The Language of Physics The Calculus and The Development of Theoretical Physics in Europe 1750 1914 PDF
The Language of Physics The Calculus and The Development of Theoretical Physics in Europe 1750 1914 PDF
The Language of Physics The Calculus and The Development of Theoretical Physics in Europe 1750 1914 PDF
Elizabeth Garber
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Contents
Preface IX
I: Introduction 1
Mathematics and Modem Physics 2
Modem Physics 9
Earlier Historical Approaches to Modem Physics 12
Mathematics as Language 20
Organization of the Text 25
IX: Physics About 1870 and the "Decline" of French Physics 307
The "Decline" of French Physics 312
Some Conclusions 317
Bibliography 363
Index 367
Preface
the operations of nature through concrete examples. The second was mathemat-
ical. Mathematical solutions to problems of mechanics were generating some
highly sophisticated work in the calculus. However, there were no mathematical
investigations into the structure and physical function of nature, and none of the
authors, save Daniel Bernoulli, used the calculus to investigate the actual workings
of nature. Enmeshed in both types of works on mechanics were disputes over its
metaphysical foundations. Some of this work needed a separate classification.
The traditional historiography of physics narrated an unbroken development for
mechanics from the seventeenth through the nineteenth century. This traditional
historiography ignored mathematics as an aspect of the development of physics.
Historians assumed that mathematics was incorporated into physics as one of its
essential elements since the seventeenth century. They also took as axiomatic that
physicists argued theoretically in ways familiar from twentieth-century examples.
Given Heilbron's work and the preliminary results of my own research, mathe-
matics in physics became the problem. Where and how did it enter physics, then
take over the expression and development of theoretical ideas?
Until recently most philosophers and historians of science did not consider the
function, role, or place of mathematics in the development of scientific ideas.
Mathematics was sidelined, downgraded to the status of a tool. Scientists took the
tool off the shelf, used, and then returned it. It had no role in shaping the solutions
to physical or any other scientific problems. However, theoretical physicists speak
of mathematics as their "language." Max Dresden passionately argued about the
interactions between mathematics and physical imagery and the depth to which
they are interdependent. Mathematics shaped how physicists thought about solu-
tions to physical problems, while at the same time physical imagery pushed their
mathematical language in some directions rather than others.
The development of the calculus was a material part of the creation of modern
theoretical physics. This meant discarding the image of mathematics as a tool and
investigating it as language. Seeing mathematics as language, and having avail-
able the recent literature on the development of calculus in the eighteenth century,
changed the history of physics. Mathematics and physics had to be defined with the
terms available during the eighteenth then the nineteenth centuries. Many papers
in mechanics in the eighteenth and early nineteenth century, traditionally taken as
physical, had to be reconsidered. Using criteria derived from what mathematics
meant in those eras, many papers historians have tried to integrate into the history
of physics belong in the history of mathematics. Physical implications could be
inferred from some of the results of these eighteenth century papers. However,
these inferences seemed never to be commented upon, developed, or seen as sig-
nificant by the authors themselves. Reactions to these papers, the issues discussed,
points over which disputes flourished and the ensuing debates confirmed the sense
that topics taken later as important in physics were discussed in terms that were
Preface xi
mathematics to theoretical and experimental physics were never linear. The end
results were a rich mixture of conceptual structures, all based on the ideas of
mechanics wedded, in a variety of ways to calculus.
This work is one of interpretation and necessarily dependent on the work of
colleagues in the history of physics and mathematics. Historians of the calculus
work in a technically demanding field that is often ignored by historians of the
other sciences. It should not be. Mathematics has become the language of choice
for many sciences. This problem alone needs historical attention. The historians
of mathematics I have relied on heavily include Judith Grabiner, Joan Richards and
particularly Ivor Grattan-Guinness. On various occasions Ivor discussed with me
many of the mathematical issues, and the absence of physical content in much of
eighteenth-century mechanical and later papers. His remarks on Liouville's work
made sensible to me nineteenth century French mathematical physics. Historians
of mathematics have cheerfully tolerated my presence and my questions at their
conferences, even though, by and large, they are concerned with the development
of physics only as it intersects their own scholarly interests in the development of
mathematics itself. In comparison to those concerns the focus of this book is on
the implications of the history of the calculus in the eighteenth and early nineteenth
centuries for the history of physics.
Other colleagues have aided and abetted this research. Until his retirement Max
Dresden argued with me regularly about issues in the history of physics in ways
that were always stimulating, although often uncomfortable. I regret that he will
be unable to give me any reactions to this book. Other colleagues include the
members of the Eighteenth Century Consortium at Stony Brook. I am grateful
to Arthur Donovan for discussions with him on the emergence of chemistry as a
modern science at the end of the eighteenth century. Arthur expressed in detail
what made chemistry with and after Lavoisier "modern" and scientific rather than
an untidy and loosely federated set of practices and ideas. Fred Weinstein taught
me how to think theoretically as a historian. Gary Marker and I have discussed
eighteenth-century cultural history, especially that of Russia and the place of the St.
Petersburg Academy in that culture and politics. He also taught me that historians
of science and historians of culture and literacy share the same theoretical problems
and the same sources for easing the solutions to those problems. I am particularly
grateful to David Cassidy who read this manuscript, criticized it and forced me
to improve it. On a less intellectual yet critical level I must thank the Interlibrary
Loan Department at Stony Brook. Donna Sommers and the staff located sources,
and in some cases pleaded my case with other librarians for loans of material that
rarely left their care. Without their help I could not have completed this.
My husband Donald has lived with this research as long as I have. He has
endured all its transformations, helped me clarify my ideas by listening, criticizing
but never asking when I would be done with the book. He never asked why it took
so long to mature and I hope that he finds the wait worthwhile.
