude toward the ontology of electromagnetism, an issue he did not address in Station 1 at all, avoiding as he did any explanation: According to our theory, the particles which form the partitions between the cells constitute the matter of electricity. The motion of these particles constitutes an electric current; the tangential force with which the particles are pressed by the matter of the cells is electromo­ tive force, and the pressure of the particles on each other corresponds to the tension or potential of the electricity.67

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This is clearly a discussion of the actual physics at the micro-level, and no analogy is drawn. Maxwell first showed that, according to his theory, each (known) electromagnetic phenomenon can be accounted for in terms of the mechanical system; and, second, that there are two independent qualities of bodies: “one by which they allow of the passage of electricity through them, and the other by which they allow of electrical action being transmitted through them without any electricity being allowed to pass.”68 Again, this is physical talk and not analogical reasoning, for the vortices are considered real and produce the phenomena of electromagnetism. Maxwell, however, appears to have changed tack; he proceeded to introduce a mechanical analogy, to be sure, a weak one, which is unrelated to the vortex hypothesis he had used in Parts I and II: A conducting body may be compared to a porous membrane which opposes more or less resistance to the passage of a fluid, while a dielectric is like an elastic membrane which may be impervious to the fluid, but transmits the pressure of the fluid on one side to that on the other.69 Again, we note the analogical structure of the argument: “X may be compared to Y,” or “X is like Y,” where “X” and “Y” are two distinct phenomena belonging to two categorically different domains in physics. In this case, using the analog of an elastic membrane, Maxwell argued that a dielectric under induction functions like a membrane under mechanical pressure, namely, it does not allow for a flow of current, but at the same time it does produce pressure on its other side; in electromagnetic terms, the induction produces “a general displacement of the electricity in a certain direction.” And Maxwell continued: This displacement does not amount to a current, because when it has attained a certain value it remains constant, but it is the commencement of a current, and its variations constitute currents in the positive or negative direction, according as the displacement is increasing or diminishing.70 This “commencement of a current,” or “the current due to displacement,”71 is undoubtedly one of the most creative moments in Maxwell’s development of electrodynamics. It is therefore worth pausing to reflect on the move which Maxwell made here. What is immediately noticeable is that the vortex hypothesis is absent. It is then evident that the concept of a current due to displacement does not depend on this hypothesis, which is presupposed throughout the paper; rather, the concept is linked to the molecular structure of the medium, to its elastic nature and inner ordering. In a dielectric under induction, we may conceive that the electricity in each molecule is so displaced that one side is rendered positively, and

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116 Station 2 (1861–1862) the other negatively electrical, but that the electricity remains entirely connected with the molecule, and does not pass from one molecule to another. The effect of this action on the whole dielectric mass is to pro­ duce a general displacement of the electricity in a certain direction.72 To be sure, this idea is mechanical, namely, that the molecular structure of the medium may exhibit a certain ordering, but it is not dependent on the vortex scheme. This fact is of great importance, for Maxwell recognized that he could disregard the hypothesis of the molecular vortices while retaining the concept of displacement current.73 Be that as it may, his next step was to formalize his account of the newly discovered phenomenon. The amount of “the current due to displacement” depends on the nature of the dielectric and on the electromotive force, and Maxwell proceeded to formulate a mathematical expression for this new term. He then remarked: These relations are independent of any theory about the internal mech­ anism of dielectrics; but when we find electromotive force producing electric displacement in a dielectric, and when we find the dielectric recovering from its state of electric displacement with an equal electro­ motive force, we cannot help regarding the phenomena as those of an elastic body, yielding to a pressure, and recovering its form when the pressure is removed.74 The analogy between a dielectric, subject to electromotive force, and an elastic body, under stress, impressed Maxwell. Indeed, he reported that he could not avoid the analogy. Although in this instance he did not call this comparison an analogy, it bears the features of a (weak) analogy: electromotive force causes stress in the dielectric resulting in a displacement which, upon recovering, produces an equal electromotive force, is similar to an elastic body under pressure which recovers its form once the pressure on the solid is removed. What is striking in this analogy is that Maxwell called phenomena the effects of displacement in a dielectric, despite the fact that these “phenomena” are not directly observable. Yet Maxwell considered these effects phenomena analogous to mechanical displacements of solid under stress—well-known phenomena that had been thoroughly studied in mechanics. We consider this analogy highly significant whose consequences have not generally been appreciated.75 Indeed, something quite dramatic occurs here which foreshadows future developments in Maxwell’s treatment of electromagnetism. Two points are worth amplifying. First, Maxwell severed the mathematical expression he found for the new term, “electric displacement,” from any hypothetical mechanism of the dielectric. The displacement current is at the level of phenomena, and phenomena are not dependent on any hypothesis. The claim is that the mathematical relations are independent of any internal mechanism. Maxwell suspected that once

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the concept of displacement current is embedded in a set of equations—a modification of Ampère’s law—a specific mechanical illustration for this concept may no longer be necessary. The second point is of no less importance, namely, Maxwell identified elasticity (and presumably the concomitant stress) as the key physical property of the medium which determines the nature of electromagnetic effects. Thus, it was sufficient to assume elasticity, implying that the commitment to a specific mechanism such as molecular vortices was superfluous. In fact, the claim is analogical: “we cannot help regarding the phenomena as those of an elastic body, yielding to a pressure, and recovering its form when the pressure is removed,” where the phrase “the phenomena . . . as those of” is the key to this claim. This is indeed the path that Maxwell chose in his subsequent study in Station 3; as we will see in the next chapter, Maxwell dropped the mechanical hypothesis of Station 2 together with physical theories, and opted for a formal presentation. However, in 1862, in Station 2, Maxwell was not yet ready to pursue this line of argument and, in proving a series of propositions in Part III of Station 2 (Props. 12 to 17) as well as in Part IV, he continued to draw consequences from the hypothesis of molecular vortices. Maxwell then returned to his main line of argumentation and remarked: “According to our hypothesis, the magnetic medium is divided into cells, separated by partitions formed on a stratum of particles which play the part of electricity.”76 He went on to describe the mechanism which he elaborated in Part II of Station 2, explicitly endowing the cells with elasticity: When the electric particles are urged in any direction, they will, by their tangential action on the elastic substance of the cells, distort each cell, and call into play an equal and opposite force arising from the elasticity of the cells. When the force is removed, the cells will recover their form, and the electricity will return to its former position.77 This is definitely a mechanical hypothesis, and a very detailed one at that. It seems that Maxwell wished to maintain his mechanical hypothesis in accounting for the phenomena but, to do so, he had to add elasticity to the physical properties of the cells. The next step was to present the second fundamental discovery including the experimental basis of the entire argument. Taking the molecular-vortex theory as given, Maxwell then deduced the relation between the statical and dynamical measures of electricity.78 He could now take advantage of the findings of Kohlrausch and Weber who had demonstrated experimentally that the ratio of electrostatic and electromagnetic units produced a number which is surprisingly close to the velocity of light.79 Maxwell referred to the value for the velocity of light, measured in 1849 by Hippolyte Fizeau (1819–1896).80 For theoretical reasons, Kohlrausch and Weber did not see a physical relation of their

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118 Station 2 (1861–1862) result to the velocity of light, although they did recognize the numerical near coincidence of the value for the ratio they found with the velocity of light. In any event, they did not ascribe any significance to this near coincidence.81 Weber’s theory of electromagnetism was corpuscular; it worked on the principle of action at a distance that presupposes no medium whatsoever. From this perspective there is nothing that could link electromagnetic phenomena to light, which was undulatory in a luminiferous ether. Here we have a clear case where a theoretical commitment interfered with the analysis—exactly the kind of bias against which Maxwell had already cautioned in Station 1. Maxwell, for his part, was actively on the lookout for relations among different phenomena, especially with respect to the medium: If we can now explain the condition of a body with respect to the sur­ rounding medium when it is said to be “charged” with electricity, and account for the forces acting between electrified bodies, we shall have established a connexion between all the principal phenomena of elec­ trical science. We know by experiment that electric tension is the same thing, whether observed in statical or in current electricity; so that an electromotive force produced by magnetism may be made to charge a Leyden jar, as is done by the coil machine.82 The experiment to which Maxwell referred is that of Weber and Kohlrausch, published in 1856 and revised in 1857. Their goals, however, were different: the two German physicists sought the quantification of electrodynamics,83 whereas Maxwell sought to place electromagnetic phenomena in a mechanical framework, specifically in an elastic medium.84 This is the background to Maxwell’s anticipatory remark concerning the main result of Part III, namely, that the elasticity of the magnetic medium in air is the same as that of the luminiferous medium, if these two coexistent, coextensive, and equally elastic media are not rather one medium.85 The radical conception that there is one medium for both electromagnetism and light constitutes a major juncture in the history of physics. It is noteworthy that elasticity was the physical property which was the subject of comparison. Maxwell claimed—to be sure, in a tentative way—that the elasticity of the medium that carries electromagnetic phenomena and the elasticity of the medium of light are the same. Therefore, it stands to reason that there is just one medium for what had previously been conceived as phenomena in different domains of physics. Maxwell thus argued that the medium for these two distinct sets of phenomena is one and the same, thereby setting the stage for the claim that light is an electromagnetic

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phenomenon. It is worth noting that in this anticipatory remark Maxwell stated his result without regard to his mechanical hypothesis, that is, the mechanical illustration of molecular vortices. In a letter to Faraday, dated 19 October 1861, Maxwell indicated that his conception led, “when worked out mathematically to some very interesting results, capable of testing my theory, and exhibiting numerical relations between optical, electric and electromagnetic phenomena.” And he continued: From the determination of Kohlrausch and Weber of the numerical relation between the statical and magnetic effects of electricity, I have determined the elasticity of the medium in air, and assuming that it is the same with the luminiferous ether I have determined the velocity of propagation of transverse vibrations. The result is 193088 miles per second (deduced from electrical & mag­ netic experiments). Fizeau has determined the velocity of light = 193118 miles per second, by direct experiment. Maxwell then added: This coincidence is not merely numerical. I worked out the formulae in the country before seeing Webers number, which is in millimetres, and I think we have now strong reason to believe, whether my theory is a fact or not, that the luminiferous and the electromagnetic medium are one.86 Again, we see that the issue is “coincidence”—how should it be interpreted? To be sure, “strong reason” is not the same as “proof,” but it is clear that Maxwell was confident he was on the verge of making a great discovery by taking the coincidence to be physically meaningful. Maxwell went on to report his theoretical attempt to examine the theory of the passage of light through a medium filled with magnetic vortices. He tried to link the nature of the vortices with optical phenomena such as polarization.87 Shortly afterward, on 10 December 1861, Maxwell wrote to Thomson: I made out the equations in the country before I had any suspicion of the nearness between the two values of the velocity of propagation of magnetic effects and that of light, so that I think I have reason to believe that the magnetic and luminiferous media are identical and that Weber’s number is really, as it appears to be, one half the velocity of light in millimetres per second.88 It is striking that the wording and the contents of these letters strongly overlap.89 No analogical reasoning was developed in these letters and the talk is essentially physical. But, above all, we have here evidence for Maxwell introducing the idea that electromagnetic phenomena are propagated at the speed of light before its

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120 Station 2 (1861–1862) publication in January 1862. Note that in his correspondence Maxwell made the bold claim: “the magnetic and luminiferous media are identical.” However, in Part III of his publication he was more cautious, only suggesting the possible identity of the magnetic and luminiferous media. Briefly, in the rest of Part III Maxwell reasoned as follows. He modified his earlier set of equations in view of the concept of the displacement current (Prop. 14). Next, he calculated the force acting between two electrified bodies, appealing to the experimental results of Weber and Kohlrausch (Prop. 15). He then determined, V, the rate of propagation of transverse vibrations through the elastic medium, and compared it to the velocity of light (Prop. 16): By referring to the equations of Part I., it will be seen that if ρ is the density of the matter of the vortices, and μ is the “coefficient of mag­ netic induction . . .” In air or vacuum μ = 1, and therefore V = E = . . . 193,088 miles per second . . . [and] the velocity of light in air, as determined by M. Fizeau*, is . . . 195,647 miles per second.90 [Footnote in Maxwell 1861–1862, 22] * Comptes Rendus, Vol. XXIX. (1849), p. 90. In Galbraith and Haughton’s Manual of Astronomy, M. Fizeau’s result is stated at 169,944 geographical miles of 1000 fathoms, which gives 193,118 stat­ ute miles; the value defined from aberration is 192,000 miles. Maxwell introduced a formula “by the ordinary method of investigation” in which V = √(m/ρ), where m is the coefficient of transverse elasticity.91 In contrast to his treatment of the displacement current, in this instance Maxwell’s determination is directly dependent on the molecular-vortex hypothesis, since he stipulated that ρ is the density of the matter of the vortices. Now, “the quantity E previously determined in Prop. XIII. is the number by which the electrodynamic measure of any quantity of electricity must be multiplied to obtain its electrostatic measure.”92 Put differently, E is the number of electrostatic units in one electromagnetic unit, and this is what Kohlrausch and Weber evaluated experimentally.93 Maxwell determined V, the rate of propagation of transverse vibrations through the elastic medium, on the assumption of the molecular vortices; he then found it equal to E (whose value had been obtained empirically by Kohlrausch and Weber), which is very nearly equal to the velocity of light, as measured by Fizeau. This was the first step toward proving that light is an electromagnetic phenomenon. At the end of Prop. 16 Maxwell proclaimed: we can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and mag­ netic phenomena.94

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Maxwell had a very good reason to italicize this conclusion, which was based on two arguments: (1) the numerical agreement of the velocities, and (2) the transverse nature of the waves in the medium. Maxwell then inferred that “the same medium” sufficed to account for waves of the same kind moving at the same velocity. We note that underlying this inference is the assumption that the elasticity of “the . . . medium of which the cells are composed . . . is due entirely to forces acting between pairs of particles.”95 Thus, in contrast to the anticipatory remark,96 Maxwell’s argument here is articulated with confidence, and it depends on the molecular-vortex hypothesis. The mechanical illustration is after all central to the argument.

5.6 Conclusion Although Maxwell changed his methodology in Station 2 from the one he had applied in Station 1, he remained faithful to his commitment, namely, that the unifying concept of lines of force is the key to accounting for electromagnetic phenomena. He noted the “beautiful illustration” afforded by strewn iron filings in the presence of magnetic force which makes one think that the lines of force are “something real.”97 We take the outline of Maxwell’s methodology in Station 2 to be the following: 1

The goal: Account for the phenomena in physical domain A within a certain conceptual framework. (1.1) Specifically, show how electromagnetic phenomena can be repro­ duced mechanically.

2

The means: Appeal to the methodology of hypothesis which supports causal argumentation. (2.1) Specifically, introduce a mechanical scheme whose dynamics in domain B1 at the micro-level is well understood and can account for the phenomena in domain A at the macro-level, that is, the mechanical scheme in domain B1 offers a causal explanation of the phenomena in domain A. (2.2) Develop a mathematical analysis of this mechanical scheme in B1; remain sensitive to the physical setting of this scheme as well as to its mathematical formulation by offering a mechanical illustration in domain B2 at the macro-level; set up this illustration so that it depicts at the macro-level the effects of the arrangement in B1 at the micro-level. (2.3) Draw mathematical consequences from the mechanical scheme in B1, but be alert to unexpected coincidences. (2.4) Link the mathematical consequences with experimental findings in domain A at the macro-level and seek novel connections among phenomena pertinent to this domain.

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122 Station 2 (1861–1862) The principle of this methodology is what might be called a “controlled hypothesis”; it is not based on substitution, a characteristic of the methodology of formal analogy. This ingenious modification of the standard methodology of the hypothesis requires control of both manipulation of symbols and physical intuition. It facilitated a wider consideration than the limited perspective of Kohlrausch and Weber. There are no analogies in Part III concerning the vortices, which are assumed to be physical. Maxwell’s argument is that the particles and the vortices as the bearers of electricity and magnetism, respectively, are the constituent parts of the electromagnetic medium, and their mechanical properties are thus the explanans of the electromagnetic phenomena. Maxwell relied on his mechanical hypothesis of the molecular vortices to the very end of this four-part paper, true to the commitment that he stated at the outset.98 Moreover, he drew a contrast between the hypotheses he introduced and those developed by his continental contemporaries. In Station 2 Maxwell’s adherence to Faraday’s concept of lines of force was vindicated, for he was able to produce an electromagnetic theory that accounted for both Faraday’s experimental results and his own new discoveries. But his appeal to “hypotheses,” specifically the molecular-vortex hypothesis, was not to endure, for in Station 3 Maxwell abandoned this methodology—to be discussed in the next chapter.

Notes 1 2 3 4 5 6 7 8

Maxwell, 1861–1862, 161–162. See ch. 2, n. 26.

See ch. 3, § 3.3, n. 44, for the passage in Stokes, [1848] 1883, 12.

See ch. 4, § 4.2, n. 29.

Maxwell, 1861–1862, 162.

Maxwell, 1861–1862, 162.

Maxwell, 1861–1862, 162, italics added.

Maxwell, 1861–1862, 282.

With the benefit of hindsight Maxwell recognized a similar transition in Faraday.

In an essay on action at a distance published in 1873 Maxwell referred to Fara­ day as progressing “from the conception of geometrical lines of force to that of physical lines of force” (Maxwell, 1873b, 52; see ch. 2, n. 21). 9 Maxwell did not refer to thermodynamics in his works on electromagnetism; he seems to have kept his research in the two domains separate. Thus, the molecules in thermodynamics were not related to the molecules in electromagnetism. To be sure, the unifying problem arises only if one takes the molecules to be real. A precedent for the hypothesis of molecular vortices is to be found in papers by Rankine from the early 1850s in which Rankine (e.g., 1851, 443) introduced “an hypothesis. . . of molecular vortices.” In Station 4 Maxwell (1873d, 2: 416, § 831) quoted at length Thomson, who referred to Rankine’s study on the mechanical action of heat, especially in gases and vapors (Rankine, 1853; the paper was read in 1850). Rankine’s Introduction to that article bears the subheading: “Summary of the principles of the hypothesis of molecular vortices, and its application to the theory of temperature, elasticity, and real specific heat.” The paper begins: “The ensuing paper forms part of a series of researches respecting the consequences of an hypothesis called that of Molecular Vortices, the object

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10 11 12 13 14 15 16

17

18 19 20 21 22 23 24 25

26 27 28 29 30 31 32 33 34

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of which is, to deduce the laws of elasticity, and of heat as connected with elasticity, by means of the principles of mechanics, from a physical supposition consistent and connected with the theory which deduces the laws of radiant light and heat from the hypothesis of undulations” (Rankine, 1853, 147). Rankine’s usage of the term “hypothesis” should not be confused with Maxwell’s invocation of “hypothesis” since—as we argue—for Maxwell “hypothesis” stood for a methodology, not just some presupposition. In 1861 Maxwell did not mention Rankine’s paper on molecu­ lar vortices, but he did note Rankine’s Manual of applied mechanics (1858): see Maxwell, 1861–1862, 167–168. For discussion of hypothesis as a methodology, see ch. 8, § 8.3.5. For further discussion on Rankine, see ch. 3, § 3.4. Maxwell, 1861–1862, 282. See Maxwell, [1890] 1965, Preface by W. D. Niven, xx. Maxwell, [1890] 1965, 291. Maxwell, [1890] 1965, 163. It is noteworthy that here Maxwell called Thomson’s methodology “mathematical analogy,” the very expression Maxwell had invoked to express his own methodology in 1856: cf. ch. 1, n. 31. See Maxwell, [1890] 1965, Plate VIII (opposite p. 488), fig. 2. Maxwell, 1861–1862, 291. Ibid., 347. Maxwell continued, arguing that “those who look in a different direc­ tion for the explanation of the facts, may be able to compare this theory. . . with that which supposes electricity to act at a distance with a force depending on its velocity, and therefore not subject to the law of conservation of energy.” Clearly, with this demonstration he was satisfied that the scheme of molecular vortices is inconsistent with the competing theories of action at a distance. In Station 2 terms such as “analogy,” “analogous” and “like” occur rarely (e.g., 1861–1862, 164, 165). While “analogy” is central to the argument in Station 1, it is mainly explanatory and invoked for didactic purposes in Station 2: see, e.g., n. 27, below. See ch. 1, § 1.3; ch. 4, n. 11, and Maxwell, 1858, 28. Maxwell, 1861–1862, 86. Maxwell, 1861–1862, 86. Maxwell, 1861–1862, 86. Maxwell, 1861–1862, 87–88. Maxwell, 1861–1862, 88. Maxwell, 1861–1862, 291. Contrary to what some scholars would lead us to expect, the term “model” is nowhere to be seen in this paper. Siegel (1991, ch. 3: The elaboration of the molecular-vortex model), for example, calls the hypothesis of molecular vortices “model.” To be sure, Maxwell’s hypothesis has some characteristics that suggest what today is called a model, but we adhere to Maxwell’s terminology. On a few occasions in Station 1 Maxwell referred to hypothesis: see ch. 4, nn. 7 and 33 but, as we have seen, hypothesis plays no methodological role in Station 1. See also, most clearly, ch. 4, n. 44. For a general discussion of hypothesis in Max­ well’s methodological odyssey, see ch. 8, § 8.3.5. Maxwell, 1861–1862, 162. Maxwell, 1861–1862, 164, italics in the original. Maxwell, 1861–1862, 165. Maxwell, 1858, 30–31. See ch. 4, n. 44. Maxwell, 1861–1862, 283. Maxwell, 1861–1862, 283. Maxwell, 1861–1862, 283. Maxwell, 1861–1862, 283. Maxwell, 1861–1862, 283.

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124 Station 2 (1861–1862) 35 36 37 38 39 40 41 42 43 44 45 46

47 48 49 50 51 52 53 54

55 56 57 58 59

60 61 62 63 64

Maxwell, 1861–1862, 286.

Maxwell, 1861–1862, 285.

Maxwell, 1861–1862, 286.

Maxwell, 1861–1862. See also Maxwell, [1890] 1965, Plate VIII, Fig. 1, opposite

p. 488. See nn. 6, 24 and 37, above. Maxwell, 1861–1862, 339, italics in the original. Maxwell, 1861–1862, 289. Maxwell, 1861–1862, 290. Maxwell, 1861–1862, 291. Maxwell, 1861–1862, 345–346. Maxwell, 1861–1862, 346. Maxwell, 1861–1862, 85–86. Maxwell referred to Helmholtz’s paper on vortices and remarked: “Helmholtz. . . has pointed out that the lines of fluid motion are arranged according to the same laws with respect to the lines of rotation, as those by which the lines of magnetic force are arranged with respect to electric currents.” See Helmholtz, [1858] 1867; see also nn. 57–59, below. Maxwell, 1861–1862, 345. Maxwell, 1861–1862, 345. Maxwell, 1861–1862, 345. Maxwell, 1861–1862, 345. Maxwell, 1861–1862, 346. Maxwell, 1861–1862, 346. Interestingly, in Station 1 Maxwell gave the reader a similar assurance: see ch. 4, n. 40. See also Harman, 1998, 102–106. Maxwell, 1861–1862, 347. Siegel (1991, 83–84) perceptively remarked that Station 1 is in the Scottish trad­ ition, while Station 2 is in the Cambridge tradition. He went on to claim that Maxwell sought to develop a unified physical electromagnetic theory within the field-theoretic framework of the British tradition, which would be as successful as that of W. Weber in the continental, action-at-a-distance tradition. For the Scottish tradition, see Olson, 1975 and Wilson, 2009. Maxwell, 1861–1862, 348. See, e.g., Siegel, 1991, 1, 85–86, 124. Maxwell, 1861–1862, 13. Maxwell, 1861–1862, 13. Cf. Siegel, 1991, 78. Maxwell, 1861–1862, 13. Rankine (1851, 441–446) proposed a connection of molecular vortices with light. See, e.g., Siegel, 1991, 1, 121–122, 209. None of Maxwell’s predecessors came close to saying that electromagnetic disturbances are propagated at the speed of light and that the medium for electromagnetic disturbances is the same as that for light. Maxwell, 1858, 51. See also Siegel, 2014, 199.

See Maxwell, 1853.

Maxwell, 1858, 28, italics in the original. For the dress metaphor see, ch. 1, n. 48.

See, e.g., Weber, [1864] 1894, 157–158; trans. in Siegel, 1991, 210 n. 5. See also

n. 86, below. The initial claim was later reinforced in various ways and by a mathematical argument. For the claim see, e.g., Maxwell, 1861–1862, 22; and for the connec­ tion to elasticity see, e.g., Maxwell, 1865, 465–466, § 20. For a detailed account of the origin of the electromagnetic theory of light, see Siegel, 1991, ch. 5. See also Siegel, 1995. In contrast to Siegel, we are not interested in the origin of Maxwell’s theory; rather, we restrict our attention to Maxwell’s presentation of the theory in his published works, specifically the argument he offered to support his claims which, in turn, reflects his methodology.

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65 Maxwell, 1861–1862, 14–15. 66 Siegel, 1991, 85. We are inclined to accept the view of Siegel (ibid., 96–100), who related the discovery of the displacement current to modifying Ampère’s law. But we avoid going into details because the path to discovery is not germane to our argument. 67 Maxwell, 1861–1862, 13. 68 Maxwell, 1861–1862, 14. 69 Maxwell, 1861–1862, 14. 70 Maxwell, 1861–1862, 14. 71 This is Maxwell’s original terminology (ibid.). As far as we can determine, the first usage of “displacement current” is to be found in FitzGerald, [1879] 1902, 92. For a thorough study of the history of the concept of displacement current, see Roche, 1998; Siegel, 1991, ch. 4. See also Chalmers, 1975; Bromberg, 1967. 72 Maxwell, 1861–1862, 14. 73 See ch. 6, nn. 3 and 37: Maxwell, 1865, 487, § 73. See also Siegel, 1991, 145–146. 74 Maxwell, 1861–1862, 15. 75 An exception is Bromberg, 1967, 224–225, where the importance of this passage and the critical juncture of the introduction of elasticity are recognized. See also; Siegel, 1991, 145: “Maxwell indicated that he was looking into the question of the magnetic effects of closed circuits as well as open circuits in the most general way. . .. [His] efforts to divorce the new form of Ampere’s law from the molecular-vortex model were to bear fruit within two years: In 1864 he presented a reformulated version that was completely independent of the theory of molecular vortices.” See ch. 6, n. 68. 76 Maxwell, 1861–1862, 15. 77 Maxwell, 1861–1862, 15. Maxwell actually repeated this mechanical account which he had already given a couple of pages earlier (p. 13). See also Chalmers, 1975, 47–48. 78 For details, see Siegel, 1991, 107–112, 129–133. 79 Kohlrausch and Weber, [1857] 1893, 648–652. This ratio has the dimension of vel­ ocity: see, e.g., Darrigol, [2000] 2002, 399. The experiment was reported in Weber and Kohlrausch, 1856, but the note about the velocity of light comes in the revised discus­ sion of the experiment in Kohlrausch and Weber (ibid., 652). In the experiment, which was very delicate, a Leyden jar was discharged by a ballistic galvanometer, and then an electrostatic measure was taken of the jar’s loss of charge. Kohlrausch and Weber compared the former quantity, namely, the electrodynamic measure of the inte­ gral current, with the latter. For details, see Darrigol (ibid., 66). 80 Fizeau, 1849; Maxwell, 1861–1862, 22, see also p. 15. Maxwell gave the precise reference to this measurement in his later publication on this subject:, 1865, 499. On Fizeau’s experiment, see Frercks, 2000. 81 See Kohlrausch and Weber, [1857] 1893, 652. D’Agostino, 1996, 22: “Weber’s and Kohlrausch’s context of ideas, embedded as it was in a particle-based approach to electrodynamics permitted a mere numerical quasi-equality between c and the light velocity to have little significance.” See also, e.g., Darrigol, [2000] 2002, ch. 2: German precision. 82 Maxwell, 1861–1862, 13. 83 On the approaches to physics of Weber and Kohlrausch, see; Darrigol, [2000] 2002, 73–74. According to Darrigol, [2000] 2002, 74 “Neumann and Weber widely extended the quantification of electrodynamics, and thus started two important tra­ ditions of German physics. On the experimental side, they focused on precision measurement, whereas Faraday and Ampère rarely measured quantities. On the the­ oretical side, they aimed at complete mathematical theories of electrodynamics.”

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126 Station 2 (1861–1862) 84 Maxwell, 1861–1862, 21 n; and Maxwell, 1865, 499 n. See nn. 5 and 6, above, for Maxwell’s stated goal at the outset of Part I of Station 2 (1861–1862). 85 Maxwell, 1861–1862, 15. 86 Harman (ed.) 1990, 685–687, italics in the original. 87 Harman (ed.) 1990, 685–687, italics in the original. 88 Harman (ed.) 1990, 695. 89 In the letter to Thomson, Maxwell had more, and somewhat different, data for the velocity of light: see ibid. 90 Maxwell, 1861–1862, 22. Interestingly, Maxwell ascribed two different values to the velocity of light that are linked to Fizeau: 193,118 miles per second (based on Galbraith and Haughton, 1855, 36 and 39) vs. 195,647 miles per second (based on Maxwell’s own computation from the data given by Fizeau). Two different values for the same constant are given without any attempt to reconcile them. 91 For further discussion, see Bromberg, 1967, 226–227. According to Siegel (1991, 135), the “ordinary method” is the mechanics of elastic media. Duhem (1902, 210), quoted this passage in Maxwell’s “On physical lines of force” verbatim, and then (on p. 212) called attention to an error in Maxwell’s formula. It should read V = √(m/2ρ), for in this equation Maxwell used “m,” rather than “m/2”; cf. Siegel, 1991, 137–138, 212–213 n. 27; Chalmers, 1975, 48. On Maxwell’s mistakes and Duhem’s criticism of them, see; Ariew and Barker, 1986, 151–152. 92 Maxwell, 1861–1862, 21. 93 Maxwell, 1861–1862, 21; cf., 1865, 491–492. Maxwell cited Kohlrausch and Weber, 1857, 260, corresponding to Kohlrausch and Weber, [1857] 1893, 647–648. 94 Maxwell, 1861–1862, 22, italics in the original. In 1865, in the next station of his methodological odyssey, Maxwell asserted without qualification (1865, 499, § 97): “light is an electromagnetic disturbance propagated through the field according to electromagnetic laws.” Undoubtedly, this new formulation reflected Maxwell’s growing confidence in his way of analyzing the phenomena of electromagnetism. 95 Maxwell, 1861–1862, 22 (Prop. 16). 96 See n. 85, above. 97 Maxwell, 1861–1862, 161–162; see also 290–291. For the quotation, see ch. 2, n. 26. 98 Maxwell, 1861–1862, 162.

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Station 3 (1865) A dynamical theory of the electromagnetic field

6.1 Introduction Maxwell’s letters to Faraday and to Thomson in the last three months of 1861 (after the publication of Station 2, Parts I and II) indicate that Maxwell was committed to the physics of the vortices; in particular, the analogical use of the vortices is no longer in evidence. For Maxwell, at that time, the mechanical hypothesis of vortices was the explanation for electromagnetic phenomena. In Station 1 Maxwell deliberately avoided any causal explanation and therefore had no reason to address the micro-level—the talk was not physical, it was formal. This was not the case in Station 2, for here the vortices at the micro­ level are considered possible explanans. It may then be asked, did Maxwell also appeal to this hypothesis to account for the displacement current (even if the vortices had nothing to do with its discovery)? As we have seen, the answer is positive: in 1862 Maxwell used vortices in conjunction with the displacement current.1 It has been cogently argued that Maxwell’s claim in 1865 (Station 3) for light as an electromagnetic phenomenon is independent of the vortex hypothesis proposed in 1861–1862 (Station 2).2 Thus, only after 1862 did Maxwell realize that he had no need for the vortex hypothesis. Given the substantial contributions to electromagnetism in Stations 1 and 2, are there references to these fundamental contributions in Station 3? Does Station 3 continue the line of research pursued in the preceding stations? The textual evidence is unambiguous: in Station 3 there is no reference to Station 1 and only a single reference to Station 2. This latter reference is made deep into the paper near the end of Part III, where Maxwell contrasted his avoidance of hypotheses in Station 3 with his appeal to them in Station 2.3 Clearly, the paper of 1865 is not a continuation of the argument in 1861–1862, even though the results of the earlier paper are, for the most part, maintained in the new paper. In Station 3 Maxwell reversed the argument, assuming that electromagnetic disturbances are propagated in the same medium as that of light and at the same velocity; he further presupposed (among other things) the existence of a kind of current for which in Station 2 he coined the expression “current due to displacement.” In other words, in Station 3 Maxwell considered his discoveries of Station 2 assumptions because they had the same status as Faraday’s

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128 Station 3 (1865) discoveries—they belong to the category “facts.” This reversal of the argument called for a different methodology from those which Maxwell employed in Stations 1 and 2. This third methodology is the subject of Chapter 6.

6.2 Part I: marking the goal—the construction of a formal theory consisting of a set of general equations Maxwell began his paper of 1865 by reflecting on the core physical issue, namely, “the mutual action by which bodies in certain states set each other in motion while still at a sensible distance from each other.”4 Once again he emphasized that, in order to “reduce the phenomena to scientific form” so that they can be treated mathematically, it is important to ascertain the magnitude and direction of the forces acting between the bodies.5 Maxwell acknowledged that the theory which best accomplished this goal at the time was that of Wilhelm Weber, a theory based on particles acting at a distance. But Maxwell maintained that a better way was to introduce a new concept—the electromagnetic field—and to render the theory dynamical. The theory I propose may therefore be called a theory of the Electromag­ netic Field, because it has to do with the space in the neighbourhood of the electric or magnetic bodies, and it may be called a Dynamical Theory, because it assumes that in that space there is matter in motion, by which the observed electromagnetic phenomena are produced.6 Maxwell called his theory of the electromagnetic field a dynamical theory because it assumes a field surrounding electrified or magnetic bodies, in which matter in motion subject to forces produces the observed electromagnetic phenomena. The task now was not to offer an explanatory mechanism by which the phenomena are produced, as Maxwell had sought in Station 2; rather, Maxwell’s goal in Station 3 was to construct a satisfactory formal theory that conforms with the phenomena.7 Given the novel concept of the electromagnetic field, Maxwell felt obliged to admit that the undulations [of any wave phenomena, be it light or radiant heat] are those of an aethereal substance, and not of the gross matter, the presence of which merely modifies in some way the motion of the aether.8 It is noteworthy that Maxwell specified that the motion of the “aetherial substance” produces electromagnetic phenomena, for it is the cause of these phenomena. This is, on all accounts, physical talk. With the insights he had obtained in Station 2, Maxwell appealed in Station 3 to the ether as the medium, the locus of undulatory phenomena be they light, heat or, indeed, electromagnetic phenomena. But Maxwell did not wish the reception of his theory to depend on accepting some

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“hypothesis.” To be sure, the medium does the work; therefore, it is better to frame the argument at the level of the phenomena rather than assuming some hypothesis at the micro-level. It appears therefore that certain phenomena in electricity and magnetism lead to the same conclusion as those of optics, namely, that there is an aetherial medium pervading all bodies, and modified only in degree by their presence; that the parts of this medium are capable of being set in motion by electric currents and magnets; that this motion is communicated from one part of the medium to another by forces arising from the connexions of those parts; that under the action of these forces there is a certain yielding depending on the elasticity of these connexions; and that therefore energy in two different forms may exist in the medium, the one form being the actual energy of motion of its parts, and the other being the potential energy stored up in the connexions, in virtue of their elasticity.9 Maxwell avoided making a strong unifying claim about the phenomena of electromagnetism and light: “It appears . . . [that certain phenomena in electromagnetism and optics] lead to the same conclusion.” To be sure, Maxwell was reasonably certain about this claim, and it is no longer a matter of “belief”;10 still, as we will see below, Maxwell formulated this empirical claim cautiously, for he did not have a proof. It is evident that despite the hesitant language, Maxwell considered the relation between electromagnetism and light as one of the results that should be presupposed in the new theory. Maxwell identified two kinds of energy that can exist in the medium: “actual” and “potential.” The former is dependent on the motions of its parts, the latter on the work the medium does in recovering from its displacement by virtue of its elasticity. In this approach a specific mechanical scheme is no longer necessary, for the medium takes on the physics of electromagnetism. Hence, the fundamental property of the medium, namely, its elasticity, becomes essential, and both kinds of energy are stored in the medium in virtue of its elasticity. This idea—introduced by Rankine—became most productive in Station 4. However, in Station 3, the two kinds of energy are mentioned, but they do not play the crucial role they were to play later on.11 In Part I of Station 3 Maxwell recalled the great discovery reported in Station 2, namely: the relation between the units employed in the two methods [i.e., dynamics and electrostatics] is shewn to depend on what I have called the “electric elasticity” of the medium, and to be a velocity, which has been experimentally determined by MM. Weber and Kohlrausch.12 Maxwell continued in this first part of the paper in the same vein, explaining the relations and values that the general equations of the anticipated formal theory are intended to capture.

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130 Station 3 (1865)

6.3 Part II: the flywheel analogy In opening Part II and getting into the details of the theory, Maxwell showed that he did not divorce himself completely from the use of analogy: Now, if the magnetic state of the field depends on motions of the medium, a certain force must be exerted in order to increase or diminish these motions, and when the motions are excited they continue, so that the effect of the connexion between the current and the electromagnetic field sur­ rounding it, is to endow the current with a kind of momentum, just as the connexion between the driving-point of a machine and a fly-wheel endows the driving-point with an additional momentum, which may be called the momentum of the fly-wheel reduced to the driving-point. The unbalanced force acting on the driving-point increases this momentum, and is meas­ ured by the rate of its increase.13 “Motions of the medium” as the seat of magnetism was part of Maxwell’s fundamental commitment, but we see that here it is not linked to any specific mechanism. Rather, the appeal to an analogy is evident. The argument is indeed analogical in the traditional sense: the effect of the connection between A and B is to endow A with some property just as the connection between D and E endows D with some property. The former relation is electromagnetic, the latter is purely mechanical. It should be noted that at this juncture Maxwell did not refer to the mechanism of a flywheel as an analogy; rather, he called it “illustration.” Nevertheless, the structure of the argument is analogical.14 6.3.1 How does the analogy work? As in Station 2, in Station 3 Maxwell introduced a mechanical scheme; in fact, a mechanical device—the flywheel. It is placed at the macro-level, without assuming a controlling role like the rack and pinion in Station 2, for it is not meant to regulate anything. Rather, the introduction of the flywheel is intended to offer insight into the working of the medium as the seat of energy; specifically, it was designed as an appeal to an analogy between mechanics and electromagnetism, where the property at issue is reduced momentum in both physical domains. In general, a flywheel is a heavy revolving wheel in a machine that is used to increase the machine’s momentum and thereby provide greater stability or a reserve of available power. It is a rotating mechanical device that is used to store rotational energy. Flywheels have an inertia called the moment of inertia and thus resist changes in rotational speed. Energy is transferred to a flywheel by the application of a torque to it, thereby increasing its rotational speed, and hence its stored energy. Conversely, a flywheel releases stored energy by applying torque to a mechanical load, thereby decreasing the flywheel’s rotational speed.

