Exploring the mathematical and interpretative strategies of Maxwell's Treatise on Electricity and Magnetism

Exploring the mathematical and interpretative strategies of Maxwell’s Treatise on Electricity and Magnetism Ronald Anderson James Clerk Maxwell’s Treatise on Electricity and Magnetism forms one of the major scientific texts of the 19th century, describing the phenomena of electricity and magnetism and the interaction between them. The sources Maxwell acknowledged as the inspiration for his own approach were the Englishman Michael Faraday and his fellow Scotsman William Thomson (later Lord Kelvin). In the Treatise Maxwell presents an approach he maintains was equivalent mathematically to the well established Continental electromagnetism but focused on an action via a medium approach to electromagnetism and located within a British experimental tradition. Exploring these features reveals the Treatise to be in accord with other deep themes in Maxwell’s writings, which ground him intellectually and personally in the world of 19th century British Natural Philosophy.

First published in 1873, Maxwell’s Treatise1 presented in a comprehensive way laws describing the phenomena of electricity and magnetism and the interaction between them. Maxwell was then 42 and the first professor of the newly established Cavendish Laboratory in Cambridge University. At that time electromagnetic theory was dominated by the extensive and well established action at a distance approach developed by physicists such as Coulomb, Ampère and Poisson in France and by Weber, Riemann and C. Neumann in Germany. In this approach, the electromagnetic interaction was taken to act directly across a distance between charged bodies and currentcarrying wires. Newton’s law of gravity, the foundational law of nature of 18th and 19th century science, had provided the model for these interactions and had seemed to imply such an action at a distance interaction, despite Newton’s own reservations about this idea. Maxwell’s Treatise contained a good deal of these theories and Maxwell praised their mathematical sophistication referring to those who developed them as ‘eminent mathematicians’ and to the German School in particular as the ‘greatest authorities in mathematical electricity’ [Preface]. At the same time, however, the Treatise contained his own unique approach, one based on a quite different physical theory that was centered around the idea that electromagnetic effects were to be understood as being mediated by electric and magnetic fields which produced stresses in a medium. Peter Tait, Maxwell’s friend, in a review of the Treatise noted the main objective of the

Ronald Anderson Is at the Department of Philosophy, Boston College, Chestnut Hill, MA 02467, USA. e-mail: [email protected]

Figure 1 James Clerk Maxwell. Courtesy of The Cavendish Laboratory.

work, everywhere evident in the work, was to ‘upset completely the notion of action at a distance’2. Moreover, Maxwell ended the Treatise with the remark that his ‘constant aim in the Treatise’ had been to construct a mental representation of the details of the action of electromagnetism in a medium [866]. Closely associated with the focus on the fields and medium between bodies, the Treatise contained two other unique features of Maxwell’s approach to electromagnetism. One was his famous ‘displacement current’,

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he noted after the publication of the Treatise that ‘I a form of current associated with the changing state of the sometimes made use of methods which I do not think electric medium whose introduction in 1862 had led to a the best in themselves, but without which the student series of equations from which flowed the prediction that cannot follow the investigations of the founders of the light was an electromagnetic wave phenomena. The other Mathematical Theory of Electricity’7. At the same time, was a somewhat complex notion of electric charge in which charge, instead of being a property associated with many of the laws of electromagnetism were the same as particles as it was understood in the Continental perhis own and rather significantly Maxwell was dependent, spective (and in our modern sense), was understood as a as we will see later, on features of Continental approach polarization phenomena manifest at the boundaries of for the path he chose in the Treatise to develop his own. fields and metals or in general as a discontinuity in fields. At the center of this relationship of approaches as The sources Maxwell acknowledged as the inspiration for presented in the Treatise is a claim by Maxwell that ‘the his own approach were the Englishman Michael Faraday theory of direct action at a distance is mathematically and his fellow Scotsman William identical with that of action by significantly Maxwell was Thomson (later Lord Kelvin). means of a medium’ [62]. The Their influence is evident throughtwo approaches are ‘mathematidependent … on features out the Treatise. Indeed, Maxwell, cally equivalent’ [59]. The claim of Continental approach in an autobiographic remark in of mathematical equivalence for the path he chose in the Treatise, noted that one of the (together with his identity as the Treatise reasons his work differs considerbeing a follower of Faraday) also ably from others, in particular, appears in an encyclopedia article those ‘published in German’, was due to the influence on electromagnetism published the same year as the first of Faraday on his thinking. He had read, he recounted, edition of the Treatise: through Faraday’s Experimental Researches in Electricity before reading any mathematical treatments of electroIn the present article, we shall follow the path pointed out by magnetism. And in a review of W. Thomson’s Reprint of Faraday, which leads to results mathematically identical Papers on Electrostatics and Magnetism, published in the with those of Ampère, but never loses sight of the phenomyear prior to the Treatise, Maxwell noted that one of ena which take place in the space between the bodies which Thomson’s papers in the collection was the ‘gem of that are observed to act on each other.8 course of speculation’ that led him to develop ‘the mathematical significance of Faraday’s idea of the physical There are also earlier references to the theme of equivaaction of the lines of force’3. lence between the two approaches in Maxwell’s writings. In his first paper on electromagnetism, Maxwell had noted One of the intriguing threads that goes through the the identical mathematical form between some of the laws Treatise consists of an intricate relationship between for the action at a distance force laws and those for the uniMaxwell’s approach and that of the Continental action at form distribution of heat throughout a body9. In the latter a distance mode of viewing interactions. Locating himself with respect to this approach was not a new task for case heat was supposed to be transferred by action through Maxwell. In his first paper on electromagnetism, ‘On a medium suggesting to Maxwell the possibility of carrying Faraday’s Lines of Force’, published nearly 20 years prior over the mathematical laws from the well established action to the publication of the Treatise he both praised the at a distance formulation of electromagnetism to those inContinental theories yet held them in reserve such to volving local action in a medium. Then in a talk to the British provide a space for his own work. As to the praise, he Association in 1870, Maxwell referred to the way in which noted Weber’s work was a ‘profoundly physical theory of the Continental approach at that time and the action via a electro-dynamics, which is so elegant, so mathematical, medium approach had a ‘large field of truth common to and so entirely different from anything in this paper, that both’ in terms of the laws of both leading to the same nuI must state its axioms, at the risk of repeating what out to merical results to all the phenomena of electricity. To be well known’4. It formed a ‘real physical theory,’ one Maxwell the reason two theories ‘apparently so fundamentally opposed’ could be in such a relationship was a matter put forward ‘by a philosopher whose experimental reof importance but not yet fully understood scientifically10. searches form an ample foundation for his mathematical investigations’5. The reservations Maxwell mentioned in Reading the Treatise, alert to ways Maxwell positioned himself in the field of electromagnetic studies, reveals a set this first paper were those that stem from considering of strategies that used the conceptual space in this remarkforces to depend on the velocity of bodies as Weber’s able type of intersection of approaches to attach and secure approach does, and several years later Maxwell added his approach to the more established Continental one, yet another value to presenting his own approach: the general present an approach with a radically different physical lack of understanding of the action of electricity at the model. The essential moves by Maxwell entailed developtime meant there was a place for two approaches6. ing Faraday’s experimental work mathematically and using It was partly for pedagogical reasons and comprehena set of practices to do with interpreting the mathematics of siveness that Maxwell had included details of the action electromagnetism. The latter topic brings one to the heart of at a distance approach in the Treatise. One of the purposes Maxwell’s mathematical physics – the practices of ascribthe Treatise picked up during its writing was to provide a ing physical significance to mathematical structures. The textbook suitable for Maxwell’s Cambridge context and