Elizabeth Garber
Stony Brook, New York
Serials
List of Abbreviations
Introduction
This book traces and explains the development of modem physics from the mid-
eighteenth century to the first world war. The focus is on how physicists used
mathematics and fused it with experimental results and physical imagery to create
a new field-theoretical physics. The emphasis in this work is on mathematics,
because consideration of this language is usually omitted from historical narratives
of the development of physics. Mathematics is taken for granted as a natural aspect
of physics and assumed to be so from the seventeenth century onwards. At the end
of the twentieth century these two disciplines draw ever closer and mathematics is
the expected language of physical theory.
Before the middle of the nineteenth century mathematics served only as a adjunct
to experiment. Speculations about the physical behavior of nature were expressed
in the vernacular. 1 Experiment and observation extended knowledge of nature
by revealing new phenomena and this expansion was a mark of the discipline's
success. Physics was both a body of knowledge and a discipline with widely
varying practices. For most of these practitioners physics was an avocation not a
profession.
By 1870 physics was redefined socially and intellectually. Socially, it became
a profession whose practitioners were located in the university systems of Europe
and the United States. Intellectually, the core of physics remained experiment.
These experiments were quantitative and performed within laboratories stocked
with technologically sophisticated equipment and instruments usually supplied by
some form of state or external support. Physicists reorganized their interpretations
of experiment around a series of principles-laws of nature-and images confined
by the scope of the mechanical concepts encompassed in those principles. Theory,
1 The term vernacular occurs in several place in the text. It means language that was familiar
to a broad, literate population of women and men, requiring little technical language or
terminology.
Physicists use mathematics in many different ways that gives theoretical physics
a range of characteristics. In some fields of physics, as in the past, physics is
mathematics. But the context within which this occurs has changed from earlier
centuries. Only since about 1850 have physicists consciously joined mathemat-
ics and physical imagery, with the understanding that physics consists of a more
complex search than the transformation of physical phenomena into mathematical
terms accompanied by an argument legitimated using purely mathematical crite-
ria. And when physicists produced results of interest to both mathematicians and
physicists, they tended to publish the mathematical portions in journals directed to
mathematicians and the physical portions in the literature directed to physicists.
Modern theoretical physics and mathematics are closely intertwined, but al-
though mathematics is the language of physics, theoretical physics is not mathe-
matics. The mathematical representation of a physical situation can symbolize a
relationship between physical entities, such as force, work and energy, for a partic-
ular physical instance. The mathematical expression can also delineate a physical
process, such as a rotation, through the mathematical operations contained within
it. Accomplishing the transformation from physical concepts applied to a partic-
ular case into a mathematical interpretation takes imagination, skill in abstraction
so that a physical situation can be given mathematical form, and analytical ability
to choose the mathematical relationship. However, what physicists then need to
do to extract physical information from the particular mathematical expression is
not clear. Just how general a mathematical solution is necessary and the ways
of connecting the solution to principles and processes to extract physically mean-
ingful results are open. The approach depends on the interplay between physical
principles, the use of imagery and the interjections of possible experimental ex-
amples into the theoretician's argument. Our hypothetical theorist may choose to
translate his mathematical solution directly into a physical interpretation, or may
do so through the intermediary of a particular model.
Developing the mathematics of physics can also be research, and this is what
some theoretical physicists do. In doing theoretical physics, when in the middle
Introduction 3
The mathematician's solution of, say, a partial differential equation of the first
order is not that of the physicist. The myriad of arbitrary functions and coefficients
that satisfy the mathematician's definition of a solution do not contain any neces-
sary physical inferences. The mathematician needs to show that a general solution
for that type of partial differential equation exists and, if possible, the number of
arbitrary functions and coefficients that make up that solution. Physicists focus
on particular solutions defined through criteria from outside of mathematics and
imposed on the behavior of mathematical functions. Understanding how to trans-
late physical criteria into limitations on the behavior of mathematical functions,
polynomials etc., and when, in the search for the mathematical solution, to inject
them and how to apply them, are again the marks of the theoretical physicist.
The mathematical exploration of a relationship, a partial differential equation
for example, is not the same as the construction of a physical theory in the math-
ematical form of partial differential equations and their physical solution. The
mathematical solution may be visualized in terms of, for example, the proper be-
havior of a function. Physically the solution may lie in seeing how the physical
conditions impact the kind of function actually sought as a solution. In this case
mathematical rigor may give way to imaginative abstractions of possible and plau-
sible experimental conditions. An understanding of the phenomena can cut into
the intricacies of the mathematical argument and confine it to particular examples.
These cases, usually mathematically uninteresting, can be the most fruitful for the
immediate solution of the physical problem.
Normally the theoretical physicist compromises the generality and purity of
the mathematical solution to fit the needs of physical explanation. Interpretation
rather than mathematical nicety takes priority. Physicists may also run rough shod
over the carefully crafted definitions and conditions of validity of mathematicians.
Exploring the linguistic structure of a particular subfield of mathematics is not a
major concern of physicists, although it may be necessary at times. The language is
usually a means of understanding certain natural processes. Within the discipline
of physics, it is appropriate to subordinate mathematics to the needs of solving a
physical problem.
Physicists assume that mathematical operations mirror or can be connected to
natural processes, and the developing mathematical logic related to physical oper-
ations. Therefore, there is a constant dialectic, a dialogue, between mathematical
operations and physical processes and a developing symbiosis between mathemati-
cal form and physical imagery. A physical interpretation of what the mathematical
operations represent is imposed on the development of the mathematics, while
the characteristics of the mathematics may shape how the physical system might
behave. Sometimes, this imposition is an intuition arising from an understanding
of the behavior of the physical system under scrutiny. An example, without get-
ting into particular physical models, is the ultra-violet catastrophe. Physically the
Introduction 5
energy available in a radiating system is finite and this must be directly reflected
in the mathematical results representing the physical behavior of the radiators.
The understanding of what a physical system would, could, or could not do, is
represented by the different paths into which the mathematical operations lead us.
Thus, some directions of development may well be mathematically interesting but
will not represent the behavior of the physical situation under discussion.
On the other hand, the arithmetization of experimental results, even though pre-
cise, is not the same as the development of a mathematical, theoretical physics.