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Given the mechanical-electromagnetic analogy, Maxwell developed a “dynamical illustration of reduced momentum” in electromagnetism in terms of mechanics.15 He assumed a body connected with two drivingpoints which puts it in motion. Maxwell proceeded with the analysis of this scheme by appealing to the Mécanique analytique of Joseph-Louis Lagrange (1736–1813) and depending on “the general equation of dynamics.”16 The analysis allows Maxwell to specify the momentum of each driving-points when forces are applied at each of them. Maxwell proceeded in this way. He framed the mechanical scheme as a dynamical illustration: Let us suppose a body C so connected with two independent drivingpoints A and B that its velocity is p times that of A together with q times that of B. Let u be the velocity of A, v that of B, and w that of C, and let δx, δy, δz be their simultaneous displacements . . .17 Introducing two forces, X and Y, which act on A and B, respectively, and applying Lagrange’s general equation of dynamics, Maxwell was able to determine the momentum of C, which is referred to A and to B, respectively. He found the effect of the force X in increasing the momentum of C referred to A and, similarly, the effect of the force Y in increasing the momentum of C referred to B. The next step was to generalize the problem by linking many bodies with A and B, and by summing the effects of all these bodies. Maxwell then calculated the momentum of the system referred to A and that referred to B, given the external forces X and Y, acting, respectively, on A and B. In this way the total momentum of the system can be calculated. Maxwell developed this purely mechanical discussion in response to an earlier remark under the heading, Mutual Action of two currents: If there are two electric currents in the field, the magnetic force at any point is that compounded of the force due to each current separately, and since the two currents are in connexion with every point of the field, they will be in connexion with each other, so that any increase or diminution of the one will produce a force acting with or contrary to the other.18 The crucial element in this remark is of course the field effect, that the two currents are connected with every point of the field, a fact which inevitably renders the two currents coupled. This is indeed what Maxwell attempted to capture with the mechanical analog: To make the illustration more complete we have only to suppose that the motion of A is resisted by a force proportional to its velocity, which we may call Ru, and that of B by a similar force, which we may call Sv, R and S being coefficients of resistance.19

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132 Station 3 (1865) Maxwell was now in a position to calculate the coupling of the forces and determine the momentum: “If the velocity of A be increased at the rate of du/dt, then in order to prevent B from moving a force, η = (d/dt) (Mu) must be applied to it.”20 In other words, the coupling leads to the question, what should be the force, η, to be applied to B in order to prevent it from moving as the velocity of A is increased? The coupling that takes place in driving the flywheel is well described in Lazaroff-Puck’s paraphrase: As a result of the increase in velocity at A there is an indirect force on B, −(d/dt)(Mu), that is to be canceled by η. Acceleration of the drivingwheel A causes an acceleration of the passive wheel B in the opposite direction. The force described that would “prevent B from moving” is then just the force we would have to apply to B to stop it accelerating as a result of the acceleration of A. The stopping force at B is the opposite of the force applied to B by our acceleration of the driving-wheel A.21 When a force is applied to the driving-wheel at A it results in the increase of the momentum of the flywheel C. This increase in the flywheel’s rotational energy is defined as the reduced momentum of A; the flywheel’s momentum is considered reduced since it decreases over time as the momentum is delivered to the passive driving-wheel B. Thus, as Lazaroff-Puck puts it: The force on B as a result of action at A is mediated . . . by the flywheel C, which Maxwell comes to see as the embodiment of the electromag­ netic field . . . This process of force transferal through the flywheel is the mechanical analogue of electromagnetic induction; in the electromag­ netic case, the field moderates the transferal of forces between circuits.22 Indeed, according to Maxwell: This effect on B, due to an increase of the velocity of A, corresponds to the electromotive force on one circuit arising from an increase in the strength of a neighbouring circuit.23 This statement is the key to understanding the connection between the mechanism of the flywheel and an electromagnetic phenomenon, namely, induction. The critical move in the analogy is the term “corresponds.” In Part I Maxwell already called attention to an important feature of the medium: According to the theory which I propose to explain, this “electromotive force” is the force called into play during the communication of motion from one part of the medium to another, and it is by means of this force that the motion of one part causes motion in another part. When electromotive force acts on a conducting circuit, it produces a current,

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which, as it meets with resistance, occasions a continual transformation of electrical energy into heat, which is incapable of being restored again to the form of electrical energy by any reversal of the process.24 “Communication of motion from one part of the medium to another” is what makes the theory dynamical, and the flywheel C captures precisely this feature in an analogical way. Lazaroff-Puck formulates the analogy thus: Just as the flywheel’s momentum decreases as it acts on B, the electro­ magnetic momentum of the field decreases as a result of its action on a circuit. The induced current is such that it produces a force which acts counter to the change in current at A that produced the induced elec­ tromotive force.25 By “electromagnetic momentum” Maxwell meant “to express that which is generated by a force acting during a time, that is, a velocity existing in a body.”26 Of course, the physical domain of electromagnetism is not that of mechanics, but insight from the latter can be projected onto the former as long as the two domains are linked by some relation, e.g., an analogical relation. Indeed, as we have seen, Maxwell constructed this linkage, but he acknowledged its problems: In the case of electric currents, the force in action is not ordinary mechan­ ical force, at least we are not as yet able to measure it as common force, but we call it electromotive force, and the body moved is not merely the electricity in the conductor, but something outside the conductor, and cap­ able of being affected by other conductors in the neighbourhood carrying currents. In this it resembles rather the reduced momentum of a drivingpoint of a machine as influenced by its mechanical connexions, than that of a simple moving body like a cannon ball, or water in a tube.27 Maxwell was certain that in the case of electric currents there is some force which he called, following Faraday, electromotive force, but it could not easily be compared to mechanical force. Maxwell explicitly stated that the force in action in electromagnetism is not ordinary mechanical force, and what is moved is not at all clear, for the moving body is not the electricity in the conductor; rather, it is something external to the conductor and can be affected by other conductors through the medium. The closest mechanical analogy he could find was that of the reduced momentum of a driving point which, he claimed, was better than comparing to a simple moving body. Clearly, Maxwell was not satisfied with this attempt to cast electromotive force in mechanical terms. Put differently, the problem Maxwell faced was that the nature of the electromotive force was undetermined and, therefore, the relation between the concepts in electromagnetism and those in

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134 Station 3 (1865) mechanics had not been determined. Interestingly, Maxwell invoked “resemblance” between the two distinct physical domains, mechanics and electromagnetism, giving the impression that he had only found a less than satisfactory connection between them. As we will see, Maxwell did not resolve this fundamental problem; rather, he found a way to circumvent it by invoking the concept of energy. This concept is well determined and it is the same whether mechanical or electromagnetic.28 6.3.2 Illustration vs. analogy At the end of the preliminary discussion in Part II of Station 3, in which the flywheel analogy was set up, Maxwell remarked that “this dynamical illustration is to be considered merely as assisting the reader to understand what is meant in mechanics by Reduced Momentum.”29 Notwithstanding Maxwell’s remark, we consider the flywheel an analogy since the structure of Maxwell’s argument is analogical. Indeed, deep into Part II Maxwell remarked: It appears . . . that if we admit that the unresisted part of electromotive force goes on as long as it acts, generating a self-persistent state of the current, which we may call (from mechanical analogy) its electromag­ netic momentum, and that this momentum depends on circumstances external to the conductor, then both induction of currents and electro­ magnetic attractions may be proved by mechanical reasoning.30 Thus Maxwell did consider the mechanics of the flywheel an analogy as well as an illustration. Either as an analogy or as an illustration, the flywheel offered proofs “by mechanical reasoning” of the role of “circumstances external to the conductor,” in other words, the role of the medium. The flywheel C is then the analog of the electromagnetic field. To be specific, the connection between the driving point and a flywheel, which endows the driving point with an additional momentum, is an illustration and not part of the proof of the way the medium works; it is designed to facilitate progress with mechanical reasoning. The illustration functions then as a plausibility argument. We note that Maxwell used the modal form: “may be proved by mechanical reasoning.” We call the mechanism of the flywheel as invoked by Maxwell at the outset of Part II of Station 3 “strong analogy”: consequences in one domain (e.g., mechanics) are applied to a different domain (e.g., electromagnetism). In this case inferences drawn from the flywheel are applied to induced current.31 For an example of the use of “weak analogy” in 1865 we turn to Maxwell’s preliminary remarks in setting up the background for the set of empirical formulations to be turned into equations (§ 18): We have here two other kinds of yielding besides the yielding of the perfect dielectric, which we have compared to a perfectly elastic

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body. The yielding due to conductivity may be compared to that of a viscous fluid (that is to say, a fluid having great internal friction), or a soft solid on which the smallest force produces a permanent alteration of figure increas­ ing with the time during which the force acts. The yielding due to electric absorption may be compared to that of a cellular elastic body containing a thick fluid in its cavities. Such a body, when subjected to pressure, is com­ pressed by degrees on account of the gradual yielding of the thick fluid; and when the pressure is removed it does not at once recover its figure, because the elasticity of the substance of the body has gradually to overcome the ten­ acity of the fluid before it can regain complete equilibrium. Several solid bodies in which no such structure as we have supposed can be found, seem to possess a mechanical property of this kind ‡; and it seems probable that the same substances, if dielectrics, may possess the analogous electrical property, and if magnetic, may have corresponding properties relat­ ing to the acquisition, retention, and loss of magnetic polarity.32 [Footnotes in Maxwell, 1865, 463] ‡ As, for instance, the composition of glue, treacle, &c., of which small plastic figures are made, which after being distorted gradually recover their shape. Maxwell was concerned with characterizing the elasticity of the medium; he sought to obtain an idea of how it functions without attempting to explain it. So the move is “to compare the medium to” X or, to liken it to X, and to speak with imprecise terms, such as “kinds of.” But no inferences are drawn from the properties of viscous fluids, soft solids, etc. These comparisons are evidently weak analogies which function as illustrations. However, the flywheel analogy is completely different, it must have a logically consistent structure so that consequences can be inferred from its properties. In Station 1 Maxwell had used an analogy from which he drew inferences, but this was contrived, appealing to the flow of an imaginary fluid; here, in Station 3, the analogy is between a physically real mechanical system and electromagnetism—it is neither contrived nor imaginary. Furthermore, in Station 1 the analogy concerns laws governing the respective phenomena; but here, in Station 3, the analogy is phenomenalist, it is restricted to the phenomena of a mechanical system and electromagnetic system and not to the governing laws. Notwithstanding these differences, the two analogies are strong, for consequences are drawn from the properties of the respective analog, namely, the flow of incompressible fluid in tubes of various crosssections and the rotation of a flywheel. Maxwell, however, remarked in Station 3 that the mechanics of a flywheel is more suitable to the task than either the motion of a projectile or the flow of fluid. The latter had been central to the physical analogy of tubes as lines of force (in Station 1), while

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136 Station 3 (1865) the former—“the reduced momentum of a driving-point of a machine as influenced by its mechanical connexions”—was relevant to induction (in Station 3).33 Maxwell wished the reader to find plausible his account of reduced momentum in electromagnetism, and the flywheel is an illustration of reduced momentum in a mechanical setting. He was certainly not claiming that there are flywheels operating in electromagnetic phenomena. This stands in strong contrast to the molecular vortices in Station 2. In a letter to Tait, dated 23 December 1867, Maxwell responded to receiving Tait’s Histories of thermodynamics & energetics: There is a difference between a vortex theory [Maxwell’s Station 2] . . . and a dynamical theory of Electromagnetics [Maxwell’s Station 3] . . . The former is built up to show that the phenomena are such as can be explained by mechanism. The nature of this mechanism is to the true mechanism what an orrery is to the Solar System. The latter is built on Lagrange’s Dynamical Equation and is not wise about vortices.34 According to Maxwell, Lagrange’s equation is indifferent to the possible existence of vortices, for it does not address the issue of vortices. It is clear, then, that in Station 3 Maxwell did not offer an explanation of phenomena; rather, he sought their formal description in a dynamical theory.35 Following the introduction of the flywheel analogy, Maxwell added: In the case of electric currents, the resistance to sudden increase or dim­ inution of strength produces effects exactly like those of momentum, but the amount of this momentum depends on the shape of the con­ ductor and the relative position of its different parts.36 The expression “produces effects exactly like those of momentum” constitutes a plausibility argument; there is no claim that flywheels are really present in electromagnetic phenomena. This is the role of illustration/ analogy for Maxwell in Station 3. In this station Maxwell made no claim for a “hypothesis” at the micro-level.

6.4 The methodology of reversing the argument At the outset of Station 3 Maxwell made it clear that he had put aside his hypothesis of molecular vortices of Station 2 and adopted a different methodology. Apparently, Maxwell was not satisfied with the fact that one of his great discoveries, namely, that the medium for light is the same as the medium for electromagnetic phenomena, was dependent in Station 2 on a specific mechanical scheme, for a proper theory of electromagnetism should be free of a mechanical scheme. Understanding that the displacement

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current need not depend on the vortex hypothesis presumably strengthened his belief that a new deductive argument could be found. The theory should be linked directly to empirical facts, expressed in symbolic language, and no contingent mechanism should be made responsible for the phenomena. Avoiding such hypotheses was part of Faraday’s legacy, to which Maxwell was committed. In Maxwell’s view, Faraday based his work on observational “facts” which Maxwell then cast in “mathematical terms.” In closing Part III of Station 3, Maxwell made the change in methodology explicit: I have on a former occasion* attempted to describe a particular kind of motion and a particular kind of strain, so arranged as to account for the phenomena. In the present paper I avoid any hypothesis of this kind; and in using such words as electric momentum and electric elasti­ city in reference to the known phenomena of the induction of currents and the polarization of dielectrics, I wish merely to direct the mind of the reader to mechanical phenomena which will assist him in under­ standing the electrical ones. All such phrases in the present paper are to be considered as illustrative, not explanatory.37 [Footnotes in Maxwell, 1865, 487] * “On Physical Lines of Force,” Philosophical Magazine, Maxwell, 1861–1862. This is a key passage. Maxwell stated his approach: (1) the theory is not dependent on any hypothesis; (2) any suggested mechanical phenomenon, e.g., elasticity, is to assist understanding; and (3) above all, phrases such as “electric momentum” and “electric elasticity” are illustrative not explanatory. In other words, and this is most important, all of Maxwell’s references to mechanical phenomena are didactic—they are not part of the argument. As we have seen, in Station 2 Maxwell reflected on his methodology in Station 1, where “mechanical illustrations [were put forward] to assist the imagination, but not to account for the phenomena.”38 Later, in 1870, he characterized “scientific illustration” as “a method to enable the mind to grasp some conception or law in one branch of science, by placing before it a conception or a law in a different branch of science.”39 An illustration, then, is designed to be an aid for understanding, and not to function as an explanans. The critical point is that the theory does not depend on any assumption concerning the micro-level. The approach of Station 3 stands in contrast to Maxwell’s contributions to electromagnetism in Station 1 and Station 2. This change resulted in abandoning explanation and relying on illustration. We are back to physics at the macro-level. The important results from Station 2 had to be derived from a new set of assumptions. Maxwell was now in possession of the solution: the problem was to find a new deductive path which is not dependent on a mechanical

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138 Station 3 (1865) hypothesis. It is then not surprising that Maxwell suggested reversing the order of the argument: The second result, which is deduced from this, is the mechanical action between conductors carrying currents. The phenomenon of the induc­ tion of currents has been deduced from their mechanical action by Helmholtz* and Thomson†. I have followed the reverse order, and deduced the mechanical action from the laws of induction.40 [Footnotes in Maxwell, 1865, 464] * “Conservation of Force,” Physical Society of Berlin, 1847; and Taylor’s Scientific Memoirs, 1853, p. 114. † Reports of the British Association, 1848; Philosophical Magazine, Dec. 1851. This is a good juncture to take stock of the argument of Station 3. Prior to introducing the illustration of the flywheel in Part II (§ 22), Maxwell listed in Part I the known electromagnetic phenomena, including induction (§§ 15–18), repeating his commitment to a medium. Maxwell acknowledged that the mechanism of the medium, the bearer of electromagnetic phenomena, is complicated and must be subject to the general laws of dynamics (§ 16). He then focused on the role of the medium for connecting the electric currents in each part of the field, and announced that he “followed the reverse order [with respect to Helmholtz], and deduced the mechanical action from the laws of induction” (§ 17). The direction then is from electromagnetic phenomena to mechanics, and not vice versa. Importantly, the two discoveries of Station 2 were now treated as facts just like Faraday’s discoveries.41 Indeed, as Maxwell claimed, he expressed “the intrinsic energy of the Electromagnetic Field as depending partly on its magnetic and partly on its electric polarization at every point.” And from this energy he determined “the mechanical force acting, 1st, on a moveable conductor carrying an electric current; 2ndly, on a magnetic pole; 3rdly, on an electrified body.”42 The introduction of the concept of energy facilitated a new perspective on electromagnetic phenomena: In speaking of the Energy of the field, however, I wish to be understood literally. All energy is the same as mechanical energy, whether it exists in the form of motion or in that of elasticity, or in any other form. The energy in electromagnetic phenomena is mechanical energy. The only question is, Where does it reside? On the old theories it resides in the electrified bodies, conducting circuits, and magnets, in the form of an unknown quality called potential energy, or the power of producing certain effects at a distance.43 Maxwell found it difficult to relate force in electromagnetism with force in mechanics. Force is a cause, whereas energy is not. Thus, invoking force

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suggests the necessity for specifying some structure at the micro-level that is causal, but energy has no such requirement. The problem then disappears when it comes to energy. Maxwell stated that energy is energy regardless of the physical domain. In other words, replacing force with energy as a fundamental concept solves the problem, and sets the stage for a novel methodology. This new approach came to fruition in Station 4. But then the introduction of the concept of energy posed a new problem, namely, that of its seat.44 At the outset of Part II (§ 22), Maxwell associated the current with a “kind of momentum.” What “kind of momentum”? This is imprecise talk. Mechanics was a highly developed discipline and momentum was a well understood fundamental concept in this physical domain. Maxwell, however, needed a plausibility argument, for it was not immediately obvious that a current can have momentum, let alone a reduced one, particularly since the nature of electricity and magnetism was not specified. It is difficult to see how consequences can be drawn from positing momentum of a current without appealing to mechanical momentum. Hence the need for an illustration of momentum, something corresponding to the medium for electromagnetic phenomena, namely, a bearer of momentum that transfers it from one part of some mechanical arrangement to another part, and the flywheel is one such mechanism. The order is reversed with respect to Helmholtz in the sense that one begins with induction—a phenomenon— and then seeks a mechanical illustration, the flywheel, that offers insights into the electromagnetic phenomenon. As we have seen (ch. 6, § 6.3), Maxwell considered the flywheel an illustration and then, deep into Part II, he applied it as a mechanical analogy to facilitate calculations. Maxwell treated his discoveries and those of Faraday as sharing the same status as phenomena. By turning his own discoveries as well as those of Faraday into assumptions, Maxwell was able to reformulate the theory. Thus, one aspect of the methodology of Station 3 is that Maxwell reversed the argument: mechanical action is not the cause; rather, it is the effect deduced from the law of induction. In Station 3 Maxwell took as assumptions (among others): (1) lines of force (as before, Faraday’s conceptual framework), (2) electromagnetic disturbances are propagated in the same medium as that of light and at the same velocity (a result in Station 2), and (3) the displacement current (a result in Station 2). There is, therefore, no need to “prove” any of this. The correctness of these assumptions is confirmed by electromagnetic phenomena for which they account, while the path to these assumptions is irrelevant for the argument. In late 1861 Maxwell was persuaded that electromagnetic disturbances were propagated at the speed of light but he had no proof, except what might have been a chance numerical agreement, and was reluctant to make a bold claim in print. In 1865 he still had no “proof” but, with the methodology of “reversal,” he no longer needed one.

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140 Station 3 (1865) The methodology of “reversal” works by inverting the argumentative structure. An argument consists of a triad: assumption(s), rule(s) of inference, and conclusion(s). Since the elements that comprise the triad are tied together by logical relations, their role in the triad can be reversed. The conclusion(s) may be presupposed and, with the appropriate rule(s) of inference, what had originally been considered assumption(s) become, after the reversal, the conclusion(s). This is a powerful methodology that can overcome profound difficulties.45 Maxwell’s methodology of reversal highlights his shifting attitude toward the fundamental concept of “lines of force.” His presentation of the concept changed from Station 1 to Station 2 and again in Station 3; in the course of time Maxwell became less reticent. Originally, he introduced the concept in Station 1 as an alternative to “action at a distance”—it works just as well, even slightly better;46 then, in Station 2 he considered “lines of force” a true account at the molecular level while arguing that “action at a distance” is problematic, for it fails to account for some of Faraday’s experimental laws.47 But it did have its supporters among the continental physicists as well as prominent British practitioners such as Thomson.48 However, in Station 3, there is no longer any need for an argument to ground the choice of “lines of force” over “action at a distance”; the concept is now taken for granted as an assumption. Consider the following passage: Now we know that the luminiferous medium is in certain cases acted on by magnetism; For Faraday* discovered that when a plane polarized ray traverses a transparent diamagnetic medium in the direction of the lines of magnetic force produced by magnets or currents in the neigh­ bourhood, the plan of polarization is caused to rotate.49 [Footnotes in Maxwell, 1865, 461] * Experimental Researches, Series 19 Here is another example: I then apply the phenomena of induction and attraction of currents to the exploration of the electromagnetic field, and the laying down systems of lines of magnetic force which indicate its magnetic properties. By exploring the same field with a magnet, I shew the distribution of its equipotential magnetic surfaces, cutting the lines of force at right angles.50 And, then, importantly, he included the concept in setting up the field equations.51 In fact, Maxwell dedicated a whole section to the theme, “On Lines of Magnetic Force,” in which he constructed the geometry of magnetic lines of forces and discussed the physical consequences of this structure.52 Throughout these passages we see that Maxwell no longer mentioned “action at a distance.” As we indicated at the outset of this section, Maxwell opened the

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paper with a reference to Weber’s theory which is based on “action at a distance,” but he rejected it right away. Maxwell then proceeded to develop his own theory purely in terms of the concept of “lines of force.” This concept is now presupposed as the fundamental, unifying concept of the theory. With his complete confidence in the unifying concept of lines of force together with the reversal of the central argument, Maxwell was ready to discuss the nature of propagation of a disturbance in the medium. He concentrated on those disturbances which are transverse to the direction of propagation and whose velocity—as found in Weber’s experimental work— is expressed in “the number of electrostatic units of electricity which are contained in one electromagnetic unit.” He then added: This velocity is so nearly that of light, that it seems we have strong reason to conclude that light itself (including radiant heat, and other radiations if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field accord­ ing to electromagnetic laws. If so, the agreement between the elasti­ city of the medium as calculated from the rapid alternations of luminous vibrations, and as found by the slow processes of elec­ trical experiments, shews how perfect and regular the elastic proper­ ties of the medium must be when not encumbered with any matter denser than air.53 In Station 2 Maxwell had claimed that light and the propagation of electromagnetic disturbances take place in the same medium, but had not said explicitly that light is an electromagnetic phenomenon.54 Here, in Station 3, Maxwell presented the argument in a novel way, namely, that the two phenomena are propagated in the same medium. In other words, light is an electromagnetic disturbance, propagated according to electromagnetic laws. However, this claim is limited by the cautious expression: “it seems we have a strong reason to conclude.” In Part IV of the paper, the same claim is repeated: The agreement of the results seems to shew that light and magnetism are affections of the same substance, and that light is an electromag­ netic disturbance propagated through the field according to electromag­ netic laws.55 Here Maxwell’s cautiousness is signaled by the expression “seems to shew” and, in another passage, “may have” serves the same purpose: Magnetic disturbances propagated through the electromagnetic field agree with light in this, that the disturbance at any point is transverse to the direction of propagation, and such waves may have all the prop­ erties of polarized light.56

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142 Station 3 (1865) It appears that Maxwell was not ready to claim certainty for his findings; however, for the methodology of reversal, it was sufficient to assume these findings as phenomena, namely, “facts” based on experimental evidence. Overall, Part I of the paper sets the scene for the new theory: it is an introduction and summary of the new theory whose presentation begins in Part II. Specifically, the term for the displacement current complements Ampère’s law, and the claim that light is propagated as transverse undulations in the same medium, which is the cause of electric and magnetic phenomena, is stated cautiously. But while the claims are maintained, the arguments that support them are different from those in his previous publications.

6.5 Intermediate summary Deep into the process of developing his theory, at the outset of Part VI of the paper, Maxwell reflected on his moves: We made use of the optical hypothesis of an elastic medium through which the vibrations of light are propagated, in order to shew that we have war­ rantable grounds for seeking, in the same medium, the cause of other phe­ nomena as well as those of light. We then examined electromagnetic phenomena, seeking for their explanation in the properties of the field which surrounds the electrified or magnetic bodies. In this way we arrived at cer­ tain equations expressing certain properties of the electromagnetic field.57 It should be noted that the reference here to hypothesis does not affect the claim which Maxwell made, namely, that no hypothesis was involved in Station 3. The so-called “optical hypothesis” merely refers to the fact that the source for the “hypothesis” of the medium had been derived from discussions of optics in Station 2. There Maxwell referred to the undulatory theory of light which requires elasticity in the luminiferous medium in order to account for transverse vibrations. This usage is quite different from Maxwell’s appeal in Station 2 to the molecular vortex hypothesis. In this intermediary summary Maxwell spelled out the steps he had taken thus far: (1) he began by finding “warrantable grounds” for assuming that electromagnetic phenomena behave like light in a medium whose property of elasticity is singled out; (2) he then examined the relevant phenomena, seeking to explain them with the properties of the medium; and finally, (3) he recast these properties into formal equations. Still, Maxwell was not entirely content with such a formal theory and, as in Station 2, here too—in Station 3—he sought an explanation, and a causal one at that: When a body is moved across the lines of magnetic force it experiences what is called an electromotive force . . . When the electromotive force is sufficiently powerful, and is made to act on certain compound bodies, it

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decomposes them, and causes one of their components to pass towards one extremity of the body, and the other in the opposite direction. Here we have evidence of a force causing an electric current in spite of resistance . . .58 Note Maxwell’s claim that he had evidence in support of a causal explanation without any reference to the micro-level. He was concerned with the forces involved that gave credence to calling his theory “dynamical.” The concept of force is causal by its very nature; there was no way to avoid this causal talk, so Maxwell—it seems—restricted it to the electromotive force at the macro-level. Although the tenor in Station 3 is essentially formal, the causal talk was inevitable. Indeed, in setting the electromagnetic phenomena as background to the formal discussion, Maxwell made several pronouncements that clearly show that there was no escape from causal talk. Consider the following passages from Part I of Station 3: This, then, is a force acting on a body caused by its motion through the electromagnetic field, or by changes occurring in that field itself; and the effect of the force is either to produce a current and heat the body, or to decompose the body, or, when it can do neither, to put the body in a state of electric polarization,—a state of constraint in which oppos­ ite extremities are oppositely electrified, and from which the body tends to relieve itself as soon as the disturbing force is removed.59 But when electromotive force acts on a dielectric it produces a state of polarization of its parts similar in distribution to the polarity of the parts of a mass of iron under the influence of a magnet, and like the magnetic polarization, capable of being described as a state in which every particle has its opposite poles in opposite conditions.60 The relation between the electromotive force and the amount of electric displacement it produces depends on the nature of the dielectric, the same electromotive force producing generally a greater electric displacement in solid dielectrics, such as glass or sulphur, than in air.61 The italicized verbs clearly indicate causality: “cause,” “produce,” “act,” “occasion.” Maxwell linked his “dynamical theory” to the groundbreaking findings in Station 2. He had been led, so he remarked, to the conception of a complicated mechanism capable of a vast variety of motions, but at the same time so connected that the motion of one part depends, according to definite relations, on the motion of other parts, these motions being communicated by forces arising from the relative displacement of the connected parts, in virtue of their elasticity. Such a mechanism must be subject to the general laws of Dynamics, and

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144 Station 3 (1865) we ought to be able to work out all the consequences of its motion, pro­ vided we know the form of the relation between the motions of the parts.62 Although Maxwell spoke of “mechanism,” indeed, a complicated one, he did not offer any details or even refer to the illustrations that accompanied these discoveries. There is no specific mechanism to illustrate the elasticity of the medium, which is its sole physical property responsible for the phenomena. In the final analysis, the dominant methodological strategy was to move from a causal mechanical explanation to a formal description of the phenomena.

6.6 A physical theory in symbolic language In Station 3 Maxwell began afresh by reviewing the state of the art of electromagnetism. He discussed several phenomena informally, which he called “results”;63 in some sense these items are conceptually driven generalizations based on observations. Most importantly, they are descriptions of the phenomena, presented in Faraday’s narrative style. The survey forms the basis for the theory of 1865; all the data are displayed without any measurements or algebraic notation, for the various descriptions are qualitative. The first step then is in the tradition of Faraday: a succinct account of several electromagnetic phenomena. For example, We may now consider another phenomenon observed in the electromag­ netic field. When a body is moved across the lines of magnetic force it experiences what is called an electromotive force; the two extremities of the body tend to become oppositely electrified, and an electric current tends to flow through the body . . .64 And for another example which is taken directly from Faraday, When electromotive force acts on a dielectric it produces a state of polarization of its parts similar in distribution to the polarity of the parts of a mass of iron under the influence of a magnet . . .65 These are all empirical observations; to be sure, they are rich in conceptual insights but, nevertheless, empirical. Maxwell then proceeded to consider a mechanism at the micro-level. The following passage is unparalleled in Station 3, and it is worthy of extensive quotation: In a dielectric under the action of electromotive force, we may conceive that the electricity in each molecule is so displaced that one side is ren­ dered positively and the other negatively electrical, but that the electri­ city remains entirely connected with the molecule, and does not pass

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from one molecule to another. The effect of this action on the whole dielectric mass is to produce a general displacement of electricity in a certain direction. This displacement does not amount to a current, because when it has attained to a certain value it remains constant, but it is the commencement of a current, and its variations constitute currents in the positive or the negative direction according as the dis­ placement is increasing or decreasing. In the interior of the dielectric there is no indication of electrification, because the electrification of the surface of any molecule is neutralized by the opposite electrifica­ tion of the surface of the molecules in contact with it; but at the bounding surface of the dielectric, where the electrification is not neutralized, we find the phenomena which indicate positive or nega­ tive electrification.66 This analysis of the dielectric is taken, almost verbatim, from a passage in Station 2, where Maxwell considered the dielectric under induction.67 As we have seen, in Station 3 Maxwell explicitly avoided any mechanical hypothesis of the kind he had earlier developed in Station 2.68 However, even in Station 2, Maxwell severed the algebraic relations from “any theory about the internal mechanism of dielectrics.”69 In Station 3 Maxwell did not consider a possible mechanism at the micro-level an “illustration”; rather, he called it a way to “conceive” a specific phenomenon of the dielectric. This leaves open the possibility that there may be other ways to conceive it. However, viewed in context, the reference to molecules is isolated, and nothing comparable is said for other phenomena. One senses hesitation on Maxwell’s part to make the theory depend exclusively on equations. To be sure, Maxwell seems to imply that this is the actual mechanism at the micro-level, but he did not pursue this scheme to see how other phenomena might be explained by appealing to molecular action. In fact, the equations do not take into account the existence of molecules or, simply stated, the discussion of molecules has no impact on the theory. What is certain and confidently expressed is that the phenomena are related to force, motion, and state, which are general mechanical concepts at the macro-level. However, in contrast to Faraday who never went beyond the verbal stage, for Maxwell the verbal formulations are a prelude to the equations of the theory. This, of course, is not surprising, for it is consistent with Maxwell’s initial motivation, as is evident in his first contribution to this domain of physics, a motivation that sustained him throughout his work on electromagnetism. What is novel is the methodology he chose to apply. Maxwell recast his survey of electromagnetic phenomena—with an emphasis on the concept of electromotive force—into a list. It consists of eight verbal descriptions of seven relations and one calculation of the electromotive force, all to be turned into a set of algebraic formulas:

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146 Station 3 (1865) In order to bring these results within the power of symbolical calcula­ tion, I then express them in the form of the General Equations of the Electromagnetic Field. These equations express— (A) The relation between electric displacement, true conduction, and the total current, compounded of both. (B) The relation between the lines of magnetic force and the inductive coefficients of a circuit, as already deduced from the laws of induction. (C) The relation between the strength of a current and its magnetic effects, according to the electromagnetic system of measurement. (D) The value of the electromotive force in a body, as arising from the motion of the body in the field, the alteration of the field itself, and the variation of electric potential from one part of the field to another. (E) The relation between electric displacement, and the electromotive force which produces it. (F) The relation between electric current, and the electromotive force which produces it. (G) The relation between the amount of free electricity at any point, and the electric displacements in the neighbourhood. (H) The relation between the increase or diminution of free electricity and the electric currents in the neighbourhood.70 We note that the elements in this textual account are diverse, including phenomena (e.g., electric displacement and conduction), assumptions (e.g., lines of magnetic force), magnitudes (e.g., total current and strength of a current), and laws (e.g., laws of induction). Maxwell claimed that his starting point for the equations was the phenomena of electromagnetism, but we see that the elements are not all purely phenomenal, for some assumptions are involved together with low level (qualitative) inferences from phenomena. Put differently, no property of the medium is a phenomenon since the medium itself is an assumption. We witness the difficulty of portraying the phenomena neutrally without any presuppositions. Furthermore, the items comprising the list are presented independently of each other, and we note that the list is not structured hierarchically. In Part III this condensed verbal description of electromagnetic phenomena is recast in symbolic form, namely, equations. In total there are 20 equations that involve 20 variable quantities.71 Maxwell assumed the traditional Cartesian scheme where the relevant variables are parallel to orthogonal axes, x, y, and z; he did not refer to vectors but these are clearly components of vectors. For the six phenomena (A) to (F) Maxwell had a separate equation for each component, with the result that every phenomenon in (A) to (F) is represented by three equations. But it is evident that each set of three equations is, in effect, one equation which represents one phenomenon.72 In the list of the 20 variables there are six sets of three variables (plus two additional variables), where each set has one variable for

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each component. It is noteworthy that the verbal formulation requires unpacking in order to render explicit in symbolic form the conceptual presuppositions. Maxwell then labeled the equations with the same alphabetical notation corresponding to their respective phenomena.73 The order of the equations is significant, for it corresponds to the order of the eight phenomena: (A) electrical currents, including the displacement currents; (B) equations of magnetic force, given the electromotive force, the electro­ magnetic momentum and induction; continuing with (C) equations of currents; (D) equations of electromotive force; then with (E) equations of electric elasticity; (F) equations of electric resistance; (G) equation of free electricity; and finally with (H) equations of continuity. Maxwell claimed repeatedly that all he did was to recast Faraday’s experimental findings into mathematical symbols. But it seems that Maxwell underrated his insight into Faraday’s results. For one thing he included in his analysis his own new discoveries, notably the displacement current, and that electromagnetic disturbances are propagated in the same medium as that of light and at the same velocity. For another, Maxwell had several ways of recasting Faraday’s results, as is clear from Maxwell’s sequence of stations in which Faraday’s results are treated differently. Here, in Station 3, Maxwell set the phenomena in some order and recast the verbal presentation in this very order into the formalism of mathematical equations. We are not aware of Faraday arranging the phenomena in any particular order to facilitate the building of a general theory. At the end of this exposition, Maxwell put the equations in a slightly different order: (B) Three equations of Magnetic Force (C) Three equations of Electric Currents (D) Three equations of Electromotive Force (E) Three equations of Electric Elasticity (F) Three equations of Electric Resistance (A) Three equations of Total Currents (G) One equation of Free Electricity (H) One equation of Free Continuity.74 In Maxwell’s summary, the equations which comprise his dynamical theory of the electromagnetic field are no longer listed in the alphabetical order he had assigned them, for equations (A) now come after equations (F). So why did Maxwell change the order, placing equations (A) after equations (F)?

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148 Station 3 (1865) Maxwell did not offer any reason for this change, but it may have to do with the order of the 20 variables: [Three variables] Electromagnetic Momentum [Three variables] Magnetic Intensity [Three variables] Electromotive Force [Three variables] Current due to true Conduction [Three variables] Electric Displacement [Three variables] Total Current (including variation of displacement) [One variable] Quantity of Free Electricity [One variable] Electrical Potential.75 It can be seen that the three variables of the total current (including variation of displacement) come after the three variables of the electric displacement. Equations (A) then include equations (C). In his first formulation Maxwell followed the order in his list of phenomena; the order in that list appears to be arbitrary, for there seems to be no reason to put one phenomenon before another. Only at the next stage did Maxwell rearrange the list to adhere to some logical order that reflects physical dependence. It is noteworthy that Maxwell did not consider this set of equations a “system.” This is an important point, one that should be borne in mind when comparing this theory with the theory that Maxwell developed in Station 4, eight years later (see ch. 7). 6.6.1 An example—the case of electric elasticity Of particular interest is the set of equations (E): Electric Elasticity: When an electromotive force acts on a dielectric, it puts every part of the dielectric into a polarized condition, in which its opposite sides are oppositely electrified. The amount of this electrifica­ tion depends on the electromotive force and on the nature of the sub­ stance, and, in solids having a structure defined by axes, on the direction of the electromotive force with respect to these axes. In iso­ tropic substance, if k is the ratio of the electromotive force to the elec­ tric displacement, we may write the Equations of Electric Elasticity; P ¼ kf Q ¼ kg ðE Þ; R ¼ kh where P, Q, and R are the components of the electromotive force at any point, and f, g, and h are the electrical displacements.76 Here we see that Maxwell linked directly the elasticity of the medium with electric phenomena without specifying

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any mechanism or appealing to any analogy. Notably, there is no reference to any molecular structure of the medium. At the end of Part III of Station 3 Maxwell emphasized his methodology at this stage in his research: On our theory [energy] resides in the electromagnetic field, in the space surrounding the electrified and magnetic bodies, as well as in those bodies themselves, and is in two different forms, which may be described without hypothesis as magnetic polarization and electric polarization, or, according to a very probable hypothesis, as the motion and the strain of one and the same medium.77 Maxwell then remarked: “The conclusions arrived at in the present paper are independent of this hypothesis, being deduced from experimental facts.”78 As we have seen, Maxwell entertained a variety of ways to explain electromagnetic phenomena, including polarization of molecular structure, not to mention the molecular vortices hypothesis of Station 2. But they were all explicitly speculative (notwithstanding the status of “very probable”) and Maxwell did not rely on them in Station 3 in recasting verbal descriptions into algebraic formulas. There appears to be a clear distinction between “hypothesis” and “assumption” in Maxwell's methodological thinking. Here we see that, while Maxwell confidently presupposed the medium as the bearer of electromagnetic phenomena, he discarded without any hesitation hypotheses at the micro-level that were designed to explain the phenomena at the macro-level.79 Maxwell’s dynamical theory of the electromagnetic field in Station 3 comprises 20 equations, which are entirely abstract: no imagery, analogy, or hypothesis is involved. In sum, the theory includes neither assumptions about the micro-level nor mechanical schemes, with the exception of elasticity which is introduced with nothing supporting it at the micro-level.80 In concluding the electromagnetic theory of light, Maxwell made a revealing remark: The equations of the electromagnetic field, deduced from purely experimen­ tal evidence, shew that transversal vibrations only can be propagated. If we were to go beyond our experimental knowledge and to assign a definite density to a substance which we should call the electric fluid, and select either vitreous or resinous electricity as the representative of that fluid, then we might have normal vibrations propagated with a velocity depending on this density. We have, however, no evidence as to the density of electricity, as we do not even know whether to consider vitreous electricity as a substance or as the absence of a substance.81 Two points are worth noting. In the first place, Maxwell stated that the equations had been deduced from purely experimental evidence: no mechanism or hypothesis intervened.82 Secondly, Maxwell freed his imagination and

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150 Station 3 (1865) speculated: “if we were to go beyond our experimental knowledge,” but he immediately brought the speculation to an abrupt end by reminding the reader that there was no evidence to support such speculation. This was the kind of question he asked in Station 2, namely, “What is an electric current”?83 But the new dynamical theory was not designed to respond to the question, What is X? Rather, the question is, How does X behave? And the answer was expected to be formulated in symbolic language. Indeed, Maxwell emphasized that he deduced the general equations of the electromagnetic field from purely experimental facts.84 Once again, Faraday’s verbal descriptions of the phenomena preceded recasting them into symbolic form expressed in algebraic equations.