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overall result meant Maxwell could situate himself as one having the mathematical sophistication of the Continental approach but grounded in a British experimental tradition. Maxwell’s mathematical mantel and interpreting mathematics as the language of Nature While praising the Continental approach as highly mathematical Maxwell also presented himself in the Treatise as the bearer of a mathematical mantel; ‘I have confined myself almost entirely to the mathematical treatment of the subject’ [Preface]. There are places where one can see him subtlety drawing attention to his own mathematical competency, a competency such to rival those who developed the Continental approaches. Examples include his extensive presentation of integral theorems in an introductory chapter in the Treatise and his hints of having good control of detailed mathematical studies such as in the following passage: There is also a considerable mass of mathematical memoirs which are of great importance in electrical science, but they lie concealed in the bulky Transactions of learned societies; they do not form a connected system; they are of very unequal merit, and they are for the most part beyond the comprehension of any but professed mathematicians. [Preface]

His use of mathematics though (unlike that of some of the existing treatments of electricity and magnetism) was to be constrained to that relevant to physical inquires; ‘I shall avoid, as much as I can, those questions which, though they have elicited the skill of mathematicians, have not enlarged our knowledge of science.’ [Preface.] In several places in the Treatise Maxwell invoked the metaphor of mathematics as being a language with the symbols of mathematics therefore when used in physical reasoning needing to be interpreted to acquire physical meaning. For example, he noted the task of expressing the theory of Faraday in a ‘mathematical language’ [83a], and that of ‘translating’ Faraday’s ideas into a mathematical form [Preface]. The mathematical formalism of Quaternions Maxwell used to express his theory was spoken of as a ‘language’ [590]11. When describing certain physicists he observed that the mathematics used in physical reasoning only comes alive for them when clothed in bodily language expressing bodily feelings. Indeed, one central feature of physical reasoning for Maxwell, and one evident throughout his writings, is the task of embodying mathematical expressions in models of visualizable physical mechanisms or processes. And in his review of Tait and Thomson’s Elements of Natural Philosophy, Maxwell praised the authors for their ‘clothing’ the symbolic language of mathematicians with words of ‘our mother tongue’ by their use of precise definitions. Even mathematicians could gain enlightenment, and get equations ‘out of their minds’ by expressing the symbols of their discipline into ordinary words12. The image of language also occurs in a more broader sense where Maxwell in places refers to ‘translating’ between different ways of looking at the phenomena. In the Preface, for example, he refered to the ‘mathematicians’ ways of looking at things, meaning those with the Continental action at a distance approach, and Faraday’s ways

as two different languages. Also when a mathematical formula is the same in different subjects, such as the example of the Italian physicist Mossotti’s use of the mathematics of Poisson’s account of magnetic induction to develop a theory of electric induction, this is a matter of translating between the language of one subject to another [62]. The phrasing used when describing this same example outside the context of the Treatise was ‘translating it from the magnetic language into the electric’ (and at the same time ‘from French to Italian’)13. References to mathematics as a language and the task of interpreting the symbols of language abound in midcentury Victorian texts on science and mathematics. One finds, for example, W. Thomson referring to expressing results in the ‘language of mathematics’14. Faraday, too, with the negative comments on his Royal Institution talk on the correlation of forms that mentioned his absence of mathematics possibly in mind15, hinted at a similar association in a letter to Maxwell: There is one thing I would be glad to ask you. When a mathematician engaged in investigating physical actions and result has arrived at his own conclusions, may they not be expressed in common language as fully, clearly, and as definitely as in mathematical formulae? If so, would it not be a great boon to such as we to express them so – translating them out of their hieroglyphics that we also might work on them by experiment16.