The vernacular remained the language of physical theory well into the nineteenth
century after the development of quantitative experiments. Precise and clear con-
ceptualization was and is not necessarily expressed mathematically. Conversely,
mathematical theories have not always clarified physical theories and can mask
muddled thinking. Yet both mathematics and quantitative experiment were as-
pects of the development of modem theoretical physics.
Physical interpretation is generally crucial in dictating the direction of the devel-
opment of the mathematical analysis. However, such theoretical explorations of
specific cases often lead to reinterpretation of the physical concepts that were used
to set up the example. One of the more dramatic conceptual transformations is that
of the second law of thermodynamics, from unavailable energy in the nineteenth to
information propagation in the late twentieth century. Physical conceptualization
of the problem and its subsequent reconceptualization are central to the enterprise
of modem theoretical physics. This condition means that its practitioners learn
early how to control and develop logical theories based on presuppositions. How-
ever, general presuppositions, while necessary to theory, are not sufficient to define
it. The same set of assumptions can lead to whole classes of theories. Theory in-
volves the use of hypotheses whose implications emerge from extended strings of
logic developed in detailed specific cases. Speculations in general terms used ad
hoc to explain small groups of phenomena, or isolated instances, do not constitute
theory. Nor are general ideas linked by analogy, metaphor, illustrative example or
other rhetorical devices, to experimental cases, counted as theory.5
5 Analogy refers here to the literary device, not the method developed in mathematical
physics in the nineteenth century. Only recently have the issues of rhetoric and language
penetrated the history of science. Nancy Leys Stepan, "Race and Gender: The Role
of Analogy in Science," Isis 77 (1986): 261-282 traces the literature on analogy and
metaphor in science. See also, Roger S. Jones Physics as Metaphor (Minneapolis, MN.:
University of Minnesota Press, 1982). Most studies ofthe languages of the sciences focus
on rhetorical purpose and devices, see Wilda Anderson, Between Library and Laboratory
(Baltimore: Johns Hopkins University Press, 1987), The Literary Structure of Scientific
Argument, Peter Dear ed. (Philadelphia: University of Pennsylvania Press, 1991), and
Persuading Science: The Art of Scientific Rhetoric, Marcello Pera and William R. Shea
eds. (Canton MA.: Science History Publications, 1990). On the more technical use of
analogy in physics, see Mary Hesse, Models and Analogies in Science (Notre Dame:
Notre Dame University Press, 1966).
6 Introduction
6 On this point see, Timothy Lenoir, "Practice, Context, and Dialogue between Theory and
Experiment," Sci. Context 2 (1988): 3-22, and Theory and Experiment: Recent Insights
and New Perspectives on Their Relations, Diderik Batens and Jean-Paul Bendegen eds.
(Dordrecht: Reidel, 1988).
7 Historians are reassessing the "theory-laden" character of experiments. Experiments
have broader uses within physics than Kuhn allowed and experimentalists more auton-
omy within the profession. For a view of the place of experiment in the historiography
of science see Frederick L. Holmes, "Do We Understand Historically How Experimental
Knowledge is Acquired?" Hist. Sci. 30 (1992): 119-136. Holmes' paper is a review
of recent literature in which historians have tried to reestablish a place for experiment
in post-Kuhnian historiography. Other attempts include, Andrew Pickering, Science as
Practice and Culture (Chicago: University of Chicago Press, 1992), 65-112, Experi-
mental Inquiries: Historical, Philosophical, and Social Studies of Experimentation in
Science, Homer E. LeGrand ed. (Dordrecht: Kluwer Academic, 1990), The Uses of
Experiment: Studies in the Natural Sciences, David Gooding, Trevor Pinch and Simon
Schaffer, eds. (Cambridge: Cambridge University Press, 1989), Peter Galison How Ex-
periments End, (Chicago: University of Chicago Press, 1987), The Development of the
Laboratory, Frank A. J. L. James ed. (New York: Macmillan, 1989), and Observation,
Experiment and Hypothesis in Modern Physical Science, Peter Achinstein and O. Han-
nawayeds. (Cambridge MA.: MIT Press, 1985). See also Allen Franklin The Neglect
of Experiment (New York: Cambridge University Press, 1986), an attempt to develop a
Introduction 7
10 Hermann von Helmholtz, "On the Origin and Significance of Geometrical Axioms," in
Science and Culture: Popular and Philosophical Essays, David Cahan ed. (Chicago:
University of Chicago Press, 1995),226-245,245.
11 However, see Peter Dear, Discipline and Experience: The Mathematical Way in the
Scientific Revolution (Chicago: University of Chicago Press, 1995) that the transcendency
claimed for mathematics was crucial in the seventeenth century to establishing the validity
of experiment.
12 Historians have begun to explore the role of quantification and mathematics in claims of
the sciences for the transcendence that surrounded mathematics. See Philip Mirowski,
More Heat than Light: Economics as Social Physics, Physics as Nature's Economics
(New York: Cambridge University Press, 1989), and Theodore Porter, Trust in Numbers:
The Pursuit ofObjectivity in Science and Public Life (Princeton NJ: Princeton University
Press, 1995).
Introduction 9
theory within mathematics. 13 Physicists did not rise to this bait. Albert Einstein,
who fretted about his lack of mathematical understanding, reminded David Hilbert
that physics was not simply mathematics. However, some physicists, notably in
the twentieth century, have worried about the mathematics appropriate for spe-
cific physical problems. 14 Ludwig Boltzmann took mathematics as the language
of physics and was concerned, as was Max Planck and Albert Einstein, with the
proper mathematics for theoretical physics. They all matched the characteris-
tics of the mathematics to the characteristics of the physical system. They used
summations for molecular and integrals for continuum systems. 15
The ability to perform the difficult task of discerning, in the mathematics, pos-
sibilities that are physical plausibilities is rare and justly celebrated. If physics
becomes mathematics (as has been true in important cases in the late nineteenth
and the twentieth centuries) members of the discipline are conscious of it. Ex-
planations of the physical meaning within mathematical symbols was required by
other members of the discipline. Physical interpretation of what is fundamentally
a physically uninterpretable theory seems to go on. 16
Modem Physics
13 Elie Zahar, "Einstein, Meyerson and the Role of Mathematics in Physical Discovery,"
Brit. 1. Phil. Sci. 31 (1980): 1-43, 2-8, considers some of the issue raised here but from
the point of view of the philosophy of mathematics rather than its history or the history
of physics.