6.7 Conclusion Once the two major discoveries of Station 2 were framed, Maxwell felt sufficiently confident to reverse the argument and to consider the two discoveries assumptions in the foundation of the dynamical theory he constructed. He cast Faraday’s empirical discoveries into mathematical equations and deduced the mechanical action between conductors carrying currents from the electromagnetic phenomenon of induction. As we have seen, Maxwell was aware that applying concepts from mechanics to electromagnetism is problematic. His solution was to reverse the direction and deduce from the laws of induction the mechanical action. The direction of the argument was then from electromagnetic phenomena to mechanics, notwithstanding the direction of the “mechanical analogy” of the flywheel. The methodology of Station 3 consists in translating the phenomena directly into mathematical equations. In general, no hypothesis was involved, and no explanatory mechanism was sought. We take the outline of Maxwell’s methodology in Station 3 to be the following: (1) The goal: Construct a formal theory, that is, a theory formulated in symbolic language, compatible with the range of phenomena in physical domain A. Put negatively, the goal is not to offer an explanatory mechanism. (2) The means: Reverse a previous argument; assume as presuppositions conclusions reached earlier; then translate the verbal description into symbolic language. (2.1) Specifically, consider the presuppositions to be inherent features of a dynamical theory with the field as the conceptual framework. (2.2) Introduce dynamical illustrations from one physical domain (mechanics) to assist in understanding the target domain (electromag­ netism). References to mechanics are to be considered illustrative, not explanatory. (2.3) Set up a list of observations related to general mechanical concepts at the macro-level such as force, motion, and state; the list should consist

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of verbal descriptions of relations and empirical values for some parameters. (2.4) Translate the verbal descriptions into symbolic language and bring the relations and empirical values within the power of calculation by turning them all into a set of algebraic formulas. The resulting theory is a set of “General equations of the electromagnetic field.”85 Once these general equations were in place, Maxwell proceeded, as he put it, “to demonstrate from these principles the existence and laws of the mechanical forces which act upon electric currents, magnets, and electrified bodies placed in the electromagnetic field.”86 Surprisingly, Maxwell even tried to apply the theory to gravitational attraction (without success).87 He admitted his inability to understand how a medium could have the requisite properties to account for gravitational attraction, but his attempt to do so shows that his commitment to the physics of the field was not restricted to electromagnetism.

Notes 1 2 3 4 5 6 7 8 9 10 11 12

13 14 15

16 17

See ch. 5, nn. 76 and 77.

Bromberg, 1967, 219, 227–228, 229–230.

Maxwell, 1865, 487, § 73. See n. 34, below.

Maxwell, 1865, 459, § 1.

The two concepts, magnitude and direction, constitute a constant theme in Maxwell’s

contributions to electromagnetism (the quantity is a vector): see, e.g., ch. 4, n. 51.

Maxwell, 1865, 460, § 3, italics in the original.

For the usage of the term “dynamical” in Station 4, see ch. 7, n. 17.

Maxwell, 1865, 460, § 4.

Maxwell, 1865, 532–533, § 15.

See ch. 5, n. 86: “strong reason to believe.” For the origin of these concepts and the use Maxwell made of them, see ch. 7, nn. 22 and 29. Maxwell, 1865, 465, § 19. In 1865 there is another reference to Weber and Kohlrausch on the velocity of electromagnetic propagation (1865, 492 and 499), and references for the velocity of light on p. 499, without a reference to the second contribution of 1862 (see also pp. 465–466, §§ 19 and 20). For references to Weber and Kohlrausch in Maxwell’s paper of 1861–1862, see Maxwell, 1861–1862, 15 and 21, and ch. 5, § 5.5. Maxwell, 1865, 466–467, § 22.

Maxwell, 1865, 468 and 471. See n. 30, below.

Maxwell, 1865, 467–468, §§ 24–25. There is no figure for the flywheel in Station

3. For the flywheel built according to Maxwell’s design, see Campbell and Garnett, 1882, 551–554; Lazaroff-Puck, 2015, 463–464. In contrast to Campbell and Garnett who associated the flywheel analogy with vortices at the micro-level (Station 2), J. J. Thomson [in Maxwell, [1873] 1892, 2: 228], and Rayleigh (1890) only associated it with electrical induction at the macro-level. Rayleigh was Maxwell’s immediate successor as head of the Cavendish Laboratory. Maxwell, 1865, 467, § 24. For the formal analysis, see Lazaroff-Puck, 2015, 465–467. See also Lagrange, 1788, 1811–1815. Maxwell, 1865, 467, § 24.

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152 Station 3 (1865) 18 19 20 21 22 23 24

25 26 27 28 29 30 31 32 33 34 35

36 37 38 39 40 41 42 43 44 45

46 47 48 49 50 51 52 53

Maxwell, 1865, 467, § 23. Maxwell, 1865, 468, § 25. Maxwell, 1865, 468, § 25. Lazaroff-Puck, 2015, 468. Lazaroff-Puck, 2015, 468. Maxwell, 1865, 468, § 25, italics added. Maxwell, 1865, 462, § 10. See also n. 18, above: “. . . the two currents are in con­ nexion with every point of the field. . ..” In Station 4, Maxwell stated that the electromotive force is the “cause” of electric currents (emphasizing the causal nature of forces): “that which is the cause of electric currents has been called Electromotive Force” (Maxwell, 1873d, 2: 196, § 570). For further discussion, see ch. 7, nn. 37 and 101–102. Lazaroff-Puck, 2015, 468. Maxwell, 1865, 469, § 27. Maxwell, 1865, 469, § 27. Cf. n. 43, below. Maxwell, 1865, 468, § 25. Maxwell, 1865, 471, § 34. For a discussion of this distinction, see ch. 1, § 1.5. Maxwell, 1865, 463–464, § 14. Maxwell, 1865, 469, § 27. Harman (ed.) 1995, 337. As Harman reports (p. 335, n. 2), Tait’s Histories of thermodynamics & energetics was printed privately for class use in 1867. The analogy of an orrery with the solar system is of interest. The solar system is governed by the laws of gravitational attraction; an orrery is a representation of the solar system usually driven by a clockwork mechanism. This suggests that in 1867 Maxwell considered the vortex theory an explanatory representation of the true mechanism of electromagnetic phenomena, not the true mechanism itself. Maxwell, 1865, 467, § 22. Maxwell, 1865, 487, § 73. Maxwell, 1861–1862, 162. See ch. 5, n. 5. Maxwell, 1870, 420. Maxwell, 1865, 464, § 17. In Station 4 (ch. 7, n. 152) Maxwell acknowledged the lack of experimental evi­ dence for the displacement current. As we see, this was not the case in Station 3. Maxwell, 1865, 465, § 19. Maxwell, 1865, 487–488, § 74. According to Maxwell, the energy resides in the electromagnetic field; for this important claim, see n. 77, below. This methodology of Maxwell had an impact on Hertz in his search for the fun­ damental equations of electromagnetism for bodies in motion. The critical point is that with reversal Hertz, like Maxwell, could appeal to equations rather than to illustrations. Hertz’s reversal of his own argument was an inspirational source for Einstein’s methodology. See Hon and Goldstein, 2005, 498–499, where we call the reversal, “methodological ‘inversion.’” Cf. Hon and Goldstein, 2006a, 105. See ch. 4, § 4.5, and nn. 18, 63 and 77. See ch. 5, nn. 4 and 47. Maxwell, 1861–1862, 87–88. Maxwell, 1865, 461, § 8. Maxwell, 1865, 464, § 18. Maxwell, 1865, 465. Maxwell, 1865, 478–479, §§ 49–50. Maxwell, 1865, 465–466, § 20.

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Station 3 (1865) 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73

74 75 76

77 78 79

80

81 82

153

See ch. 5, nn. 85 and 94. Maxwell, 1865, 499, § 97. Maxwell, 1865, 499, § 95. Maxwell, 1865, 497, § 91. See the corresponding passage in Station 2: Maxwell, 1861–1862, 12–13; see also ch. 5, nn. 55–57 and 75. Maxwell, 1865, 461–462, § 9. Maxwell, 1865, 462, § 9, italics added. Maxwell, 1865, 462, § 11, italics added. Maxwell, 1865, 463, § 11, italics added. Maxwell, 1865, 464, § 16. Maxwell, 1865, 465, § 18. Maxwell, 1865, 461–462, § 9. Maxwell, 1865, 462, § 11. Maxwell, 1865, 462–463, § 11. “Molecule” is also invoked in the discussion of electrical displacement, see ibid., 554, § 55. Maxwell, 1861–1862, 14–15. Cf. ch. 5, nn. 70–73. Maxwell, 1865, 487, § 73. Cf. nn. 3 and 37, above. Maxwell, 1861–1862, 15. Cf. ch. 5, n. 74. Maxwell, 1865, 465, § 18. Maxwell, 1865, 485–486, § 70. See, for example, equations (E), discussed below. Equations (A) are on Maxwell, 1865, 480; equations (B) on p. 482, equations (C) on p. 482, equations (D) on p. 484, equations (E) on p. 485, equations (F) on p. 485, equation (G) on p. 485, and equation (H) on p. 485. For a detailed analysis of equations (E) as an illustration of Maxwell’s approach, see below. In the Treatise the general equations are treated in vol. 2, §§ 604–617, summarized in §§ 618–619: see ch. 7. Maxwell, 1865, 486, § 70. Maxwell, 1865, 486, § 70. Maxwell, 1865, 485, § 66; for the electrical displacements, see ibid., 480, § 55; and for the electromotive force, see ibid., 480–481, § 56. This set of equations, (E), is directly related to a passage in Station 4, the Treatise, where Maxwell (1873d, 1: 60, § 60) drew attention to the analogy between the action of electro­ motive intensity in producing electric displacement and of ordinary mechanical force in producing the displacement of an elastic body. Cf. ch. 7, n. 143. See also Bromberg, 1967, 230. Maxwell, 1865, 488, § 74. Maxwell, 1865, 488, § 75. We recall that in Station 3 Maxwell “made use of the optical hypothesis of an elas­ tic medium through which the vibrations of light are propagated.” This is the undulatory theory of light to which Maxwell had referred in Station 2 (Maxwell, 1861–1862, 13; ch. 5, n. 59), and it facilitated the admission of “elasticity in the luminiferous medium, in order to account for transverse vibrations.” The argu­ ment is auxiliary, offering “warrantable grounds for seeking, in the same medium, the cause of other phenomena as well as those of light”: see n. 57, above. To be sure, Maxwell did refer to analogies (see 1865, 464, § 14) to help in under­ standing what he meant by the elasticity of the medium for electric and magnetic phenomena. However, the tone is very hesitant (see n. 32, above: “it seems prob­ able that. . . may have. . .”) and, in any event, Maxwell did not pursue this approach further. Throughout the paper elasticity remained a central feature of the medium without a specific mechanism. Maxwell, 1865, 500–501, § 100. See also, Maxwell, 1865, 488, § 75.

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154 Station 3 (1865) 83 84 85 86

Maxwell, 1861–1862, 283. See n. 78, above. Maxwell, 1865, 480, Part III. Maxwell, 1865, 488, § 75. Maxwell made an important distinction between elec­ tromotive force and ordinary mechanical force. Later on, in Station 4, he made it even clearer: “the only force which acts on electric currents is electromotive force, which must be distinguished from the mechanical force. . .” (1873d, 2: 145, § 501). But, then, “the work done by an electromotive force is of exactly the same kind as the work done by an ordinary force. . .” (Maxwell, 1873d, 2: 196, § 570). Interestingly, Maxwell noted that “if we ever come to know the formal relation between electricity and ordinary matter, we shall probably also know the relation between electromotive force and ordinary force” (ibid.). We discuss in detail the issue of different kinds of forces and a single concept of energy in ch. 7. 87 Maxwell, 1865, 570–571, § 82. Maxwell was interested in applying Faraday’s con­ cept of lines of force to gravitation early on: see his comments to Faraday and Thomson in November 1857 (Harman (ed.) 1990, 552 and 556).

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Station 4 (1873) A treatise on electricity and magnetism

7.1 Introduction In his preface to the third edition of Maxwell’s Treatise (1892), Joseph J. Thomson (1856–1940), Nobel laureate in physics (1906), noted: It is now nearly twenty years since this book was written, and during that time the sciences of Electricity and Magnetism have advanced with a rapidity almost unparalleled in their previous history; this is in no small degree due to the views introduced into these sciences by this book: many of its paragraphs have served as the starting-points of important investigations.1 Undoubtedly, Maxwell’s Treatise had much more impact on the physics community than the three papers which preceded it. Tait alluded to this point in his article “Maxwell” in the ninth edition of the Encyclopaedia Britannica: The first paper of Maxwell’s in which an attempt at an admissible physical theory of electromagnetism was made was communicated to the Royal Society in 1867 [sic; read: 1865]. But the theory, in a fully developed form, first appeared in 1873 in his great treatise on Electricity and Magnetism (1873). This work, already in a second edition, is one of the most splendid monuments ever raised by the genius of a single individual. Availing him­ self of the admirable generalized co-ordinate system of Lagrange, Maxwell has shown how to reduce all electric and magnetic phenomena to stresses and motions of a material medium, and, as one preliminary, but exces­ sively severe, test of the truth of his theory, has shown (if the electromag­ netic medium be that which is required for the explanation of the phenomena of light) the velocity of light in vacuo should be numerically the same as the ratio of the electromagnetic and electrostatic units.2 Tait called attention to the new perspective that Maxwell introduced into physics, namely, the reduction of all electric and magnetic phenomena to

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156 Station 4 (1873) stresses and motions of a material medium. This new approach to electromagnetism was in fact the starting point for Einstein, who remarked on the occasion of the centenary of Maxwell’s birth in 1931: The greatest alteration in the axiomatic basis of physics—in our concep­ tion of the structure of reality—since the foundation of theoretical physics by Newton, originated in the researches of Faraday and Max­ well on electromagnetic phenomena . . . Before Maxwell, Physical Real­ ity, in so far as it was to represent the processes of nature, was thought of as consisting in material particles, whose variations consist only in movements governed by partial differential equations. Since Maxwell’s time Physical Reality has been thought of as represented by continuous fields, governed by partial differential equations, and not capable of any mechanical interpretation. This change in the conception of Reality is the most profound and fruitful that physics has experienced since the time of Newton . . .3 Einstein clearly indicated that his two most important predecessors for establishing the foundations of physics were Newton and Maxwell. Indeed, Maxwell’s electrodynamics profoundly changed the way physical reality is conceived. As Tait noted, the theory of electrodynamics, in a fully developed form, first appeared in Maxwell’s Treatise on electricity and magnetism (1873)— Station 4. The three preceding stations comprise research papers in this physical domain, while the fourth station is both cutting-edge research and a textbook, an introduction to the study of electricity and magnetism.4 What was the methodology, or the methodologies, which facilitated this profound change in the conception of physical reality? The methodological practices of the first three stations found their way into Station 4. In this sense, Station 4 is a summary of Maxwell’s past contributions to this domain of physics. Indeed, Maxwell’s fundamental conceptions were recast into the foundations of the Treatise. Thus, the unifying concept of lines of force as well as the dual approach to physics, that is, the duality of expression in symbolic and textual language, are strongly present in Station 4. The common conceptual background of the three papers as well as the Treatise suggests continuity: Maxwell continued to hold to his strong belief in the productivity of Faraday’s unifying concept of lines of force, and to his own longstanding commitment, namely, that physics has to be formulated both mathematically and verbally. The crucial point is that in addition to the conceptual framework that Maxwell had proposed previously, Station 4 contains several mathematical innovations that Maxwell adopted and developed for applying them to electrodynamics. While Maxwell took advantage of recent advances in mathematics, he was always guided by physical reasoning rather than by algebraic or purely geometric analysis.5 To be specific, it is the concept of

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energy—and the various mathematical tools for expressing it—that dominates Station 4 methodologically. This completely novel approach to electromagnetism, in which energy is presupposed as the fundamental physical concept for electromagnetic phenomena, marks a break from his own previous contributions in this domain of physics. Undoubtedly, the change is dramatic in terms of physical presuppositions, and it required a change in methodology as well. The success of Station 4 was due, in large measure, to a consistent and productive combination of a physical concept—energy—together with mathematical tools designed by Maxwell specifically for treating electromagnetic phenomena.

7.2 Framework As his researches into electromagnetism developed, Maxwell’s methodological versatility is clearly expressed in the structure of Station 4. The Treatise does not begin with an account of the phenomena of electromagnetism—a move which was taken in Station 3; rather, the point of departure of Station 4 is a preliminary discussion, preceding Part I, on the measurement of quantities, in which several mathematical innovations are introduced and their application to physics is discussed. This way of framing the research, namely, starting with the introduction of a variety of mathematical operations, is different from the approaches of his predecessors; indeed, it is very different from what he himself had done previously. Although the first move is mathematical, Maxwell did not proceed in deductive steps, as was the case for many continental physicists. In fact, Maxwell’s procedure has often been seen as opaque, motivated by several methodologies which at times seem to contradict one another.6 An overview of Station 4—consisting of Part I Electrostatics; Part II Electrokinematics; Part III Magnetism; and Part IV Electromagnetism— confirms two points: (1) as previously, Maxwell included many references to observations of the pertinent phenomena as well as to their measurements and the relevant units; and (2) Maxwell built theories at many different levels, ultimately seeking general dynamical equations for the electromagnetic field. Observation and measurement (including discussion of units) together with constructing theories, correspond to Maxwell’s general conception of what science is all about. Maxwell articulated this conception two years before the publication of the Treatise: The first part of the growth of a physical science consists in the discovery of a system of quantities on which its phenomena may be conceived to depend. The next stage is the discovery of the mathematical form of the relations between these quantities. After this, the science may be treated as a mathematical science, and the verification of the laws is effected by a the­ oretical investigation of the conditions under which certain quantities can be most accurately measured, followed by an experimental realisation of these conditions, and actual measurement of the quantities.7

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158 Station 4 (1873) According to these three steps, physical science begins by deriving physical quantities inherent in the phenomena under study. Then comes the stage of finding the mathematical formalism for expressing effectively the relations that hold among the quantities. And, finally, the third step is a theoretical investigation to explore consequences that may lead to unforeseen physical connections which can then be tested experimentally. At the end of Chapter 1 in Part I Maxwell dedicated four subsections in narrative form to describing the plan of the Treatise, including a characterization of the theory together with its peculiar features and consequences.8 In the plan Maxwell conceived the physical relation between the electrified bodies in two “mathematically equivalent” ways,9 namely, “either as the result of the state of the intervening medium, or as the result of a direct action between the electrified bodies at a distance.” He then compared the two approaches: If we adopt the latter conception [direct action at a distance], we may determine the law of the action, but we can go no further in speculating on its cause. If, on the other hand, we adopt the conception of action through a medium [lines of force], we are led to enquire into the nature of that action in each part of the medium.10 Evidently, in Station 4 Maxwell was intent on explaining the “ordinary theory of electrical action”11 by “speculating on its cause.” The plan then was to investigate the medium in a state of mechanical stress. I have therefore thought it a warrantable step in scientific procedure to assume the actual existence of this state of stress, and to follow the assumption into its consequences.12 One consequence is that “the motions of electricity are like those of an incompressible fluid.”13 And Maxwell continued to list, as he put it, “the peculiar features of the theory.”14 Thus, what began in Station 1 as a bold move, namely, an analogy between a physical system of electromagnetism and an imaginary system of incompressible fluid, which was abandoned in Stations 2 and 3, proved in Station 4 to be of ongoing interest in a somewhat different context.15 However, as we will see, in Station 4 Maxwell no longer used analogies and illustrations as research tools; rather, they served the didactic purpose of making parts of the theory seem plausible to the reader. For this reason Maxwell did not require the analogies and illustrations in different sections of this work to be consistent with one another.16

7.3 Novel methodologies We have indicated that Station 4 contains several methodologies that Maxwell had already applied in the preceding stations, which may give the

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impression of eclecticism. However, there is no question that Station 4 abounds with novelties; indeed, there is a strong sense of originality, the mark of genius of the likes of Newton and Einstein. What characterizes the novel parts of Station 4 are insightful connections across scientific domains that no one before Maxwell had recognized, even though most of the elements of these connections were already known either in physics or in mathematics. It seems that Maxwell was on the lookout for new ideas, concepts, and mathematical methods, and there were many mathematical and scientific innovations in the mid-nineteenth century upon which he could draw. Maxwell imported some of these novelties into electromagnetism and recast them for application to this specific domain of physics, thereby turning it from electromagnetism into electrodynamics.17 The realization that the precise underlying mechanism of electromagnetic phenomena had not been determined, called for a concept to account for the phenomena without depending on any specific mechanism. What was needed was an overarching concept with which one could comprehend the phenomena without being committed to a particular structure at the micro­ level and, most importantly, it should be amenable to mathematical analysis. Maxwell opted for the concept of “energy” whose application across several physical domains came to be widely used in the midnineteenth century.18 Energy exhibits precisely the features which Maxwell sought. Maxwell recognized that a system of material conductors carrying currents is a dynamical system—the seat of energy, part of which may be kinetic and part potential. This recognition is the key to understanding the novel methodologies which Maxwell applied ingeniously in Station 4. In what follows, we show how Maxwell systematically selected physical concepts and mathematical methods from the available literature and adapted them to electromagnetism. The goal was to construct a general theory of electrodynamics, formulated in mathematical terms complete with a consistent mental imagery. 7.3.1 Energy as a key concept in electrodynamics As we have seen, at the outset of Station 3 Maxwell singled out a well known mechanical phenomenon that takes place in any electrical and magnetic experiment, namely, “bodies in certain states set each other in motion while still at a sensible distance from each other.”19 He then stated that the first step needed to render these mechanical phenomena amenable to scientific analysis is to ascertain the magnitude and direction of the force acting between the bodies. In contrast to the theories of Weber and Neumann which are based on the fundamental concept of action at a distance, Maxwell introduced a dynamical theory that is based on the concept of an electromagnetic field (Maxwell italicized this expression).20 The field is a portion of space that contains and surrounds bodies in electric and magnetic states. Clearly, in this conception of electromagnetism

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160 Station 4 (1873) the force that acts between the bodies finds its seat in the medium. Now comes a key claim in the dynamical theory of Station 3: The medium is . . . capable of receiving and storing up two kinds of energy, namely, the “actual” energy depending on the motions of its parts, and “potential” energy, consisting of the work which the medium will do in recovering from displacement in virtue of its elasticity.21 As noted (in ch. 3, § 3.4), the expressions “actual energy” and “potential energy” were introduced by Rankine.22 In Station 4, Maxwell combined Rankine’s conceptual insight with the formal analysis developed by Thomson and Tait whose investigation of 1867 promoted the link between dynamics and the conservation of energy: The language of dynamics has been considerably extended by those who expounded in popular terms the doctrine of the Conservation of Energy, and it will be seen that much of the following statement is suggested by the investigation in Thomson and Tait’s Natural Philosophy, especially the method of beginning with the theory of impulsive forces.23 In a paper read in February 1873, “On the proof of the equations of motion of a connected system,” within days of the date of the Preface of Maxwell’s Treatise,24 Maxwell outlined his methodological preferences that characterize Station 4. In the first place the science is that of dynamics: “to deduce from the known motions of a system the forces which act on it.”25 He then reported that his goal was to obtain the final equations of motion independently of any explicit consideration of the motion of any part of the system. The fact that the final equations do not include the symbols by which the dependence of the motion of the parts on that of the variables was expressed, indicates that Maxwell’s goal could be realized. With these equations Maxwell sought to present “to the mind in the clearest and most general form the fundamental principles of dynamical reasoning.” He then continued: In forming dynamical theories of the physical sciences, it has been a too frequent practice to invent a particular dynamical hypothesis and then by means of the equations of motion to deduce certain results. The agreement of these results with real phenomena has been supposed to furnish a certain amount of evidence in favour of the hypothesis. However, this form of the hypothetical-deductive method was not the “true method” in Maxwell’s view. In fact, he was not interested in such hypotheses: The true method of physical reasoning is to begin with the phenomenon and to deduce the force from them by a direct application of the equa­ tions of motion. The difficulty of doing so has hitherto been that we

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arrive, at least during the first stages of the investigation, at results which are so indefinite that we have no terms sufficiently general to express them without introducing some notion not strictly deducible from our premises. The suggested hypothesis, “some notion not strictly deducible from our premises,” was merely an expedient, not part of “the true method.” Maxwell concluded his short talk with this advice: It is therefore very desirable that men of science should invent some method of statement by which ideas, precise so far as they go, may be conveyed to the mind, and yet sufficiently general to avoid the introduc­ tion of unwarrantable details.26 Maxwell called for a plan of presentation that points the way for conducting research in general terms while avoiding specific details which are not supported by the evidence. The search in Part IV of the Treatise for “the general equations of dynamics,” “the application of dynamics to electromagnetism,” and then “the general equations,” is a consequence of this methodology. What was novel in Maxwell’s methodology is the application of mathematical techniques developed in rational mechanics to electromagnetism. One can safely say that in Station 4 Maxwell gave himself the task of rewriting electromagnetism as the electrodynamics of energy. This move is not at all obvious. It is not merely an extension of Thomson and Tait’s methodology, for it required a judicious assessment of the importance of energy in the domain of electromagnetism. Presumably, only after Maxwell had become convinced of this claim, did he turn to Lagrange and Hamilton for the tools to make it work in electrodynamics. Characteristic of Maxwell’s stations preceding Station 4 is a physical approach, rather than reliance on abstract symbolism. Indeed, he thought in physical terms: imagined tubes with “physical” features in Station 1, a mechanical hypothesis in Station 2, and a dynamical medium in Station 3. However, in Station 4 this “physical” point of view required, as Maxwell termed it, “retranslation,” for the dynamical analysis in terms of the Lagrangian and the Hamiltonian are purely abstract, carried out by symbolic operations. Maxwell formulated this position succinctly: Our aim . . . is to cultivate our dynamical ideas. We therefore avail our­ selves of the labours of the mathematicians [Lagrange and Hamilton], and retranslate their results from the language of the calculus into the language of dynamics, so that our words may call up the mental image, not of some algebraical process, but of some property of moving bodies.27

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162 Station 4 (1873) In Station 4 Maxwell’s discussion of magnetic induction did not depend on Lagrange and Hamilton; rather, it was based on physical reasoning, and then translated into symbolic/algebraic language. The Lagrangian and the Hamiltonian provide effective mathematical symbolism but they had to be (re-)translated into words so that a mental image of the dynamics can be formed. Clearly, Maxwell was fully aware of the power of symbolic argumentation, but he was not content, as he had already indicated in Station 1, with just “analytical subtleties.”28 Maxwell expressed much enthusiasm for his new approach to the study of electromagnetism. In a letter to Mark Pattison, dated 13 April 1868, he wrote: Energy is of two kinds, Kinetic and Potential. Energy is the capacity which a body has of doing work.[*] A moving body has energy due to its motion called kinetic energy and measured by ½ mass × (velocity)2 … Energy of both kinds is capable of exact measurement, and the progress of science at present is in the direction of measuring additional forms of energy . . . .29 [Footnote in Harman (ed.) 1995, 365] * Compare Thomson and Tait, [1867], 177–178 Evidently, in 1868 Maxwell was fully aware of the new way of expounding physics based on the concept of energy which had then been developed by Thomson and Tait. In the Preliminary in vol. 1 of the Treatise Maxwell offered the following definition: “The energy of a system, being its capacity of performing work, is measured by the work which the system is capable of performing by the expenditure of its whole energy.”30 This general definition can admit products of different pairs of magnitudes; indeed, as Maxwell noted in the first chapter of the Treatise, energy is the product of the following pairs of factors: A A A A

Force Mass Mass Pressure

× × × ×

A distance through which the force is to act.

Gravitation acting through a certain height.

Half the square of its velocity.

A volume of fluid introduced into a vessel at that

pressure. A Chemical Affinity × A chemical change, measured by the number of electro­ chemical equivalence which enter into combination.31

Among these definitions of energy as products of physical quantities, there is the product of “mass” times “half the square of its velocity,” m·(½v2), known as kinetic energy.32 We will return to this product shortly. And Maxwell added:

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If we ever should obtain distinct mechanical ideas of the nature of electric potential, we may combine these with the idea of energy to determine the physical category in which “Electricity” is to be placed.33 Interestingly, the early discussions of energy in Station 4 are not focused on the issue of dynamics; rather, they concern the nature of electricity, namely, Is it a substance?34 Maxwell then suggested that a mechanical conception of electric potential combined with the concept of energy might help in deciding the physical category of “Electricity.” In contrast to the above definition of kinetic energy, E as “mass” times “half the square of its velocity,” in the second volume of the Treatise Maxwell presented a modified version. Under the heading, The Kinetic Energy expressed in Terms of the Momenta and Velocities,35 Maxwell argued that The kinetic energy is . . . half the sum of the products of the momenta into their corresponding velocities. When the kinetic energy is expressed in this way . . . it is a function of the momenta and velocities only, and does not involve the variables themselves.36 The definition of kinetic energy as E = ½p·v (where the momentum p = mv)— instead of E = m·(½v2)—renders the components of kinetic energy physically meaningful and amenable to the mathematical treatment of Lagrange and, more productively, that of Hamilton. What is the background to this novel insight in the second volume? Maxwell argued analogically: an ordinary force, acting on a body in the direction of its motion, increases its momentum, and communicates to it kinetic energy, or the power of doing work on account of its motion. In like manner the unresisted part of the electromotive force has been employed in increasing the electric current. Has the electric current, when thus produced, either momentum or kinetic energy?37 The question appears to be rhetorical, for Maxwell continued his argument: But a conducting circuit in which a current has been set up has the power of doing work in virtue of this current, and this power cannot be said to be something very like energy, for it is really and truly energy . . . A system containing an electric current is a seat of energy of some kind; and since we can form no conception of an electric current except as a kinetic phenomenon*, its energy must be kinetic energy, that is to say, the energy which a moving body has in virtue of its motion.38 [Footnote in Maxwell, 1873d, 2: 182, § 552] * Faraday, Exp. Res. 283

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164 Station 4 (1873) As the footnote indicates, Maxwell’s conclusion was inspired by Faraday. Thus, from a logical point of view, the sequence, roughly, is that Maxwell got the notion from Faraday that kinetic energy is the same whether it takes place in a mechanical or an electromagnetic system. Maxwell began to think seriously about energy in electromagnetism in Station 3, and then sought a way to treat it mathematically. This search was among the considerations that led Maxwell to turn to the new approach of Thomson and Tait, where energy in mechanical systems was discussed. The way to the application of Lagrange and Hamilton was then open. It is, however, important to note that the conceptual issue preceded the mathematical treatment; in fact, clarifying the issue of energy motivated the mathematical search. In what sense are the above two definitions of energy equivalent, and in what sense are they not? Formally, the two expressions are identical. However, m times v2 makes no physical sense since v2 is not physically meaningful; but p times v is meaningful, namely, it is a product of momentum and velocity, both of which are vectors. In this way Maxwell adhered to his methodological precept of the duality of expression, namely, that there has to be a correspondence of symbolic language and physical concepts.39 Maxwell’s new definition of kinetic energy as the scalar part of half the product of two vectors, momentum and velocity, corresponds to the distinctions he had already introduced in the Preliminary: A Vector, or Directed quantity, requires for its definition three numerical specifications, and these may most simply be understood as having refer­ ence to the directions of the coordinate axes. Scalar quantities do not involve direction. The volume of a geometrical figure, the mass and the energy of a material body, the hydrostatical pressure at a point in a fluid, and the potential at a point in space, are examples of scalar quantities. A vector quantity has direction as well as magnitude, and is such that a reversal of its direction reverses its sign. The displacement of a point, rep­ resented by a straight line drawn from its original to its final position, may be taken as the typical vector quantity, from which indeed the name of Vector is derived. The velocity of a body, its momentum, the force acting on it, an electric current, the magnetization of a particle of iron, are instances of vector quantities.40 Energy is treated very differently in vol. 2 of Station 4 compared to the discussion in vol. 1. The change seems to be due in part to Maxwell’s taking advantage of Hamilton concerning vector notation.41 We see a distinct development in the treatment of energy which was undoubtedly motivated in part by new advances in mathematics. However, the conception that the seat of energy is in the medium was already in evidence in Station 3, and it was well embedded in Station 4:

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In the theory of electricity and magnetism adopted in this treatise, two forms of energy are recognised, the electrostatic [i.e., potential] and the elec­ trokinetic … and these are supposed to have their seat, not merely in the electrified or magnetized bodies, but in every part of the surrounding space, where electric or magnetic force is observed to set. Hence our theory agrees with the undulatory theory in assuming the existence of a medium which is capable of becoming a receptacle of two forms of energy.42 7.3.2 Mathematical tools in the treatment of electrodynamics In 1871 in an essay entitled “On the mathematical classification of physical quantities.” Maxwell introduced some of the mathematical techniques that he later put to use in Station 4. These techniques—when applied to electrodynamics—turned out to be methodological. Maxwell aimed at a fundamental feature: If we had a true mathematical classification of quantities, we should be able at once to detect the analogy between any system of quantities pre­ sented to us and other systems of quantities in known sciences, so that we should lose no time in availing ourselves of the mathematical labours of those who had already solved problems essentially the same.43 In Maxwell’s view scientific discoveries and their diffusion would be greatly helped by such a classification. The first distinction which Maxwell introduced within his mathematical classification was taken from Hamilton, namely, the distinction between vector and scalar. Maxwell then invoked Hamilton’s calculus of quaternions whose discovery Maxwell compared to the discovery of analytical geometry by Descartes. Hamilton, according to Maxwell, facilitated the treatment of dynamical quantities.44 Maxwell proceeded to draw what he regarded as “very important distinction among vector quantities.” And he continued: Vectors which are referred to unit of length I shall call Force . . . Vectors which are referred to unit of area I shall call Fluxes.45 Maxwell considered his distinction between force and flux “one of the most important mathematical results” for the “directions [of force and flux] may be different.”46 This fundamental distinction allowed him to take a novel approach to electromagnetism: In statical electricity the resultant force at a point is the rate of vari­ ation of potential, and the flux is a quantity, hitherto confounded with the force, which I have called the electric displacement.