In addition there was a sustained discussion of the meaning of abstract algebra that had been developed in the 1830s, one that had continued the discussion in the 18th century of the meaning of negative and imaginary numbers17. One can find, for example, encyclopedia articles by the mathematician De Morgan on ‘symbol’, ‘interpretation’, and ‘imaginary numbers’18. Recent scholarship on the emergence of the study of language in 19th century England has pointed out the power of the language metaphor in scientific reasoning such as in Darwin’s theory of evolution19. Maxwell’s own references to the meaning and interpretation of the mathematics of electromagnetism most probably have resonances to these discussions of the status of mathematical symbols and the nature of language. Representing Faraday’s approach as mathematical One feature of the work of Michael Faraday was its experimental nature and almost complete absence of mathematics. Maxwell was quite explicit in acknowledging Faraday’s absence of mathematics compared to that of the Continental electromagnetic figures: ‘Open a Poisson and Ampère, who went before him, or a Weber or Neumann, who came after him, and you find their pages full of symbols, not one of which Faraday would have understood’20. Maxwell even made a virtue out of Faraday’s absence of explicit mathematics. It meant Faraday was not tempted to be sidetracked into research in pure mathematics that his discovers would have suggested to him if he had placed them into mathematical form and he was left to coordinate his ‘ideas with facts’ and to express them in an natural untechnical language [529]. Endeavour Vol. 25(4) 2001

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However, one of Maxwell’s central themes was that Faraday’s ideas were both implicitly mathematical and could be translated into mathematical form. In the Treatise Maxwell noted that Faraday’s ‘method of conceiving the phenomena was also a mathematical one’ even if not in the ‘conventional form of mathematical symbols’. And that his task was one to make the ideas in Faraday’s ‘natural untechnical language’ the basis of a ‘mathematical method’ [528]. Maxwell’s presentation of the mathematical dimension of Faraday’s work takes up a theme in Thomson’s writings21. By stressing both the experimental origins and nature of Faraday’s ideas as well as their implicit mathematical nature we see Maxwell positioning himself as one to have created or unearthed a mathematical tradition of electromagnetism on a basis that was independent of the Continental tradition. When Maxwell first raised the issue of presented Faraday’s way of conceiving the phenomena in a mathematical way in the Treatise it was in the context of a distinction between what could be labeled as global and local ways of mathematical representation. In Volume 1 of the Treatise in a chapter on integral theorems and potentials Maxwell elaborated the distinction, calling the method associated with action exerted by contiguous parts of the medium and to do with differential equations the ‘inverse method’ [95a,95b]. It is a local method associated with Poisson’s partial differential equation where the derivatives of V at a point are related to the value of  at that point: d 2V d 2V d 2V + + + 4πρ = 0 dx 2 dy 2 dz 2

(1)

The appropriate mathematical expression for the action at a distance theory is an integral where the potential is calculated by a process of integration. It is the ‘direct method’. V=

+∞ +∞ +∞

ρ dx ’dy’dz’ r −∞

∫ ∫ ∫

−∞ −∞

(2)

To Maxwell that the results of integration satisfy the differential equation in a unique way when certain conditions are satisfied meant not only that the mathematical equivalence of the two expressions was established but that our minds were prepared ‘to pass from the theory of direct action at a distance to that of action between contiguous parts’ [95a]. Here we have an example of the first of the ways in which Maxwell used the mathematical structure of electromagnetism to argue for a form of mathematical equivalence between the action at a distance approach and his own. Moreover for Maxwell this meant the Continental mathematical methods to the approach of Faraday was possible: The whole theory, for instance of the potential, considered as a quantity which satisfies a certain partial differential equation, belongs essentially to the method which I have called that of Faraday. According to the other method, the potential, if it is to be considered at all, must be regarded as

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the result of a summation of the electrified particles divided each by its distance from a given point. Hence many of the mathematical discoveries of Laplace, Poisson, Green and Gauss find their proper place in this Treatise, and their appropriate expressions in terms of the conceptions mainly derived from Faraday. [Preface]

Or as he expressed it in the Preface, he had found that the most ‘fertile methods of research discovered by mathematicians could be expressed much better in terms of ideas derived from Faraday than in their original form.’ Maxwell also noted in the Preface that Faraday’s approach, focused on the actions in the medium and on lines of force, was such to resemble methods where ‘we begin with the whole and arrive at the parts by analysis.’ On the other hand ‘ordinary mathematical methods’ began at the parts and built up the whole by synthesis. These were associated with mathematicians seeing centers of force acting at a distance. Later in the Treatise Maxwell strengthened this association of Faraday’s methods with a mathematical method by the following rather complex argument. In a section where he was considering Faraday’s methods he remarked that there two ways or viewing the relationships between wholes and parts. In the ‘most natural’ way mathematicians consider the universe as made up of parts and begin with a single particle and consider its relationship to the rest. For Maxwell though this involves a process of abstraction, suggesting another way of looking at matters: To conceive of a particle, however, requires a process of abstraction, since all our perceptions are related to extended bodies, so that the idea of the all that is in our consciousness at a given instant is perhaps as primitive an idea as that of any individual thing. Hence there may be a mathematical method in which we proceed from the whole to the parts instead of from the parts to the whole. [529; emphasis in original.]