14 See Eugene Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural
Sciences," Comm. Pure and Appl. Math. 13 (1960): 1-14. For earlier attempts to
examine the relationship between mathematics and physics see Maxwell, "Address to the
Mathematical and Physical Section," Rep. British Assoc. 40 (1870): 1-9.
15 See Christa Jungnickel and Russell McCormmach, Intellectual Mastery ofNature: The-
oretical Physics from Ohm to Einstein (Chicago: University of Chicago Press, 1986) vol.
2,13,188,336-337.
16 Particularly noteworthy are Ludwig Boltzmann's attempts to give mechanical meaning
to his statistical derivation of entropy and Niels Bohr's attempts to interpret the quantum
mechanics of Werner Heisenberg.
17 This study focuses on Europe because physics in the United States followed older patterns
of research into the twentieth century. See John Servos, "Mathematics and the Physical
Sciences in America, 1880-1930," Isis 77 (1986): 611-629. For a discussion of the
reorientation of physics in the United States in the 1920s, see Spencer Weart, "The
Physics Business in America 1919-1940: A Statistical Reconnaissance," in The Sciences
in the American Context: New Perspectives Nathan Reingold ed. (Washington DC:
Smithsonian Press, 1979),295-358. However, S. S. Schweber, "The Empiricist Temper
Regnant: Theoretical Physics in the United States, 1920-1950," Hist. Stud. Phys. Sci. 17
(1986): 55-98, maintains the uniqueness of American physics well beyond the 1920s.
10 Introduction
18 See Paul Forman, John Heilbron and Spencer We art, "Physics circa 1900: Personnel,
Funding, and Productivity of the Academic Establishments," Hist. Stud. Phys. Sci. 5
(1975): 30-32. Christa Jungnickel and Russell Mccormmach, Intellectual Mastery of
Nature make the same important point although they do not always differentiate the title
of a chair and the actual research of its occupant. Some physicists taught theoretical
physics while their research was experimental.
19 The kinetic theory of gases is the first place in which these relationships occur consistently.
20 See James Clerk Maxwell's surprise at the predicted behavior of the viscosity of gases,
and then his investigation of the transport coefficients, in, Maxwell on Molecules and
Gases Elizabeth Garber, Stephen G. Brush and Francis Everitt, eds. (Cambridge MA.:
MIT Press, 1986), 282-283, 359-386. This is in contrast to the earlier uses of experiment
in the work of Pierre Simon Laplace, Denis Poisson and earlier mathematicians.
21 There are even claims that pure mathematics does not exist. See Nordon Didier Les
mathematiques pures n'existent pas! (Paris: Actes Sud, 1981).
Introduction 11
when the invaders annex languages whose power and structure were revealed only
after considerations of definition and careful examination of the language. From
the point of view of mathematicians the rigor, precision of expression and gener-
ality, the core of their discipline, is betrayed. Mathematicians see physicists using
mathematics as sloppy, and making unjustifiable assumptions about how to use
their language.
On the other side of the disciplinary and professional boundary, physicists see
mathematicians as obsessed with the wrong issues. They spend too much time
over whether a proof is adequate or whether another, more general one might be
necessary. The point is to use the results of these theorems, with an intuitive sense
that the theorems seem correct. Given the same problem, the mathematician's and
the physicist's solutions are developed with different criteria of what those solutions
might be. This makes for lively, if acrimonious, joint sessions of two departments
with members that have overlapping research interests. If mathematicians begin
with a physical problem, they do not have to reflect on the implications of their
mathematical findings for the physical particulars of that problem. This closure
is the point of the physicist's exploration of the same issue and separates the
practitioners of the two disciplines. The numerically correct answer to particular
problems is crucial for the physicist because of its significance for ideas about
nature. In a certain sense it is irrelevant to the mathematician. Mathematics
and physics remain as distinct disciplines, expressed institutionally as separate
academic departments, curricula, professional societies, journals, and research
methods. Yet, as disciplines and professions, they remain locked in an historical
symbiosis that demands investigation.
Several elements join to characterize modern, theoretical physics. Any history
of its development must consider all these factors and their fusion into this modern
form. This is not merely a matter of intellectual history. Theoretical physics is
an academic discipline, most of whose members are systematically training the
next generation of theoretical physicists. The university also serves as the locus of
research, much of it done by the same faculty and their students. The results of such
research appears in the footnoted pages of specialist journals usually published by
the professional societies, to which all claiming to be members of the discipline,
belong. As aspiring members of the discipline, applicants to such societies must
meet minimum requirements of certification. In turn, membership in such a society
legitimates claims of practicing as a physicist and belonging to the profession. The
practices of the societies' members are expressed and developed within the pages
of its journals, and reinforced by a system of prizes, medals and other honors.
The readership of these journals is small, usually restricted to those understanding
the technical methods, language, and problems being addressed in their pages.
Interpreting such publications for a broader public requires a battery of mediators.
Access to training and practicing within physics is controlled by physicists as
12 Introduction
Of the various ways of treating the history of physics, the most prominent is
still intellectual. Conceptual changes are generally used to demarcate stages in the
development of physics. Thomas Kuhn reenforced this approach.22 This emphasis
is unchanged whether the historian attributes the development of ideas to forces of
change within the scientific disciplines or the larger cultural nexus. Concepts join
science to society and culture, and for social constructivists and cultural historians
only serve to demonstrate ideological activities. 23 Only recently have historians
turned to other factors under the rubric of practices. 24
Using conceptual change to mark the development of physics has severe limi-
tations. The "origin" of modem physics depends on those concepts the historian
25 For recent examples see H. Floris Cohen, The Scientific Revolution: A Historiographical
Inquiry (Chicago: University of Chicago Press, 1994), in which the sciences are still
defined intellectually such that, "the seventeenth century marks the origin of modern
science." For use of the concept revolution to explain change in the sciences see, I. B.