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166 Station 4 (1873) In magnetism the resultant force is also the rate of variation of the potential, and the flux is what Faraday calls the magnetic induction . . .47 Undoubtedly, the distinction Maxwell drew between force and flux, and identifying the latter as the displacement current in electricity and magnetic induction in magnetism, was one of Maxwell’s fundamental contributions to electrodynamics. This distinction and its nomenclature received much attention and amplification in Station 4.48 Another mathematical innovation that Maxwell introduced in the essay is Hamilton’s quaternions. The mathematical classification facilitates the study of “a ratio of a force to a flux in its most general form.”49 A further distinction which Maxwell included in his mathematical classification characterizes vectors of physical magnitudes with respect to translation and rotation. For Maxwell the former relates to electricity while the latter to magnetism.50 Maxwell’s appeal to the distinction between scalar and vector quantities, and the use of quaternions, is a novelty; it does not appear in Stations 1 to 3, written prior to this essay of 1871; however, this distinction becomes an integral part of the argument in Station 4.51 When general characterizations of mathematical elements, such as vectors and scalars, are identified as bearers of physical magnitude, the formal method becomes methodological—the mathematical formalism becomes a physical theory. The critical step, of course, is to identify correctly a certain magnitude as scalar or vector, and to indicate whether the vector is defined with respect to translation or with respect to rotation. Once the identification is completed, the formalism casts the relations among the various magnitudes into physical domains where unforeseen connections can be explored. Maxwell imposed a strong constraint on the mathematical methods he extended to this new domain in physics, namely, electromagnetism: In the theory which I propose to develope, the mathematical methods are founded upon the smallest possible amount of hypothesis, and thus equations of the same form are found applicable to phenomena which are certainly of quite different natures, as, for instance, electric induc­ tion through dielectrics; conduction through conductors, and magnetic induction. In all these cases the relation between the force and the effect produced is expressed by a set of equations of the same kind, so that when a problem in one of these subjects is solved, the problem and its solution may be translated into the language of the other subjects and the results in their new form will also be true.52 This brief remark is worth unpacking. In the first place, Maxwell recommended the search for equations of the same form which are applicable to phenomena of different natures. This is akin to the formal analogies between different physical domains introduced by Thomson which

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Maxwell, as we have seen, found most useful.53 However, the analogy here is across phenomena within the same domain, namely, electromagnetism. Furthermore, Maxwell expressed his reluctance to form “hypotheses.” He spoke of “the smallest amount of hypothesis”; this call for minimal presuppositions at the micro-level is reminiscent of Maxwell’s disposition in Station 1, in contrast to his vortex hypothesis of Station 2. Here is a rare case of continuity in methodology from Station 1 to 4. It is notable that Maxwell included his previous methodologies in Station 4. But we return to specific mathematical methods. The recasting of electrodynamics into Lagrangian and Hamiltonian forms is introduced in the middle of volume 2 of the Treatise. By contrast, the discussions of vectors, scalars, quaternions, line- and surface-integrals, and the geometry of position are placed near the beginning of vol. 1 in the Preliminary. This suggests that these mathematical methods are more fundamental for Maxwell than the Lagrangian and the Hamiltonian. Clearly, advanced mathematics played a much more important role in Station 4 than in the preceding stations. In fact, the presentation in Station 4 is different from that in Station 3, where no discussion of mathematical methods is developed before the Lagrangian is introduced.54 In Station 4, in the chapter where Maxwell discussed the induction of a current on itself, he acknowledged that he had no specific idea about the nature of the forces active in an electromagnetic system. However, he did assume that these forces cause motions, and this led to the question, is there a methodology that can account for motions without appealing to forces? This is what the Lagrangian (and the Hamiltonian) offered Maxwell: What I propose . . . to do is to examine the consequences of the assump­ tion that the phenomena of the electric current are those of a moving system, the motion being communicated from one part of the system to another by forces, the nature and laws of which we do not yet even attempt to define, because we can eliminate these forces from the equations of motions by the method given by Lagrange for any connected system.55 The powerful method of Lagrange (and Hamilton) allowed Maxwell to forge a new approach to the theory of electromagnetism, turning this mathematical method into a methodology, thus replacing the theory of Weber and Neumann. This could be effected by deducing the main structure of the theory from purely dynamical considerations. Maxwell was explicit: I have chosen this method56 because I wish to show that there are other ways of viewing the phenomena which appear to me more satisfactory, and at the same time are more consistent with the methods followed in the preceding parts of this book than those which proceed on the hypothesis of direct action at a distance.57

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168 Station 4 (1873) Once again Maxwell reiterated his critical position vis-à-vis the conceptual framework of action at a distance and his complete confidence in the concept of lines of force—a distinctive mark in all his work on electromagnetism. What was the fundamental advantage in the new method(s) which attracted Maxwell’s attention? According to Maxwell, Lagrange and most of his followers, to whom we are indebted for these methods . . . in order to devote their attention to the symbols before them . . . have endeavoured to banish all ideas except those of pure quan­ tity, so as not only to dispense with diagrams, but even to get rid of the ideas of velocity, momentum, and energy, after they have been once for all supplanted by symbols in the original equations. In order to be able to refer to the results of this analysis in ordinary dynamical language, we have endeavoured to retranslate the principal equations of the method into language which may be intelligible without the use of symbols.58 As indicated, the power of the Lagrangian (and the Hamiltonian) lies in elimination and avoidance. This methodology of eliminating and avoiding physical quantities by formal means circumvents complicated physical issues and simplifies matters greatly. But then the demand persists to retranslate the empty symbols back into physically meaningful entities. This is consistent with the dominant feature of Station 4, namely, the dual approach of symbolic language (mathematical equations) and natural language (physical illustrations). According to Maxwell the aim of Lagrange was to apply the calculus to dynamics. Lagrange, Maxwell continued, began by expressing the elementary dynamical relations in terms of the corresponding relations of pure algebraical quantities, and from the equations thus obtained he deduced his final equations by a purely alge­ braical process.59 The power of this method—“from a mathematical point of view”—as Maxwell put it,60 derives from the elimination of certain quantities from the final equations. In the course of this elimination the mathematician is engaged in calculations and is therefore free from physical considerations such as dynamical ideas. But “from a physical point of view” one conceives the system with a suitable mechanism which connects the moving parts. This is an ideal mechanism, for it is “free from friction, destitute of inertia, and incapable of being strained by the action of applied forces.” And Maxwell added: The use of this mechanism is merely to assist the imagination in ascrib­ ing position, velocity, and momentum to what appear, in Lagrange’s investigation, as purely algebraical entities.61

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Evidently, there is a strong interplay here between mathematics and physics. It is noteworthy that the two perspectives, namely, physical and mathematical, are presented in adjacent passages (§§ 553, 554). In contrast to his practice in the preceding stations, Maxwell considered Station 4 a textbook in addition to being cutting-edge research on electromagnetism.62 Here he had the student in mind, and his advice reflected the methodologies he developed: It is doubtless important that the student should be able to trace the connexion of the motion of each part of the system with that of the variables, but it is by no means necessary to do this in the process of obtaining the final equations, which are independent of the particular form of these connexions.63 Lagrange’s technique made available efficacious procedures with equations that then have to be retranslated back to physics. However, Maxwell favored Hamilton’s technique for the expression of the kinetic energy of the system.64 He explained, as Harman reported, that Newton’s law preserves “quantities of motion” (momenta) and not velocities; so, if one wishes to be true to Newtonian mechanics, one should express the kinetic energy in terms of momenta (the Hamiltonian) and not velocities (as is the case with the Lagrangian).65 Maxwell summarized the advantage of the Hamiltonian: By this method we have avoided the consideration of the change of con­ figuration during the action of the forces . . . The variables, and the cor­ responding velocities and momenta, depend on the actual state of motion of the system at the given instant, and not on its previous history.66 The fact that the Hamiltonian disregards the way the system reached its instantaneous state is seen as a great advantage. “We have kept out of view,” Maxwell remarked, “the mechanism by which the parts of the system are connected.”67 And he added, We have not even written down a set of equations to indicate how the motion of any part of the system depends on the variation of the vari­ ables. We have confined our attention to the variables, their velocities and momenta, and the forces which act on the pieces representing the variables. Our only assumptions are, that the connexions of the system are such that the time is not explicitly contained in the equations of condition, and that the principle of the conservation of energy is applic­ able to the system.68

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170 Station 4 (1873) The methodology based on the Lagrangian and the Hamiltonian proved to be powerful by circumventing physical issues that could not be solved and, indeed, need not be solved, as this methodology requires only minimal assumptions with respect to variables and forces. Already in Station 1 Maxwell was of the opinion that mathematical connections may suggest to physicists the means of establishing physical connections such that symbolic speculations may lead to new experimental results.69 This is indeed the goal of applying the Lagrangian and the Hamiltonian in Station 4. In fact, Maxwell’s position hardened, namely, there are connections that can be comprehended only with “mathematical training”: As the development of the ideas and methods of pure mathematics has rendered it possible, by forming a mathematical theory of dynamics, to bring to light many truths which could not have been discovered with­ out mathematical training, so, if we are to form dynamical theories of other sciences, we must have our minds imbued with these dynamical truths as well as with mathematical methods.70 He then added some words of caution: In forming the ideas and words relating to any science, which, like elec­ tricity, deals with forces and their effects, we must keep constantly in mind the ideas appropriate to the fundamental science of dynamics, so that we may, during the first development of the science, avoid incon­ sistency with what is already established, and also that when our views become clearer, the language we have adopted may be a help to us and not a hindrance.71 Both the “mathematical method” and “the language we have adopted” should work coherently, consistently, and in tandem in pursuit of scientific knowledge. Put differently, Maxwell noted that Lagrange and Hamilton took the physics out of the equations of dynamics (empty vessels, so to speak, ready to be filled with physical meaning). Maxwell wished to restore the physics so that the systems of equations of the phenomena of electrodynamics are not just empty symbols. 7.3.3 Dimensionality of units As we have remarked, Station 4 is both cutting-edge research and a textbook, an introduction to the study of electricity and magnetism.72 This conception of the Treatise can immediately be seen in the opening sections. Maxwell began with basic remarks on the measurement of quantities. He defined the “Unit” as the standard quantity and the “numerical value” as the number of times the standard quantity is to be taken, and then made the claim that:

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There must be as many different units as there are different kinds of quantities to be measured, but in all dynamical sciences it is possible to define these units in terms of the three fundamental units of Length [L], Time [T], and Mass [M].73 It is immediately clear that this claim, namely, that there is a distinction between the three fundamental units and other units which are derivative, has far-reaching consequences. In framing a mathematical system of a certain physical phenomenon, these fundamental units are presupposed and all derivative units are deduced from them by a set of definitions and nomological relations. This requirement underlies Maxwell’s incisive remark: In all scientific studies it is of the greatest importance to employ units belonging to a properly defined system, and to know the relations of these units to the fundamental units, so that we may be able at once to transform our results from one system to another.74 This mathematical transformation is most relevant, as we will see, to the ratio of electrostatic to electromagnetic units. In fact, as Maxwell explained, A knowledge of the dimensions of units furnishes a test which ought to be applied to the equations resulting from any lengthened investigation. The dimensions of every term of such an equation, with respect to each of the three fundamental units, must be the same. If not, the equation is absurd, and contains some error, as its interpretation would be differ­ ent according to the arbitrary system of units which we adopt.75 Dimensional analysis thus offers a procedure for checking the physical consistency and coherence of the equations representing the phenomenon in question. For example, in a section titled, “Derived units”, Maxwell defined the unit of force as the force which produces a unit of momentum in a unit of time, so that its dimensions are [MLT–2]. According to Maxwell, this is the absolute unit of force; its dimensional definition is implied in every equation on dynamics. Clearly, the strength of this approach lies in its universality.76 Against this mechanical background, Maxwell developed a discussion of force in electromagnetism and demonstrated how the units in this domain could be transformed into the fundamental (mechanical) units. In particular, he developed the dimensions of electric units. We note that in Station 4 Maxwell first developed a dimensional analysis of units in mechanics, and subsequently extended this analysis to units of electricity and magnetism. He was, in effect, going from the familiar appeal to length [L], time [T], and mass [M] in mechanics to the unfamiliar appeal to these units in electromagnetism. Following the presentation of the general equations of the electromagnetic field, Maxwell dedicated an entire chapter (ch. 10) to the dimensions of electric

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172 Station 4 (1873) units. Based on what is taken as the unit of electricity, one obtained, according to Maxwell, the electrostatic system. If, however, one began with the unit strength of a magnetic pole, a different system is obtained, namely, the electromagnetic system.77 Maxwell drew a table of the dimensions of the units according to each system. True to the claim that he had made at the outset of his discussion of measurements of quantities, Maxwell specified the different units for the different kinds of quantities to be measured in the domain of electromagnetism. For each unit Maxwell assigned a symbol and the corresponding dimensions in the electrostatic system and the electromagnetic system. Fixing the dimensionality of each unit in terms of powers of [L], [M], and [T], facilitated the dimensional calculation for the ratios of units which are, as Maxwell remarked, “in certain cases of scientific importance.”78 He then made the significant claim that reflected his successful discovery in Station 2: If the units of length [L], mass [M], and time [T] are the same in the two systems, the number of electrostatic units of electricity contained in one electromagnetic unit is numerically equal to a certain velocity, the absolute value of which does not depend on the magnitude of the fun­ damental units employed. This velocity is an important physical quan­ tity, which we shall denote by the symbol v.79 Maxwell thus established a relation between the two systems, namely, the number of electrostatic units in one electromagnetic unit is either proportional or inversely proportional to a velocity, depending on the unit (with some cases where the square of the velocity is required). In a subsequent chapter (ch. 19), Maxwell compared directly the electrostatic units with the electromagnetic units, and remarked that If, therefore, we determine a velocity which is represented numerically by this number, then, even if we adopt new units of length and of time, the number representing this velocity will still be the number of electro­ static units of electricity in one electromagnetic unit, according to the new system of measurement. This velocity, therefore, which indicates the relation between electro­ static and electromagnetic phenomena, is a natural quantity of definite magnitude, and the measurement of this quantity is one of the most important researches in electricity.80 Maxwell showed how to obtain a physical conception of this velocity which is, in effect, the velocity of light, denoted v by Maxwell. According to Maxwell, “the first numerical determination of this velocity was made by Weber and Kohlrausch.”81 The path was then paved for the transition from this abstract analysis of the dimensions of electric units to a practical system. Maxwell remarked:

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The electrical units derived from these fundamental units have been named after eminent electrical discoverers. Thus the practical unit of resistance is called the Ohm, and is represented by the resistance-coil issued by the British Association . . . It is expressed in the electromag­ netic system by a velocity of 10,000,000 metres per second.82 Maxwell went on to refer to the Farad and the Volt. In this way Maxwell put the practice of measurements in electromagnetism and its nomenclature on solid theoretical ground. Most importantly, he demonstrated the importance of velocity as the dimension that relates the two systems, the electrostatic and the electromagnetic. This analysis is entirely novel in Maxwell’s corpus on electromagnetism. There is no such discussion in any of the preceding stations. Surprisingly, the origin of this analysis can be traced back to 1863, when Maxwell was engaged with the issue of standards of measurement, but these results showed up in the study of electromagnetism only in 1873, in Station 4. In 1863, as part of Maxwell’s contribution—coauthored with the engineer Fleeming Jenkin (1833–1885)—to the report by the committee appointed by the British Association on standards of electrical resistance, the claim was made that given the two systems, electrostatic and electromagnetic, the relations between the different units remain unchanged when passing from the one system to the other.83 The constant ratio between units in the two systems, designated v, has the dimensions [L/T], which is a velocity. Maxwell and Jenkin then made a historical claim and acknowledged the findings of Weber and Kohlrausch: The first estimate of the relation between quantity of electricity meas­ ured statically and the quantity transferred by a current in a given time was made by Faraday*. A careful experimental investigation by MM. Weber and Kohlrausch† not only confirms the conclusion that the two kinds of measurement are consistent, but shows that the velocity v = q/Q is 310,740,000 meters per second—a velocity not differing from the esti­ mated velocity of light more than the different determinations of the latter quantity differ from each other. v must always be a constant, real velocity in nature, and should be measured in terms of the system of fun­ damental units adopted in electrical measurements . . . A redetermination of v . . . will form part of the present Committee’s business in 1863–1864. It will be seen that, by definition, the quantity transmitted by an electro­ magnetic unit current in the unit time is equal to v electrostatic units of quantity.84 [Footnotes in Maxwell and Jenkin, 1864, 149; and 1865 509.] * Experimental Researches, series iii. § 361, &c. † Abhandlungen der König. Sächsischen Ges. vol. iii. (1857) p. 260; or, Poggendorff’s Annalen, vol. xcix. p. 10 (Aug. 1856).

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174 Station 4 (1873) In 1864, Maxwell and Jenkin simply reported the findings of Faraday as well as those of Weber and Kohlrausch, but did not elaborate on the physical consequences; they also stressed the constancy of this ratio with the dimensionality of velocity but, again, without elaborating further. They did not then reflect on the significance of this constant ratio which is of the dimension of velocity and is strikingly similar numerically to the velocity of light, measured independently of this ratio. To understand in what way the dimensionality of units is a methodology and not just a method, we turn to a brief examination of the report. We seek to show that dimensionality of units generates new knowledge, a feature that renders this analysis a methodology. Maxwell and Jenkin made claims independent of any physical theory which typically comes with an accompanied ontology. In acknowledging the achievements of Weber and Kohlrausch, Maxwell and Jenkin concluded that it could be seen that “by definition, the quantity transmitted by an electromagnetic unit current in the unit time is equal to v electrostatic units of quantity.”85 We draw attention to the claim, namely, that this fundamental result is obtained “by definition” and is not theory dependent. Maxwell and Jenkin specify the macro-level phenomena on which they developed their analysis of dimensionality of units: All our knowledge of electricity is derived from the mechanical, chem­ ical, and thermal effects which it produces, and these effects cannot be ignored in a true absolute system. Chemical and thermal effects are, however, now all measured by reference to the mechanical unit of work; and therefore, in forming a coherent electrical system, the chemical and thermal effects may be neglected, and it is only necessary to attend to the connexion between electrical magnitudes and the mechanical units . . .. The four equations now given are sufficient to measure all electrical phenomena by reference to time, mass, and space only, or, in other words, to determine the four electrical units by reference to mech­ anical units.86 The key point for our argument is that measurement relates to effects not to causes; hence, the reality at the micro-level (whatever it may be) is irrelevant in this context. It was critical that the analysis should not fall prey to some controversy and that it would simply be accepted by the community of practitioners of electricity and magnetism as independent of any idiosyncratic commitment. In short, the claim is that the analysis is natural, and neutral among theories: In the opinion of the most practical and the most scientific men, a system in which every unit is derived from the primary units with deci­ mal subdivisions is the best whenever it can be introduced. It is easily learnt; it renders calculations of all kinds simpler; it is more readily

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accepted by the world at large; and it bears the stamp of the authority, not of this or that legislator or man of science, but of nature.87 In Station 4 Maxwell offered a systematic treatment of units both in mechanics and in electromagnetism in terms of L, T, and M. This systematic discussion of the dimensionality of units displays relationships between two different systems that may not be obvious otherwise. In fact, Maxwell went beyond the practical needs for measurement as he sought connections inherent in nature. This is a contribution to knowledge and thus the underlying procedure is methodological. Consider the following contrast: in 1873 in the Treatise Maxwell ordered all the various units in terms of L, T, and M, whereas in 1867 in their Treatise, Thomson and Tait did not consider any relation among units of different kinds.88 We consider Thomson and Tait’s practical account a method, unlike the contribution of Maxwell which we consider a methodology. In Station 2 Maxwell depended on a mechanical hypothesis to come to his conclusion concerning the ratio between the two systems of units as equal to the velocity of light. But in Station 4 the argument that the ratio is a velocity is independent of any hypothesis. This is the result of the methodology of dimensionality of units. And then Maxwell argued for the physical significance of this ratio.89 The lesson is implicit: Dimensional analysis supports the claim for the ratio as a velocity independent of any hypothesis or theory. This is an important aspect of Maxwell's overall methodology. As Chrystal remarked in his review of the second edition of the Treatise: It begins by divesting the facts of all hypothetical raiment, and express­ ing them in language appropriate to themselves, suggesting nothing but what Nature has indicated, indicating nothing that Nature has denied, supposing as little as may be where nothing has been revealed. Above all banishing from the catalogue of physical conceptions the imponder­ able electrical fluids that have worked such mischief in indolent minds, and poisoned electrical literature so long.90 In sum, the dimensionality of units helped to liberate Maxwell from dependence on the hypothesis. 7.3.4 Analogies, illustrations, and working models as mechanical representations In Station 4 Maxwell applied a variety of conceptual tools: “mechanical illustrations,”91 “working models”92 as well as “mental images.”93 He introduced these tools to assist the mind in comprehending the (strange) phenomena of electricity and magnetism. The appeal to them was mostly

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176 Station 4 (1873) didactic.94 Thus, for the most part they did not partake in the argumentative structure of Station 4 and therefore were not required to be consistent.95 Maxwell developed illustrations and analogies for research purposes and then recast them as didactic tools. For example, he occasionally appealed to taut ropes and rigid rods as illustrations and analogies: If we now proceed to investigate the mechanical state of the medium on the hypothesis that the mechanical action observed between electrified bodies is exerted through and by means of the medium, as in the famil­ iar instances of the action of one body on another by means of the ten­ sion of a rope or the pressure of a rod, we find that the medium must be in a state of mechanical stress. The nature of this stress is, as Faraday pointed out*, a tension along the lines of force combined with an equal pressure in all directions at right angles to these lines. The magnitude of these stresses is proportional to the energy of the electrification per unit volume, or, in other words, to the square of the resultant electromotive intensity multiplied by the spe­ cific inductive capacity of the medium.96 [Footnote in Maxwell, 1873d, 1: 59, § 59] * Exp. Res., series xi, 1297 According to Maxwell, electric tension is a tension of exactly the same kind, and measured in the same way, as the tension of a rope, and the dielectric medium, which can support a certain tension and no more, may be said to have a certain strength in exactly the same sense as the rope is said to have a certain strength. Thus, for example, Thomson has found that air at the ordinary pressure and temperature can support an electric tension of 9600 grains weight per square foot before a spark passes.97 To be sure, drawing similarities between distinct phenomena, e.g., tension in a medium and tension in a rope, helps develop an intuition for understanding the phenomenon under study, but it is not necessarily the case that consequences drawn in the familiar domain apply to the unfamiliar domain. We have sought to retain this distinction between a methodology that aids intuition and thus enhances comprehension and a methodology that advances research. We have therefore characterized analogies as weak and strong, respectively.98 We surmise that Maxwell’s appeal to ropes, muscles, etc. in Station 4 is didactic and illustrative: no formal consequences are drawn from these analogies—these are weak analogies.99 There was no expectation that using these analogies would help in making progress toward the goal of casting

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Faraday’s verbal descriptions of his experimental results into mathematical symbolism. There are a few strong analogies in Station 4. Consider, for example, Maxwell’s discussion of the dynamical theory of electromagnetism. The framework of the discussion is clearly mechanical: energy comes in two forms, kinetic and potential, and current—however problematic—has to be conceived kinetically: The electric current cannot be conceived except as a kinetic phenomenon. Even Faraday, who constantly endeavoured to emancipate his mind from the influence of those suggestions which the words electric current and electric fluid are too apt to carry with them, speaks of the electric current as something progressive, and not a mere arrangement*.100 [Footnote in Maxwell, 1873d, 2: 195, § 569] * Exp. Res., 1648 This is a most revealing remark. We noted Maxwell’s warning against the allure of analogies; and here we see him acknowledging the problem but assuming it nonetheless in order to develop a strong analogy to advance his research. But all that we assume here is that the electric current involves motion of some kind. That which is the cause of electric currents has been called Electromotive Force. This name has long been used with great advantage, and has never led to any inconsistency in the language of science. Electromotive force is always to be understood to act on electri­ city only, not on the bodies in which the electricity resides. It is never to be confounded with ordinary mechanical force, which acts on bodies only, not on the electricity in them. If we ever come to know the formal relation between electricity and ordinary matter, we shall probably also know the relation between electromotive force and ordinary force.101 It is noteworthy that Maxwell is fully aware of the problem of inconsistency when developing such an analogy between mechanical current and electric current. He claimed that if we were to restrict the discussion to each domain, the electric and the mechanical, no confusion or incoherence would arise. And if the formal, mathematical relation between electricity and matter were to be discovered, the relation between electromotive force and ordinary force would be known. Having secured his assumptions regarding the consistency of the analogy, he proceeded: When ordinary force acts on a body, and when the body yields to the force, the work done by the force is measured by the product of the force into the amount by which the body yields. Thus, in the case of water forced through a pipe, the work done at any section is measured

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178 Station 4 (1873) by the fluid pressure at the section multiplied into the quantity of water which crosses the section. In the same way the work done by an electromotive force is measured by the product of the electromotive force into the quantity of electricity which crosses a section of the conductor under the action of the electro­ motive force. The work done by an electromotive force is of exactly the same kind as the work done by an ordinary force, and both are measured by the same standards or units. We . . . know enough about electric currents to recognise, in a system of material conductors carrying currents, a dynamical system which is the seat of energy, part of which may be kinetic and part potential. The nature of the connexions of the parts of this system is unknown to us, but as we have dynamical methods of investigation which do not require a knowledge of the mechanism of the system, we shall apply them to this case. We shall first examine the consequences of assuming the most general form for the function which expresses the kinetic energy of the system.102 We have italicized two key expressions: “In the same way” and “exactly the same kind”; these expressions indicate that Maxwell developed at this juncture of Station 4 a strong analogy. Indeed, he explicitly stated that having drawn this analogy he turned to “examine the consequences,” and then proceeded to develop the dynamical theory in formal terms.103 This approach is reminiscent of the appeal to physical analogy in Station 1. Later in this discussion of the dynamical theory, Maxwell remarked: All this would be true, if, instead of electric currents, we had currents of an incompressible fluid running in flexible tubes. In this case the vel­ ocities of these currents would enter into the expression for T, but the coefficients would depend only on the variables x, which determine the form and position of the tubes.104 Evidently, Maxwell thought that the analogy between the flow of electricity and the flow of incompressible fluid under these constraints is strong. Maxwell’s mechanical thinking is clearly in evidence in Station 4, especially in the many weak analogies. He regarded the lines of force as tubes; for example, he called the surface generated by the motion of line of force, “a tubular surface,” and then referred to it as “a tube of induction.”105 Moreover, when Maxwell offered a “mechanical illustration of the properties of a dielectric,” his illustration came complete with “tubes” filled partly with mercury and partly with water, as well as with stopcock and piston:

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To represent the case in which there is true conduction through the dielectric we must either make the piston leaky, or we must establish communication between the top of [one] tube . . . and the top of [another] tube. In this way we may construct a mechanical illustration of the properties of a dielectric of any kind, in which the two electricities are represented by two real fluids, and the electric potential is represented by fluid pres­ sure. Charge and discharge are represented by the motion of . . . [a] piston, and electromotive force by the resultant force on the piston.106 Maxwell’s ingenuity is displayed here in the way he conceived the properties of a dielectric mechanically. We note the modal expression, namely, “we may construct a mechanical illustration.” Maxwell developed these weak analogies cautiously. In Station 4 Maxwell sought “a complete dynamical theory” to account for the physics of the phenomena, not just a formal description of them. However, this methodology had to be used judiciously, and Maxwell warned the reader of possible pitfalls. Indeed, on several occasions Maxwell cautioned the reader about the misleading nature of analogies in general.107 In the section, “On the induction of a current on itself,” Maxwell drew attention to Faraday’s remark that “the first thought that arises in the mind is that the electricity circulates with something like momentum or inertia in the wire.” And Maxwell added, When we consider one particular wire only, the phenomena are exactly analogous to those of a pipe full of water flowing in a continued stream. If while the stream is flowing we suddenly close the end of the pipe, the momentum of the water produces a sudden pressure, which is much greater than that due to the head of water, and may be sufficient to burst the pipe.108 On the face of it, this example expresses strong analogy: electromagnetic and hydrodynamic phenomena are exactly analogous. Indeed, the analogy of the flow of fluid in a pipe with the flow of electricity in a wire had been well impressed on the minds of many researchers. However, the analogy is in fact misleading. The inertia of the fluid in the tube depends solely on variables pertaining to what is inside the tube; it does not depend on anything outside it or on the form into which the tube may be bent. Maxwell contrasted the physical analysis of the fluid flowing in a pipe to the wire conveying a current and stated that in the latter case variables external to the tube/wire are critical to the flow phenomena. He then added an insightful remark: It is difficult, however, for the mind which has once recognised the ana­ logy between the phenomena of self-induction and those of the motion

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180 Station 4 (1873) of material bodies, to abandon altogether the help of this analogy, or to admit that it is entirely superficial and misleading. The fundamental dynamical idea of matter, as capable by its motion of becoming the recipient of momentum and of energy, is so interwoven with our forms of thought that, whenever we catch a glimpse of it in any part of nature, we feel that a path is before us leading, sooner or later, to the complete understanding of the subject.109 Maxwell cautioned the physics community against such misleading conceptions that come from uncritical habits of thinking. This was probably one reason for his increasing use of weak analogies in Station 4. Previously, analogy was part of the argument; now it is principally didactic and illustrative. Here the focus is on the misleading analogy between electric current and the current of a fluid: It appears to me, however, that while we derive great advantage from the recognition of the many analogies between the electric current and a current of a material fluid, we must carefully avoid making any assumption not warranted by experimental evidence, and that there is, as yet, no experimental evidence to shew whether the electric current is really a current of a material substance, or a double current, or whether its velocity is great or small as measured in feet per second.110 Still, this was a beginning: A knowledge of these things would amount to at least the beginnings of a complete dynamical theory of electricity, in which we should regard electrical action, not, as in this treatise, as a phenomenon due to an unknown cause, subject only to the general laws of dynamics, but as the result of known motions of known portions of matter, in which not only the total effects and final results, but the whole intermediate mech­ anism and details of the motion, are taken as the objects of study.111 Maxwell entertained the possibility that electricity might, after all, be a current of some kind of “material substance.” Moreover, he envisaged a complete theory of electrodynamics that would cover comprehensively the entire chain from cause to effect, that is, the intermediate mechanisms and the known motions of the known portions of matter, as well as the governing laws. But meanwhile he was not committed to any mechanism at the micro-level. This is the epistemic context in which Maxwell commented on his “working model,” as he now, in Station 4, called his “mechanical hypothesis” of Station 2.112 In a note (§ 831) Maxwell quoted at length Thomson’s discussion of Faraday’s discovery of the magnetic influence on

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light.113 As Thomson remarked, “the magnetic influence on light discovered by Faraday depends on the direction of motion of moving particles.”114 And Thomson continued: I think it is not only impossible to conceive any other than this dynam­ ical explanation . . . but I believe it can be demonstrated that no other explanation of that fact is possible.115 We have omitted the physical content regarding the direction of motion of the particles in order to emphasize the grammatical structure of this claim: the explanation stands in a one-to-one relation to the phenomenon—so certain was Thomson of his proposal. It appeared to Thomson that Faraday’s optical discovery affords a demonstration of the reality of Ampère’s explanation of the ultimate nature of magnetism; and gives a definition of magnetization in the dynamical theory of heat . . . Whether this matter is or is not electricity, whether it is a continuous fluid interpermeating the spaces between molecular nuclei, or is it itself molecu­ larly grouped; or whether all matter is continuous, and molecular heterogeneousness consists in finite vortical or other relative motions of contiguous parts of a body; it is impossible to decide, and perhaps in vain to speculate, in the present state of science.116 For Thomson, then, the exact mechanism cannot be determined—the hypothesis is unproven; but the theory holds. This argument impressed Maxwell who quoted it at length. At this juncture Thomson referred to the hypothesis of molecular vortices which Rankine put forward and to which both Thomson and Maxwell appealed in their explanations of various phenomena of magnetism. At the end of the quotation of Thomson’s text Maxwell referred to his own work on molecular vortices.117 Maxwell ended the quotation, but Thomson continued, “I append the solution of a dynamical problem for the sake of the illustration it suggests for the two kinds of effect on the plane of polarization . . .” He then described a mechanical arrangement that consists of a cord attached to the two ends of a horizontal arm which is made to rotate around a vertical axis at its midpoint; a second cord is tied at the midpoint of the first cord, bearing a weight. The two cords are assumed to be “perfectly light and flexible” and the problem is to determine the motion of this artificial device “when [it is] infinitely little disturbed from its position of equilibrium.”118 According to Thomson, the various solutions offer illustrations for the possible mechanism of magnetism: From these illustrations it is easy to see in an infinite variety of ways how to make structures, homogeneous when considered on a large enough scale, which (1) with certain rotatory motions of component

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182 Station 4 (1873) parts having, in portions large enough to be sensibly homogeneous, resultant axes of momenta arranged like lines of magnetic force, shall have the dynamical property by which the optical phenomena of transpar­ ent bodies in the magnetic field are explained; (2) with spiral arrange­ ments of components parts, having axes all ranged parallel to a fixed line, shall have the axial rotatory property corresponding to that of quartz crystal; and (3) with spiral arrangements of component groups, having axes totally unarranged, shall have the isotropic rotatory property possessed by solutions of sugar and tartaric acid, by oil of turpentine, and many other liquids.119 Thomson was convinced that magnetism is to be explained mechanically; he realized, however, that there are infinite modes of mechanical explanations. This is the background to Maxwell’s opening remark in § 831 of Station 4: “the whole of this chapter may be regarded as an expansion of the exceedingly important remark of Sir William Thomson in the Proceedings of the Royal Society, June 1856.” We consider the remark that follows a key to understanding Maxwell’s view of working models: I think we have good evidence for the opinion that some phenomenon of rotation is going on in the magnetic field, that this rotation is per­ formed by a great number of very small portions of matter, each rotat­ ing on its own axis, this axis being parallel to the direction of the magnetic force, and that the rotations of these different vortices are made to depend on one another by means of some kind of mechanism connecting them.120 This is the celebrated mechanical conception of molecular vortices which Maxwell introduced in Station 2 and to which Maxwell explicitly referred at this juncture of Station 4.121 Curiously, in Station 2 Maxwell referred to this mechanism as a hypothesis, whereas here, in Station 4, he called it a “working model”: The attempt which I then made to imagine a working model of this mechanism must be taken for no more than it really is, a demonstration that mechanism may be imagined capable of producing a connexion mechanically equivalent to the actual connexion of the parts of the elec­ tromagnetic field. The problem of determining the mechanism required to establish a given species of connexion between the motions of the parts of a system always admits of an infinite number of solutions. Of these, some may be more clumsy or more complex than others, but all must satisfy the conditions of mechanism in general.122 This usage of “working model” is most idiosyncratic. At the time of Maxwell, “working model” was primarily related to submissions to the

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patent office and the like (a miniature machine which is able to perform the same functions as the full-scale machine it is intended to represent).123 The expression “working model” in the sense of “working hypothesis” does not seem to be attested before 1873. The context makes it clear that Maxwell did not consider the scheme of molecular vortices to be a false theory: the scheme is just unproven and one of many (mechanical) possibilities for assisting in understanding the phenomenon, namely, the relation between light and magnetism.124 Maxwell commented on the limitations of what he called his “working model,” namely, a “mechanism may be imagined,” which is capable of accounting for the phenomena. But at the same time, he recognized, echoing the view of Thomson, that an “infinite number of solutions [are possible].” Still, once the theory is complete and confirmed, its scaffolding —be it a “model” or a “mechanical illustration”—becomes superfluous.125 Maxwell’s remark is both autobiographical and philosophical. In the first place he recalled his attempt in Station 2 “to imagine a working model,” which at the time he called a “mechanical illustration,” composed of ordinary materials, based on his molecular-vortex hypothesis.126 So Maxwell took account of his earlier work. But then he added a philosophical note of great importance, arguing that his theory does not depend on any specific mechanism. Nevertheless, he seemed to adhere to the view that there is an underlying mechanism, involving vortices and stress, which linked electromagnetic phenomena. For him, however, it was an open question whether the mechanism underlying these phenomena could be discovered and experimentally verified. Despite the absence of a clear statement to this effect, it is likely that Maxwell did not believe that such a mechanism could be found. As was the case with Thomson, the exact mechanism cannot be determined; the hypothesis is unproven, but the theory holds. 7.3.5 Mental imagery In the last section of Station 4 Maxwell commented on Neumann’s mathematical concept of potential: In the theory of Neumann, the mathematical conception called Poten­ tial, which we are unable conceive as a material substance, is supposed to be projected from one particle to another, in a manner which is quite independent of a medium, and which, as Neumann has himself pointed out, is extremely different from that of the propagation of light. In the theories of Riemann and Betti it would appear that the action is sup­ posed to be propagated in a manner somewhat more similar to that of light.127 According to Maxwell, Neumann failed to offer any “mental image” of his concept of potential, and this is a real deficiency in Neumann’s theory.

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184 Station 4 (1873) Maxwell believed it was incumbent on Neumann to explain how his potentials can be conceived (mechanically or otherwise). Maxwell, however, did not require that Neumann come up with a consistent interpretation of his various equations. Presumably, this inability to construct a consistent mental image of the theory was one reason for Maxwell’s dissatisfaction with Neumann’s proposed theory.128 At the very end of Station 4 Maxwell elaborated his view on mental representation: All these theories [e.g., Neumann’s] lead to the conception of a medium in which the propagation takes place, and if we admit this medium as an hypothesis, I think it ought to occupy a prominent place in our investigations, and that we ought to endeavour to construct a mental representation of all the details of its action, and this has been my con­ stant aim in this treatise.129 These are the very last words of the Treatise. Whether this was Maxwell’s “constant aim” is not at all clear; in any event, he did not accomplish it. Perhaps he intended to get back to this issue in the second edition, which Maxwell had not completed at the time of his death.130 Be that as it may, Maxwell regarded the continental theories as leading to the concept of a medium, despite the fact that these theories presuppose action at a distance, which does not require a medium. The demand “to bring electrical phenomena within the province of dynamics,”131 that is, turning electromagnetism into electrodynamics, convinced Maxwell that a mental image of the seat of action, the medium, was mandatory. In the case of the properties of the dielectrics the move was to form a “mechanical illustration,” an arrangement that could “serve to represent the state of a dielectric acted on by an electromotive force.”132 Indeed, in Station 4 Maxwell applied the formulation of dynamics by Lagrange and Hamilton, but he sought to translate “some algebraical process” into a physical property.133 Maxwell emphasized this critical point in a paper he read in 1873 at a meeting of the Cambridge Philosophical Society: “In this way our words will call up the mental image, not of certain operations of the calculus, but of certain characteristics of the motion of bodies.”134 Evidently, the required mental image refers to a physical effect, in this case the motion of bodies, rather than some symbolic operation. At another juncture in Station 4, where Maxwell developed a theory of magnetization, he appealed to the same criterion, namely, the value of a theory lies in its capacity to facilitate mental imagery: The scientific value of a theory . . . in which we make so many assump­ tions, and introduced so many adjustable constants, cannot be estimated merely by its numerical agreement with certain sets of experiments. If it

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has any value it is because it enables us to form a mental image of what takes place in a piece of iron during magnetization.135 Here we have an explicit requirement: the theory should facilitate mental imagery of a physical phenomenon. We note the dual approach of verbal descriptions and mathematical symbolism. In the previous stations Maxwell translated Faraday’s verbal descriptions into symbolic form; in Station 4, however, the process is sequential, from verbal account to symbolic language, and then from mathematical symbolism to verbal form. The second stage in the sequence, namely, the retranslation, is from symbolic language to ordinary language, which facilitates the formation of a mental image. In Maxwell’s view, the second stage is essential for completing a proper argument in physics and, clearly, a mental image is not a matter of mathematical symbolism.