The first way was like conceiving of the potential of a material system as a function found by integration along lines and over surfaces throughout finite spaces, while the second was the method of differential equations and one where masses ‘have no other meaning than the volumeintegrals of (1/4π)2ψ, where ψ is the potential’. The integration for Maxwell are over all space. Maxwell cautiously notes that Faraday’s approach ‘seems’ to be intimately related to the second of these ways as he conceives of all of space a field of force and never considers bodies as existing with nothing between them. Maxwell then has proposed an idea of a ‘mathematical approach’ other than that associated with the action at a distance theories and associated it with Faraday thus there is a subtle extension of his previous association of Faraday’s approach (and his own) with local differential equations to now include a global way of viewing the phenomena, and an integral over all space. Later it will be evident that Maxwell has tailored this manner of taking Faraday to be mathematical to the mathematical ways he used the action at a distance approach to develop his own.

In his obituary on Faraday, Maxwell mentioned another way in which he saw Faraday’s approach to be mathematical. It could be liken to aspects of the ‘geometry of position’ that he had referred to in the Treatise, a mathematical science established without the ‘aid of a single calculation’22. Faraday’s lines of force were like the pencils of lines in this type of mathematics, and their ‘new symbolism’ provided a mental image of the thing reasoned about. A further feature Maxwell promotes about Faraday is the manner in which he provides the reader with a step by step path through his experiments [529]. This enables the student to identify with Faraday the discoverer and match his own thinking with that of Faraday’s. Such a process was not possible in reading Ampère who simply presents the final formulae giving the results of the discovery. Reading Faraday was a way to cultivate a scientific spirit. Maxwell promoted the same value about Cavendish’s writings – they show us the path by which he arrived at his experiment23. The theme of following the value of following steps in a process of reasoning or discovery occurs elsewhere in the Treatise in particular in interpreting the physical significance of mathematical formulae. After referring to results that are derived in steps that are purely mathematical even though they involved terms with physical meanings Maxwell adds that the ‘physicist, when he has to follow a mathematical calculation, will understand it all the better if each of the steps of the calculation admits of a physical interpretation’ [132]. Years earlier in his Inaugural lecture at King’s College London, in 1860, Maxwell also referred to the value of following the steps of a calculation. But as we are engaged in the study of Natural Philosophy we shall endeavour to put our calculations into such a form that every step may be capable of some physical interpretation, and thus we shall exercise powers far more useful than those of mere calculation – the application of principles and the interpretation of results.24

The virtue is similar to the one Whewell had promoted in a tract on education at Cambridge University in arguing for the value of a geometrical approach to problems as opposed to an abstract algebraic approach25. The ‘mathematical equivalence’ of energy expressions In Maxwell’s electromagnetism there are two main energy expressions, that associated with static charges, the potential energy, and that associated with currents and electromagnetism, a form which Maxwell referred to as the ‘electrokinetic’ energy. To Maxwell the expressions of both forms in the action of at a distance approach are equivalent to those in his own approach. As an illustration of Maxwell’s approach I will present the outline of his derivation of the energy due to electric currents. It is slightly more complicated than that for electrostatic energy but draws on one of Faraday’s notions that has an important experimental role in his discovery of electromagnetic induction in 1831. For Maxwell the kinetic energy of a system of currents takes the form of the following sum:

T=

1 ∑ ( pi) 2

(3)

where p is electromagnetic momentum of a circuit and i the strength of the current around that circuit. Expressing p as: p = ∫ A • dl

(4)

where A represents Maxwell’s ‘electromagnetic momentum’ a term he was to label later in the Treatise as the vector potential. It is the ancestor of the term with the same name in contemporary electromagnetic theory, and the origins in Maxwell’s work is Faraday’s notion of an electrotonic state. I have chosen to represent Maxwell’s symbols by their modern descendants and to use modern notation for his formulae. In terms of a current density, J, Maxwell then expressed the total energy as: T=

1 8π

∫∫∫ ( A • J )dxdydz

(5)

where Maxwell noted that the integration is to be extended to every part of space where there is currents. Now a key move in translating this formula into one in terms of the magnetic field, B, and magnetic intensity, H, occurs by first of all using the field equation expressing J in terms of H, viz., J =  × H to give: T=

1 8π

∫∫∫ [ A • (∇ × H )]dxdydz

(6)

and by integrating by parts and taking H to decrease in the order of r −3 at great distances Maxwell obtained, using B =  × A, T=

1 8π

∫∫∫ ( B • H )dxdydz

(7)