Cohen, Revolutions in Science (Cambridge MA: Belknap Press, 1987). Here revolutions
are related to conceptual change only.
26 There are exceptions. See Geoffrey Cantor, "The Reception of the Wave Theory of Light
in Britain: A Case Study Illustrating the Role of Methodology in Scientific Debate," Hist.
Stud. Phy. Sci. 6 (1975): 109-132 and some of the recent literature on practices.
27 See Robert Fox, The Caloric Theories of Gases from Lavoisier to Regnault (Oxford:
Clarendon Press, 1971). See also Conceptions of the Ether: Studies in the History
of Ether Theories, 1740-1900 G. N. Cantor and M. J. S. Hodge, eds. (Cambridge:
Cambridge University Press, 1981).
14 Introduction
If ideas about caloric were connected to the results of experiment, it was through
illustrative examples, figures of speech, and arguments about reasonable expec-
tations, not through any species of logical strings, even excluding any forms of
mathematics.
The usual starting place for the development of theory in modern physics is not
from postulating the existence of a general principle or substance but through the
solutions to problems. A particular set of physical circumstances, that is, particular
problems associated with a narrowly defined physical system, is the locus of the
theorist's attention. Upon these specific cases general principles may be brought
to bear. However, these principles are in a form that relates directly to the specific
case at hand. The starting point for Einstein's theory of relativity was the motion of
a charged particle moving through an electromagnetic field, not the nature of space
and time. In modern theoretical physics specific physical circumstances are joined
to mathematics, then judgments are made about how much of the latter to use to
reveal the workings of a particular physical system. The range of interpretations is
limited by the structure and use of the available language, the mathematics. These
practices were not used in physics until well into the nineteenth century. Forms of
argument need integration into the history of physics.
The shortcomings of histories that deal only with foundational assumptions are
also compounded by another. Intellectual historians of science usually assume
that all problems, ideas, and methods that are now accepted as part of theoretical
physics have been used consistently within physics since their first appearance. 28
The usual assumption is that mathematics has been a permanent aspect of physics
ever since Galileo's mathematization of the fall of terrestrial bodies. Isaac Newton
in his Principia was dealing with a problem in physics, and we can expect amidst
the struggle to develop his ideas the same mix of observation and mathematics
that we find in theory in the twentieth century. The usual account of the history
of physics narrates the spread of this method from mechanics to the study of
electricity, magnetism and light. Extension of the use of mathematics has followed
the successful development of physical ideas.
Because mathematics is now a permanent factor within physics, intellectual
historians tend to regard the work of Newton, the Bernoullis, Denis Poisson, or
George Gabriel Stokes as forming a continuum with our own. For these historians,
the issue is not whether these men produced physical solutions to problems we take
to lie within physics, but what are the physical constituents of those solutions. This
approach is apparent in recent histories of nineteenth-century physics. Authors
assume the relationship between physical imagery, mathematics, and experiment
in the early nineteenth are those of the twentieth century. They also treat theories
29 See, Crosbie Smith and Norton Wise, Energy and Empire: A Biographical Study ofLord
Kelvin (Cambridge: Cambridge University Press, 1989), Nahum Kipnis, History of the
Principle of Interference (Boston: Birkhauser, 1990). Jed Buchwald, The Rise of the
Wave Theory of Light: Optical Theory and Experiment in the Early Nineteenth Century
(Chicago: University of Chicago Press, 1989).
30 This is the approach of Rene Dugas, Histoire de la Mecanique (Paris: Edition Dunod,
1950) that set the pattern of discussion by historians of physics.
31 See Clifford Truesdell, "Reactions of Late Baroque Mechanics to Success, Conjecture,
Error, and Failure in Newton's Principia," Texas Quart. 10 (1967): 238--258, The Annus
Mirabilis of Sir Isaac Newton, 1666-1966 Robert Palter ed. (Cambridge MA.: MIT
Press, 1967), and Henry Guerlac, "The Early Reception of his Physical Thought," in
Guerlac, Newton on the Continent, 41-73.
16 Introduction
and schematic systems were plentiful. Theories that centered upon experimental
observations were phenomenological, or classified phenomena into rational lists
tbat reduced large bodies of data to manageable size. 32 Within physics, speculations
about the operations of nature were expressed in language and images available to
a general, literate audience using devices of argument familiar from sources out-
side of science. This kind of physics does not fit into twentieth century categories.
Harnessing metaphysics to logic, even in the vernacular, did not appear in physics
until the nineteenth century, after physicists had definitively narrowed the forms
and focus of the discipline.
The shortcomings of some histories spring partly from being unaware that the
label may stay the same, but the discipline can change profoundly. If we do
trace the changes in the meaning and content of the terms "physics" since the
seventeenth century, we find that mathematics only enters into its methodology
in the latter half of its history.33 How then, did modern physics develop out of
its experimental and metaphysical practice? We can no longer see physics as the
residue left after chemistry and other specialities separated from the general body
of natural philosophy. Nor can we argue that modern physics is the fusion of
mathematics and physics without some understanding of the process of fusion. 34
The one extensive recent work on the development of modern theoretical physics
leaves us with some dilemmas. This study by Christa Jungnickel and Russell
McCormmach is one for which other historians should be extremely grateful. 35
However, the subject of their study, theoretical physics, is never defined. While
appreciating that leaving the definition moot avoids the temptation of whiggish
32 Examples are of Charles Dufay in electricity, Karl Linnaeus in botany and William
Herschel in observational astronomy.