7.4 The impact of the new methodologies on the construction of the theory 7.4.1 The electric displacement and the displacement current Maxwell addressed the issue of the displacement current in the plan of Station 4, and his approach here is consistent with what we have seen in the preceding stations. He offered this verbal description of what he called “an electric displacement”: We may conceive it [i.e., the electric polarization of a dielectric] to con­ sist in what we may call an electric displacement, produced by the elec­ tromotive intensity. When the electromotive force acts on a conducting medium it produces a current through it, but if the medium is a non­ conductor or dielectric, the current cannot flow through the medium, but the electricity is displaced within the medium in the direction of the electromotive intensity, the extent of this displacement depending on the magnitude of the electromotive intensity, so that if the electromotive intensity increases or diminishes, the electric displacement increases or diminishes in the same ratio. Maxwell then proceeded to draw a mechanical analogy: The analogy between the action of electromotive intensity in producing electric displacement and of ordinary mechanical force in producing the displacement of an elastic body is so obvious that I have ventured to call the ratio of the electromotive intensity to the corresponding electric displacement the coefficient of electric elasticity of the medium. The coefficient is different in different media, and varies inversely as the spe­ cific inductive capacity of each medium.136

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186 Station 4 (1873) For Maxwell the mechanical analogy was “obvious” and suggested a way to develop a mathematical formalism similar to that of elasticity. It is noteworthy that Maxwell chose first to exhibit the electric displacement verbally, in the tradition of Faraday, although this was his own discovery, and not one by Faraday. This means that Maxwell probably inferred the electric displacement from the verbal analogy to elasticity and not from some symbolic language. In other words, it is most unlikely that the discovery came from looking at equations. Maxwell remarked further that the ordinary theory (e.g., Ampère’s theory) does not take into account this electric displacement and is restricted to the bounding surfaces of the conductors and the dielectrics, that is, to closed circuits.137 The discussion is essentially descriptive with a detailed account of the electric displacement. The reader is then informed that a fuller account, with a mechanical illustration, is given in Part I, Chapter V, “Mechanical action between electrified bodies.” Indeed, in this chapter Maxwell distinguished between two different issues in the theory of the action of the medium. First was the fundamental assumption that the medium is in a state of stress and that, in turn, called attention to the issue of the cause of this stress: but we have not in any way accounted for this stress, or explained how it is maintained. This step, however, seems to be an important one, as it explains, by the action of the consecutive parts of the medium, phenom­ ena which were formerly supposed to be explicable only by direct action at a distance.138 Maxwell acknowledged that he was unable to account for the stresses in the dielectric by mechanical considerations. He proceeded to offer a verbal description of electric displacement: When induction is transmitted through a dielectric, there is in the first place a displacement of electricity in the direction of the induction. For instance, in a Leyden jar, of which the inner coating is charged posi­ tively and the outer coating negatively, the direction of the displacement of positive electricity in the substance of the glass is from within outwards. According to Maxwell, The whole quantity of electricity displaced through any area of a sur­ face fixed in the dielectric is measured by the quantity which we have already investigated . . . as the surface-integral of induction through that area, multiplied by1/(4π)K, where K is the specific inductive capacity of the dielectric.139

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Earlier in Station 4, in a section entitled “Resultant force at a point,” Maxwell made the following claim: When we wish to express the fact that the resultant force is a vector, we shall denote it by the German letter E.140 If the body is a dielectric, then, according to the theory adopted in this treatise, the electricity is displaced with it, so that the quantity of electricity which is forced in the direction of E across unit of area fixed perpendicular to E is D¼

1 KE ð4πÞ

where D is the displacement, E the resultant force, and K the specific inductive capacity of the dielectric. For air, K = 1.141 The vector formulation became particularly useful when Maxwell treated a polarized dielectric medium as the seat of electrical energy.142 The dominant role of the new mathematics is amply present in this formulation, especially the concepts of vector and surface integral, by which the calculation of a quantity forced across a unit area perpendicular to the direction of the force can be carried out. The question now presents itself, did Maxwell formulate the electric displacement back in Station 3 in a similar way? A comparison with the analysis Maxwell had presented less than a decade earlier is instructive, for it may shed light on his changing views in response to recent developments. As we have seen, in Station 3 the set of equations (E) was designated as the equations of electric elasticity. Maxwell was consistent in claiming that the amount of electrification when an electromotive force acts on a dielectric depends on the intensity of the electromotive force and on the nature of the substance. If k is the ratio of the electromotive force to the electric displacement, then Equations of Electric Elasticity; P ¼ kf Q ¼ kg R ¼ kh

ðE Þ;

where P, Q, and R are the components of the electromotive force at any point, and f, g, and h are the electric displacements.143 Maxwell linked the elasticity of the medium directly with electric phenomena; this is reflected in the three-dimensional representation of a vector, without vector notation. This set of equations (E) is closely related to a passage in Station 4 where Maxwell drew attention to the analogy between the action of electromotive intensity in producing electric displacement and of ordinary mechanical force in producing the displacement of an elastic body.144

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188 Station 4 (1873) On the face of it, the equations of Station 3 are different from those of Station 4. But the proportionality of D to E in the equation in Station 4, D =1/(4π)KE, corresponds to equations (E) in Station 3. In other words, the k of Station 3 has become [1/(4π)]K in Station 4. To be sure, in Station 4 Maxwell had vectorial notation and did not confine his attention to the components. Furthermore, in Station 4 Maxwell presented the electric displacement in terms of the energy of the medium, thus linking the phenomenon directly to the tension of the medium, thereby providing a mechanical-causal explanation for the phenomenon. However, while Maxwell acknowledged on both occasions the analogy with the mechanical phenomenon of stress, in Station 4 he was explicit about his inability to offer insight into the mechanism responsible for the phenomenon of displacement. Still, he was keen to illustrate the phenomenon mechanically.145 We now turn to the formulation of the electric displacement in the general equations in Station 4. In Part IV, Chapter IX, “General equations of the electromagnetic field,” Maxwell returned to the electric displacement, but this time in the context of the entire domain of the electromagnetic phenomena. Maxwell presented this array of equations under the heading: “Quaternion Expressions for the Electromagnetic Equations.”146 He remarked that he avoided demanding from the reader knowledge of the calculus of quaternions, but had no hesitation introducing a vector when the physics so required. In this set of equations there are 11 vectors, 4 scalar functions, and 3 quantities indicating physical properties of the medium at a point. One of the eleven vectors is the electric displacement and Maxwell presented it, as before, without any change, namely,147 D¼

1 KE ð4πÞ

Now, as Maxwell remarked, one of the chief peculiarities of this treatise is the doctrine which it asserts, that the true electric current C, that on which the electromagnetic phenomena depend, is not the same thing as K, the current of conduc­ tion, but that the time-variation of D, the electric displacement, must be taken into account in estimating the total movement of electricity.148 Thus, the equation of true current is given by the electric current plus the displacement current. In developing the general equations Maxwell first “determined the relations of the principal quantities concerned in the phenomena discovered by Ørsted, Ampère, and Faraday.” Then, Maxwell continued, “to connect these with the phenomena described in the former parts of this treatise, some additional relations are necessary.”149 The displacement current is one such additional term required to produce the

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equation of the true current; it is the time-variation of the electric displacement, which is a differential.150 This is essentially equivalent to the formulation in Station 3 where, in Part III of the paper under the heading “General equations of the electromagnetic field,” Maxwell had formulated the electrical currents and the electrical displacements, with one fundamental difference: in Station 3 it is a vector without vector notation, unlike the vectorial notation in Station 4.151 In the final analysis the equations in Station 3 and in Station 4 for the electric displacement and the displacement current convey the same proportionality; the difference is rather in the presentation: other than vector notation the equations are the same. Still, there is another important aspect that distinguishes the discussion of Station 4 from the earlier presentation of the equation in Station 3, namely, the role of experiment. In Station 4 Maxwell insisted that he had no direct experimental evidence for this phenomenon; rather, it was a result of his reasoning: The only experimental fact which we have made use of in this investiga­ tion is the fact established by Ampère that the action of a closed circuit on any portion of another circuit is perpendicular to the direction of the latter. Every other part of the investigation depends on purely math­ ematical considerations depending on the properties of lines in space. The reasoning therefore may be presented in a much more condensed and appropriate form by the use of the ideas and language of the math­ ematical method specially adapted to the expression of such geometrical relations—the Quaternions of Hamilton.152 In this remark Maxwell indicated that he went beyond “experimental fact” and depended on “purely mathematical considerations.” Here Maxwell engaged in innovative thinking, no longer bound by the project of casting Faraday’s experimental findings into mathematical language. And the new mathematical method facilitated the expression of geometrical relations in a “much more condensed and appropriate form.” The Cartesian representation of vectors is mathematically equivalent to vector notation, but not physically; that is, the vector is a quantity with a direction that has physical meaning whereas the components separately have no physical significance. This was a triumph of pure thought, assisted by a novel mathematical formulation. Maxwell underlined the lack of experimental evidence: We have very little experimental evidence relating to the direct electro­ magnetic action of currents due to the variation of electric displacement in dielectrics, but the extreme difficulty of reconciling the laws of elec­ tromagnetism with the existence of electric currents which are not closed is one reason among many why we must admit the existence of transient currents due to the variation of displacement.153

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190 Station 4 (1873) Maxwell made this statement just before he presented the formulas for the electric displacement and the displacement current.154 It is noteworthy that in his previous publications he was not explicit about the absence of experimental evidence to support the equation of the displacement current; however, now he had confidence in the mathematics to guide him correctly in the treatment of the phenomena. Vector notation and the accompanying analysis proved highly productive, for this was not just a mathematical method of great convenience, but a reliable methodology for generating new knowledge. One of Maxwell’s achievements in Station 4 was the integration of the displacement current into a mathematical electrodynamic theory, together— and this is the critical point—with a physical picture that was completely faithful to the field conception. His new theory was not dependent on some mechanical scheme at the micro-level and could fully rely on mathematics at the macro-level. As Siegel argued, central to this success was “a redefinition of the relationship between ‘electricity’ and ‘charge.’”155 In the new theory Maxwell sought to avoid the microscopic level for an explanation. One consequence of this avoidance was the transformation of the concept of charge. According to Siegel, in Station 4, There were to be no primitive charges or microscopic charge carriers. “Electricity” was to be viewed as an abstract, mobile, incompressible fluid that did not consist of electric charges and that did not have accumulation points. “Charge” thus could not be regarded as an accumulation of this fluid, both because the fluid did not consist of electric charges in the first place and because the fluid could not accumulate . . . Electric “charge,” finally, would arise from the effects that the flow of “electricity” would have on the medium through which it flowed.156 Likening electricity to an incompressible fluid rather than to charged particles is, to be sure, methodological. The analogy between electricity and an incompressible fluid is suggestive, which, in our terms, is a weak analogy, and it enhances the development of arguments at the macro-level of the phenomena.157 7.4.2 General equations of the electromagnetic field In his opening remarks to Part 4, Ch. IX, “General Equations of the Electromagnetic Field,” Maxwell explained that his reasoning is purely dynamical: In our theoretical discussion of electrodynamics we began by assuming that a system of circuits carrying electric currents is a dynamical system, in which the currents may be regarded as velocities, and in which the coordinates corresponding to these velocities do not

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themselves appear in the equations. It follows from this that the kinetic energy of the system, in so far as it depends on the currents, is a homo­ geneous quadratic function of the currents, in which the coefficients depend only on the form and relative position of the circuits. Assuming these coefficients to be known, by experiment or otherwise, we deduced, by purely dynamical reasoning, the laws of the induction of currents, and of electromagnetic attraction.158 What is striking in this passage is that once the system is assumed to be dynamical, Maxwell was able to proceed with symbolical reasoning devoid of empirical content. For instance, the coordinates of the current as velocities do not appear in the equations, nor are the coefficients of the functions necessarily known by experiment—as Maxwell put it, “assuming these coefficients to be known, by experiment or otherwise.” This is the power of “rational mechanics.” Maxwell put forward three sets of equations he had already presented in the previous chapter, “Exploration of the Field by means of the Secondary Circuit”:159 (A), are those [equations] of magnetic induction, expressing it in terms of the electromagnetic momentum.160 (B), are those [equations] of electromotive intensity, expressing it in terms of the motion of the conductor across the lines of magnetic induction, and of the rate of variation of the electromagnetic momentum.161 (C), are the equations of electromagnetic force, expressing it in terms of the current and the magnetic induction.162 It is noteworthy that in Station 4 Maxwell assigned the equations to three distinct groups: (A) magnetic, (B) electromotive, and (C) electromagnetic. Altogether these three sets of equations provide the formal framework for the treatment of all electromagnetic phenomena. But before displaying the equations, he listed the principal vectors that had to be considered (11 in number), the scalar functions (4 in number) and, finally, quantities of physical properties of the medium (3 in number).163 In the spirit of exhibiting explicitly knowledge of this domain in symbolic language, Maxwell concluded with a set of equations.164 Maxwell presented his theory in a set of equations, 12 in number, marked (A) to (L):165 (A) . . . Relations of magnetic induction (B) . . . General equations of electromotive force (C) . . . Electromagnetic force on an element of a conducting body (D) . . . Equations of magnetization (E) . . . Equations of electric currents

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192 Station 4 (1873) (F) . . . Equations of electric displacement (G) . . . Equations of electric conductivity (H) . . . Equations of true currents (I) . . . Currents in terms of electromotive force (J) . . . Volume-density of free electricity (K) . . . Surface-density of free electricity (L) . . . Equations of magnetic permeability This set of equations is reminiscent of the list that Maxwell had already presented in Station 3,166 where he called it the “General equations of the electromagnetic field”167—the very title for the set of these equations in Station 4. The comparison of the set of equations in Station 3 with the general equations which Maxwell formulated in Station 4 throws light on the changes that occurred in Maxwell’s thinking both at the theoretical and the methodological level. Our goal here is to highlight both differences and similarities in the methodologies. We begin by assessing external features. It is evident that the number of the equations is different in the two theories, due in part to vector notation. As we have seen, the dynamical theory of Station 3 has six sets of three equations for some phenomena. Hence there were 18 equations plus 2 equations of one variable each, making a total of 20 equations.168 However, if Maxwell had kept the equations of Station 3 and only converted them to vector notation, he would have had 8 equations instead of 20, namely, 6 equations for each of the three orthogonal coordinates in Cartesian notation, x, y, and z, plus two equations of one variable each. This is not what Maxwell did. In the new dynamical theory of Station 4 there are 12 equations; this means that there are 4 additional equations that make the theory richer and more nuanced. In the new theory Maxwell listed 11 principal vectors, four scalar functions, and three quantities indicating physical properties of the medium at each point.169 These 18 variables compare with the 20 variables which Maxwell listed in Station 3. But, again, the old theory, when converted to vectorial notation, has only eight variables. Hence, there is no doubt that the dynamical theory of Station 4 is more developed than the earlier theory of Station 3: 12 equations vs. 8 equations, and 18 variables vs. 8 variables, respectively. We now turn to internal features, and we note that from a methodological point of view the theories of both Station 3 and Station 4 are constructive, that is, the equations are not set in an axiomatic framework and they serve as building blocks that can be combined in various ways. Thus, in both theories the phenomena are represented by equations (that are not ordered hierarchically) and then combined in various ways to solve a set of problems. In principle, the methodology of construction is similar in the two theories; indeed, as in Station 3, the set of

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equations Maxwell introduced in Station 4 in Part 4, Ch. IX, was not part of a deductive system.170 Against this background, the different use of the term “dynamics” in the two theories of Station 3 and Station 4 is instructive. As we have seen, the dynamics of the old theory referred to matter in motion by which the observed electromagnetic phenomena are produced, whereas the dynamics of the theory in Station 4 draws on “abstract dynamics.”171 Yet the equations in the new theory were combined together in the same constructive way as in the old theory. The formal framework is thus similar in both theories, but what were the reasons for the changes and the increase in the number of equations? And were these changes motivated by methodological considerations? Let us look closely at two examples. We have already discussed induced magnetization which offers a vivid illustration of the continuity in Maxwell’s approach from Station 3 to Station 4 in developing the physics of electricity and magnetism. We have shown how the three equations of the magnetic force (equation B, Station 3) are in fact equivalent to the relation of magnetic induction (equation A, Station 4).172 But, as we already indicated, there are additional equations in Station 4 that have no counterparts in the theory of Station 3. One such development is the distinction between the volume-density and the surface-density of free electricity articulated in Station 4 (equations J and K, respectively).173 In order to see what is novel in Station 4 we return to Maxwell’s short, but seminal, paper of 1871, “Remarks on the mathematical classification of physical quantities.”174 In this paper Maxwell introduced, inter alia, an important distinction between two kinds of vectorial expressions: Vectors which are referred to unit of length I shall call Forces . . . The operation of taking the integral of the resolved part of a force in the direction of a line for every element of that line, has always a physical meaning. In certain cases the result of the integration is independent of the path of the line between its extremities. The result is then called a Potential.175 This was the case in Station 3, at which time Maxwell had not yet recognized the topological issue of directionality. Put differently, in Station 3 Maxwell did not distinguish between force and flux. Evidently, in developing the mathematical arguments in 1871 Maxwell depended on analogies. This is also true in Station 4 but then Maxwell expressed himself cautiously and was amply clear that this is just an analogy: there was no claim that electricity is a substance like water, or a state of agitation akin to heat: Potential, in electrical science, has the same relation to Electricity that Pressure, in Hydrostatics, has to Fluid, or that Temperature, in

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194 Station 4 (1873) Thermodynamics, has to Heat. Electricity, Fluids, and Heat all tend to pass from one place to another, if the Potential, Pressure, or Tempera­ ture is greater in the first place than in the second. A fluid is certainly a substance, heat is as certainly not a substance, so that though we may find assistance from analogies of this kind in forming clear ideas of formal electrical relations, we must be careful not to let the one or the other analogy suggest to us that electricity is either a substance like water, or a state of agitation like heat.176 Indeed, Maxwell was adamant that electricity and magnetism constitute a domain of their own, distinct from any other known phenomena: Those writers who supposed electricity to be a material fluid or a collec­ tion of particles, were obliged . . . to suppose the electricity distributed on the surface in the form of a stratum of a certain thickness . . . There is, however, no experimental evidence either of the electric stratum having any thickness, or of electricity being a fluid or a collection of particles.177 If anything, the closest analogical option was an incompressible fluid: There is nothing . . . among electric phenomena which corresponds to the capacity of a body for heat. This follows at once from the doctrine which is asserted in this treatise, that electricity obeys the same condi­ tion of continuity as an incompressible fluid.178 Indeed, fairly early in Station 4 Maxwell remarked: We are thus led to a very remarkable consequence of the theory which we are examining, namely, that the motions of electricity are like those of an incompressible fluid, so that the total quantity within an imagin­ ary fixed closed surface remains always the same.179 Thus, according to the analysis in Station 4 electricity is akin to incompressible fluid; in particular, the motions of these two “entities” are taken to be similar. This analogy is only suggestive and it is not invoked in the demonstration of the theory. To take stock of the situation: In 1871 Maxwell discussed fluids, heat, and electricity for illustrating the distinction between force and flux. He invoked analogies (fluid and heat) and drew inferences from them. To be sure, in 1873 in Station 4, he also appealed to analogies but this time he used them cautiously.180 In fact, in Station 4, Part 4, Ch. IX, “General Equations of the Electromagnetic Field,” Maxwell dropped all references to analogies and illustrations. These were tools for research that Maxwell discarded when it came to formulating the equations. In sum, the analogies that led to the equations are not part of the theory.

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It is noteworthy that, for the additional equation K in Station 4, Maxwell was not motivated by any new electromagnetic phenomenon. Put differently, it is not the case that between Station 3 and Station 4 new electromagnetic phenomena were discovered which called for additional equations. Evidently, the new equation is due to more refined mathematical distinctions; specifically, given a vectorial expression, what could be its physical meaning? Maxwell distinguished four such vectorial expressions with distinct physical meanings, namely, unit of length (= force), unit of area (= flux), translation (= electricity) and rotation (= magnetism).181 The discovery of structure in a set of equations is undoubtedly an advantage that can lead to the elimination of redundancy and result in compactness. But Maxwell did not seek mathematical compactness. Well into Station 4, indeed in its last part, Maxwell’s preference was to keep a formula, even if it might seem redundant, rather than risk eliminating something that could be useful: These may be regarded as the principal relations among the quantities we have been considering. They may be combined so as to eliminate some of these quantities, but our object at present is not to obtain com­ pactness in the mathematical formulae, but to express every relation of which we have any knowledge. To eliminate a quantity which expresses a useful idea would be rather a loss than a gain in this stage of our enquiry.182 This explicit statement shows that Maxwell was not thinking in terms of a deductive system of equations. Maxwell had to account for very complex phenomena, and he did so by representing them in several sets of equations. The formal similarities between different physical domains, such as heat and electricity, had been noticed earlier by Thomson, and Maxwell was well aware of these similarities. But in Station 4 Maxwell charted the physical domain of electromagnetism with novel methodologies that he devised for this purpose.183 7.4.3 From mathematics to physics: vortices and mechanism Given Maxwell’s methodology, mathematics (equations) has to be followed by physics—physical ideas expressed in words. Thus, Station 4 does not end with the general equations; rather, both the chapter on the electromagnetic theory of light and a discussion of magnetic action on light are placed after the chapter on the general equations.184 Maxwell now concentrated on a critical aspect of the theory, namely, the possible relations between electric and magnetic phenomena and those of light. Maxwell reported that Faraday had not detected a relation between electric phenomena and light, but “he succeeded . . . in establishing a relation between light and

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196 Station 4 (1873) magnetism.”185 This discovery served as the starting point for further investigation in Station 4 into the nature of magnetism, based on the hypothesis of molecular vortices which Maxwell had already considered in Station 2. Here too, as in the discussion of the electromagnetic theory of light, Maxwell noted that “a theory of molecular vortices, which I worked out at considerable length, was published in the Phil. Mag. for March, April, and May, 1861, Jan. and Feb. 1862.”186 Maxwell, once again, referred to a preceding station. Evidently, in Station 4 Maxwell intended to take advantage of all his previous works on electromagnetism including his speculations. Indeed, in the case of the relation of light and magnetism Maxwell had nothing new to offer, for he reverted to the vortices to which he had appealed in Station 2 (but not in Station 3), despite his limited commitment to them.187 It seems that his physical ideas did not develop to the same extent as his mathematical analysis. He therefore returned to his previous set of physical ideas rather than find a completely new set to correspond to the new mathematical tools that he introduced in Station 4. In fact, Maxwell depended on the same theoretical source as he had in Station 2, namely, Helmholtz’s memoir on vortex motion (1858), adding a reference to the English translation which Tait published in 1867.188 It is evident that Maxwell sought an explanation, and a physical one at that: The physical explanation of the phenomenon [magnetic action on light] presents considerable difficulties, which can hardly be said to have been hitherto overcome, either for the magnetic rotation, or for that which certain media exhibit of themselves. We may, however, prepare the way for such an explanation by an analysis of the observed facts.189 The explanation, as Maxwell remarked, proceeds by analyzing “the observed facts.” Seeking the resemblance between optical and magnetic phenomena, Maxwell reached the conclusion that “at each point of the medium something exists of the nature of an angular velocity about an axis in the direction of the magnetic force.”190 Throughout the discussion Maxwell insisted that “the disturbance which constitutes light, whatever its physical nature may be, is of the nature of a vector, perpendicular to the direction of the ray.”191 Being a vector, the disturbance—of whatever nature—could be resolved into components, allowing the mathematical analysis to proceed. And since the analysis was spatial, Maxwell applied tools taken from the geometry of position. Maxwell expanded what he called “the Geometry of Position”—a reference to Geometria Situs in contrast to Geometria Magnitudinis.192 He noted the origin of this subject in Leibniz’s work and its further development by Gauss but, as Maxwell reported, Gauss lamented how little progress had been made in “the Geometry of Position since the time of Leibniz, Euler and Vandermonde.” Maxwell then added, “we have now,

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however, some progress to report, chiefly due to Riemann, Helmholtz, and Listing.”193 Maxwell depended on the work of Johann B. Listing (1808–1882), citing his “Census räumlicher Complexe” (1861).194 Indeed, as Tait indicated, “[Listing’s papers] seem scarcely to have been noticed in this country [Great Britain], until attention was called to their contents by Clerk-Maxwell.”195 In 1847 Listing coined the term Topologie in German for this new mathematical domain, but it was not immediately adopted in English. In fact, Listing considered the appropriateness of the expression (in Latin) Geometria Situs as well as the expression (in French) géométrie de position, but decided to introduce the new term (in German) Topologie. Maxwell did not adopt Listing’s term, preferring a literal translation of the French, namely, geometry of position.196 Typically, topological concepts and results belong to pure mathematics but, as Epple argued, “Maxwell viewed topology as linked to natural philosophy for conceptual reasons.” In this view, “topology was the science investigating the properties of physical continuity in actual space.”197 Indeed, at stake were the relations between the directions of linear and rotational motions due to an electric current flowing in a circuit and the associated magnetic field. In applying topology to electrodynamics Maxwell depended on new results of his own in this branch of pure mathematics as well as those obtained by his predecessors. He was motivated by issues in electrodynamics, and searched for novel ways to recast effectively his physical reasoning in mathematical form. This concern made its way into Station 4: If the actual rotation of the earth from west to east is taken positive, the direction of the earth’s axis from south to north will be taken posi­ tive . . . If we place ourselves on the positive side of a surface, the posi­ tive direction along its bounding curve will be opposite to the motion of the hands of a watch with its face towards us.198 Evidently, it was important for Maxwell to determine directionality. Later in Station 4 he turned it into a rule: Hence we have the following rule for determining the electromotive force on a wire moving through a field of magnetic force. Place, in imagination, your head and feet in the position occupied by the ends of a compass needle which point north and south respectively; turn your face in the forward direction of motion, then the electromotive force due to the motion will be from left to right.199 Maxwell concluded the chapter on the magnetic action on light (ch. 21) with a section “On the Hypothesis of Molecular Vortices” (§§ 822–831). The leading physical conception is that since

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198 Station 4 (1873) the angular velocity cannot be that of any portion of the medium of sensible dimensions rotating as a whole . . . we must . . . conceive the rotation to be that of very small portions of the medium, each rotating on its own axis. This is the hypothesis of molecular vortices.200 The consequence is inescapable: the explanans must be at the micro-level and thus of a speculative nature. Maxwell ended the analysis with what he called “results of the theory” which he considered of “higher value”: (1) Magnetic force is the effect of the centrifugal force of the vortices. (2) Electromagnetic induction of currents is the effect of the forces called into play when the velocity of the vortices is changing. (3) Electromotive force arises from the stress on the connecting mechanism. (4) Electric displacement arises from the elastic yielding of the connecting mechanism.201 Together, the four results offer a unified view of electromagnetic phenomena, based on mechanical concepts. All four aspects of electrodynamics, namely, magnetic and electromotive forces as well as electromagnetic induction and electric displacement, are associated in these four “results of the theory” with a mechanical feature of the field. It is noteworthy that in this presentation the theory is linked to a variety of mechanisms: the “centrifugal force of the vortices,” the “velocity of the vortices” as well as “the stress on the connecting mechanism,” and the elasticity of the medium. Above all, the “results” are expressed verbally, an essential stage according to Maxwell’s scientific methodology. However, and this is crucial for understanding Maxwell’s insightful scientific methodology, he was fully aware that here, with the hypothesis of molecular vortices, he was engaged in ontological speculation at the micro­ level: The [proposed] theory . . . is evidently of a provisional kind, resting as it does on unproved hypotheses relating to the nature of molecular vor­ tices, and the mode in which they are affected by the displacement of the medium. We must therefore regard any coincidence with observed facts as of much less scientific value in the theory of the magnetic rela­ tion of the plane of polarization than in the electromagnetic theory of light, which though it involves hypotheses about the electric properties of media, does not speculate as to the constitution of their molecules.202 It is a sign of Maxwell’s genius that he was not persuaded by the mere agreement of observed facts with theory. While he had complete confidence in his identification of light as an electromagnetic disturbance which was indeed based on such a coincidence, he did not allow his hypothesis of

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molecular vortices at the micro-level to overwhelm his reluctance to adopt a mechanism which was utterly speculative. It seems that the introduction of the “new” abstract concept of energy into the physics of electromagnetism made it more difficult for Maxwell to come up with a novel “picture.” And we recall that Maxwell would not accept a purely formal theory, for mathematical formalism (even if successful) was not sufficient in the absence of a corresponding mental image of the physics of the phenomenon.203 Perhaps for this reason he reverted to the old “picture,” trying as he did to match the new mathematics with the old mental image.204 In the Preface to Station 4, Maxwell outlined the goals of the project: In the following Treatise I propose to describe the most important of these [electromagnetic] phenomena, to shew how they may be subjected to measurement, and to trace the mathematical connexions of the quan­ tities measured. Having thus obtained the data for a mathematical theory of electromagnetism, and having shewn how this theory may be applied to the calculation of phenomena, I shall endeavour to place in as clear a light as I can the relations between the mathematical form of this theory and that of the fundamental science of Dynamics, in order that we may be in some degree prepared to determine the kind of dynamical phenomena among which we are to look for illustrations or explanations of the electromagnetic phenomena.205 The concluding section of Station 4 addresses the third stage in Maxwell’s overall argument. As is clear from the final sentence in Station 4, in Maxwell’s view he had achieved what he set out to do.206 In other words, Maxwell sought to discover hitherto unknown physical connections that follow from the manipulation of symbols in the mathematical form of the theory.

7.5 Conclusion The fundamental feature which has emerged from our analysis of Station 4, the culmination of Maxwell’s odyssey, is that the mathematics of abstract dynamics applied to the physical concept of energy served as a new foundation for electrodynamics. Maxwell’s novel methodological approach required appropriate mathematical tools such as quaternions, line- and surface-integrals, Lagrangians and Hamiltonians, as well as the geometry of position. This set the scene for our examination of selected instances of Maxwell’s new approach. Three cases were discussed, namely, displacement phenomena, the general equations of the electromagnetic field, followed by a mechanical explanation of the magnetic action on light as part of a unified view of electromagnetic phenomena.

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200 Station 4 (1873) We take the outline of Maxwell’s methodology in Station 4 to be the following: (1) The goal: Construct a formal theory, namely, a theory formulated in symbolic language, compatible with the range of phenomena in physical domain A. Make sure the theory has intuitive mental imagery express­ ible verbally. (1.1) Construct the theory so that it can be understood in terms of a variety of mechanisms at the micro-level. (1.2) Seek, with the theory, unforeseen physical connections that can then be tested experimentally and may lead to the discovery of new phenomena. (2) The means: Apply a multi-layered set of methodologies. (2.1) Include as many references to observations as possible of the per­ tinent phenomena in domain A as well as to their measurements and the relevant units. (2.2) Maintain presuppositions and results that were established in pre­ vious research, but recast them in mathematical form which may be dif­ ferent from that presented in previous studies. Specifically, adhere to a unifying concept (lines of force) recast mathematically and assume its condition (that is, the medium which is the same for electromagnetic disturbance as well as for light and, likewise, assume the existence of the displacement current). In sum, reverse the argument, which means turning consequences into assumptions. (2.3) Exploit general physical concepts (such as “energy”) and adapt them to the pertinent phenomena of domain A, using the methodology of “abstract dynamics” and advanced mathematical methods (e.g., the Lagrangian and vectorial analysis). (2.4) Construct theories at many different levels, ultimately seeking a general dynamical theory comprising a set of equations of the bearer of the phenomena (i.e., the electromagnetic field). (2.5) Group the empirical observations in domain A in relation to the explanatory mechanical features at the micro-level of the unifying con­ cept (in this case, the electromagnetic field: “centrifugal force of the vor­ tices,” “velocity of the vortices” as well as “stress on the connecting mechanism,” and “elasticity of the medium”). (2.6) Express the relations that hold among the quantities found in (2.5) in appropriate mathematical formalism, while making sure that mental images correspond to the symbolic presentation. This multi-layered set of methodologies aimed at achieving explanation of the known phenomena as well as the discovery of new phenomena. There are several distinct features that should be emphasized. In the first place,

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the construction of the theory is piecemeal, that is, the theory is not axiomatic. As long as deductions are drawn from the set of equations, the scheme must be consistent; however, there are several methodological aspects, mainly didactic for facilitating understanding the phenomena, which do not require consistency. Typically, they are pedagogical and not part of the theory. The appeal to mental imagery may be supported, then, by weak analogies. And experimental results (including measurements) are cited frequently. But what is truly innovative is the conceptual framework of the field as the seat of energy and adapting available mathematical methods to render electromagnetism into electrodynamics. Maxwell’s contributions to electrodynamics culminated in Station 4. All the various methodologies, which he had pursued in electromagnetism and electrodynamics, are represented in this station. Analogy, algebraic formalism, mechanical illustration, and working model, all make their appearance in Station 4 without, however, being woven into a consistent fabric, either autobiographical or philosophical: on the one hand, Station 4 is rich in a variety of ingenious methodologies but, on the other hand, this wealth of ideas for scientific research did not coalesce into a single coherent methodology. Yet, Station 4 constitutes the first comprehensive theoretical treatment of electromagnetic phenomena, cast in a set of equations, the result of a variety of ingenious complementary methodologies. Throughout Station 4 one senses that methodologies were the driving force for Maxwell, and his success depended on his flexibility in applying them. However, Maxwell’s goal was not limited to presenting a set of equations, for the equations had to be translated into ordinary language to be physically meaningful by invoking mental images of the physics of electromagnetism.

Notes 1 Maxwell, [1873] 1892, 1: xv. For a striking example, consider the discovery of the Hall effect. Hall began his paper with a quotation from the Treatise: “Sometime during the last University year, while I was reading Maxwell’s Elec­ tricity and Magnetism in connection with Professor Rowland’s lectures [at the Johns Hopkins University], my attention was particularly attracted by the fol­ lowing passage in Vol. II, p. 144” (Hall, 1879, 287; cf. Maxwell, 1873d, 2: 144– 145, § 501). For the central argument of the theory, see Fisher, 2014. 2 Tait, 1883a, 647. 3 Einstein, 1931, 66–71. 4 In Station 4 Maxwell addressed the student directly. See, e.g., Maxwell, 1873d, 1: Preface xiii–xiv, 94 § 91, 143 § 117, 196 § 158, 324 § 271, 405 § 351. Cf. n. 62, below. For purposes of focusing on Maxwell’s views here and elsewhere we cite the first edition of 1873 which was published during his lifetime. 5 Cf. Harman, 1998, 155.

6 See ch. 2, n. 24.

7 Maxwell, 1871b, 224.

8 See Maxwell, 1873d, 57–65, §§ 59–62.

9 Maxwell, 1873d, 58, § 59.

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202 Station 4 (1873) 10 Maxwell, 1873d, 58, § 59. On Maxwell’s view of the contrast between the con­ cepts advocated by Ampère and Faraday, respectively, see Maxwell, 1873d, 2: 162–163, § 528. 11 Maxwell, 1873d, 1: 57, § 59.

12 Maxwell, 1873d, 1: 59, § 59.

13 Maxwell, 1873d, 1: 62, § 61, italics in the original.

14 Maxwell, 1873d, 1: 63, § 62.

15 For the discussion on incompressible fluid, see ch. 3, nn. 33–34, and ch. 4, nn.

33 and 35. 16 For further discussion, see ch. 8, § 8.3.4. 17 Other physicists had already invoked the term electrodynamics, notably Ampère ([1826] 1883) and W. Weber ([1846] 1893). Indeed, Maxwell himself had already called his previous theory “dynamical” (1865): see ch. 6, nn. 6 and 7. 18 See ch. 3, § 3.4, and nn. 21–22, below. 19 Maxwell, 1865, 459, § 1. See ch. 6, n. 4. 20 Maxwell, 1865, 460, § 3. See ch. 6, § 6.2. 21 Maxwell, 1865, 461, § 6. For earlier instances of “actual energy” in Maxwell’s publications, see 1861–1862, 286; and 1865, 460–461 (§ 6), 464 (§ 15). See also ch. 6, nn. 9–11. 22 Rankine, 1853, 106. Rankine is mentioned in Station 4 in two places (1873d, 1: 139, §115, and 2: 416, § 831). See Harman (ed.) 1995, 190 n. 7; Smith, 1998, 225. 23 Maxwell, 1873d, 2: 185, § 554. See also, Maxwell, 1873d, 2: 64, § 437; Maxwell, 1873d, 2: 280, § 676. For the Treatise by Thomson and Tait, see Thomson and Tait, 1867. 24 Compare Maxwell, 1876, Maxwell, 1873d, 2: 185, § 555.

25 Maxwell, 1876, 292.

26 Maxwell, 1876, 294. As Olson (1975, 307–308) has persuasively argued, Max­ well—under the influence of Scottish Common Sense Philosophy—objected to the French molecularist school of Lagrange, Fourier, and Laplace, and empha­ sized a phenomenal approach to physical reasoning. See also Wilson, 2009. 27 Maxwell, 1873d, 2: 184–185, § 554. Maxwell invoked “mental image” and “mental representation” interchangeably; we adhere to “mental image” since Maxwell frequently used the expression “mechanical representation” which has a different methodological purpose. 28 Maxwell, 1858, 27. See ch. 4, n. 7.

29 Harman (ed.) 1995, 365–366.

30 Maxwell, 1873d, Preliminary, 1: 5, § 6.

31 Maxwell, 1873d, 1: 37, § 35.

32 According to Harman (1995, 2: 365 n. 13), the expression “kinetic energy” was

introduced in Thomson and Tait, 1867, 163. It is also in Tait, 1867, 280. 33 Maxwell, 1873d, 1: 37, § 35. 34 Maxwell, 1861–1862, 283. See also ch. 5, n. 7. 35 Maxwell, 1873d, 2: 190, § 562. 36 Maxwell, 1873d, 2: 191, § 562. Maxwell discussed this issue in his correspond­ ence in 1868: see Harman (ed.) 1995, 364–365. 37 Maxwell, 1873d, 2: 182, § 551. 38 Maxwell, 1873d, 2: 182, §§ 551–552. 39 Cf. Harman (ed.) 1995, 2: 744–745 (Maxwell, Text 419): “Draft on the inter­ pretation of Lagrange’s and Hamilton’s equations of motions” (dated ca. July 1872): “[The earlier definition of energy] involves the. . . unimaginable concept of the square of the velocity,. . . whereas the factors in the [new] definition are. . . both vectors. . . . In the dynamics of a system their directions are, in

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40 41

42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62

63 64

65 66 67

203

general, different, and the multiplication must be performed on Hamiltonian principles, and the scalar part taken. . . [which is] the kinetic energy of the system.” See also Maxwell, 1876. Maxwell, 1873d, 1: 9, § 11. Cf. nn. 44–48, 51, 140–151 and 181, below. The modern notation for distinguishing between the dot product of two vectors (a scalar quantity) and their cross product (a vector quantity) was introduced in the 1880s, after Maxwell’s death. For some historical details concerning this notation, see Crowe, 1967, 155–156. Maxwell, 1873d, 2: 384, § 782. Maxwell, 1871b, 225. Maxwell, 1871b, 226. See, e.g., Hamilton, 1866. Maxwell, 1871b, 227. Maxwell, 1871b, 228. Maxwell, 1871b, 228–229. For discussion, see ch. 8, §§ 8.2.4 and 8.3.1. Maxwell, 1871b, 229. Maxwell, 1871b, 229. For example, Maxwell, 1873d, Preliminary, 1: 9–10, § 11. Maxwell, 1873d, 1: 65, § 62. This paragraph was modified in the second edition of the Treatise ([1873] 1881, 1: 67), and the modified version served as the basis for the corresponding paragraph in the third edition (Maxwell, [1873] 1892). For Thomson’s formal analogy, see ch. 3, § 3.2. See ch. 6, n. 16. Maxwell, 1873d, 2: 183, § 552. On the use of “method” and “methodology,” see ch. 4, nn. 33–34. Maxwell, 1873d, 2: 183, § 552. Maxwell, 1873d, 2: 194, § 567. See n. 27, above. Maxwell, 1873d, 2: 184, § 554. For Lagrange’s Méchanique analitique, see Lagrange, 1811–1815 and Lagrange, [1788/1811] 1997. For an analysis of the Méchanique analitique, see Pulte, 2005. Maxwell, 1873d, 2: 184, § 554. Maxwell, 1873d, 2: 185, § 555. In Station 4 Maxwell frequently referred to the student; he had a beginner in mind. In the Preface he indicated that he had not attempted an exhaustive account of electrical phenomena, experiments, and apparatus, and recom­ mended the interested student to consult other texts. Most of all he felt that: “If by anything I have here written [in Station 4] I may assist any student in understanding Faraday’s modes of thought and expression, I shall regard it as the accomplishment of one of my principal aims—to communicate to others the same delight which I have found myself in reading Faraday’s Researches.” (Maxwell, 1873d, 1: xii.) Cf. n. 4, above. Maxwell, 1873d, 2: 185, § 554. Harman, 1998, 122. Hamilton developed this technique in his memoir, “Gen­ eral methods in dynamics” (see Hamilton, 1834; 1835). As Harman explained (pp. 122–123), “in this method, the kinetic energy is expressed in terms of the variables (q) and momenta (p) of the particles constituting a material system. . .. The kinetic energy of the system changes as a result of the action of an infini­ tesimal force impulse. . .; but the instantaneous state of the system is independ­ ent of the cause of motion. . .. [Thus] the hidden machinery of the system is not specified.” Harman, 1998, 123. See Maxwell, 1873d, 2: 188, § 560. Maxwell, 1873d, 2: 189, § 560. Maxwell, 1873d, 2: 193–194, § 567.

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204 Station 4 (1873) 68 69 70 71 72 73 74 75 76

77 78 79 80 81

82 83 84 85

86 87 88 89 90 91 92 93 94

Maxwell, 1873d, 2: 194, § 567.

Maxwell, 1858, 52. See ch. 4, n. 20.

Maxwell, 1873d, 2: 194, § 567.

Maxwell, 1873d, 2: 194, § 567.

See nn. 4 and 62, above.

Maxwell, 1873d, 1: 1, § 1.

Maxwell, 1873d, 1: 2, § 2.

Maxwell, 1873d, 1: 2, § 2.

Maxwell, 1873d, 1: 5, § 6. For a unit of force in a different system altogether,

see Thomson and Tait (1867, 322, italics in the original): “Force.—Weight of a Pound or Kilogramme, etc., in any particular locality (gravitation unit); kinetic unit.” Cf. pp. 164–166, 325–326. Contrast this definition with that of Maxwell’s. These two definitions of the unit of force are worlds apart. Maxwell took advantage of Newton’s second law and made use of it in defining the dimen­ sions of momentum. For another case where Maxwell differed from Thomson and Tait, see n. 88, below. Maxwell, 1873d, 2: 239, § 620.

Maxwell, 1873d, 2: 243, § 627.