Maxwell noted that the integration was to be extended to all space where B and H are non zero. The first integral in equation (5) Maxwell noted is the natural expression for a theory where currents act on each other at a distance. The integrand is only non-zero in places where there are currents. The second integral in equation (7), however, is that of a theory that explains the action in terms of action in a medium between the currents. Given that this latter method of investigation was the one adopted in the Treatise for Maxwell it was the most significant expression for the kinetic energy. Implicit in Maxwell’s reasoning is an interpretation of the mathematical equality of the two integrals in physical terms, a move he proposed in his ‘Plan of this Treatise’ [59]. A derivation also exists for the electrostatic energy where, by a mathematical manipulation Maxwell moves from a volume integral over places where free electricity (an integral in terms of charges and the ‘scalar’ potential) to one over all space and in terms of the electric force and electric displacement. Maxwell interprets equation (7) as giving an actual energy per volume element of (1/8π)(B • H) which is everywhere the magnetic field exists. A distinctive feature of Maxwell’s approach was the localization of energy in places in space. The equivalence only entails the entire integral so Endeavour Vol. 25(4) 2001

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Maxwell is proceeding in a way that interprets the expression within the integral sign as giving a localizable physical entity to be associated with each volume element. Maxwell does this at several places in the Treatise and it is in accord with his way of talking about integrals. For example, in the General Theorems section of Volume 1 of the Treatise he speaks of classifying the mathematical status of the ‘quantity under the integral sign’ [95b]. Also, when reasoning out the mathematical transformation between a line integral to a surface integral that forms Stoke’s theorem [see 587] Maxwell implicitly treats elements within a line integral as a reality in that he takes two of them adjacent to each other but with circuits pointing in different directions to cancel each other out. This is evident as well in the way he speaks physically of lines of integration when considering Ampère’s theory of magnets such as in the following: If we support our mathematical machinery to be so coarse that our line of integration cannot thread a molecular circuit, and that an immense number of magnetic molecules are contained in our element of volume… but if we suppose our machinery to be of a finer order and capable of investigation… [638]26

As well, to give a physical significance for the integrand is the reverse of the reasoning where an integral is taken as an actual sum of elements of physical significance, such as that which occurs in the construction of integral in equation (5). One can see here the procedure Maxwell follows is matching his comparison of the approaches of the ‘mathematicians’ to that of Faraday. The analytic part he ascribed to the mathematicians was one of building up the whole by summing parts – this matches that of equation (3) above, while that of Faraday, of starting with the whole and moving to the parts, matches that of starting with the integral expression for the energy in equation (7) and obtaining an expression for the energy at a point. Maxwell in his presidential address to a section of the British Association meeting in 1870 also referred to the way mathematicians by their operations on symbols of number and quantity are able to ‘express the same thing in many different ways’. This means ‘the mathematician’ can transform an expression which is puzzling to us into one which explains its ‘meaning in more intelligible language’27. This understanding accords with the way Maxwell used relationships between integral expressions in the Treatise such as those considered above. The transition between the expressions associated with the two approaches is a purely mathematical one, but for Maxwell one was more intelligible, especially in the light of his own approach. Maxwell’s claim that the two approaches are ‘mathematically identical’ was to cause confusion later on for the Maxwellians, the group who came after him and developed his ideas, such as to FitzGerald in 1879 when considering the possibility of electromagnetic waves28. Heaviside wrote to FitzGerald 10 years later to point out how FitzGerald had been misled by these statements of Maxwell. To Heaviside, Maxwell had only meant an equality regards ‘phenomena to be got by assuming displacement ignorable’29.

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It was to become clear later that the transformation between the two forms of electrokinetic energy given by Maxwell is only valid strictly in the case of steady currents and charges. H.M. MacDonald, for example, in an Adams prize essay at the University of Cambridge on Maxwell’s theory in 1902, organized around the significance of the two energy expressions above that Maxwell gives for the electrokinetic energy, remarks on the inadmissibility of the transformation Maxwell makes on going from the expression in equation (5) to that in equation (6)30. In particular, if one allows for the propagation of electric waves the components of the magnetic force and electrokinetic momentum are such that their contribution to the surface integral at large distances is non negligible, undermining an assumption necessary to make the transition. For MacDonald the first expression is the only admissible energy expression that can be derived from Faraday’s theory and the use of the second led to a wrong assumption that MacCullagh’s optical theory and Maxwell’s electromagnetic theory are the same since both have identical Lagrangian functions31. The matter of how potentials may be seen to propagate was also of some complexity for the those who followed Maxwell. Even at the end of the Maxwellian period it remained an issue as correspondence between Larmor, Lodge and FitzGerald indicates. Larmor in 1901 remarked to Lodge that he had just come across an article by Levi-Civita of Padua which ‘aims to do away with the mystery of potentials being propagated – or rather the uncanniness, not mystery’. Lodge forwarded Larmor’s letter to FitzGerald who noted in reply the difference between Larmor’s treatment of potentials as ‘analytic functions’ for calculating the electric from the magnetic forces and the use of the potentials by Levi-Civita32. The transformation of Faraday’s ‘electrotonic state’ Both Maxwell’s treatment of Faraday’s electrotonic state and his remarks on Faraday’s approach illustrate directly one of Maxwell’s aims in the Treatise to demonstrate that within the tradition of action within a local medium there was possible a mathematical treatment that matched the Continental action at a distance approaches. In the Treatise Maxwell’s treatment of induction between two current carrying wires led him to a quantity, M, that determined the electro-magnetic action between two circuits. It was a function of the relative shape and relation of the two circuits and for Maxwell it represented the ‘potential of one circuit over another’ [543]. It was directly related to the ‘electromagnetic momentum’ of equation (3) above and corresponded to the condition of matter Faraday had identified as the ‘electrotonic state’ in his study of electromagnetic induction. One half of Maxwell’s first paper on electromagnetism was devoted to providing a theoretical and mathematical analysis of Faraday’s electrotonic state. This quantity, by virtue of its structure as a type of ‘momentum’ whose time rate of change provided the electric force was pivotal to his development of a dynamic approach to the field based on Lagrangian and Hamiltonian formulations of the dynamics of connected systems. To Maxwell it forms the fundamental quantity of electromagnetism and was moreover a ‘mathematical quantity’.