33 For changes in the meaning of the term physics see, Hans Schimank, "Die Wandlung des
Begriff 'Physik' wahrend der ersten Halfe des 18 Jahrhunderts," WissenschaJt, WirtschaJt
und Technik: studien zur Geschichte Munich (1969): 453-468, Fritz Krafft, "Der Weg
von den Physikern zur Physik an deutschen Universitaten," Ber. Wissen. 1 (1978): 123-
167, and, "Alte und neue Physik," in Disciplinae Novae Christoph Scriba, ed. (G6ttingen:
Vandenhoek und Ruprecht, 1979), 45-63, R. Hookyas, "Von Physica zur Physik," in Hu-
manismus und NaturwissenschaJten Rudolf Schnitz and Fritz Krafft, eds. (Boppard am
Rhein: Harald Bopt, 1980), 9-38, and John Heilbron, "Experimental Natural Philoso-
phy," in Ferment of Knowledge: Studies in Eighteenth Century Science, R. S. Rousseau
and Roy Porter, eds. (Cambridge: Cambridge University Press, 1980),357-387. The
same is true of the term "element" in chemistry where Boyle's definition can be taken
over into a modern, elementary textbook without change, until the reader becomes aware
of Boyle's theory of matter. There are of course numerous others.
34 This fusion is assumed in Kuhn, Structure, "Postscript," 179. Kuhn later posited two
traditions within physics, without considering that mathematics might also be a problem-
atic category. See, Kuhn, "The Mathematical and Experimental Traditions in Physics,"
J. Interdic. Hist. 7 (1976): 1-31.
35 Jungnickel and McCormmach, Intellectual Mastery of Nature.
Introduction 17
purpose, we are left with that group appointed to positions as theoretical physicists
in the universities of Germany and middle Europe. The institutional and social
definition does not touch on the intellectual content of physics. The authors accept
the content of the research of these men, appointed to certain chairs in the above
universities, as physics wherever it was placed for publication. Some of their
research appeared in mathematics journals and yet counts as physics because the
authors are physicists from their position in the university structure. And, the
measure of the development of theoretical physics is in terms of ideas and theories,
not practices. They miss the ways in which criteria of solutions and standards
of argument change across time. They are assumed to be those we accept today.
Mathematics is simply a "tool" although central to what theoretical physicists did
even in the last decade of the nineteenth and the first decade of this century when
the required tools were changing.
These omissions mean that we cannot probe crucial differences between math-
ematics and physics, mathematicians and physicists. They leave unexplained how
the new mathematical "tool" of absolute differential calculus could transform the
physicists' image of the world. This crucial connection between mathematics and
physical imagery needs more investigation. While noting that some physicists
did refer to mathematics as a language, they probe no further. However, this is
an important key to exploring how theoretical physics came into existence as an
intellectual discipline. To explore this process we need criteria that are broader
than subject matter and ideas and to follow them across space and time to trace
how physics and mathematics were distinguished.
It seems necessary at this juncture to ask how mathematics has been used in
physics. Most historians of physics ignore mathematics in their narratives, be they
scholarly studies or textbooks. At best they cite formulae and derivations without
paying any attention to their origins. Ivor Grattan-Guinness attributes this to a
dislike of mathematics on the part of historians of science. 36 However, there is a
more pervasive problem here. Most historians and philosophers of physics treat
mathematics as a tool, an instrument to be applied to a task, like a hammer to a
nail, then replaced on the shelf. It is unchanging and always available. And even if
mathematics has a history, the type of mathematics used has no impact upon pos-
sible physical interpretations drawn from the mathematics. Thus, the mathematics
in physics can be rewritten in modern form; the physics remains the same. Edmund
Whittaker complained that physicists for whom vector analysis was not available
did not see certain implications of their work. Whittaker and later historians then
translated physicists' mathematics into vector form and drew implications about
their work directly from the modern mathematical forms, not the original ones. 37
36 See Ivor Grattan-Guinness, "Does History of Science Treat of the History of Science?
The Case of Mathematics," Rist. Sci. 28 (1990): 149-173.
37 This is the assumption in Buchwald, The Rise of the Wave Theory and The Creation of
18 Introduction
However, until the end of the nineteenth century, the mathematics of physics was
in algebraic form and Cartesian coordinates. Historians have not considered how
physicists have proceeded through mathematical forms to conclusions, and how
the available mathematics might shape, or limit, those conclusions.
Mathematics has interacted with, and become integral to physics. And, as in
other cases, this transformation became invisible. In retrospective examinations,
eighteenth-century mathematics was subsumed by physicists as physics. However,
problems that arose in mechanics and astronomy were the means by which math-
ematicians developed the calculus itself in the eighteenth century. The resulting
papers were mathematical, not physical. Mechanics in its mathematical form was a
field within mathematics. Yet, in the mid-nineteenth century mechanics became the
explanatory core for physics. For mid-nineteenth century physicists, eighteenth-
century rational mechanics became physics. Mathematical expressions were given
physical meanings derived from methodological and conceptual commitments not
present in the eighteenth century. In their definitive textbook on mechanics of
the 1870s, William Thomson and Peter Guthrie Tait used Lagrangian generalized
coordinates extensively. They accomplished the considerable task of explaining
Lagrangian methods to engineers and physicists and of harnessing them to solve
physical problems. In the process, Lagrange's original mathematical goals were
largely lost. 38 This retrospective treatment of mathematics in the history of physics
has appeared consistently since the mid-nineteenth century because mathematics
is such an integral part of modern physics.
New mathematical languages such as non-arithmetic algebras or non-Euclidean
geometries had profound effects on theoretical physics, including the ways in which
physicists envision reality. Seeing the vector quality of physical entities makes
visualizing their behavior in space immediate in ways that their expression in alge-
braic form do not. Vectors make visible the spatial aspects of the motion of a wave
front and of moving particles but were available only after waves became a central
feature of nineteenth-century physics. The same is true for electrodynamics. 39
Aspects of the theory of relativity became apparent after Hermann Minkowski put
Einstein's equations into non-Euclidean geometrical form. Mathematics is repeat-
edly claimed by physicists as their language. It is also seen as the one characteristic
Scientific Effects: Heinrich Hertz and Electric Waves (Chicago: University of Chicago
Press, 1994) among many others. It began with Edmund Whittaker A History of the
Theories ofA ether and Electricity (New York: Thomas Nelson and Sons, 1962),2 vols.