Maxwell, 1873d, 2: 243, § 628. On the history of the introduction of v, see

Mitchell, 2017a, 2017b, and 2019, espec. §§ 2 and 8. Maxwell, 1873d, 2: 368, § 768. Maxwell, 1873d, 2: 369–370, §§ 769–771. Maxwell inserted the following refer­ ence to the work of Weber and Kohlrausch: “Elektrodynamische Maasbestim­ mungen; and Pogg. Ann. xcix, (Aug. 10; 1856).” Maxwell referred somewhat tersely—in this order—to two papers: Kohlrausch and Weber, 1857 (in Abhan­ dlungen der mathematisch-physikalischen Klasse der k. sächsischen Gesellschaft der Wissenschaften), and Weber and Kohlrausch, 1856 (in Annalen der Physik und Chemie). In Station 2 Maxwell (1861–1862, 21) cited only Kohlrausch and Weber, [1857] 1893; cf., ch. 5, n. 79. In Station 3 Maxwell (1865, 465, § 19) men­ tioned Weber and Kohlrausch—in this order—but did not include a reference. Maxwell, 1873d, 2: 244, § 629. Notice that the unit of resistance is velocity. Maxwell was a member of the committee of the British Association in 1863 on standards of electrical resistance. The report (Maxwell and Jenkin, 1864) was partially reprinted in Maxwell and Jenkin, 1865. Maxwell and Jenkin, 1864, 149; reprinted in Maxwell and Jenkin, 1865, 509. Maxwell and Jenkin, 1864, 149 Essentially, Weber and Kohlrausch interpreted their experimental results in terms of their theory on the nature of electricity. For a discussion of the work of Weber and Kohlrausch, see Station 2, ch. 5, § 5.5. As we see here, with the dimensionality of units Maxwell and Jenkin obtained results independent of any specific theory. Maxwell and Jenkin, 1864, 114. Maxwell and Jenkin, 1864, 131; reprinted in Maxwell and Jenkin, 1865, 437. For further discussion see Hunt, 2015, and Mitchell, 2017a and 2017b. Thomson played a key role in setting up the committee of the British Associ­ ation (Hunt, 2015, 310; Maxwell and Jenkin, 1864, 111). Maxwell, 1873d, 2: 368ff, §§ 768. Chrystal, 1882, 237. Maxwell, 1873d, 1: 319, 385, 387, §§ 266, 334. Maxwell, 1873d, 2: 416, § 831; see nn. 112 and 122–123, below. Maxwell, 1873d, 2: 184–185, § 554. Maxwell intended the Treatise to be both cutting-edge research and a textbook: see n. 4, above.

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95 For the criticisms of Poincaré and Duhem on the issue of consistency, see ch. 2, n. 24.

96 Maxwell, 1873d, 1: 59, § 59.

97 Maxwell, 1873d, 1: 59, § 59.

98 See ch. 1, §1.5; ch. 2, § 2.1, ch. 4, n. 10.

99 For the images of rope and muscles, see also Maxwell, 1873b, 54.

100 Maxwell, 1873d, 2: 195, § 569. 101 Maxwell, 1873d, 2: 196, § 569. Notice the contrast between energy and force: energy is energy (no matter what generates it: see ch. 6, n. 43), but force is not the same for, unlike energy, it is a cause of change. Maxwell distinguished between “ordinary mechanical” force which acts on bodies only and “electro­ motive” force which acts on electricity only. 102 Maxwell, 1873d, 2: 196–197, § 570, italics added. 103 Maxwell, 1873d, 2: 197, § 571. 104 Maxwell, 1873d, 2: 198, § 572. 105 Maxwell, 1873d, 1: 385, § 82. In § 82 Maxwell indicated that “lines of force” is an expression used by Faraday and others, but that “in strictness. . . these lines should be called Lines of Electric Induction.” For Maxwell’s introduction of this idea, see ch. 4, nn. 33, 48 and 59. 106 Maxwell, 1873d, 1: 385–387, § 334. 107 See nn. 176 and 177, below. 108 Maxwell, 1873d, 2: 180, § 547. 109 Maxwell, 1873d, 2: 181, § 550. 110 Maxwell, 1873d, 2: 201–202, § 574. 111 Maxwell, 1873d, 2: 201–202, § 574. 112 Maxwell, 1873d, 2: 416, § 831. See n. 123, below. 113 Maxwell, 1873d, 2: 415–416, § 831. Cf. Thomson, 1856–1857; reprinted in Kelvin [1884] 1904, Appendix F. 114 Thomson, 1856–1857, 151; reprinted in Kelvin (Lord), [1884] 1904, 570. 115 Thomson, 1856–1857, 152; reprinted in Kelvin (Lord), [1884] 1904, 570–571. 116 Thomson, 1856–1857, 152; reprinted in Kelvin (Lord), [1884] 1904, 570–571. 117 Maxwell, 1873d, 2: 416, § 831. 118 Thomson, 1856–1857, 152–153; reprinted in Kelvin (Lord), [1884] 1904, 571–572. 119 Thomson, 1856–1857, 157–158, italics in the original; reprinted in Kelvin (Lord), [1884] 1904, 582–583. 120 Maxwell, 1873d, 2: 416, § 831. 121 See n. 189, below. 122 Maxwell, 1873d, 2: 416–417, § 831; for the mechanism, see the following note. 123 For an example of a typical usage of “working model” at the time, see ch. 8, § 8.3.6. For the requirement of working models and their specifications by the United States Patent Office, see Conner and Brenner, 1965; for a critical assess­ ment, see Sang, 1833. In Station 1 Maxwell invoked another idiosyncratic usage of model, namely, “geometrical model”: see ch. 4, nn. 29 and 42; cf.; Hon and Goldstein, 2012, 242. Maxwell did not use this expression ever again. 124 Maxwell, 1873d, 2: 415, § 830: “The [proposed] theory. . . is evidently of a pro­ visional kind, resting as it does on unproved hypotheses relating to the nature of molecular vortices, and the mode in which they are affected by the displace­ ment of the medium.” 125 As we have seen (ch. 6, n. 78), in concluding Part III of his paper on the dynamical theory, Maxwell claimed to have deduced the results from experi­ mental facts.

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206 Station 4 (1873) 126 For “mechanical illustration,” see Maxwell, 1861–1862, 162; for “hypothesis,” see ibid., 162, 169, 170, 282, 285, 286, 287, 290, 291, 346. 127 Maxwell, 1873d, 2: 437, § 866. 128 Neumann, 1868. For discussion of Neumann’s work, see Schlote, 2004. 129 Maxwell, 1873d, 2: 438, § 866. See n. 27, above, for the equivalence of “mental representation” and “mental image” For discussion of hypothesis and mental imagery as methodologies, see ch. 7, § 7.3.5, ch. 8, §§ 8.3.3, and 8.3.5. 130 See Niven’s Preface to the second edition:, [1873] 1881, xv–xvi. 131 Maxwell, 1873d, 2: 184, § 553. 132 Maxwell, 1873d, 1: 385, § 334. 133 Maxwell, 1873d, 2: 184–185, § 554. 134 Maxwell, 1876, 293. For this idea in Station 4, formulated somewhat differently, see n. 27, above. 135 Maxwell, 1873d, 2: 83, § 446. 136 Maxwell, 1873d, 1: 60, § 60, italics in the original. 137 Maxwell, 1873d, 1: 63, § 61. 138 Maxwell, 1873d, 1: 132, § 110. 139 Maxwell, 1873d, 1: 132–133, § 111. 140 The decision to replace Maxwell’s German Gothic font with boldface font is due to Heaviside (1892, 429): “I. . . put the vectors in the plain black type, known. . . as Clarendon type, rejecting Maxwell’s German letters on account of their being hard to read.” 141 Maxwell, 1873d, 1: 70, § 68. 142 Maxwell, 1873d, 1: 134, § 111. 143 Maxwell, 1865, 485, § 66. See also Bromberg, 1967, 230. See ch. 6, n. 76. 144 Maxwell, 1873d, 1: 60, § 60. 145 For the discussion in Station 3, see ch. 6, § 6.6.1. 146 Maxwell, 1873d, 2: 236–237, § 618. 147 Maxwell, 1873d, 2: 232, 238, §§ 608, 619. 148 Maxwell, 1873d, 2: 232–233, § 610. 149 Maxwell, 1873d, 2: 232, § 608. 150 Maxwell, 1873d, 2: 228, § 604: “The current in all these cases is to be under­ stood as the actual current, which includes not only the current of conduction, but the current due to variation of the electric displacement.” 151 Maxwell, 1865, 480, §§ 53–55. 152 Maxwell, 1873d, 2: 158–159, § 522, italics in the original. See also Maxwell, 1873d, 2: 231, § 607. 153 Maxwell, 1873d, 2: 231, § 607. 154 Maxwell did not say how he discovered the displacement current, and this has led to much speculation in the secondary literature: see ch. 5, n. 66. As we have seen, in Station 4 Maxwell was explicit about the analogy to elasticity, so much so that he ventured “to call the ratio of the electromotive intensity to the cor­ responding electric displacement the coefficient of electric elasticity of the medium” (n. 136, above). But this analogy is not mentioned in Station 2. 155 Siegel, 1991, 152. 156 Siegel, 1991, 150–151. 157 In Station 1, however, the analogy between electricity and an incompressible fluid was strong: see ch.1, § 1.5 and ch. 4, n. 10. 158 Maxwell, 1873d, 2: 225, § 604. 159 Maxwell, 1873d, 2: 214, 219, 226, §§ 591, 598, 603. 160 For (A) see Maxwell, 1873d, 2: 215, § 591. 161 For (B) see Maxwell, 1873d, 2: 221, § 598. 162 For (C) see Maxwell, 1873d, 2: 226, 228 §§ 603, 605.

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Station 4 (1873) 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192

193

194 195 196 197

207

Maxwell, 1873d, 2: 236–237, § 618. Cf. Achard, 2005, 580. Maxwell, 1873d, 2: 237–238, § 619. Maxwell, 1873d, 2: 237–238, § 619. Cf. Achard, 2005, 581. For the list in Station 3, see Maxwell, 1865, 465, § 18, and Part III, 480–488, §§ 53–75 (see also ch. 6, § 6.6). What are now known as Maxwell’s equations are different from the equations that comprise the list. Maxwell, 1865, 465, § 18, and Part III, 480. Maxwell, 1865, 485–486, § 70. See ch. 6, § 6.6. Maxwell, 1873d, 2: 236–237, § 618. See, for example, the theorems associated with line- and surface-integral: Max­ well, 1873d, 1: 17–27, §§ 19–22, 24. For the differences in the key concepts, namely, force in Station 3 and energy in Station 4, see nn. 29–30, 102–103, above. See ch. 6, § 6.6.1; and espec. ch. 7, § 7.4.1. See n. 165, above. For a discussion of this paper, see ch. 7, § 7.3.2. Maxwell, 1871b, 227. Maxwell, 1873d, 1: 74, § 72. Cf. n. 107, above. Maxwell, 1873d, 1: 67, § 64. Maxwell, 1873d, 1: 298, § 245. Maxwell, 1873d, 1: 62, § 61, italics in the original. Cf. nn. 13 and 15, above. Cf. nn. 107 and 109, above. Another important distinction which Maxwell introduced in 1871 concerns vec­ torial expressions of translation and rotation: see Maxwell, 1871b, 229–230. Cf. n. 50, above. Maxwell, 1873d, 2: 234, § 615. Hertz and Heaviside sought to achieve compact­ ness in recasting Maxwell’s equations by considering their symmetrical struc­ ture: see Hon and Goldstein, 2006b, 643. For these novel methodologies, see ch. 7, § 7.3. For the framework of Station 4, see ch. 7, § 7.2. Maxwell, 1873d, 2: 400, § 806. Maxwell, 1873d, 2: 416, § 831. For Maxwell’s reluctance to make micro-level claims, see nn. 18 and 155–157, above. Maxwell, 1873d, 2: 409, § 823; cf. ch. 5, n. 46. Maxwell, 1873d, 2: 402, § 811. Maxwell, 1873d, 2: 408, § 821. Maxwell, 1873d, 2: 404, § 816. Maxwell, 1873d, Preliminary, 1: 16, § 18. In a letter to Tait in May 1873, Max­ well indicated that what he called “geometry of position” is what Gauss had called Geometria Situs, as opposed to Geometria Magnitudinis (Harman, 1998, 154; cf. Gauss, 1867, 605 § 4). However, as Gauss noted, this distinction goes back to Leibniz: see Leibniz’s letter to Huygens, 8 September 1679, in Ger­ hardt, 1899, 568–569. Maxwell, 1873d, 2: 41, § 421. See also Maxwell, 1873d, Preliminary, 1: 16, § 18. Maxwell’s references to Leibniz, Euler, and Vandermonde are taken directly from Gauss’s notes on electrodynamics of 1833, which were only published posthumously: see Gauss, 1867, 605, § 4; see also Nash, 1999, 362, Harman, 1998, 155, and Epple, 1998, 377. See Maxwell, 1873d, Preliminary, 1: 16, § 18.

Tait, 1883b, 316.

Listing, [1847] 1848, 6.

Epple, 1998, 381, italics in the original.

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208 Station 4 (1873) 198 199 200 201 202 203 204

Maxwell, 1873d, Preliminary, 1: 24, § 23. Maxwell, 1873d, 2: 167, § 532. Maxwell, 1873d, 2: 408, § 822. Maxwell, 1873d, 2: 417, § 831. Maxwell, 1873d, 2: 415, § 830. See ch. 7, § 7.3.5. Recall Maxwell’s critique of Neumann’s theory: see nn. 20, 57, 127 and 128, above. See also ch. 8, n. 11. 205 Maxwell, 1873d, 1: v–vi. 206 See n. 129, above.

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Philosophical reflections on Maxwell’s methodological odyssey

8.1 Commitment From the beginning of his study of electromagnetism, Maxwell was committed to Faraday’s conceptual framework of lines of force. The commitment was based on Maxwell’s confidence that the concept of lines of force was the proper way to unify the phenomena in this domain. And indeed Maxwell adhered to this concept—against the dominant concept of action at a distance—but kept changing the discussion by placing it in different methodological contexts from which further consequences were inferred. Thus, in Station 1, the lines of force are imagined to be tubes; in Station 2 they are considered to be physically real, the product of some hypothetical mechanical scheme at the micro-level; in Station 3, they are set in a field; and, finally, in Station 4 they are embedded in a medium which is the seat of energy. The fundamental concept does not change in the course of Maxwell’s journey in electromagnetism, but the settings and the methodologies vary as Maxwell moved from one station to the next.1 On the basis of the evidence we have presented, we conclude that changing methodologies characterizes the move from one station to the next, hence the appellation “methodological” odyssey. This consistent approach is in fact an expression of commitment to two related concepts, namely, lines of force together with a medium as the agent of electromagnetic phenomena. The transition from one station to the other reflects fundamental changes in the practice of physics. Nevertheless, Maxwell adhered consistently to a set of presuppositions which constitutes an essential methodological feature of his work in electromagnetism. It is characteristic of Maxwell’s corpus in electromagnetism that both lines of force and a medium as the agent of electromagnetic phenomena are set in different methodological contexts. We will return to this feature and reflect on it philosophically.2

8.2 Modifications of methodologies We have analyzed in great detail the four stations that mark the trajectory which Maxwell followed in his researches in electromagnetism. We have identified in each station a different methodology, but all of Maxwell’s

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210 Reflections on Maxwell’s methodologies methodologies in this domain were based on and, indeed, enhanced Faraday’s unifying concept of lines of force. It is tempting to assert that Maxwell invented four kinds of methodologies to suit the goal of his research—casting Faraday’s empirical findings into symbolic form. But such an assertion would be misleading, for we have shown that in each station Maxwell modified an existing methodology. Indeed, the versatility that Maxwell displayed in modifying existing methodologies is astounding.3 It is clear that of all the sources to which Maxwell referred Faraday’s works are the most prominent. It is not surprising that in the title of his first major contribution to this domain Maxwell invoked the name of Faraday and the conceptual framework he had offered, namely, lines of force. But it is remarkable that after 15 years, Faraday’s ideas and experimental findings still figure prominently in Maxwell’s research agenda. In fact, in the Preface to Station 4, Maxwell recommended that the student of electromagnetism begin by thoroughly studying the works of Faraday.4 Indeed, a central feature of the entire trajectory of Maxwell’s methodological odyssey is the work of Faraday, conceptually, experimentally, and methodologically. In each station Maxwell sought to address Faraday’s work with different analytical tools that came from different methodologies. This cannot be said of any of his other predecessors to whom he referred in his studies of electromagnetism. In the next subsections we reflect philosophically on the methodologies Maxwell pursued in each station of his odyssey and highlight the nature of the transitions. We emphasize the modifications he introduced at critical junctures. The section ends with a general philosophical overview. 8.2.1 Station 1: analogy Maxwell’s overall goal was to express Faraday’s theoretical framework in mathematical terms and to seek physical consequences that were otherwise unexpected. In Station 1 the creative act was to modify Thomson’s methodology of formal analogy of physical domains by extending it to the imaginary realm, where lines are imagined to be tubes. Imagining tubes filled with incompressible fluid and regulating its motion in abstract terms transform the system from geometry which expresses only directionality to physics which includes the intensity of the force. The resulting physical system, however imaginary, is then subject to formal manipulation—the vehicle of mathematical reasoning. Faraday’s verbal concept of lines of force was thus cast into symbolic language. Physical consequences could now be drawn deductively from the mathematical formulation which had not been anticipated in Faraday’s verbal formulation. Importantly, Maxwell did not claim to have introduced a new concept; rather, he found a way to express Faraday’s concept of lines of force formally. It is noteworthy that no explanatory scheme of the pertinent phenomena is offered; on the contrary, Maxwell was explicit that he was not interested at all in offering an explanation. Accounting for the phenomena would

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Reflections on Maxwell’s methodologies 211 require substantial presuppositions about which he wished to stay agnostic. The goal was set clearly, and the methodology was tailored to achieving this goal, namely, when formulated mathematically lines of force can represent the phenomena as well as action at a distance, if not better. And what is more, in its symbolic formulation as “artificial notation,” to quote Maxwell, it could be applied in solving mathematically various physical problems without appeal to any theory. Imagination, both in physics and in mathematics, functions in this context as a technical term, instrumental in extending the methodology of analogy to include imaginary physical systems. The key philosophical move is to consider an imaginary physical system with properties that are entirely at the disposal of the physicist, in order to obtain physically meaningful results. In this case, Maxwell imagined lines as tubes and analyzed formally the flow of incompressible fluid in these tubes. This was an extraordinary move that facilitated the casting of Faraday’s verbal formulation into mathematical symbolism. The methodology in Station 1 takes advantage of a strong analogy that works in one direction from the imaginary physical system to the physics of electromagnetic phenomena. The contrived analogy served as a vehicle of mathematical reasoning. No ontology on either the micro- or the macro-level is discussed or implied. The goal of demonstrating the effectiveness of the unifying concept of lines of forces was achieved successfully. 8.2.2 Station 2: hypothesis In Station 2 Maxwell’s stated goal was to account for electromagnetic phenomena within Faraday’s conceptual framework. Maxwell discarded the imaginary extension of formal analogy of Station 1, and opted—in Station 2—for explaining the phenomena mechanically. From a philosophical perspective this is a dramatic change for, in Station 1, Maxwell explicitly avoided any attempt at explanation, preferring to be agnostic about the causal processes that produce the phenomena. By contrast, in Station 2 he was ready to make assumptions about the micro-level which were not testable directly at the time. Accounting for the phenomena requires causal argumentation, and Maxwell put forward a mechanical hypothesis, the socalled molecular vortex hypothesis. The methodological principle was then that mathematical deductions from the hypothesis were linked to experimental findings at the macro-level—the phenomena. Maxwell’s assumptions about molecular particles and vortices were motivated by experimental evidence, namely, the association of translational and rotational motions with electricity and magnetism, respectively. The justification of the assumptions at the micro-level comes from the inferences drawn from them at that level which can be tested at the macro-level. Maxwell referred systematically throughout Station 2 to “hypothesis.” This methodology is now called hypothetical–deductive (H–D). Evidently, Maxwell was not content with the standard hypothetical approach, for he

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212 Reflections on Maxwell’s methodologies introduced a well-understood mechanical system at the macro-level, the rack and pinion, that represented the assumed mechanism at the micro­ level. The purpose of adding an external mechanical system, and a wellunderstood one at that, is clear: with this representation he was able to retain physical intuition which would otherwise be lost in the formal deduction. Maxwell was fully aware of the tension between symbolic manipulation and physically meaningful claims, and negotiated a path to take advantage of both approaches. Thus, in addition to the mechanical assumptions at the micro-level from which inferences can be drawn to the macro-level, this third, external mechanical system provided control of the formal deduction from the micro- to the macro-level. How does this control work? Maxwell drew inferences at the macro-level from this mechanical system and applied them to the hypothetical micro-level, which in turn led safely to explanations of electromagnetic phenomena at the macro-level. What was the nature of the modification in Station 2? Maxwell did not follow the traditional practice of relying solely on a direct mathematical deduction from a set of conjectures at the micro-level to the macro-level, namely, the phenomena. Rather, what is striking about Station 2 is that the micro-level is not only informed by the phenomena at the macro-level but also by a seemingly irrelevant mechanical system that has nothing to do with electromagnetic phenomena. The connection is that the mechanical system exhibits translational and rotational motions; for Maxwell this made it relevant for understanding the combined translational motion of particles and rotational motion of molecular vortices at the micro-level. It is important to emphasize that Maxwell did not believe there was such a mechanical system at the micro-level. Nevertheless, in Station 2 he elaborated a mechanical explanation at the micro-level to account for electromagnetic phenomena.5 The key philosophical move then is to foster physical intuition with a mechanical illustration at the macro-level to complement the inferences drawn formally from mechanical assumptions made to account for the phenomena at the micro-level. This “mixed” approach is what made Maxwell’s use of hypothesis different from the application of hypotheses by contemporary physicists (either in mechanics or electromagnetism) who drew consequences directly from assumptions in a formal way without attending to physical intuition. In this station Maxwell reported two fundamental discoveries, the displacement current and that light is an electromagnetic disturbance. It is not entirely clear if these two fundamental discoveries were the product of Maxwell’s modification of the hypothetical approach.6 Nevertheless, it is undoubtedly the case that Station 2 had a greater impact on the community of physicists than Station 1, certainly on Maxwell himself who proceeded in Station 3 to presuppose the results of Station 2—a development that contrasts sharply with the transition from Station 1 to Station 2.

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Reflections on Maxwell’s methodologies 213 8.2.3 Station 3: textual description In Station 3 Maxwell commenced his discussion with an announcement that he will propose a theory, a dynamical theory as he referred to it, which will explain the electromagnetic phenomena by the action of excited bodies and their surrounding medium, where the action is transmitted by lines of force and not by action at a distance. Maxwell attained this goal by discarding the methodology of hypothesis of Station 2 while retaining its results, namely, presupposing the displacement current and that light is an electromagnetic disturbance. In fact, this amounts to reversing the argument of Station 2: assume phenomena; discard hypothesis. And, clearly, there is no return to the methodology of Station 1 either. In Station 3 Maxwell started from verbal descriptions of electromagnetic phenomena and translated them into equations. This translation from the verbal to the symbolic transformed Faraday’s methodology which is solely descriptive; Faraday offered only a verbal account of the phenomena. To be sure, Faraday’s descriptions are made precise by assuming distinct concepts; moreover, he described phenomena as well as regularities and empirical laws. Still, Faraday’s discussions are entirely qualitative.7 In Maxwell’s transformed version of Faraday’s methodology certain phenomena serve as assumptions of the theory. Once these assumptions are recast into symbolic language, formal inferences can be drawn with respect to the physics of electrodynamics, namely, in quantitative terms, energy, force, motion, and state. Maxwell did not begin with abstract principles; the equations were not deduced from such principles and they are not presented in a deductive form, for the resulting set of equations does not exhibit an axiomatic structure. Indeed, the equations do not form a discernable hierarchy. The theory—a set of equations—is constructed directly from verbal descriptions of electromagnetic phenomena which are essentially Faraday’s, translated into mathematical symbolism. As formulated, the mathematical equations facilitate quantitative calculation of the intrinsic energy of the electromagnetic field as well as the demonstration of the laws of mechanical forces which act upon electric currents, magnets, and electrified bodies placed in the field.8 Since the goal is a theory comprising the general equations of the electromagnetic field, the function of a mechanical arrangement as an explanatory scheme is now secondary, i.e., illustrative, and appealed to as an aid for understanding. Maxwell thus proposed the mechanism of Station 3, the flywheel, as a mechanical illustration of the field; this mechanism offers insights into the transmission of forces in the medium. In fact, the flywheel functions as a strong analogy, a well-understood mechanism from which consequences can be drawn and applied to the theory of the electromagnetic medium. Such a strong analogy is not to be found in Faraday. With this analogy Maxwell carried out the modification of Faraday’s methodology. Interestingly, Maxwell considered the methodology of Station 3 powerful enough to account also for gravity, but he acknowledged his failure to do

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214 Reflections on Maxwell’s methodologies so. Notwithstanding his failure, the proposed application is insightful and throws light on Maxwell’s general approach in Station 3. What emerged clearly in this discussion of gravity is the appeal to the intrinsic energy of the medium.9 Indeed, in Station 3 Maxwell began to realize that the concept of energy was so important for his discussion of electromagnetism that it became the central concept in Station 4. The key philosophical move here is the realization that a mechanical illustration may be effective as a strong analogy from which inferences may be drawn. Yet, at the same time, it also functions as an aid for understanding. The focus in this station is on the equations and on the consequences that can be drawn from them with respect to the physics of the dynamical medium. Here too, as in Station 2, insights are offered from a mechanical perspective, though Maxwell explicitly cautioned that the two domains, mechanics and electromagnetism, are distinct.10 A closer look at the different roles the mechanical arrangement of Station 2 and of Station 3 play, throws light on the different methodologies which Maxwell pursued in his study of electromagnetism in these two stations. In Station 2 the arrangement of rack and pinion serves to explain how the molecular vortices work at the micro-level so that an “imitation” of electromagnetic phenomena arises. In Station 3 Maxwell introduced the flywheel to address the phenomenon of induction at the macro-level—a strong analogy to make the action of induction through the medium plausible and to draw inferences from the mechanism. Whereas in Station 2 the mechanism supports the claim that the hypothesis can do the job it was designed to do, in Station 3 it addresses a phenomenon with an illustration of the transmission of force(s) and thus constitutes a strong analogy. In other words, the mechanical arrangement at the macro-level in Station 2 enhances physical intuition at the micro-level, whereas that of Station 3 offers a strong analogy as part of the argumentative structure. The methodological difference between Station 2 and Station 3 can be looked at from yet another perspective. We have claimed that the modification in Station 2 concerns the introduction of a mechanical arrangement at the macro-level—rack and pinion—external to the traditional hypothetical-deductive (H–D) sequence of inferences from micro- to macrolevel. As we have seen, this mechanical arrangement at the macro-level made plausible the mechanical arrangement at the micro-level, namely, particles and molecular vortices. Similarly, here, in Station 3, Maxwell introduced at the macro-level a mechanism—the flywheel—external to the sequence of deductions from the presupposed phenomena formulated in algebraic equations and the inferences drawn from them. The external mechanical arrangement was designed to offer a strong analog to the transmission of force through the electromagnetic field. But here the similarity with Station 2 ends, for in Station 3 there is no micro-level. Indeed, the deductions in Station 3 are from phenomena that serve as assumptions. Maxwell did not appeal to the micro-level in Station 3. Thus, the general equations of the electromagnetic field are essentially phenomenal.

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Reflections on Maxwell’s methodologies 215 8.2.4 Station 4: abstract dynamics Station 4 is the culmination of Maxwell’s methodological odyssey in electromagnetism. It is a mélange of methodologies and scientific ideas intended for the student of physics as well as for the expert physicist at the cutting edge of this scientific domain. Yet, however eclectic the Treatise may appear, it has a unifying theme which reflects a systematic methodology: the unifying concepts are energy and lines of force, and the methodology is essentially a modification of the methodology of Thomson and Tait that they developed in their Treatise on natural philosophy (1867). In their Treatise Thomson and Tait applied the methodology of “abstract dynamics” to the domain of mechanics. For his part, Maxwell tailored Thomson and Tait’s principles of abstract dynamics to the domain of electromagnetism; he thereby turned electromagnetism into electrodynamics. Maxwell considered the medium the seat of energy and endowed it with dynamical properties by transmitting forces through the medium along lines of force. This was a novel approach whose origin can be traced back to Station 3; but it is only eight years later, in Station 4, that Maxwell brought this creative approach to fruition with wide-ranging ramifications. To make this approach viable, Maxwell needed to introduce ingenious modifications to a set of mathematical tools available at the time, such as the Lagrangian and the Hamiltonian, that had never been applied directly to the phenomena of electromagnetism. As a mélange of methodologies and scientific ideas Station 4 is rich in discussions at both the micro- and macro-level. Theories are offered at different levels of generality that were intended to encompass all the known phenomena of electromagnetism. It is apparent that Maxwell insisted on maintaining an intuitive physical grasp of the theory over and above the powerful formalism he introduced. He was therefore interested in translating formal expressions into mental imagery; indeed, he rendered this requirement a criterion for viable theories. Evidently, Maxwell was willing to reject outright any theory that could not be expressed verbally and that lacked mental imagery.11 The key philosophical move in this station is the development of the mathematical treatment of energy in electrodynamics based on the Lagrangian and the Hamiltonian. There are a few references to energy in Station 3; to be sure, the distinction between two forms of energy, the actual and the potential, is noted, but this is nothing like what is found in Station 4. It appears that the replacement of force with energy as the key concept is the prime motivation for the transition from Station 3 to Station 4. The mathematical techniques introduced in Station 4 are related to the assumption that the medium is the seat of energy. At the outset of his discussion of the dynamical theory of electromagnetism, Maxwell referred to an earlier discussion in Station 4 in which he had shown that when an electric current exists in a conducting circuit, it has the capacity for doing mechanical work, independent of any external electromotive force maintaining the current.12 Maxwell then continued:

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216 Reflections on Maxwell’s methodologies Capacity for performing work is nothing else than energy, in whatever way it arises, and all energy is the same in kind, however it may differ in form. The energy of an electric current is either of that form which consists in the actual motion of matter, or of that which consists in the capacity for being set in motion, arising from forces acting between bodies placed in certain positions relative to each other. The first kind of energy, that of motion, is called Kinetic energy, and when once understood it appears so fundamental a fact of nature that we can hardly conceive the possibility of resolving it into anything else. The second kind of energy, that depending on position, is called Potential energy, and is due to the action of what we call forces, that is to say, ten­ dencies towards change of relative position. With respect to these forces, though we may accept their existence as a demonstrated fact, yet we always feel that every explanation of the mechanism by which bodies are not in motion forms a real addition to our knowledge.13 Evidently, Maxwell realized the advantages of basing the theory on energy rather than on force, for energy puts electrodynamics on a par with mechanics in its implications for work. Energy may differ in form but it is always of the same kind—a capacity for performing work. The move then from Station 3 to Station 4 seems to be a methodological extension but, in effect, the change is revolutionary since the traditional appeal to force is replaced here with energy. Another significant change in methodology from Station 3, which is linked to the appeal to energy, has to do with the logical status of the nature of light. In Station 3 light as an electromagnetic disturbance was an assumption which, therefore, required no proof. By contrast, in Station 4 Maxwell stated: “We have now to shew that the properties of the electromagnetic medium are identical with those of the luminiferous medium.”14 In other words, what had been considered as a presupposition in Station 3, in Station 4 had to be inferred. In Station 4 Maxwell developed an argument in support of the claim that light is an electromagnetic phenomenon. The proof of the reality of the medium had to be part of this argument. In supporting these claims Maxwell reached the conclusion that during the passage of light through the medium the luminiferous medium becomes a receptacle of energy.15 This, of course, comes as no surprise since in Station 4 Maxwell identified the luminiferous medium with the electromagnetic field. While Maxwell’s contribution in Station 4 was essentially theoretical in turning electromagnetism into electrodynamics, he remained closely attuned to the phenomena and emphasized the importance of observations, experiments, and indeed measurements. This demand, part of Maxwell’s overall methodological approach, proved critical in making his theory successful.16 Maxwell cautioned against unfounded generalizations. He spelled out the limitation of inductive inference in the current case:

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Reflections on Maxwell’s methodologies 217 We are so little acquainted with the details of the molecular constitution of bodies, that it is not probable that any satisfactory theory can be formed relating to a particular phenomenon, such as that of the mag­ netic action on light, until, by an induction founded on a number of different cases in which visible phenomena are found to depend upon actions in which the molecules are concerned, we learn something more definite about the properties which must be attributed to a molecule in order to satisfy the conditions of observed facts.17 Despite this limitation at the micro-level due to a lack of knowledge of molecular properties, the theory was successful; and, what is more, it offered, as Maxwell required, mental imagery. For Maxwell the theory is not just a set of equations. He reached his goal: the theory is formal and accounts for the phenomena and, at the same time, it facilitated mental imagery of the phenomena which can be described verbally. This is an extraordinary achievement: the theory does not depend on any specific hypothesis at the micro-level, yet it enhances physical intuition in accounting for the phenomena. Maxwell’s mechanistic interpretation in terms of the elasticity of the medium offered one explanation, but it was not exclusive. It provided a close linkage between observed facts at the macro-level and the properties of a possible explanans at the micro-level. In a nutshell, the equations stand on their own, irrespective of any interpretation.18 Put differently, the path to discovery can be discarded, keeping only the results. 8.2.5 Transitions from one methodology to the next Maxwell’s remarks on philosophy of science, his reflections on methodological procedures—as insightful as they are—do not explain the transition from the methodology of one station to another; rather, they are general and may apply to any methodology. Nevertheless, Maxwell did comment on these transitions, however briefly; moreover, he included several cross references in the stations, indicating that he was fully aware of the transitions. He explicitly noted the dramatic change from Station 1 to Station 2 where he stated in the latter that at this station he would pursue a completely different methodology from that in the previous station (i.e., Station 1).19 In Station 3 he explicitly remarked that he intended to reverse the methodological procedure of Station 2.20 In 2015 Hunt addressed the very issue that we have raised here and discussed the transition from Station 2 to Station 3. He based his argument on Maxwell’s scientific engagement subsequent to the publication of Station 2. According to Hunt, Maxwell was working at the time in close collaboration with engineers in developing an analysis of dimensionality in the context of setting standards for electrical measurement. That, in turn, led Maxwell to adopt—according to Hunt—an “engineering approach” that

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218 Reflections on Maxwell’s methodologies focused on establishing relationships between measurable quantities rather than devising hypothetical mechanisms.21 However, Maxwell’s move from Station 2 to Station 3 was not entirely based on practical considerations, as Hunt claimed. It is striking that dimensional analysis is missing altogether in Station 3,22 and this suggests that other issues played a major role in leading the research of Station 3 as well as in deciding on the methodology to be applied. Indeed, Maxwell entertained a variety of methodologies and so his decision to opt for a new methodology in Station 3 is not surprising and conforms to his scientific practice. The decision in Station 3 to avoid specific hypothetical mechanisms at the micro-level need not have been dependent on Maxwell’s collaboration with members of the group of the British Association on standards of measurement. The last leg of the odyssey, leading to the general equations of the electromagnetic field, is characterized by two extraordinary moves. These moves offer a clue to the character of the last transition from Station 3 to Station 4, the culmination of Maxwell’s study of electromagnetism. Thomson and Tait’s treatment of abstract dynamics in mechanics in 1867 served as the springboard for Maxwell’s creative application of this methodology to electromagnetism. Within this framework of abstract dynamics, Maxwell emphasized the role of energy (rather than force), and identified the medium as the seat of energy; these two related moves led to the set of general equations of the electromagnetic field in a way that differed from what he had presented in Station 3.23 Maxwell’s (relative) lack of attention to issues of transition reflects his lack of commitment to any specific methodology. To be sure, when he embarked on his study of electromagnetism he did express criticism of previous methodologies. In fact, criticisms of methodologies appear throughout the odyssey, especially in Station 4. However, he did not criticize his own previous methodologies, and he never really abandoned them. Indeed, they resurface in one way or another in Station 4.

8.3 Methodologies in Maxwell’s practice Maxwell’s odyssey in electromagnetism exhibits a variety of methodologies that are worthy of philosophical reflection. In the following subsections we analyze six kinds of methodology which play a substantial role in the odyssey, all of which were intended to recast Faraday’s conceptual and empirical findings into a comprehensive formal system. These methodologies are: (1) applying novel mathematical methods, (2) translating the verbal to the formal and then back from the symbolic language to the description of the phenomena at the macro-level, (3) creating mental imagery of the physics of the phenomena, (4) appealing to analogy (strong and weak), (5) assuming hypotheses at the micro-level for explanatory purposes, and (6) introducing working models to enhance intuition.