By adopting a course of experiments, 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 of electromagnetism [540].

Maxwell then remarked that other ‘eminent investigators’, namely, F.E. Neumann, had developed a mathematical theory of induction based on a concept of a potential between two circuits, and moreover that this potential of Newmann corresponded to Faraday’s notion of an electrotonic state. For Maxwell, Neumann had ‘completed for the induction of currents the mathematical treatment which Ampere had applied to their mechanical action’ [542]. Maxwell noted, however, that none had recognized via this mathematical path Faraday’s idea which he had done. Thus just as Maxwell has provided a mathematical theory to capture action by a local medium that he claimed, via equivalence of energy expressions, to be mathematically identical to action at a distance theories by the eminent continental mathematicians (such as Ampère and Weber), he had now, in one more area (electromagnetic induction) provided a mathematical development of one of Faraday’s ideas to obtain a quantity that was identical with that of Neumann’s analysis of induction. It can be seen as part of his hopes in the Treatise to make Faraday’s ‘ideas the basis of a mathematical method’ [528]. The irony here is that Faraday’s electrotonic state gives rise to an energy expression that is the one appropriate to an action at a distance approach rather than his own. Maxwell also showed that by introducing a vector, B (defined by B =  × A) in combination with Stoke’s theorem relating surface and line integrals he could obtain a surface integral of B over a surface enclosed by a circuit such that it could be interpreted to represented another notion of Faraday’s, namely, that of the number of lines of force through the circuit. Maxwell stressed the mathematical manner in which B was introduced, and that this procedure introduced no ‘new fact into the theory’ but was justified by the agreement of the ‘relations of the mathematical quantity with those of the physical quantity indicated by that name’ [593]. It enabled him to claim that mathematical equivalence of two of Faraday’s key ideas: ‘the number of lines of force which at any instant pass through the circuit is mathematically equivalent to Faraday’s earlier concept of the electrotonic state of that circuit…’ [542]. It was also a powerful demonstration of the presence of a mathematical glue holding Faraday’s ideas together. Maxwell’s use of Stoke’s theorem in this manner meant another thread was woven in the mathematical tapestry of the Treatise. Faraday’s lines of force concept figured prominently in the Treatise, indeed Tait remarked in his review that one of the curious features of the Treatise was the ‘amount of labour bestowed upon the exceedingly useful, but dry and uninteresting, pursuit of accuracy in the tracing of the Lines of Force’33. When Maxwell introduced Faraday’s lines of force (induction) in a section on equipotential surfaces he remarked on their association

with the mathematical theory of the potential that figured in the Continental tradition, as well as their productivity compared to that tradition: There is therefore no contradiction between Faraday’s views and the mathematical results of the old theory, but, on the contrary, the idea of lines of forces throws great light on these results, and seems to afford the means of rising by a continuous process from the somewhat rigid conceptions of the old theory of notions which may be capable of greater expansion, so as to provide room for the increase of our knowledge by further researches [122].

These lines of force were related mathematically to the conditions of electric and magnetic forces Maxwell emphasized [541]. In his claim that the results of the old theory were met and even improved by his own approach coupled together with his reference to the mathematical nature of his own treatment of the lines of force approach, Maxwell added a further example to his energy analysis, of the mathematical identical nature of his approach to that of the continental approach. The alternative derivation of the inverse square law to that of Coulomb Maxwell’s account of electrostatics that begins the Treatise has a series of experiments woven into a presentation of the concepts and phenomena of electrostatics. The style of exposition is not unlike that of Newton’s Optics. Several of the experiments involve the situation of a hollow metal vessel and the various phenomena associated with placing charged bodies within the container [28–32]. Maxwell’s reference here is to Faraday’s work on electrostatics in Series 11 of the Experimental Researches. One of the experiments (experiment VII) describes the phenomena of how a charged body loses all of its charge when it touches the inside of a closed vessel. On the basis of this experiment Maxwell provided a mathematical analysis to show that the result entails an inverse square law for electric attraction between charged bodies. If the inverse square law did not hold then the electrified conductor introduced into the container would remain electrified after touching the surface. For Maxwell this result established the ‘perfect accuracy’ of the inverse square law and was a far more accurate verification of this law than the more direct means Coulomb had used. Between editions of the Treatise, Maxwell had become familiar with the work of H. Cavendish and included mention in the later edition of Cavendish’s use of this experiment to prove the inverse square law. Maxwell, himself had reproduced the experiment at the Cavendish Laboratory [74b]. Given the centrality of the law of attraction to the theory of electricity and Maxwell’s remarks that his approach via the experiments of Faraday and Cavendish was better than that of Coulomb we have a further subtle underlining by Maxwell that his exposition builds on and extends a British experimental tradition. It complements his demonstration that the dynamic theory of electromagnetism arises from Faraday’s notion of an electrotonic state. Endeavour Vol. 25(4) 2001