38 William Thomson and Peter Guthrie Tait, Treatise on Natural Philosophy (New York:
Dover reprint of 1879 edition, 1962) 2 vols. For the development and writing of this
volume see Crosbie Smith and Norton Wise, Energy and Empire.
39 Vector algebra developed in the 1880s. See Michael Crowe,A History of Vector Analysis:
The Evolution of the Idea of a Vectorial System (Notre Dame: Notre Dame University
Press, 1967).
Introduction 19
of the discipline that provides any unity to the current sprawl of physics. 4o
Mathematics also has a history, as complex and contested as any other scientific
discipline. Meanings of the fundamentals of mathematics, and in particular, the
foundations of the calculus, have changed through time and across space. 41 As
the content of mathematics mutated, the mathematics available to physicists has
also changed. The language forms with which physics is expressed have become
dependent on available mathematics. Yet issues that arose in the physical world
established the problems that mathematicians used to explore mathematics, ex-
tend its range, change its character and reach. Physics has also been a source for
mathematics. For mathematicians of the eighteenth century, their starting point
in problems in the real world guaranteed that solutions existed. Mathematicians
needed physics for their own purposes as surely as physicists have molded math-
ematics to their own uses. For a more successful history of physics to emerge, we
need to scrutinize this interdependence.
The intellectual questions surrounding the relationship between mathematics
and physics are inevitably coupled to social ones. New institutional forms were as
integral to the creation of modern physics as its new intellectual ones. While one
transformation need not involve the other, in this case both social context and in-
tellectual content changed almost simultaneously. However, the immediate causes
for both were not the same. In Britain, industrialization opened up opportunities
for livelihoods in science, yet this did not dictate changes in the methodology, form
of expression or content of physics. Those came later, under the social pressures of
successful industrialization on older academic institutions. The changes in practice
in the middle third of the nineteenth century followed the intellectual leadership
of Paris. 42 In the German States the social opportunities brought by the reform of
the Prussian universities, driven by an educational ideology serving the political
purposes of the state, involved the transformation of faculty from collegial teaching
bodies of local importance into scholars with international research reputations. In
this case, research into esoteric questions and issues without visible connection to
40 James A. Krumhand, "Unity in the Science of Physics," Phys. Today, March (1991):
33-38, and David Mermin, "What's Wrong with these Equations," same journal (1989):
9-11, p. 9.
41 See Judith Grabiner, "Is Mathematical Truth Time-Dependent," Amer. Math. Monthly 81
(1974): 354-363. See also Joseph Dauben, "Conceptual Revolutions and the History
of Mathematics,"in Transformation and Tradition in the Sciences: Essays in Honor of
I. B. Cohen, Everitt Mendelsohn, ed. (New York: Cambridge University Press, 1984)
81-103. For a more extended discussion see, Ivor Grattan-Guinness, The Development
of the Foundations ofMathematical Analysis from Euler to Riemann (Dordrecht: Reidel,
1970). See also Hans Niels Jahnke, "Mathematics and Culture: The Case of Navalis,"
Sci. Context 4 (1991): 279-295.
42 For social transformation without intellectual change see Rachel Laudan, "Ideas and
Organization in British Geology: A Case Study in Institutional History," Isis 68 (1977):
527-538.
20 Introduction
social use, led to particular forms of theoretical discourse in physics that depended
on prior philosophical commitments. The educational ideology of the state did not,
however, predetermine that Parisian scholarship in the sciences and mathematics
be the model for German academics. Nonetheless, the French offered the most
successful model to emulate and then transcend. During this process in Britain and
the German States physics, as discipline and profession, was put into the forms we
have inherited and hence labelled here, for convenience, as modern.
Mathematics as Language
43 For our purposes we are exploring in detail only those meanings of the languages of
mathematics as they affected and were used by physicists rather than mathematicians.
44 Ludwig Wittgenstein, Remarks on the Foundations ofMathematics. R. Rhees, G. H. von
Wright and G. E. M. Anscombe, eds. (London: Blackwell, 1967). See New Directions
in the Philosophy of Mathematics, Thomas Tymoczko, ed. (Boston: Birkhiiuser, 1986),
for the range of views on this point.
45 Brian Rotman, "Towards a Semiotics of Mathematics," Semiotica (1988): 1-35, Rotman,
Signifying Nothing: The Semiotics of Zero (New York: St Martins Press, 1987), and
David R. Lachtermann The Ethics of Geometry: A Genealogy of Modernity (New York:
Routledge, 1989).
46 However, see, P. Hugly and C. Sayword, "Can a Language have Indenumerable Many
Expressions?" Hist. Phil. Logic (1984).
Introduction 21
changes through time. Aesthetics is also a major factor in judging any particular
product. Subjects and forms can be reworked any number of times, as aesthetics and
meanings attached to those subjects and themes change. 47 Within both disciplines,
individuals can be spotted in their products. When solving the same problem,
or using the same musical form, individuals express their own "style.,,48 Even as
d' Alembert, Euler, and Lagrange all wrote on the wave equation and its solution,
their style is apparent in their characterization of the equation and its solution. This
is in addition to national traditions that are obvious in mathematics throughout our
period. 49
In regarding mathematics literally as language one can draw parallels between
sentences and mathematical expressions, the linguistic roles played by nouns and
terms, adjectives and coefficients, verbs and operators. 50 As in ordinary language
much is left understood and is a major source of ambiguity. There are implicit
assumptions that one writer may treat as shared among all practitioners but are not.
There may be deliberate omissions and ambiguities for moral or political purposes.
Careers and reputations depend on the judgments of other mathematicians with
known preferences and prejudices. Lacuna or obscurity may also hide difficulties,
or spring from not understanding the significance of aspects of the problem under
discussion, or from implicit assumptions of a deeper kind that make discussions
of certain issues impossible as they may destroy the foundations of the discourse.
This of course is true of eighteenth-century calculus.