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Reflections on Maxwell’s methodologies 219 8.3.1 The role of mathematics In 1870, in his “Address to the Mathematical and Physical Sections of the British Association,” Maxwell argued that the process of scientific inquiry had become the measurement and registration of quantities, combined with a mathem­ atical discussion of the numbers thus obtained. It is this scientific method of directing our attention to those features of phenomena which may be regarded as quantities which brings physical research under the influence of mathematical reasoning . . . I wish . . . to direct your attention to some of the reciprocal effects of the progress of science on those elementary con­ ceptions which are sometimes thought to be beyond the reach of change.24 When Maxwell spoke of the progress of science in 1870 he definitely had in mind his own work on electromagnetism. His account offers a summary of the development of his own scientific thought (a path he recommended to his reader: study Faraday, which is essentially qualitative, and then render his findings quantitative): The student who wishes to master any particular science must make himself familiar with the various kinds of quantities which belong to that science. When he understands all the relations between these quan­ tities, he regards them as forming a connected system, and he classes the whole system of quantities together as belonging to that particular science. This classification is the most natural from a physical point of view, and it is generally the first in order of time. But when the student has become acquainted with several different sci­ ences, he finds that the mathematical processes and trains of reasoning in one science resemble those in another so much so that his knowledge of the one science may be made a most useful help in the study of the other. When he examines the reason for this, he finds that in the two sciences he has been dealing with systems of quantities, in which the mathematical forms of the relations of the quantities are the same in both systems, though the physical nature of the quantities may be utterly different. He is thus led to recognize a classification of quantities on a new prin­ ciple, according to which the physical nature of the quantity is subordin­ ated to its mathematical form. This is the point of view which is characteristic of the mathematician; but it stands second to the physical aspect in order of time, because the human mind, in order to conceive of different kinds of quantities, must have them presented to it by nature.25 Maxwell generalized in this passage his understanding of Thomson’s discovery of the analogy between physical systems based on their identical mathematical structures which are reflected in the same formulation.26

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220 Reflections on Maxwell’s methodologies While the physical nature of the quantity is subordinated to its mathematical form, generally the physics is the first in order of time. We see here that Maxwell recognized a strong mutual relation between physics and mathematics. The pronouncement in Station 1—that methodology serves “as the vehicle of mathematical reasoning”—is a recurring theme in Maxwell’s odyssey. From one station to the next, methodologically, the claim remains the same, namely, mathematics offers unforeseen connections which simply cannot be conceived without the formal approach. The advantages of mathematical formulation are evident; it offers both a way to comprehend the phenomena as well as procedures for calculation whose results can then be tested experimentally. It is a characteristic of Maxwell’s odyssey that as the methodologies changed, the mathematical techniques changed as well. To be sure, the role of mathematics remained constant, but the techniques changed substantially. In particular, in Station 4 Maxwell included a great variety of mathematical tools which he adapted to the domain of electromagnetism such as the recasting of electrodynamics into Lagrangian and Hamiltonian forms, the introduction of vectors, scalars, quaternions, line- and surface-integrals, and the geometry of position (i.e., topology). Advanced mathematics played a much more important role in Station 4 than in the preceding stations.27 Maxwell warned against the lure of manipulating mere symbols of the formal language, detached from physical meanings. In keeping with his commitment to Faraday’s ideas, he sought “to present the mathematical ideas to the mind in an embodied form” so that the formal language would adapt itself to the phenomena to be explained.28 Thus, it is no surprise that Maxwell expected a physical theory to be expressed both verbally and symbolically, and that the one could be translated to the other, complying with the demand for reciprocity. 8.3.2 Reciprocity of formulation: translation It is a philosophical truism that the cognitive faculties of human thinking create abstract patterns of thought that correspond—within some conceptual framework—to a certain natural phenomenon. These patterns of thought are categorically different and quite distinct from the phenomenon in question. What is not a truism is the extent to which a thinker is aware of the relations that establish the connections between the phenomenon in question and the corresponding patterns of thought. Indeed, insights into these patterns of thought and their relations to a phenomenon could be used to forge methodological tools to advance scientific research. Maxwell did not engage in purely philosophical discussions; however, he was receptive to the manipulation of these relations in order to generate novel scientific knowledge. Maxwell demanded reciprocity between physics and mathematics which he then transformed into a methodology relating the verbal to the symbolic, where the verbal is textual and the symbolic is mathematical. In Station 1

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Reflections on Maxwell’s methodologies 221 this means that Faraday’s verbal account was turned into a symbolic account by means of a “mathematical analogy.” Maxwell began with a verbal description of the imaginary physics and then cast it into symbolic formulation. Clearly, Maxwell intended Faraday’s verbal and his own symbolic formulation to be reciprocal, although some consequences were expected to be drawn only from the symbolic. In Station 2 the means of transforming the verbal description of the phenomena into the symbolic is the molecular vortex hypothesis. The transition is from verbal description of the micro-level to symbolic formulations of the macro-level. Consider the following verbal description: Every vortex is essentially dipolar, the two extremities of its axis being distinguished by the direction of its revolution as observed from those points . . . We shall suppose at present that all the vortices in any one part of the field are revolving in the same direction about axes nearly parallel, but that in passing from one part of the field to another, the direction of the axes, the velocity of rotation, and the density of the substance of the vor­ tices are subject to change. We shall investigate the resultant mechanical effect upon an element of the medium, and from the mathematical expression of this resultant we shall deduce the physical character of its different component parts.29 Maxwell described the procedure he intended to undertake. Initially, some properties of vortices are verbally described in general terms before turning attention to the micro-level. In this station Maxwell was not translating phenomena from verbal to symbolic, but he did translate his goal expressed verbally to the symbolic. Here the primary concern is the deduction from the micro-level to the macro-level and so the underlying methodology does not involve an explicit translation. To be sure, this procedure is different from what was done in the other stations but, as a methodology, it bears a family resemblance. In Station 3, the translation is explicit: Maxwell began with Faraday’s verbal descriptions of the phenomena and then turned them into symbolic formulations—equations. The flywheel in Station 3 serves as a means of constructing the symbolic formulation. Again, the reciprocity is between the verbal and the symbolic. Station 4 is richer than all the preceding stations in instances of translation with an emphasis on the role of mental imagery.30 Perhaps the strongest plea for recasting the symbolic into the verbal, and calling it translation, had been made by Faraday in response to Maxwell’s first contribution to electromagnetism in Station 1. In late 1857, upon receipt of a preprint of Maxwell’s paper which was published in 1858, Faraday sharply criticized Maxwell for not presenting his mathematical results in ordinary language:

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222 Reflections on Maxwell’s methodologies When a mathematician engaged in investigating physical actions and results has arrived at his own conclusions, may they not be expressed in common language as fully, clearly, and definitely as in mathematical for­ mulae? If so, would it not be a great boon to such as I to express them so?—translating them out of their hieroglyphics, that we also might work upon them by experiment. I think it must be so, because I have always found that you could convey to me a perfectly clear idea of your conclu­ sions which, though they may give me no full understanding of the steps of your process, give me the results neither above nor below the truth;— and so clear in character that I can think and work from them. If this be possible would it not be a good thing if mathematicians, writing on these subjects, were to give us their results in this popular, useful, working state, as well as in that which is their own and proper to them?31 It is noteworthy that, somewhat sarcastically, Faraday considered symbolic language “hieroglyphics,” not conducive to experimental testing. Faraday complained that the mathematical, symbolic language may be useful to mathematicians, but not to experimenters like him, who need the language to be “popular, useful,” and in a “working state.”32 Evidently, Maxwell took this request seriously. Indeed, he extended the usage of the term translation. For example, in 1873 he invoked this demand in a technical context, where the transformation of a physical problem in one domain to another was associated with a translation from one natural language to another.33 A good example of this lesson, which Maxwell drew from Faraday’s demand, is Maxwell’s assessment of Thomson’s early work when he reviewed it at the time he published Station 4: The first paper, communicated to the Royal Society in 1849 and 1850, is the best introduction to the theory of magnetism that we know of. The discussion of particular distributions of magnetisation is altogether ori­ ginal, and prepares the way for the theory of electro-magnets which fol­ lows. This paper on electro-magnets is interesting as having been in manuscript for twenty-three years, during which time a great deal has been done both at home and abroad on the same subject, but without in any degree trenching upon the ground occupied by Thomson in 1847. Though in these papers we find several formidable equations bristling with old English capitals, the reader will do well to observe that the most important results are often obtained without the use of this mathematical apparatus, and are always expressed in plain scientific English.34 In the spirit of Faraday, Maxwell clearly indicated that physics could be developed and indeed conveyed in “plain scientific English,” without invoking the symbolic language of mathematical formulation. The demand for translatable systems arose from Maxwell’s strong belief which he expressed in several places, namely, that scientific truth may be

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Reflections on Maxwell’s methodologies 223 formulated in equivalent ways. As the motto of our book indicates, according to Maxwell “there is no more powerful method for introducing knowledge into the mind than that of presenting it in as many different ways as we can.”35 Elsewhere Maxwell articulated this idea with greater specificity: to divest scientific truths of symbolic language in which the mathematicians have left them, and to clothe them in precise terms in natural language.36 Thus, there can be no doubt that Maxwell considered it a paramount goal to have two kinds of formulations of the same scientific truth, specifically verbal and symbolic. The two modes of expression, the verbal and the symbolic, were not independent paths to physical knowledge; rather, it is clear, especially in Station 4, that they follow sequentially, from textual to symbolic (i.e., mathematical) language and back, for the results in symbolic language are to be retranslated into ordinary language. Conjuring up mental images of the concepts involved is part of this process of retranslation from the symbolic back to the verbal.37 Thus, according to Maxwell, words, unlike symbols, call up mental images, not of formal operations, but of physical characteristics of, say, the motion of bodies.38 For Maxwell algebraic representations of dynamics, such as the Lagrangian and the Hamiltonian, were powerful tools that he was happy to apply but, in his view, physics requires more than mere formalism, it must have mental imagery capable of being rendered verbally. Indeed, Maxwell sought to retranslate—his expression—the equations into a natural language so that the physics may be intelligible without the use of symbols.39 Interestingly, the required criterion of translation (and retranslation) is explicit in Station 4 and well elaborated. In Station 3 translations from verbal to symbolic are evident, but the reverse is absent. In Stations 1 and 2 “translation” is not invoked explicitly, although the whole enterprise, of course, is about translating the newly discovered phenomena into symbolic language. This suggests that as the odyssey progressed, Maxwell became more aware of his inclination to grasp a certain scientific truth in different “languages.” By considering the odyssey from this perspective, it becomes clear that the issue of translation took on an ever-larger role in his arsenal of methodologies. 8.3.3 Mental imagery At the very end of Station 4—we can therefore say at the end of his methodological odyssey—Maxwell strongly defended the view that a theory which cannot offer a consistent mental imagery of the phenomena is inadequate. Indeed, he was unwilling to accept a theory that is inaccessible to mental imagery. In other words, mathematical formalism (even if successful) was not sufficient in the absence of a corresponding mental image.40 It is not immediately clear what Maxwell meant by “mental image.” He explicitly demanded the development of a mental image of the medium’s action, as the medium transmits the disturbances that take place in it.41 To

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224 Reflections on Maxwell’s methodologies be sure, transmission is the phenomenon in question; but what does it mean to form a mental image of the action of the medium which, in addition to the symbolic expression in formulas, has to be recast verbally? The demand is sequential, tied up, as we have seen, with translation: from a verbal description of the phenomena, through symbolic representation—equations—and then back to mental imagery, amenable to verbal description. By contrasting Maxwell's negative assessment of the continental theories with his own conception of transmission of light through the medium, one can gain insight into what Maxwell meant by having a mental image, the result of retranslating the symbolic formulation into a verbal description of transmission phenomena. We first present the challenge of retranslation which Maxwell set himself, and then the way he grappled with it both negatively and positively. The aim, Maxwell remarked, was to retranslate the mathematical results into “the language of dynamics” so that the verbal description will trigger mental imagery of bodies in motion and not some equations.42 Thus, a description will be produced which is intelligible without the use of symbols.43 The challenge is clear: to create mental imagery of dynamics expressed verbally. This is in fact a response to what we may call “Faraday’s demand for translation.”44 As part of his critique of the continental theories, which were based on action at a distance, Maxwell explicitly remarked that he was incapable of forming a mental image of the transmission of potential: We have seen that the mathematical expressions for electrodynamic action led, in the mind of Gauss, to the conviction that a theory of the propagation of electric action in time would be found to be the very key-stone of electrodynamics. Now we are unable to conceive of propa­ gation in time, except either as the flight of a material substance through space, or as the propagation of a condition of motion or stress in a medium already existing in space.45 Against his positive view of the conditions required for an action to be transmitted across space, Maxwell found himself incapable of conceiving what it means, in theories based on action at a distance, for an action to be transmitted. For Maxwell, the only possibilities for such transmission were either something like a projectile or a wave in a medium. And he continued to question the possibility of mental imagery of the mathematical conception called potential which was central to continental theories such as Gauss’s and Neumann’s. According to Maxwell, unlike the transmission of light through a medium, potential was supposed to be projected from one particle to another without an intervening medium. Maxwell objected to potential because he could not form a mental image of its propagation without a medium.46 Clearly, the subject is the propagation of electromagnetic disturbances, and Maxwell drew a sharp contrast between Newmann’s theory based on the Newtonian concept of action at a distance and his own theory founded

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Reflections on Maxwell’s methodologies 225 on Faraday’s concept of lines of force. To repeat: Maxwell openly expressed his inability to conceive, let alone understand, what is meant by transmitting potential across space, independent of any medium. The problem was grave: In all these [action-at-a-distance] theories the question naturally occurs: —If something is transmitted from one particle to another at a distance, what is its condition after it has left the one particle and before it has reached the other? If this something is the potential energy of the two particles, as in Neumann’s theory, how are we to conceive this energy as existing in a point of space coinciding neither with the one particle nor with the other? In fact, whenever energy is transmitted from one body to another in time, there must be a medium or substance in which the energy exists after it leaves one body and before it reaches the other.47 Maxwell focused on the transmission of potential energy and remarked that he was unable to conceive how potential energy could be transmitted without any material localization. The problem is one of material as well as localization, for matter must be the carrier of energy. Maxwell concluded on a positive note that one could form a mental image of transmission through a medium whose action results in propagation.48 We note that the subject is propagation, the transmission of action, rather than the existence of a medium or its nature. Maxwell advocated that one should construct a mental image of propagation as well as of the details of the action, where the issue is how transmission of electromagnetic disturbances and energy takes place. But did Maxwell address the mental imagery of the medium and the way it serves as a mediator in the transmission of energy? At this juncture we turn to Maxwell’s own conception of transmission in the medium, namely, the undulatory theory of light: According to the theory of undulation, there is a material medium which fills the space between the two bodies, and it is by the action of contiguous parts of this medium that the energy is passed on, from one portion to the next, till it reaches the illuminated body.49 This is a clear statement: there is a material medium which functions as a mediator between two bodies for the transmission of energy. The claim is unambiguous: the medium offers a contiguous, uninterrupted link between the two bodies. This is a description of a material medium that can easily be conceived. And Maxwell continued: The luminiferous medium is therefore, during the passage of light through it, a receptacle of energy. In the undulatory theory, as devel­ oped by Huygens, Fresnel, Young, Green, &c., this energy is supposed

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226 Reflections on Maxwell’s methodologies to be partly potential and partly kinetic. The potential energy is sup­ posed to be due to the distortion of the elementary portions of the medium. We must therefore regard the medium as elastic. The kinetic energy is supposed to be due to the vibratory motion of the medium. We must therefore regard the medium as having a finite density.50 Maxwell associated himself with a long tradition, beginning with Christiaan Huygens (1629–1695). In this theory the medium is a receptacle of energy. And energy—as Rankine had suggested—comes in two forms, potential and kinetic. So, in principle, Maxwell had no objection to potential energy; what he objected to is the way its transmission was conceived in the continental theories. The key for mental imagery is the medium’s property of elasticity which can be easily conceived as the seat for both kinds of energy. Maxwell then brought his conundrum to a resolution: his dynamic theory is consistent with the requirements of the undulatory theory of light offering easily conceived mental imagery of the transmission of energy. Two forms of energy are recognized, the electrostatic and the electrokinetic, and they have their seat, in Maxwell’s theory, in every part of the surrounding space, where electric or magnetic force is observed to act.51 Maxwell confirmed that his theory indeed complies with the requirement for a medium that functions as the seat of energy and the carrier of action by its contiguous parts. Let us next determine the conditions of the propagation of an electro­ magnetic disturbance through a uniform medium, which we shall sup­ pose to be at rest, that is, to have no motion except that which may be involved in electromagnetic disturbances.52 Thus, having established the requirement for a medium in a dynamic theory, Maxwell could now turn to specific physical problems. His challenge, as we have seen, was to retranslate the formalism of abstract dynamics into verbal language which is intelligible without the use of symbols; in other words, the challenge was to retranslate the equations into words that call up mental images of properties of moving bodies. It appears that Maxwell was convinced that he had succeeded in casting the physics of electrodynamics as action transmitted in a medium. The symbolic and the verbal were thus put into a reciprocal relation. Maxwell met the challenge he had set himself. There were two related goals: (1) retranslation, and (2) mental imagery. For (1) Maxwell offered a verbal account of the transmission of electromagnetic phenomena without recourse to symbols. For (2) the mental image of transmission was the medium, conceivable as the bearer of undulatory motion. Both the medium and its essential property—elasticity—were conceivable. Maxwell did not address the nature of the medium beyond the property of elasticity. He was satisfied that with his dynamic theory he had attained both goals. But,

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Reflections on Maxwell’s methodologies 227 would Faraday have been satisfied? Probably not, because he had hoped for a translation from the equations to ordinary language so that he could apply them in experiments.53 In his laudatory remarks on Faraday, Maxwell expressed this view of the progress of scientific research succinctly: For the advance of the exact sciences depends upon the discovery and development of appropriate and exact ideas, by means of which we may form a mental representation of the facts, sufficiently general, on the one hand, to stand for any particular case, and sufficiently exact, on the other, to warrant the deductions we may draw from them by the appli­ cation of mathematical reasoning.54 Throughout his methodological odyssey, Maxwell considered mental imagery—however he conceived it—useful for enhancing physical intuition as well as for precise formal deduction. 8.3.4 Analogy Maxwell invoked analogy in the opening passage of the abstract to his first contribution to electromagnetism (1856).55 His initial step in the study of electromagnetism, recorded in this abstract, was methodological: it was an appeal to Thomson’s use of analogy. He proceeded to modify Thomson’s methodology, calling this revised methodology “mathematical analogy.”56 At the outset of Station 1, Maxwell reflected on the available methodologies. He addressed a scientist who enters a new physical domain as a researcher seeking simplification of the results to a form that is easily grasped. According to Maxwell two options are available: either a “purely mathematical formula” or a “physical hypothesis.” As we have seen, for Maxwell neither option was satisfactory and he sought a new way —“physical analogy.” It appears that this kind of analogy is in fact the analogy Maxwell had called “mathematical” in the abstract that had appeared previously. This analogy functions in Station 1 as a research tool. It is part of the argumentative structure of the paper which facilitates the development of a mathematical formulation of Faraday’s findings in electromagnetism. Thus, the analogy is “strong,” to be used for drawing consequences; therefore, the analogy must be consistent. Maxwell was clearly impressed by the methodological power of strong analogy and he applied it as his principal tool of research.57 Moving on to Station 2, we note that Maxwell changed his approach altogether and opted for a traditional methodology—a physical hypothesis. Analogy, specifically strong analogy as the principal tool of research, is abandoned. To be sure, he did appeal to analogy, but this time to a weak analogy—it did not play a role in the argumentative structure. The appeal to the analogy of rack and pinion at the macro-level is qualitative only, to

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228 Reflections on Maxwell’s methodologies aid in comprehending and controlling the two mechanical motions underlying the electric and magnetic phenomena—translational and rotational motions, respectively.58 In Station 3 Maxwell returned to strong analogy. However, what he called “mechanical” analogy—the flywheel—initially played the role of illustration. But there is no question that the analogy is part of the argumentative structure of the paper. It made possible the construction of a dynamical theory of the field; it was not just an aid for understanding how the medium functions. In other words, the analogy is quantitative. One of the fundamental differences between Maxwell’s earlier stations and Station 4, is that analogies were no longer used primarily as tools of research; rather, they mostly served a didactic purpose. In Station 1 Maxwell invoked a “geometrical model” which he later considered “mechanical illustrations”;59 then, in Station 2, he appealed to “a mechanical point of view”;60 and in Station 3 he drew an illustration which he then applied as a “mechanical analogue.”61 These expedients were introduced, as we have seen, to aid in advancing research. The critical development of Maxwell’s theory took place in Station 3 where the properties of the vortices in Station 2 were transferred to the medium; this transfer is retained in Station 4. As in Stokes,62 the issue had to do with the (strange) properties of the medium, namely, the ether. It was a challenge to make these properties plausible. The medium was considered the bearer of electromagnetic disturbances. How is this new perspective on electromagnetism to be understood? And what is the role of analogies? Of interest is whether Maxwell was consistent in appealing to analogies, or whether they were simply ad hoc. It appears that Maxwell changed his methodological approach in Station 4, for strong analogies were not frequently invoked as part of the argument; rather, a number of weak analogies were introduced as aids for understanding. Station 4 is rich in a variety of analogies but in this station Maxwell, for the first time, was explicit about the potentially misleading character of analogy citing, for example, the analogy of self-induction with the motion of material bodies.63 Thus, he explicitly warned the reader that, despite similarities in formal structure, fluid and heat are categorically different and electricity should not be identified with either one. Electricity is neither a substance like a fluid, nor a state of agitation like heat.64 In other words, one must be on guard against misleading analogies. That does not mean that Maxwell gave up on analogy. Not at all; but then most of the analogies to which Maxwell appealed in Station 4 are of the “weak” variety, for they are qualitative, designed to aid in understanding, and to assist intuition in grasping the physical nature of the phenomenon. After all, Station 4 is also a textbook.65 For example, Maxwell called the ratio of the electromotive intensity to the corresponding electric displacement the coefficient of electric elasticity of the medium, since the mechanical analogy between the action of electromotive intensity in producing electric displacement and of

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Reflections on Maxwell’s methodologies 229 ordinary mechanical force in producing the displacement of an elastic body was so “obvious.”66 For Maxwell the mechanical analogy was “obvious” and it suggested a way to develop a mathematical formalism. The analogy was suggestive—qualitatively. For another example, Maxwell occasionally appealed to taut ropes and rigid rods as analogies, comparing lines of force to ropes that transmit tension and pressure at the right angle to them.67 He further developed the analogy of mechanical action in animals which offers an explanation for the work that a living body can do, but with no insight into the causes of muscular tensions. The analogies of ropes and muscles are clearly weak analogies and not part of the argument. With these analogies, Maxwell sought to make his theory plausible to the reader. In Station 4 there are a few strong analogies that Maxwell exploited to advance his research. For example, in his discussion of the dynamical theory of electromagnetism Maxwell claimed that “the work done by an electromotive force is of exactly the same kind as the work done by an ordinary force, and both are measured by the same standards or units.”68 This offered the possibility of drawing consequences from the mechanical domain to the electromagnetic domain in quantitative terms, exactly as strong analogies function. In the course of his odyssey Maxwell was increasingly reluctant to rely on strong analogies. Instead, he appealed to a variety of analogies, most of them weak, to strengthen an intuitive appreciation, as well as to increase the intelligibility of his novel dynamical theory. In Station 4 Maxwell demonstrated that he was both a physicist at the cutting edge of scientific research and a master of pedagogy. 8.3.5 Hypothesis: the micro-level and explanatory claims In Station 1 Maxwell contrasted two methodologies, namely, the appeal to purely mathematical formulas on the one hand, and conjecturing a physical hypothesis on the other. Adopting the latter, according to Maxwell, leads researchers to see the phenomena mediated by some scheme that may keep them from focusing on the properties of those phenomena. Therefore, researchers are liable to blindness to facts and may be carried away from the truth by a commitment to a convenient perspective.69 Maxwell thus considered physical hypothesis a methodological position that may obscure the phenomena, inadvertently creating obstacles to research. Hypothesis here means a claim or assumption about the nature of the micro-level underlying the phenomena from which the laws governing the phenomena can be deduced. Indeed, in the scheme of Station 1 Maxwell referred to action at a distance as hypothesis, while refraining from considering the imaginary properties he endowed the flowing incompressible fluid hypothetical.70 This was not the case in Station 2 where Maxwell based his methodology on the hypothesis of molecular vortices.71 The hypothesis was “purely mechanical”:72

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230 Reflections on Maxwell’s methodologies according to our hypothesis, an electric current is represented by the transference of the moveable particles interposed between the neigh­ bouring vortices. We may conceive that these particles are very small compared with the size of a vortex, and that the mass of all the par­ ticles together is inappreciable compared with that of the vortices, and that a great many vortices, with their surrounding particles, are contained in a single complete molecule of the medium. The particles must be con­ ceived to roll without sliding between the vortices which they separate, and not to touch each other, so that, as long as they remain within the same complete molecule, there is no loss of energy by resistance. When, however, there is a general transference of particles in one direction, they must pass from one molecule to another, and in doing so, may experience resistance, so as to waste electrical energy and generate heat.73 In fact, Maxwell illustrated the hypothesis with a figure in which “large spaces” represent vortices and small circles separating the vortices represent the layers of particles placed between them which, according to Maxwell, represent electricity.74 I have shewn how the forces acting between magnets, electric currents, and matter capable of magnetic induction may be accounted for on the hypothesis of the magnetic field being occupied with innumerable vor­ tices of revolving matter, their axes coinciding with the direction of the magnetic force at every point of the field.75 ... According to our hypothesis, the magnetic medium is divided into cells, separated by partitions formed of a stratum of particles which play the part of electricity. When the electric particles are urged in any direction, they will, by their tangential action on the elastic substance of the cells, distort each cell, and call into play an equal and opposite force arising from the elasticity of the cells. When the force is removed, the cells will recover their form, and the electricity will return to its former position.76 This was Maxwell’s answer to his motivating question, namely, “Is there any mechanical hypothesis as to the condition of the medium indicated by lines of force, by which the observed resultant forces may be accounted for?”77 Maxwell’s view of his hypothesis in Station 2 differed from that which he had expressed in Station 1 concerning what he had called “physical hypothesis.” Essentially, hypothesis is explanatory for the purpose of drawing consequences from the micro-level to the macro-level. On the one hand, Maxwell indicated his dissatisfaction with “the explanation founded on the hypothesis of attractive and repellent forces.” And on the other, he pointed to the mechanical consequences of a new hypothesis; he wished to strengthen the position of those, like him, who opted for the concept of lines of force against that of action at a distance, without however any

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Reflections on Maxwell’s methodologies 231 experimental support.78 Maxwell’s goal in introducing his hypothesis was to offer an alternative to the Newtonian concept. He therefore sought to make his argument dependent on experimental evidence.79 Maxwell sought consequences of the mechanical hypothesis in the hope that it would help in determining what he characterized as “the true interpretation of the phenomena.”80 In contrast to Station 1, in Station 2 Maxwell deemed it appropriate to introduce a mechanical hypothesis, use it as an explanatory tool, and draw consequences from it. Indeed, he argued, the hypothesis will not hinder the researcher and it may lead the way to the “true interpretation.” And all this without any commitment to the physical reality of the hypothesis: it is “mechanically conceivable.” Maxwell thus offered a plausible physical scheme. Moving on to Station 3, Maxwell announced that he would avoid “any hypothesis of this kind.”81 This is a dramatic departure from the view expressed in Station 2. Maxwell stressed the difference between illustration and explanation. Here, in Station 3, he was not seeking an explanation. And, in fact, the conclusions reached in Station 3 are, as Maxwell remarked, “independent of . . . hypothesis, being deduced from experimental facts.”82 The odyssey continues—we move on to Station 4. The appeals to hypothesis in the Treatise are complex and function on multiple levels. In this station Maxwell distanced himself from such scientists as W. Weber, Gauss, Riemann, C. Neumann, and Lorenz, all of whom founded their theories on action at a distance. The great success which these eminent men have attained in the applica­ tion of mathematics to electrical phenomena gives, as is natural, add­ itional weight to their theoretical speculations, so that those who, as students of electricity, turn to them as the greatest authorities in math­ ematical electricity, would probably imbibe, along with their mathemat­ ical methods, their physical hypotheses. These physical hypotheses, however, are entirely alien from the way of looking at things which I adopt, and one object which I have in view is that some of those who wish to study electricity may, by reading this trea­ tise, come to see that there is another way of treating the subject.83 It is noteworthy that at this juncture Maxwell did not characterize his theory as based on a “physical hypothesis.” Rather, he called it “dynamical hypothesis”: In the next five chapters of this treatise I propose to deduce the main structure of the theory of electricity from a dynamical hypothesis of this kind, instead of following the path which has led Weber and other investigators to many remarkable discoveries and experiments, and to conceptions, some of which are as beautiful as they are bold. I have chosen this method because I wish to shew that there are other ways of viewing the phenomena which appear to me more satisfactory, and at

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232 Reflections on Maxwell’s methodologies the same time are more consistent with the methods followed in the preceding parts of this book than those which proceed on the hypoth­ esis of direct action at a distance.84 In Station 4 it is a recurring theme to spell out the disadvantages of theories of electricity and magnetism based on the physical hypothesis of action at a distance. The undulatory nature of light was typically regarded in the literature as a hypothesis.85 Maxwell was not an exception in this respect: We have been hitherto obliged to use language which is perhaps too suggestive of the ordinary hypothesis of motion in the undulatory theory. It is easy, however, to state our result in a form free from this hypothesis. Whatever light is, at each point of space there is something going on, whether displacement, or rotation, or something not yet imagined, but which is certainly of the nature of a vector or directed quantity, the direc­ tion of which is normal to the direction of the ray. This is completely proved by the phenomena of interference.86 It is clear that Maxwell considered it an advantage that his novel physical approach, based on a dynamical medium, allowed him to treat the phenomena of light without depending on its undulatory characterization. A survey of the usages of hypothesis as a methodology in Maxwell’s research in electromagnetism reveals then that he distinguished between schemes based on real physical phenomena (Station 2) and schemes which are based on imaginary physics (Station 1). The former he called physical hypothesis, but he did not call the latter hypothesis, for there was no claim at the micro-level. However, in both cases the scheme presupposed a set of assumptions that allows consequences to be drawn that can be tested experimentally. As the odyssey progressed, Maxwell relaxed the distinction—which, needless to say, is not observed in modern terminology—and characterized any set of assumptions as hypotheses (Station 4). In particular, he considered both “action at a distance” and “lines of force” hypotheses. But he did not adhere to this methodology in Station 3, and was critical of its allure in Station 1. Two years after the publication of the Treatise, in a paper “On the dynamical evidence of the molecular constitution of bodies,” in a different physical domain from electrodynamics, Maxwell spelled out in detail what he thought of the methodology of hypothesis and the modifications which he introduced. Hypothesis was the common practice: The method which has been for the most part employed in conducting such inquiries [determining, from the observed external actions of an unseen piece of machinery, its internal structure] is that of forming an hypothesis, and calculating what would happen if the hypothesis were

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Reflections on Maxwell’s methodologies 233 true. If these results agree with the actual phenomena, the hypothesis is said to be verified, so long, at least, as some one else does not invent another hypothesis which agrees still better with the phenomena.87 In Station 2 Maxwell handled his hypothesis of molecular vortices in precisely this way by “calculating what would happen if the hypothesis were true.” Maxwell went on to spell out the reason for the popularity of the methodology of hypothesis: The reason why so many of our physical theories have been built up by the method of hypothesis is that the speculators have not been provided with methods and terms sufficiently general to express the results of their induction in its early stages. They were thus compelled either to leave their ideas vague and therefore useless, or to present them in a form the details of which could be supplied only by the illegitimate use of the imagination. In the meantime the mathematicians, guided by that instinct which teaches them to store up for others the irrepressible secretions of their own minds, had developed with the utmost generality the dynamical theory of a material system.88 Was Maxwell reflecting on his own recent experience in electrodynamics? We need not dwell on this biographical question, but one thing is clear: mathematics came to the rescue. Why is it, we ask, that in Station 4 Maxwell called the hypothesis of Station 2 a “working model”?89 We claim that this change in nomenclature is an indication of an important methodological development. We explore this development in the next subsection. 8.3.6 Model and modeling Modeling has become a characteristic feature of modern science. Indeed, it seems that almost all scientific papers today, no matter in what domain, include discussions of a model of one kind or another. One difficulty, which is widely recognized, is that “model” has been invoked for a variety of concepts; indeed, the term has had many different meanings and its usages have changed over time. In contemporary usage modeling has become a concept applied to concrete objects as well as to abstract thoughts; for scaling; and for representing phenomena and data. It can also function as a thought experiment, a simulation, or an idealization of a general theory. Moreover, it may consist of set-theoretic structures, descriptions, as well as equations, and can offer physical interpretations of a mathematical structure.90 Thus, although the term remains, the underlying concepts have changed. It is no wonder that, when asked what they mean by model, scientists give a remarkable variety of responses.91 We need then to be cautious when making any claim about model and modeling in Maxwell’s methodological odyssey.

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234 Reflections on Maxwell’s methodologies As we have seen, in Station 2, when Maxwell referred to the figure with the mechanical scheme of the molecular vortices,92 he used various grammatical forms of “represent.”93 But he did not refer to his “imaginary system of molecular vortices” as a model.94 The term “model” is nowhere to be seen. In fact, as we have pointed out, Maxwell considered his mechanical arrangement, a “hypothesis.”95 It appears that Maxwell’s usage of “hypothesis” in Station 2 followed that of Rankine for, in 1855, Rankine claimed that, since the theory of motions is the only complete physical theory, it is not surprising that mechanical hypotheses were adopted in theories in other branches of physics. Rankine explained that A mechanical hypothesis is held to have fulfilled its object, when, by applying the known axioms of mechanics to the hypothetical motions and forces, results are obtained agreeing with the observed laws of the classes of phenomena under consideration . . .96 For Maxwell in Station 2 consistency was a requirement and “hypothesis” implied a logical structure, precisely what was needed.97 In line with the common usage of the time, Maxwell called his mechanical scheme of the molecular vortices a hypothesis. However, some historians and philosophers of science considered this hypothesis a model; but, as we argue below, in the mid-nineteenth century hypothetical entities had not yet entered the domain of models.98 In 1858, in Station 1, Maxwell invoked an idiosyncratic usage of the term model, namely, “geometrical model”: We should thus obtain a geometrical model of the physical phenomena, which would tell us the direction of the force, but we should still require some method of indicating the intensity of the force at any point.99 The expression “geometrical model” had been used long before Maxwell invoked it, referring to a tangible representation of a three-dimensional object, e.g., an octahedron, either solid or partly dissected, used for didactic purposes.100 But Maxwell’s usage is unprecedented, for Maxwell’s “geometrical model” does not refer to a material object that serves a didactic purpose; rather, it refers to a unifying abstract scheme at the cutting edge of scientific research which facilitated linking various electromagnetic phenomena. For Maxwell this was a way of referring to Faraday’s “process of reasoning” as a methodology.101 He wished to characterize Faraday’s innovative method but did not come up with a satisfactory expression in 1858; in fact, Maxwell did not use the “geometrical model” for this purpose ever again.102 However, at the same time Maxwell also used modeling in its traditional meaning. In his prize-winning essay on the rings of Saturn, Maxwell invoked the expression “mechanical model,” and used it in the traditional sense of a device that can be built with the goal of exhibiting in miniature

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Reflections on Maxwell’s methodologies 235 the phenomenon in question (i.e., the rings).103 Maxwell reported that he had “made an arrangement by which 36 little ivory balls are made to go through the motions belonging to the . . . series of waves.”104 And he indicated that the design was illustrated by two figures.105 He then continued: By considering these figures, and still more by watching the actual motion of the ivory balls in the model, we may form a distinct notion of the motions of the particles of a discontinuous ring, although the motions of the model are circular and not elliptic. The model, repre­ sented on a scale of one-third in figs. 7 and 8, was made in brass by Messrs. Smith and Ramage of Aberdeen.106 Clearly, the model was idealized, for the motions in the model were circular and not elliptical and the 36 balls were intended to represent a large number of components of Saturn’s rings. The scaled-down version thus also illustrates Maxwell’s theoretical claims concerning the stability of the rings. It was used to “form a distinct notion,” that is, the purpose was illustrative. A few years later, in 1862, in a note on theories of the constitution of Saturn’s rings, Maxwell recalled this usage of model: The motion of one of these waves was exhibited to the Society by means of a small mechanical model made by Ramage of Aberdeen.107 In a letter to Thomson, dated 30 January 1858, Maxwell remarked that: “I have had a model of my theory of the wave in a ring of satellites made by Ramage.”108 We see that at this time Maxwell appealed to the usual meaning of model. Some 15 years later, in Station 4 of his odyssey, Maxwell referred to the hypothesis of molecular vortices as “a working model” which, in his view, was “a demonstration that mechanism may be imagined capable of producing a connexion mechanically equivalent to the actual connexion of the parts of the electromagnetic field.”109 This is a revealing statement: in 1873 Maxwell considered the vortex hypothesis a working model, thereby introducing a new meaning of model. Maxwell allowed the model itself to have hypothetical elements. This move was new. We are not aware of anyone else appealing to a model with hypothetical elements prior to 1873. The Oxford English Dictionary (OED) highlights two different usages of the term “model” in the period before 1800. It indicates that before the late eighteenth-century model was invoked (1) in technical contexts in the sense of a representation of a structure, something which accurately resembles or represents something else, especially in miniature or on a small scale. The OED further reports that model also had the meaning of (2) “an example to be emulated, a paragon.” Indeed, the term was used in scientific contexts in the early decades of the nineteenth century in the sense of paragon as well as of a scale model (or design) for a machine or an instrument. These older meanings were not displaced when new senses of model were introduced.

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236 Reflections on Maxwell’s methodologies In the first half of the nineteenth century there were three relevant usages of model: (1) small-scale model, (2) geometrical (didactic) model, and (3) working model. A small-scale model is proportional to the large object it is intended to represent; the origin of this usage can be traced back to the making of architectural models. A geometrical (didactic) model is one that is intended to help someone visualize a complex object (especially in crystallography), or to illustrate a physical effect. The didactic model is intended to represent a class of objects; it can be either a small-scale or a large-scale representation, but this feature is secondary. Here the idea can be traced back to model as a paragon, an exemplar for other instances. A working model is still different since it is not intended to represent a class of objects; it combines both scale and function. It illustrates a device on a small scale such that the small-scale model has the same proportions, relative distribution, and—most importantly—it functions as the device itself.110 But we observe that in Station 4 Maxwell used “working model” to refer to a large-scale representation of invisible entities at the micro-level, for he increased the scale of the intended mechanism together with its detailed mechanical functioning rather than reducing it, as was the standard practice with working models at the time.111 It may come as a surprise to many philosophers of science that the term model—as it is used today—was late in entering scientific discourse.112 To be sure, the term had long been invoked in scientific contexts, but not in the modern sense as a methodology. Although modern commentators have singled out the pioneering work of Maxwell in this respect, the results of our study call for caution when claims are made about historical actors. In our case, Maxwell seemed to hint at the modern usages of modeling only in Station 4, where he recast the hypothesis of Station 2 into a working model. Thus, the introduction of the modern concept of “model” tends to obscure Maxwell’s methodology, for it carries with it a lot of “baggage” that is alien to Maxwell’s practice of physics. Our analysis of methodologies indicates that Maxwell changed his view regarding his hypothesis of Station 2. In fact, in Station 4 Maxwell transformed the methodology of hypothesis, recasting it as a “working model,” and then applied it retrospectively to Station 2. This retrospective account became a new methodology.113 Model was still not abstract in the nineteenth century even after 1873—but the shift was from introducing a miniature version of a machine to a hypothetical mechanism at the micro-level. This claim is worth amplifying. Our focus on methodology suggests that there were two different approaches to the molecular vortex scheme. In Station 2 Maxwell considered the scheme a hypothesis, while in Station 4 he recast the scheme as a working model, extending thereby the usage of the expression from a concrete mechanical scheme to a hypothetical mechanism at the micro-level. In the first approach the logical structure is deductive without any analogical element, while in the latter the scheme is analogical, based on exploiting its representational aspect. It is critical to bear in mind

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Reflections on Maxwell’s methodologies 237 that the molecular vortex scheme is the same in Station 2 and in Station 4. But in Station 2 the scheme is hypothetically facilitating a deduction from the micro-level to the macro-level, which is the level of phenomena. By contrast, in Station 4 the same scheme offers an analogical representation of the phenomena, thus functioning as a working model. What changed then is the methodology in which the scheme is embedded. Once again, Maxwell took a known methodology as his point of departure and transformed it to suit his research needs. The transformation of the hypothesis of Station 2 into a working model in Station 4 is an example of the introduction of a new methodology, where analogy plays a key role that was not the case with hypothesis. In sum, hypothesis in Station 2 is replaced by model in Station 4. According to the methodology of Station 4, the micro-level is an analogy to the macro-level of phenomena, where Maxwell was explicit that other “models” at the micro-level may be just as successful (and should replace the existing one if they agreed better with the phenomena). There was thus less commitment in Maxwell’s approach in Station 4 than there had been in his approach in Station 2. Evidently, the model in Station 4 is much more flexible than the hypothesis of Station 2— a methodological means which suited Maxwell’s needs. Maxwell gave a new meaning to “working model”; this is another example of his ingenuity in forging a new methodology from previous methodologies. He recast a hypothesis, transforming it into a working model. Specifically, the details of the vortex hypothesis do not change, only the status of the vortex “model” has changed: instead of assumptions in a deductive scheme there is an analogy between the micro-level and the macro-level. But, again, the methodologies are very different. To be sure, the meaning of the term model continued to evolve after Maxwell, notably in the twentieth century when it became ubiquitous in scientific discourse.