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Maxwell also hints that the priority of making the concept of potential the basis of electrical science belongs to George Green, whose essay on these matters had not been known until 1846, by which time his important theorems had been rediscovered by others such as Gauss [16]. In his notes to the edited papers of Cavendish, Maxwell remarks that Cavendish implicitly had the concept of potential in the modern sense in his analysis of electrostatic phenomena, and moreover, that there was no hint of that idea in Coulomb’s papers34. While I am drawing on a source other than the Treatise, overall Maxwell’s intent in his writings on electromagnetism at the time of the Treatise are clear: the origins of what was a central term in the mathematics of central force situations analysed in the mathematics within the Continental contexts lies within a British context. Representing the stress in the medium As indicated above Maxwell used integrals expressions to transform energy expressions involving quantities that expressed charges and currents to those quantities that existed in the space between charges and currents. When considering the electric force between two electrified systems in the Treatise Maxwell first arrived at an expression that involved a volume integral and a potential that arises from both systems. Using one of his general integral theorems Maxwell transforms this to a surface integral enclosing one and only one of the electrical systems [105]. Maxwell noted of the terms in the integrand that if one has an action at a distance theory then these terms can only be taken as symbolic expressions having no physical significance, but if one is to take the mutual action to be by ‘means of a stress in the medium’ between the two systems then these components can actually represent the stress existing in a medium. Thus in a particular context the mathematics can be interpreted as referring to supporting an action via a medium approach. Then after quoting Faraday’s description of induction in a dielectric Maxwell notes that Faraday’s account is the exact conclusion of what he has been led to via a ‘mathematical investigation’ [109]. Maxwell acknowledged, however, that he had not been able to account for such stresses by mechanical considerations. Still, for Maxwell, when considering the topic of stress in the medium in the context of electrodynamic matters the task of explaining this stress was a separate and independent part of the theory and one that did not affect his electrodynamic theory [645]. In his Encyclopaedia Britannica article on ‘Attraction’ Maxwell admits he was more successful in accounting for electrodynamical stress by a mechanism involving rotation rather than what he was able to do in explaining electrostatic stress35. Nevertheless, the very idea of stress in the medium in the electrostatic context leads to one of the distinctive features of Maxwell’s theory, in particular, that charge is an effect of the polarization of a medium and only becomes apparent when the medium meets a surface other than itself such as metal. Maxwell’s closing remarks in the Treatise critique the action at a distance theories of Gauss, Weber, C. Neumann,

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and others. His main point of criticism is that while the mathematical implication of these theories is that effects are propagated between bodies they provide no ‘consistent representation’ of that propagation. Such a representation to Maxwell entails a medium for this propagation to take place in. The critique is entirely in line with one of the arguments for the medium Maxwell has provided by the energy and force considerations: a mathematical exploration with the concepts and expressions of action at a distance theories leads to expressions which naturally suggest action in a medium. Such local action in a medium provides a natural mathematical expression of Faraday’s notions. There are also other places in the Treatise where Maxwell demonstrates his grounding in a non-Continental world, both physical and mathematical. While Lagrange’s equations of motion form a connected system and are a part of Maxwell’s dynamic approach to the field in Part IV of the Treatise, it was the dynamic form given to them by W. Hamilton and in particular the use given to them by W. Thomson and P.J. Tait in their recently published, Natural Philosophy, that is at the heart of the form Maxwell praises and uses in the Treatise [554]. In addition, the value for ‘physical reasoning’ of the mathematical notation and system of representing directed magnitudes in space by the quaternions created by W. Hamilton and developed in texts by Hamilton and Tait was extolled by Maxwell compared to the methods and ‘geometry of Des Cartes’ [10]. It is the Englishman, George Green who dominates his formal mathematical development of integral theorems. When developing methods of solving the distribution of electric forces due to charged bodies, it was not any of the ‘mathematicians’ such as Poisson who noticed certain mathematical solutions of equations provided a method of ‘electric images’ as a way to develop solutions, rather it was William Thomson [155]. There are exceptions, such as Maxwell’s account of topological notions where he refers to the work of Lebniz, Gauss and Listing, however, it features little in any physical reasoning in the Treatise [18]. Steadily then throughout the Treatise Maxwell’s various strategies have woven interpretative practices to do with giving physical significance to mathematical expressions into a skillful theorizing on the status of his approach. The result is to present his own position as equivalent mathematically to the well established Continental electromagnetism yet grounded in an action via a medium approach to electromagnetism and more broadly, in a British experimental and to a lesser extent mathematical tradition. The results are in accord with other deep themes in Maxwell’s writings, ones that ground him intellectually and personally in the world of 19th century British Natural Philosophy and the early formative influences of his Scottish education36. Acknowledgement Conversations with Jed Buchwald, Bruce Hunt and Andrew Warwick on the subject of this paper are gratefully acknowledged.