8.4 Concluding remarks Here are the four stations that comprise Maxwell’s research in electromagnetism which began in 1856 and ended in 1873. The stations are presented in outline, informally and sequentially, juxtaposed in a concise fashion: this is a methodological odyssey. Station 1: (1856–1858) Maxwell set the goal of casting Faraday’s dis­ coveries in electromagnetism into mathematical formalism; the task was to translate Faraday’s descriptive text into symbolic language; the idea was that mathematical formulas could point to unforeseen phys­ ical relations that could in turn be demonstrated experimentally; for that purpose Maxwell developed the novel methodology of mathemat­ ical analogy (which he later called physical analogy)—an explicit modification of Thomson’s analogical methodology—in which an imaginary physical system is constructed; he did not draw inferences

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238 Reflections on Maxwell’s methodologies from an analogy between two different physical domains; rather, he contrived an analogy between a physical domain and an imaginary one; since the phenomena were recast in purely symbolic form, this approach (intentionally) lacked explanatory power; thus, in this station Maxwell did not attempt to establish a physical theory; both unifying concepts, namely, “action at a distance” and “lines of force,” underlie Maxwell’s procedure; it appears that Station 1 had no discernable impact on the physics community. Station 2: (1861–1862) After three years, Maxwell discarded the ingeni­ ous methodology of Station 1 and turned to the methodology of hypothesis; he pursued the traditional methodology of assuming some mechanical hypothesis at the micro-level from which physical conse­ quences at the macro-level can be drawn; in this station Maxwell sought a physical explanation of the phenomena: a mechanical illustra­ tion taking the lines of force to be physically real; he set aside the instrumental approach of Station 1 and assumed a hypothesis at the micro-level to account for the phenomena at the macro-level; the change is categorical, from analogical argumentation to causal reason­ ing; Maxwell had a clear preference for the concept of lines of force, for the hypothesis “imitates” electromagnetic phenomena on the assump­ tion of lines of force; indeed, he openly expressed his dissatisfaction with action at a distance; two fundamental discoveries were made but it is not entirely clear if the molecular vortex hypothesis played a product­ ive role in making these discoveries; nevertheless, Maxwell continued to explore the consequences of this hypothesis. Station 3: (1865) Three years later, Maxwell assumed yet another novel per­ spective on electromagnetism; he expressed complete confidence both in the two discoveries reported in Station 2 (in fact, he considered them phenom­ ena on a par with Faraday’s discoveries) and in Faraday’s unifying concept of lines of force; he made a renewed attempt at the methodology of transla­ tion, this time without any imaginary physical system; he translated Fara­ day’s descriptive text of electromagnetic phenomena into mathematical symbols formulating a new set of equations, with the discoveries in Station 2 taken as assumptions; thus, the equations of the theory had been deduced from purely experimental evidence: no mechanism or hypothesis intervened; rather, it was entirely based on Faraday’s verbal account of the phenomena plus Maxwell’s own two fundamental discoveries; Maxwell appealed to a strong mechanical analogy in which the equations represent descriptions of the phenomena; he specified that the motion of the assumed “aetherial sub­ stance” which constitutes the field, produces, or indeed causes, electromag­ netic phenomena; as the medium does the work, the mechanical scheme of the molecular vortices (of Station 2) becomes superfluous; the overriding task was not to offer an explanatory mechanism by which the phenomena are produced; rather, the goal was to construct a satisfactory formal theory compatible with the phenomena; the concept of lines of force is combined

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Reflections on Maxwell’s methodologies 239 now with the concept of field as the medium; the concept of action at a dis­ tance is therefore no longer to be seen, even in the context of gravitation; the field expresses the dynamics of electromagnetic phenomena; Maxwell identified two kinds of energy that can exist in the medium, “actual” and “potential”; the fundamental property of the medium, namely, its elasticity, is then essential, since in virtue of this elasticity both kinds of energy can be stored in the medium; there were no new discoveries, but energy became fundamental to the analysis of electromagnetic phenomena. Station 4: (1873) Eight years passed before Maxwell chose a whole new approach which consisted of several methodologies; in many ways, this is an extraordinary work, for it is a textbook on electromagnetism addressed explicitly to a student of physics and, at the same time, it includes cutting edge research; it is also an intricate work that appeals to several method­ ologies; two fundamental elements are called for: novel mathematical approaches combined with a new physical conceptualization of the domain of electromagnetism in terms of the field as the seat of energy; the concept of energy then became one of the principal concepts for handling the dynamics of the field for which several known mathematical techniques were adapted and applied; the mathematical tools that Maxwell adapted and ingeniously applied belong to what Thomson and Tait had called “abstract dynamics”; at its core, Maxwell’s new dynamical theory pre­ sented in this last station consists of a set of equations which again trans­ lated Faraday’s verbal descriptions of the phenomena into mathematical symbolism; Maxwell demanded for the first time that translation should go both ways, namely, from text to symbol and vice versa; symbols had therefore to be explicitly recast in physical imagery; he further reintroduced the hypothesis of molecular vortices of Station 2 but now considered it a “working model” adding an analogical imaginary aspect to the hypothesis, thus preparing the ground for the modern methodology of modeling; finally, the way was paved for a comprehensive formal theory that is phys­ ically meaningful, and this, as Einstein put it, profoundly changed the way physical reality is conceived.114 There are several recurring features in these four distinct stations. We have noted the commitments to the concepts of lines of force and of a medium— the seat of energy—which make it coherent to speak of an odyssey. We note further that Maxwell did not design any methodology de novo; rather, he modified existing ones. This we take to be an important clue to Maxwell’s practice of physics, at least in electromagnetism. Maxwell’s creative responses principally to Faraday but also to Thomson, Helmholtz, Stokes, Tait, Rankine, Lagrange, and Hamilton, among others, show him to be an extraordinary thinker capable of building on the works of his predecessors, both theoretical and experimental, in unforeseen ways. The integration of past achievements with his innovative ideas paved the way for the construction of a novel theory of electrodynamics that has stood the test of

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240 Reflections on Maxwell’s methodologies time. The advantage for historians and philosophers of science of taking into account the entire trajectory from 1856 to 1873 is that it reveals Maxwell’s methodological ingenuity as well as the critical relation between formal means and goals. Maxwell’s great interest in methodology throughout his study of electromagnetism indicates that methodology played for him a crucial role in advancing the frontier of knowledge. In 1856, when Maxwell commenced his study of electromagnetism, he had nothing to add to Faraday’s well-founded body of knowledge. In fact, at that time Maxwell intended neither to engage in experiments nor to construct a novel physical theory;115 rather, the goal he set himself was to translate the known empirical facts into symbolic, formal language in the hope of discovering with this novel representation new physical, experimentally testable, connections. Given this interest, how then to proceed? That was the question. And the answer was definitive, namely, methodology! Find a suitable methodology to facilitate the required translation. That was the challenge. That realization left a lasting impact on Maxwell’s scientific thinking. Methodology was a means to approach Faraday’s theoretical and empirical findings in a different way—integrating them in mathematical physics rather than just leaving them in descriptive-qualitative physics. Methodology was for Maxwell “the vehicle of mathematical reasoning,” to use his own expression when he began his study of electromagnetism.116 But then Maxwell knew that methodology—as a means—was not a single path and that there were other ways to approach Faraday’s findings. Indeed, each methodology led to consequences that could be tested experimentally. Methodologies are not mutually exclusive; therefore, Maxwell did not reject one methodology when he opted for another. In all four stations we find Maxwell, first and foremost, devoting his attention to constructing and then applying different methodologies, suitable to attaining his well-defined goal—a formal description of the empirical facts within a consistent conceptual framework. Maxwell was not committed to a specific methodology but to methodology as a way to generate a greater understanding of the phenomena. He thereby sought to discover new “mathematical connexions” which “may suggest to physicists the means of establishing physical connexions.”117 Maxwell pursued this approach to scientific knowledge in all four stations, culminating in Station 4. The construction of an entirely novel methodology is a rare event. Maxwell was aware of this fact; hence his reflection on past methodologies which he then adapted to his needs. This is probably a principal reason for his success. In his insightful reflections on methodologies he did not seek to separate the heuristic aspect from the argumentative structure. Put in modern terms, it is not helpful to think of methodologies in terms of the context distinction. In other words, it is not productive to sever the goal of discovering new physical connections and the way to justify the search procedure. For Maxwell methodologies comprise heuristics and justification seamlessly. Methodology is the engine of scientific change; this is the way forward to generating new scientific knowledge.

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Reflections on Maxwell’s methodologies 241

Notes 1 According to Siegel (1991, 27–28), Maxwell was exposed both in Edinburgh and Cambridge to a variety of methodologies. Siegel referred to Forbes and Thomson as well as to Herschel and Whewell; however, Siegel did not discuss the Treatise in detail, and so he did not mention Thomson and Tait whose influence on Maxwell was very different from that of Thomson some 20 years earlier. On Herschel and Whewell, see Siegel, 1991, 19–23. For the role of Forbes and the tradition of Scottish skepticism which contrasted with Whe­ well’s approach, see Siegel, 1991, 23–26. See also Whewell, 1831. For further discussion, see Olson, 1975, and ch. 8, § 8.4. 2 See ch. 8, § 8.3.

3 See ch. 1, § 1.1.

4 Maxwell, 1873d, 1: Preface xiii–xiv. See also ch. 7, nn. 4, 62.

5 See Maxwell, 1861–1862, 164–165 for a clear example of mechanical

explanation. 6 See Maxwell, 1861–1862, 15, where the expression for the electric current due to displacement is found to be independent of any theory about the internal mechanism of dielectrics. 7 For a study of Faraday’s methodology, see Steinle, [2005] 2016, pp. 295–300. Steinle offers a detailed comparison of Faraday’s practice with that of Ampère. 8 Maxwell, 1865, 465, 488, §§ 19, 75. 9 Maxwell, 1865, 492–493, § 82. Already in late 1857 Maxwell had entertained the possibility of applying the concept of lines of force to gravitation. While continental physicists had taken gravitational force to be the “model” for elec­ tromagnetic force, Maxwell considered the reverse, namely, the possibility that electromagnetic force may be the “model” for gravitational force. Maxwell’s approach evolved probably from Faraday’s position as it was expressed in their correspondence. For Faraday it was a mystery how like bodies attract one another by gravitation, while for Maxwell the mystery could be resolved if one were to apply lines of force in a gravitational context. See Maxwell’s discussion of lines of force in his letter to Faraday, 9 November 1857 (Harman (ed.) 1990, 548–552, especially p. 550). See also Maxwell’s remarks on Faraday’s views in his letter to Thomson, 14 November 1857 (Harman (ed.) 1990, 548–556, espe­ cially p. 556). For Faraday’s letter to Maxwell, 13 November 1857, see n. 31, below. In Station 2, about 4 years after this exchange of letters, Maxwell returned to gravitational phenomena in the context of his theory of molecular vortices applied to magnetic phenomena. Considering stress in the medium, Maxwell spoke of calculating the lines of force in the neighborhood of two gravitating bodies. However, here the mechanical effect is that of attraction instead of repulsion as is the case for two magnetic poles “of the same name.” Maxwell noted this difference and the corresponding different arrangement of the lines of force, and did not pursue the idea further (see Maxwell, 1861–1862, 164–165). Evidently, the comparison between the two domains had continued to be on his mind. 10 Maxwell, 1865, 469, § 27: “In the case of electric currents, the force in action is not ordinary mechanical force, at least we are not as yet able to measure it as common force. . ..” This claim stands in contrast to the usual mechanical approach (e.g., the definition of momentum), expressed in the preceding pas­ sage: see ch. 6, n. 26. 11 For Maxwell’s rejection of Neumann’s theory on these grounds see, Maxwell, 1873d, 2: 436, § 863: “In order to understand the theory of Neumann, we must form a very different representation of the process of the transmission of poten­ tial from that to which we have been accustomed in considering the

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12 13

14 15 16 17 18

19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

propagation of light. Whether it can ever be accepted as the ‘construirbar Vor­ stellung’ of the process of transmission, which appeared necessary to Gauss, I cannot say, but I have not myself been able to construct a consistent mental representation of Neumann’s theory.” Maxwell, 1873d, 2: 182–183, § 552. Maxwell, 1873d, 2: 195, § 568. For Maxwell’s technical definition of energy, see ch. 7, § 7.3.1. It is characteristic of Maxwell’s methodology to provide recipro­ cal expressions, symbolic and descriptive; in this case, formal and verbal defin­ itions of energy. For Rankine’s conceptions of “actual” and “potential” energy, see ch. 3, n. 55. See Moyer, 1977, 255 and 257. Maxwell, 1873d, 2: 383, § 781. Maxwell, 1873d, 2: 384, § 782. The discovery of the Hall effect is a case in point: see ch. 7, n. 1. Maxwell, 1873d, 2: 414–415, § 830. Indeed, this is certainly the case for Hertz and Heaviside, two of Maxwell’s most significant successors. Heaviside’s remark is particularly apt (1894, 394): “If I should claim (which I do) to have discovered the true method of establish­ ment of current in a wire. . . I might be told that it was all ‘in Maxwell.’ So it is; but entirely latent. . .. Maxwell’s theory and methods have stood the test of time, and shown themselves to be eminently rational and developable.” See also Heaviside (1893, vii): “Speaking for myself, it was only by changing [Maxwell’s theory’s] form of presentation that I was able to see it clearly.” For Hertz’s pos­ ition, see ch. 1, n. 54. On the Maxwellians see, e.g., Buchwald, 1985. See Maxwell, 1861–1862, 162. See also, ch. 5, nn. 5 and 6. For reversing the argument in Station 3 with respect to Station 2, see Maxwell, 1865, 464, § 17. Maxwell (1865, 487, § 73) referred to the passage in his previ­ ous paper (op. cit., pp. 451–452). See also ch. 6, n. 40. Hunt, 2015, 305. The analysis of dimensionality is central in Maxwell and Jenkin, 1864; 1865. For discussion, see ch. 7, § 7.3.3. Cf. Siegel, 1991, 27. For details, see ch. 7, § 7.3.1. Maxwell, 1870, 419. Maxwell, 1870, 419–420. For Thomson’s concept of analogy, see ch. 3, § 3.2. For discussion of the mathematical tools in Station 4, see ch. 7, § 7.3.2. Maxwell, 1858, 51. For Thomson’s view, see ch. 3, n. 13. Maxwell, 1861–1862, 165. See ch. 7, § 7.3.5. Faraday in a letter to Maxwell, dated 13 November 1857. See James (ed.) 2008, 304–306; Campbell and Garnett, 1882, 145. See, Campbell and Garnett, 1882, 290. Maxwell, 1871b, 225, where the example of translation from one natural lan­ guage to another is from French to Italian. Maxwell, 1873a, 220. Maxwell, 1871a, 13. See, Maxwell, 1873e, 399. See also, ch. 6, n. 66, for an illustration of Maxwell’s appeal to a verbal description of a mechanism at the micro-level. See ch. 7, § 7.3.5 and ch. 8, § 8.3.3. Maxwell, 1876, 293. See also ch. 7, n. 134. Maxwell, 1873d, 2: 194, § 567. See also ch. 7, n. 58. Maxwell, 1873d, 2: 436, § 863. For the text, see, n. 11, above. Maxwell, 1873d, 2: 438, § 866. See also ch. 7, n. 129. Maxwell, 1873d, 2: 184–185, § 554. See also ch. 7, n. 27.

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Reflections on Maxwell’s methodologies 243 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73

74 75 76 77 78 79 80 81 82 83 84 85 86 87

Maxwell, 1873d, 2: 194, § 567. See also ch. 7, n. 58. See n. 31, above. Maxwell, 1873d, 2: 437, § 866. Maxwell, 1873d, 2: 437, § 866. See also ch. 7, n. 127. Maxwell, 1873d, 2: 437–438, § 866. Maxwell, 1873d, 2: 438, § 866. See ch. 7, n. 129. Maxwell, 1873d, 2: 384, § 782. Maxwell, 1873d, 2: 384, § 782. Maxwell, 1873d, 2: 384, § 782. See ch. 7, n. 42. Maxwell, 1873d, 2: 384, § 783. Faraday in a letter to Maxwell, dated 13 November 1857. See n. 31, above. Maxwell, 1873c, 399. Note that for consistency we opt for the expression “mental imagery” while Maxwell invoked both “mental image” and “mental representation” interchangeably; see ch. 7, n. 27. Maxwell, 1856, 404. See ch. 1, n. 30. Maxwell, 1856, 316. See ch. 1, n. 31. For the distinction between “strong” and “weak” analogy, see ch. 1, § 1.5. For the appeal to weak analogy, see ch. 5, nn. 27, 37, 69 and 74. Maxwell, 1858, 30; see also ch. 4, n. 29. Maxwell, 1861–1862, 162; see also ch. 5, n. 5. Maxwell, 1861–1862, 162. See also ch. 5, n. 6. Maxwell, 1865, 471, § 34. See also ch. 6, n. 22. See ch. 3, § 3.3. Maxwell, 1873d, 2: 181, § 550. See also ch. 7, n. 109. Maxwell, 1873d, 1: 74, § 72. See also Maxwell, 1873d, 2: 201–202, § 574; and ch. 7, n. 176. See ch. 7, n. 4. Maxwell, 1873d, 1: 60, § 60, italics in the original. See also ch. 7, n. 136. Maxwell, 1873d, 1: 59, § 59. See also ch. 7, nn. 96–99. Maxwell, 1873d, 2: 196, § 570, italics added. See also ch. 7, n. 102. Maxwell, 1858, 27–28. Maxwell, 1858, 30; see also ch. 4, n. 33. For the imaginary properties of the incompressible fluid, see Maxwell, 1858, 30–31; ch. 4, n. 44. For example, Maxwell, 1861–1862, 283. See ch. 5. Maxwell, 1861–1862, 287. Maxwell, 1861–1862, 285–286. In Station 4 Maxwell explicitly indicated that the “molecular vortices” was Rankine’s hypothesis (Maxwell, 1873d, 2: 416, § 831); for a full account of this hypothesis, see Maxwell, 1873d, 2: 408–409, § 822. Maxwell, 1861–1862, 291. Maxwell, 1861–1862, 12. Maxwell, 1861–1862, 15. Maxwell, 1861–1862, 282. Maxwell, 1861–1862, 162. See also ch. 5, n. 4. Maxwell, 1861–1862, 162. See also ch. 5, n. 26. Maxwell, 1861–1862, 346. See also ch. 5, n. 52. Maxwell, 1865, 487, § 73. See also ch. 6, n. 37. Maxwell, 1865, 488, § 75. See also ch. 6, n. 78. Maxwell, 1873d, 1: xi–xii. Maxwell, 1873d, 2: 183, § 552. E.g., Herschel, [1830] 1851, 196 and 251. Maxwell, 1873d, 2: 407, § 821. Maxwell, 1875, 357.

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244 Reflections on Maxwell’s methodologies 88 89 90 91 92 93 94 95 96 97 98

99 100

101 102

103 104 105 106 107 108

109 110

Maxwell, 1875, 357.

Maxwell, 1873d, 2: 416–417, § 831.

Frigg and Hartmann, 2009.

Bailer-Jones, 2002.

Maxwell, [1890] 1965, Plate VIII (opposite p. 488), fig. 2.

Maxwell, 1861–1862, 291. See ch. 5, n. 15.

Maxwell, 1861–1862, 347.

See ch. 5, § 5.3.

Rankine, 1855, 383–384, italics in the original. See ch. 3, n. 59.

Maxwell, 1861–1862, 88–89.

Nersessian (1984a, 177 n. 2, 188 n. 29, 191), for example, considered Maxwell’s

“vortex hypothesis” of Station 2 a model. See also Nersessian, 1984b, and 2008. For a few more discussions of this issue, see Hesse, 1966; Chalmers, 1973, 1975, and 1986; and Morrison, 2000. Maxwell, 1858, 30. See also ch. 4, nn. 29 and 30. Cf. Hon and Goldstein, 2012, 242. The use of models for didactic purposes was well established in the early nine­ teenth century. For example, models of geometrical solids were meant to help students visualize complex shapes in crystallography; they were made of ordin­ ary materials (Accum, 1812, 1813). This didactic practice continued well into the twentieth century. For an illuminating account of the development of geo­ metrical models made of plaster for didactic purposes in the late nineteenth century, see Rowe, 2013. Maxwell, 1858, 29. See also ch. 4, n. 16. In his discussion in 1873 of the same point regarding Faraday’s innovation, Maxwell did not invoke the expression “geometrical model,” remarking that “[Faraday] went on from the conception of geometrical lines of force to that of physical lines of force” (Maxwell [1873/1890] 1965, 320). Maxwell was awarded the Adams Prize in 1857 for this essay (see Maxwell, 1859). Maxwell, 1859, 59. Maxwell, 1859, Figs. 7 and 8. Maxwell, 1859, 62. For a photograph of the actual model, see Harman (ed.) 1990, facing p. 578. Maxwell, 1862, 100. Harman (ed.) 1990, 578; see also Jones, 1973. For comparison, consider that Faraday rarely used the term model but, when he did, he clearly intended it to illustrate some physical effect. Faraday constructed his models from ordinary materials for illustrative purposes. The elements which comprise Faraday’s model are not hypothetical; they are tangible objects set in some arrangement to facilitate the comprehension of a theoretical claim. It is important to note that while the goal is theoretical, perhaps even hypothetical, the model itself is concrete in conformity with the common view of the time. With his model Faraday presented a didactic tool for assisting the reader in grasping the phe­ nomenon under discussion. Importantly, for Faraday models are not tools of research at the cutting edge of science; they are merely aids for understanding. See Faraday, 1832a, 155, § 116; reprinted in Faraday, [1839–1855] 1965, 1: 33 as well as, Faraday, 1846, 5; reprinted in Faraday, [1839–1855] 1965, 3: 6, § 2161. There are a few other occurrences of “model”: see espec., Faraday, [1839–1855] 1965, 3: 11, § 2200, which includes a cross-reference to § 2161. Maxwell, 1873d, 2: 416–417, § 831. This usage of “working model” became increasingly common in the nineteenth century. In a text of 1828 which describes a newly invented steam carriage, it is

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Reflections on Maxwell’s methodologies 245

111 112

113

114

115 116 117

clear that the working model performs the required functions as well as being constructed to scale (Newton, 1828, 374–375). For another example, Whewell ([1837] 1858, 2: 173) referred to Watt’s “working model” of a steam engine. In the nineteenth century working models of machines that were small enough to fit on a shelf had to be part of the submission to the US patent office: see US Patent Act of 1836 in Peters (ed.) 1846, 5: 119, 125. In sum, the important point is that the expression “working model” did not refer at that time to something hypothetical. See ch. 7, nn. 122 and 123. Boltzmann’s essay “Model” ([1902] 1911) offers a benchmark for the meaning of the term at the turn of the last century. According to Boltzmann, “Model: a tangible representation, whether the size be equal, or greater, or smaller, of an object which is either in actual existence, or has to be constructed in fact or in thought.” For Boltzmann, then, a model was still a tangible object, not an abstract scheme, akin to theory, suitable for formal manipulation. For Von Neumann ([1955] 1976, 492), some 50 years later, model meant “a mathemat­ ical construct which, with the addition of certain verbal interpretations, describes observed phenomena”—a far cry from Boltzmann’s model of a con­ structed object either in “actual existence” or “in thought.” This new methodology was accepted and extended by Oliver Lodge (1851– 1940) and George FitzGerald (1851–1901). See, e.g., Lodge, 1876; FitzGerald, 1885. Indeed, Maxwell himself applied his new methodology to what Josiah Willard Gibbs (1839–1903) had called a “method” in 1873. The case of Gibbs is of particular interest since (with respect to Gibbs) Maxwell considered a geo­ metrical representation of phenomena a model as well: see Gibbs, 1873. For Einstein’s assessment of Maxwell’s theory, see ch. 7, n. 3. It should be recalled that Maxwell’s theory was recast in ways which Maxwell had not anticipated. In the hands of Hertz and Heaviside, his immediate successors, the theory was stripped of its physical meaning and became entirely formal (see ch. 1, n. 54, and ch. 7, n. 182). Maxwell, 1858, 29. See also ch. 4, n. 5. Maxwell, 1856, 404. See also ch. 1, n. 30. Maxwell, 1858, 52. See also ch. 4, n. 20.

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256 References Thomson, W. 1872. Reprint of papers on electrostatics and magnetism. London: Macmillan. Thomson, W. 1882. Mathematical and physical papers, collected from different scientific periodicals from May 1841 to the present time, 6 vols, vol. 1. Cambridge: University Press. Thomson, W. 1889. On ether, electricity, and ponderable matter. Inaugural address to the Institution of Electrical Engineers. The Electrical Engineer New Series, vol. 3 (January 18, 1889), 45–51. Thomson, W. 1890a. Ether, electricity, and ponderable matter, part of the Presidential Address to the Institution of Electrical Engineers, delivered 10 January 1889. Reprinted in Thomson 1890c, 484–515. Thomson, W. 1890b. Motion of a viscous liquid; equilibrium or motion of an elastic solid; equilibrium or motion of an ideal substance called for brevity ether; mechanical repre­ sentation of magnetic force, published for the first time in Thomson 1890c, 436–465. Thomson, W. 1890c. Mathematical and physical papers, collected from different scien­ tific periodicals from May 1841 to the present time, 6 vols, vol. 3. Cambridge: Uni­ versity Press. Thomson, W. [1847] 1882. On a mechanical representation of electric, magnetic, and galvanic forces. Cambridge and Dublin Mathematical Journal 2, 61–64. Reprinted in Thomson 1882, 76–80. Thomson, W. [1854b] 1872. On the mathematical theory of electricity in equilibrium. Philosophical Magazine Series 4, 8: 42–62. Reprinted in Thomson 1872, 15–37. Thomson, W. [1884] 1987. Notes of lectures on molecular dynamics and the wave theory of light. Delivered at the Johns Hopkins University, Baltimore. stenographically reported by A. S. Hathaway. Baltimore: Johns Hopkins University. Reprinted in Kargon and Achinstein (eds.), 1987, pp. 9–263. See also (Lord) Kelvin [1884] 1904. Thomson, W. and Tait, P. G. 1867. Treatise on natural philosophy, vol. 1. Oxford: Clar­ endon Press. Thomson, W. and Tait, P. G. [1867] 1879. Treatise on natural philosophy. Cambridge: University Press. Turner, J. 1955. Maxwell on the method of physical analogy. The British Journal for the Philosophy of Science 6, 226–238. Vaihinger, H. [1911] 1968. The philosophy of “As if”. C. K. Ogden (tr.). London: Routledge. Verdet, E. 1854. Mémoires sur la physique publiés à l’étranger: Démonstration des lois de Ohm fondée sur les principes ordinaires de l’electricité statique; par M. Kirchhoff. Annales de Chimie et de Physique Series 3, 41, 496–500. Verdet, E. 1856. Mémoires sur la physique publiés à l’étranger: Mémoire sur la propa­ gation de l’électricité dans les plaques métalliques; par M. G. Quincke. Annales de Chimie et de Physique Series 3, 47, 203–206. Von Neumann, J. [1955] 1976. Method in the Physical Sciences. In L. Leary (ed.), The unity of knowledge. New York: Doubleday, pp. 157–164. Reprinted in A. H. Taub (ed.), John von Neumann: Collected Works, 6 vols. Oxford: Pergamon Press, 6: 491–498. Weber, H. (ed.). 1893. Wilhelm Weber’s Werke, Vol. 3. Galvanismus und Elektrodyna­ mik. Berlin: Springer. Weber, H. 1894. Wilhelm Weber’s Werke, vol. 4. Galvanismus und Elektrodynamik. Berlin: Springer. Weber, W. [1846] 1893. Elektrodynamische Maassbestimmungen über ein allgemeines Grundgesetz der elektrischen Wirkung. In H. Weber (ed.) 1893, 25–214.

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References 257 Weber, W. [1864] 1894. Elektrodynamische Maassbestimmungen insbesondere über elektrische Schwingungen. In H. Weber (ed.) 1894, 107–241. Weber, W. and Kohlrausch, R. 1856. Ueber die Elektricitätsmenge, welche bei galva­ nischen Strömen durch den Querschnitt der Kette fliesst. Annalen der Physik und Chemie Series 4, 9, 10–25. Weisbach, J. [1845] 1848. Principles of the mechanics of machinery and engineering. Vol. 1: Theoretical mechanics. First American edition, edited by W. R. Johnson. Philadelphia: Lea and Blanchard. Whewell, W. 1831. [Review of:] J. F. W. Herschel. A preliminary discourse on the study of natural philosophy. London: 1830. Quarterly Review 45 (90), 374–407. Whewell, W. [1837] 1858. History of the inductive sciences: From the earliest to the pre­ sent time, 3 vols. New York: Appleton. Wilson, D. B. 2009. Seeking nature’s logic: Natural philosophy in the Scottish enlighten­ ment. University Park, PA: Pennsylvania State University Press. Wise, M. N. 1979. The mutual embrace of electricity and magnetism. Science 203, 1310–1318. Wise, M. N. 2005. William Thomson and Peter Guthrie Tait, Treatise on natural philosophy, First edition (1867). An essay by M. N. Wise in Grattan-Guinness 2005, pp. 521–533. Woodruff, A. E. 1962. Action at a distance in nineteenth century electrodynamics. Isis 53, 439–459.

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Index

A Achinstein, P. 8; Evidence and method 9 action at a distance 14, 224; see also lines of force, Newton Ampère, A.-M. 4, 14, 68, 87, 89, 102, 181, 186, 188, 189 analogy 51–52, 55, 60, 76, 103 (motion of a screw), 104 (rope), 115–16, 130–32, 176, 205 n. 99, 229 (taut rope, rigid rod and muscle); contrived 15–16, 57, 78, 80–81, 86, 210–11, 236–37; formal 210–11; illustration 139; mathematical 10, 79, 92, 227; mechanical 186, 228; misleading 179–80, 228; physical 15, 227; strong vs. weak 19, 35–36, 49–50, 76, 91–92, 134, 135, 176, 178–80, 210–11, 213, 214, 227, 228, 229; unidirectional/bidirectional 11, 91–92; physical and formal see Thomson axiom 6, 17, 63, 64, 72 n. 62, 156, 192, 201, 213, 234 B Bacon, F. 4, 13 Betti, E. 44, 183–84 Boltzmann, L. 245 n. 112 British Association 173 C Cavendish, H. 42 coincidence 111–12; numerical 118–19 Collingwood, R. G. 13 context distinction, justification/ discovery 1, 240 Coulomb, C.–A. de 42, 51

D Descartes, R. 13; analytical geometry 165–66 dimensionality 171, 217 Duhem, P. 46 n. 24, 126 n. 91 dynamics 160–61, 193; dynamical system 190; abstract see Thomson and Tait E Einstein, A. 156 elasticity 17, 18, 27, 44, 54–55, 59, 64, 89, 97 n. 74, 111–19, 122 n. 9, 128–29, 135, 143–44, 147, 148ff, 160, 187, 217, 226, 228 electricity: charge 190; current 114, 177, 178 (flow), 186, 190, 230; displacement current 16–17, 19, 113, 114–16, 128–29, 164, 185–90; nature of 82–83, 93, 100, 149–50, 179–80, 181, 194, 228; representation 105; rotational motion 102, 108, 212; substance 163; tension 87; translational motion 102, 108, 212 energy 13, 18, 66, 134, 148, 157–58, 205 n. 101, 225, 230; actual (kinetic) and potential 62–63, 128–29, 159, 160–62, 163, 193–94, 215–16, 224, 225 (transmission), 226; conservation 160–61, 169; electrodynamic 160–61; kinetic, definition 163–64, 169, 178, 215; mechanical 138; seat of 138–39, 159 equations: Cartesian form 2; equations and list of phenomena 147; general 188, 191; see also representation ether (aethereal) 59 (jelly, glue, water), 128–29, 228 Euler, L. 196 evidence: experimental 104

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Index experiment: facts, deduction from 149; testing 222 explanation 82–83, 111–12, 211–12, 229, 231; causal 143–44, 211–12; mechanical 144; physical 196; see also Thomson F Farad 173 Faraday, M. 4, 11, 14, 26–27, 91, 173,

188, 221, 227; lines of force 37–38;

process of reasoning 77, 234, 239

field 18, 128–29, 138, 156,159; general

equations of the electromagnetic 146 FitzGerald, G. F. 245, n. 113 Fizeau, H. 117, 118–19 fluid: equations of conduction 86; flow 178; imaginary 10, 84; incompressible 10, 15, 56, 79, 84, 86, 95 n. 35, 96 n. 54, 178, 190, 194, 229; perfect 60; source and sink 84 flywheel 130–32, 136, 139, 213, 214, 221, 228 Forbes, J. D. 241 n. 1 force 18, 138–39, 215; and energy 205 n. 101, 216; and flux 165–66; direction and intensity 15, 80, 210–11; electromotive and mechanical 133; 154 n. 86, 177, 241 n. 10; gravitation 241 n. 9 Fourier, J. 51, 57 Fresnel, A.–J. 225

259

18, 44, 165–66, 189; varying action 68 heat (temperature) 194, 230 Heaviside, O. 2, 206 n. 140, 242 n. 18 Helmholtz, H. v. on Faraday 26–27, 124 n. 46, 138, 139, 196, 197, 239 Herschel, J. F. W. 4, 241 n. 1, Hertz, H. 20, 152 n. 45, 242 n. 18 heuristics 1, 240 Huygens, C. 225, 226 hypothesis 17, 98, 107, 122–23 n. 9; assumption 149; axiom 62–3, 72 n. 62; H-D (hypothetical deductive) 211–12; mechanical 99, 179–80 (as a working model), 229; micro- macro-level 16, 190, 198, 229–30; molecular vortices 17, 100–1, 109–10, 115–16, 181–82, 230, 233, 235; optical 142; physical 75, 227, 229, 231; molecular see Maxwell I idealization 15, 48, 57ff, 70, 96 nn. 52–53 idle wheel 105 illustration 130–32, 231; mechanical 99, 103, 137, 175, 179–80, 213–14, 228 image: mental 160–61, 175, 183–85, 202 n. 27, 215, 223–24, 226 imagination 61, 79, 210–11, 233 inference: inductive 216 integral: line- and surface- 18, 167, 187 interpretation 231 (true); 217 (mechanistic)

G Gauss, C. F. 7, 14, 44, 196, 207 nn. 192–93, 224, 231, 242 n. 11, geometry, analytical see Descartes geometry of position 28, 45 n. 10, 167, 196–97; see also directionality 193, 197; Geometria situs 196; Geometria magnitudinis 196; Listing, J. B.: Topologie, 45 n. 10, 197; topology 19, 193 Gibbs, J. W. 245 n. 113 Green, G. 60, 225

J Jenkin, F. 173–74 Joule, J. P. 66

H Hall, E. 24 n. 50; effect 201 n. 1, 242 n. 16 Hamilton, W. 18, 22 n. 11, 66, 68, 69, 160–62, 163–66, 183–84, 189, 239; Hamiltonian 19, 160–62, 167–70, 215, 223; quaternions

L Lagrange, J.-L. 18, 66, 69, 136, 160–62, 163–64, 183–84, 239; Lagrangian 18, 160–62, 167–69, 170, 215, 223; Mécanique Analytique 67, 203 n. 59 language: hieroglyphics, symbolic 222; see also verbal vs. symbolic

K Kelvin 48; see Thomson, W.

Kirchhoff, G. 87

knowledge, scientific 1

Kohlrausch, R. and Weber, W. 20,

113–14, 117, 118–19, 120, 122–23; Weber and Kohlrausch 151 n. 12, 128–29, 172–74

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260 Index Laplace, P.–S. 57 least action see principle; Maupertuis Leibniz, G. W. 196 light 196, 216; electromagnetic disturbance 16, 17, 114, 118–20, 141; undulatory hypothesis (theory) 52, 58, 112, 142, 165, 225–26, 232; velocity 20, 117–19 lines of electric induction 205 n. 105 lines of force, 4, 10, 19, 21, 37–38, 84, 229–30; commitment to see Maxwell; gravitation 241 n. 9; method of investigation 10; tubes 15, 80–81, 84, 178; vs. action at a distance 16, 31–32, 140 Listing, J. B.: Topologie see geometry of position Lodge, O. 245 n. 113 Lorentz, H. 44, 231 M Mach, E. 2; idealization 96 n. 53 magnetism: magnetic matter, imaginary 86; magnets 230; induced magnetization 193; induction 196, 230; mechanical explanation of 182; mathematical theory 12; nature of 181, 191 Maupertuis, P. L. 68 Maxwell, J. C.: abstract 9; commitment 78; dynamical theory of gases 6, 23 n. 14; goal 199; molecular hypothesis 23 n. 14; odyssey 5, 6, 20; philosophy 21, 29, 111, 220; preference for lines of force 21; stations 5; Theory of heat 5 mechanism 183–84, 144; macro-level 145 medium 10, 16–20, 27, 29, 32, 35, 51, 59, 98, 105, 112–13, 118, 129, 130–33, 186, 217, 230 methodology: abstract dynamics 17–18; abstractive and hypothetical 62–63, 65; engine of scientific change 14; mathematical reasoning 220; mechanical reasoning 134; method 1; method of investigation 75; method/ methodology distinction 2–4, 29, 174; 140; physical reasoning (the true method) 160–61; process of reasoning 77, 234, 240; reversing the argument 16, 128–29; science three steps 157–58; scientific 1, 219; term 4; see also analogy, Hamilton,

hypothesis, model, translation/ retranslation model 80–81, 103, 122–23 n. 25, 233; didactic 242 n. 100; geometrical 80, 228, 234–37 (definition); working 17, 175, 179–80, 182, 183–84, 205 n. 123, 235, 242 n. 110; mechanical 234; see also Thomson momentum: bearer 139; reduced 106, 133–34, 136 N Newton, I.: action at a distance (Newtonian) 12; laws of motion 66; Newtonian mechanics 5, 6, 169; Philosophiae naturalis principia mathematica 6 Neumann, C. 44, 89, 159, 167, 206 n. 128, 208 n. 204, 224, 231; potential theory 183–84, 224, 241 n. 11 Neumann, J. 44 O Ohm, G. 10, 87, 96 n. 66, 173 optics 128–29 orrery 136, 152 n. 35 Ørsted, H. C. 4, 42, 188 P particle 156, 230 pedagogy: didactic purpose 75, 228–29; student 19, 219; textbook 228 physics: imaginary 16; quantity and intensity 11; reciprocity with mathematics 220 piston 179–80 Poincaré, H. 46 n. 24 Poinsot, L. 7 point of view: mechanical 100, 103, 228 Poisson, S. D. 42 principle, extremum 68; least action 68 progress 219 Q quaternions 2, 167; see Hamilton Quincke, G. 87, 94 R rack and pinion 105–8, 212, 214, 227 Rankine, W. J. M. 17, 122–23 n. 9, 128–29, 181, 226, 234, 239, 242 n. 13; see also energy (potential and

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Index kinetic) 226; hypothesis (molecular vortices) 17, 182

reality 156

representation: Cartesian 189;

mechanical 53, 101; see also

Thomson

researcher, advanced 19

Riemann, B. 44, 183–84,

197, 231

S Saturn, rings 234, 235; see also Maxwell

scalar 165–166

Siemen’s governor 105

skepticism, Scottish 241 n. 1

Stokes, G. 99, 228, 239

system: connected 219; translatable 222

T Tait, P. 2, 136, 155, 239; see also

Thomson and Tait

theory: constructive vs. principle 72 n. 63,

200; continental 183–84; dielectrics 86,

179–80; dynamical 128–29, 159, 213,

228; dynamical theory of gases 6, see

also Maxwell; potential 183–84, see

also Neumann, C.; results 198

Thomson, J. J. 155

Thomson, W. 4, 10, 79, 101, 103,

122–23 n. 9, 138, 181, 222, 239;

analogy, physical 77, 227; analogy,

formal 91; explanation, mechanical

182; model, mechanical 55

Thomson, W. and P. Tait 17, 18,

160–61, 175, 215, 218, 241

261

n. 1; abstract dynamics 215;

representation, mechanical 101;

Elements 5–6, 7; Treatise on natural

philosophy 13

topology see geometry of position translation 19, 213, 221–22;

retranslation 160–61, 169,

185, 223, 226

transmission: potential, action 224–25

tube: of induction 178; tubular surface

178; unit 84

V

Vandermonde, A.–T. 196

vector 164–66, 187, 195 (vectorial

expressions)

verbal vs. symbolic 18, 213, 220, 223

Verdet, E. 94

Volt 173

Von Neumann, J. 245 n. 112

W Weber, W. 4, 14, 44, 68, 89, 102,

118–19, 128–29, 141, 159,

167, 231

Weber, W. and Kohlrausch, R. see Kohlrausch and Weber

wheel, toothed 106

Whewell, W. 4, 241 n. 1

Y Young, T. 225

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