Notes and references 1 Maxwell, J.C. A Treatise on Electricity and Magnetism, 2 Vols (3rd edition Dover Reprint) In the following references will be given in the text to paragraphs of this edition 2 Tait, P.J. (1873) Clerk-Maxwell’s Electricity and Magnetism. Nature 7, 478–80 3 Maxwell, J.C. (1890) Review of Reprint of papers on Electrostatics and Magnetism by W. Thomson reproduced in The Scientific Papers of James Clerk Maxwell (Niven, W.D., ed.) (2 vols, Cambridge), Vol. 2, p. 304, hereafter referred to as SP 4 Maxwell, J.C. On Faraday’s Lines of Force. In SP Vol. 1, p. 207 5 Ibid p. 208. Similar praise for Weber’s work may be found in ‘A Dynamic Theory of the Electromagnetic Field’. In SP Vol. 1, p. 527 6 Maxwell, J.C. ‘On Physical Lines of Force’ in SP Vol. 1, p. 208 7 Maxwell, J.C. (1881) An Elementary Treatise on Electricity (Garnett, W., ed.) (Oxford, Clarendon Press) Preface 8 Maxwell, J.C. (1873) ‘Electromagnetism’, English Cyclopaedia Supplementary volume on ‘Arts and Sciences’ columns 854–857 9 Maxwell, J.C. ‘On Faraday’s Lines of Force’ in SP Vol. 1, p. 156 10 Maxwell, J.C. ‘Address to the Mathematical and Physical Society Sections of the British Association,’ in SP Vol. 2, p. 228 11 Also e.g. Maxwell, J.C. On the Mathematical Classification of Physical Quantities. In SP Vol. 2, p. 260 12 Maxwell, J.C. Review of Elements of Natural Philosophy. In SP Vol. 2, p. 328 13 Maxwell, J.C. op. cit. note 11, p. 258 14 Thomson, W. (1872) ‘II. On the Mathematical Theory of Electricity in Equilibrium’ reproduced in Reprint of papers on Electrostatics and Magnetism, MacMillan, London 15 Review of ‘On Conservation of Force A Lecture delivered by Prof. Faraday at the Royal Institution February 27, 1857,’ The Athenaeum No. 1535 March 28, 1857 16 M. Faraday to J. Clerk Maxwell 13 November 1857 in The Selected Correspondence of Michael Faraday L. Pearce Williams (Cambridge, Cambridge University Press 1971) Vol. 2, p. 884 17 See e.g. Richards, J.L. (1980) ‘The Art and the Science of British Algebra: A Study in the Perception of Mathematical Truth,’ Historia Mathematica 7, 343–365 18 De Morgan, A. Penny Cyclopaedia, 27 Vols, London 1833–1843 19 e.g. For S. Alter the metaphor of the evolution of languages represented in a tree structure provided an analogy for Darwin to understand and express the evolution of species see Atler, S.G. (1999) Darwinism and the Linguistic Image: Language Race and natural Theology in the Nineteenth Century, Johns Hopkins University Press, Baltimore 20 Maxwell, J.C. ‘Faraday.’ In SP Vol. 2, p. 355

21 Thomson, W. Op. cit. note 14, p. 29. Thomson notes e.g.: ‘…that Faraday arrives at a knowledge of some of the most important of the general theorems which from their nature seemed destined never to be perceived except as mathematical truths, p. 31 22 Maxwell, J.C. Op. cit. note 20, p. 360 23 Maxwell, J.C. (1879) in ‘The electrical researches of the Honourable Henry Cavendish: written between 1771 and 1781 edited from the original manuscripts in the possession of the Duke of Devonshire’ (Maxwell, J.C., ed.), p. 410, Cambridge University Press, Cambridge 24 Maxwell, J.C. ‘Inaugural Lecture at King’s College London October 1860’ . In The Scientific Letters and Papers of James Clerk Maxwell, (Harman, P.M., ed.) Vol. 1, pp. 662–674, 672 25 Whewell, W. (1845) On a Liberal Education in General: and with Particular Reference to the Leading Studies of the University of Cambridge, John W. Parker, London. To Whewell, in geometrical reasoning ‘we treat the ground ourselves, at every step feeling ourselves firm, and directing our steps to the end aimed at. In the other case, that of analytical calculation, we are carried along in a rail-road carriage, entering it at one station, and coming out of it at another, without having any choice in our progress in the intermediate space’. For Whewell this provided no exercise of our reasoning powers. See p. 41. 26 Later in the Treatise Maxwell speaks of ‘extending our mathematical vision into the interior of the molecules’ when considering Ampère’s theory [835] 27 Maxwell, J.C. ‘Address to the Mathematical and Physical Society Sections of the British Association.’ In SP, Vol. 2, pp. 215-229 28 FitzGerald, G.F. (1902) In The Scientific Writings of the Late George Francis FitzGerald (Larmor, J., ed.), pp. 90–92, Dublin University Press, Dublin 29 For a discussion of this point with reference to Heavside’s letter to FitzGerald see Hunt, B.J. (1991) The Maxwellians, p. 34f, Cornell University Press, Ithaca 30 MacDonald, H.M. (1902) Electric Waves: Being an Adams Prize Essay in the University of Cambridge, Cambridge University Press, Cambridge 31 Ibid. p. 34 32 Letters Larmor to Lodge January 8 1901 and FitzGerald to Lodge January 16 1901 (University College London Lodge Papers Ms.Add.89) 33 Tait, P.J. op. cit. note 2. p. 479 34 Maxwell, J.C. Op. cit. note 23, p. xlix 35 Maxwell, J.C. ‘Attraction’ in SP Vol. 2, p. 488 36 For recent studies of the formative influences on Maxwell’s world see Cat, J. (2001) On Understanding: Maxwell on the Methods of Illustration and. Scientific Metaphor. In Studies in History and Phililosophy of Modern Physics 32, pp. 395–441 and Harman, P. (1998)The Natural Philosophy of James Clerk Maxwell. Cambridge University Press, Cambridge

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