Ambiguity and omission in mathematics can serve the same purposes as in any
other languages. Similarly, delaying clarification may simply be a way of getting
something expressed and out in the open. Linguistic niceties can wait. Such
obscurities are cleared up only when implicit assumptions are forced into the open.
The calculus was one such language that developed this way. Initially calculus
was the algebraic description of geometric relationships: It was a language whose
roots lay in another language. The differential represented the tangent to a curve,
the integral the area under the curve. Calculus began to solve problems that older
languages were inadequate to tackle. The mode of its development was the solution
of problems, not the formal exploration of the structure and function of the calculus
47 For example, the same musical forms have been reworked and Euclidean geometry
reformulated many times.
48 The concept of style is problematic. However, it is still useful. For its problems see Sci.
Context 4, no. 2 (1991). For style in physics see, N. David Mermin, BoojumsAll the Way
Through (Cambridge: Cambridge University Press, 1990).
49 Joan Richards, "Rigor and Clarity: The Foundations of Mathematics in France and
England, 1800-1840," Sci. Context 4 (1991): 297-319.
50 For short introductions to mathematics as language and its uses in literature and as litera-
ture see Helen M. Pycior, "Mathematics and Prose," and John Fauvel, "Mathematics and
Poetry," in Companion Encyclopedia of the History and Philosophy of the Mathematical
Sciences Ivor Grattan-Guinness, ed. (New York: Routledge, 1994).
22 Introduction
51 What is stated here about mathematics can be applied to theoretical physics. Most works
on physics as literature focus upon the rhetorical purposes, deduced from conceptual
presuppositions.
52 In the eighteenth century language was regarded as a form of calculus, Turgot, "Ety-
mologie," in Encyclopedie in Oeuvres vol. 2, 473-516, 506-507. For mathematics and
language in the eighteenth century, see Robin E. Rider, "Measures of Ideas, Rule of
Language: Mathematics and Language in the Eighteenth Century," in The Quantifying
Spirit in the Eighteenth Century, Tore Frangsmyr, John Heilbron and Robin Rider, eds.
(Berkeley CA: University of California Press, 1990).
53 Other narrative forms developed in the nineteenth century, including that of the metaphys-
ical novel, i.e., the search for an understanding of foundations of mathematics. Disputes
over these issues sometimes made, broke, or redirected careers.
Introduction 23
a springboard into the exploration of the more general linguistic structures latent
in the particular algebraic expressions of mechanics, mathematicians constructed,
then explored those general linguistic structures. They used meanings and forms
familiar later to nineteenth-century physicists. However, mathematicians did not
use the results of the mathematical enquiries to reflect back upon interpretations
of the physical entities with which they began their problem solving. In the eigh-
teenth century interpretations and reinterpretations of force, and vis viva, for ex-
ample, were metaphysical matters. Their existence as theoretical entities might
emerge from experiment, observation or philosophical contemplation but having
mathematical expression they lost all physical significance. Their function was
mathematical, not physical. Experiment set up the terms from which mathemati-
cians linguistic explorations began, and to confirm the correctness of the syntax
and grammar that developed out of those explorations: experiment confirmed the
mathematics, not the physics.
Yet, the notion of mathematics as language is insufficient to explain the timing
of the transformation process in the middle third of the nineteenth century. Why
did this not happen in the eighteenth century? Both physicists and mathematicians
had a common interest in problems of mechanics. However, in this era disci-
plines were defined through practices exploited in the study of nature. Physicists
used experiment and observation to explore nature and these defined their terms
of engagement with problems, and of their solutions. Mathematicians used their
own terms of engagement and methodologies to explore the same phenomena, and
defined the solutions to those questions using criteria developed within mathemat-
ics. At the end of the eighteenth century, within the French scientific community,
this methodological core for physics was coupled with an attempt to repudiate the
search for essences. This predisposition truncated any search for meanings and
confined explorations of nature to description. Therefore, physique-mathematique
was an extension of the calculus as a descriptive, not an interpretative, language
for understanding the operations of nature. Given the standards and practices of
mathematicians it was also mathematics, not physics. In these same decades, the
implicit assumptions at the foundations of the calculus were forced into the open.
This process of disclosure was initiated partly by external criticism and internal
dissatisfactions with the syntax of the calculus, and challenges to the implicit limits
of the language imposed by its practitioners. The foundations were reworked and
the first formal and logically defensible form for the calculus emerged from that
work. In France at least, these changes were not accompanied by the redrawing of
the boundaries between mathematics and physics.
Within the scientific communities of the German States and Britain, a search
for essences was legitimate. Mechanics carried physical meanings deliberately
avoided by the French. Because of these predispositions physique-mathematique
seemed to physicists in Britain and Germany to promise solutions to physical prob-
Introduction 25
lems. Learning how to understand, use, then develop this language for the purposes
of exploring nature viewed as a mechanism, meant the exploitation of this language
in domains outside of its own self-reference. Learning included misunderstanding
then replicating the standards of demonstration in physique-mathematique. Syn-
tax, grammar and the method of extracting meaning from the language remained
the same. The meanings themselves were changed by the needs of the new mechan-
ical views of nature. The solutions offered by French mathematical methods were
finally unsatisfactory in Germany and Britain. Practitioners of physics required
explanations rather than descriptions of appearances. Mathematical solutions were
given physical meaning.
In other contexts historians have noted the transformation of language from one
set of purposes into another. 56 In physics, the purposes of the language changed,
from the expansion of the language to the interpretation of nature. The language,
the calculus, was redirected outwards from the solution of problems set up within
to the description of processes external to itself. The terms of expression were
established by physical processes that also set the limits of development of the
language. (This in tum opened to mathematicians further opportunities to explore
aspects of the calculus that they had previously bypassed.) These borrowings
between languages have continued into this century.
From this process of annexation emerged a new language and literature, that of
theoretical physics. Plots had changed. External elements, experiments, added
sometimes unexpected twists to the narratives and as they developed the meanings
of the initial elements could also change. Many of the characteristics of the math-
ematical scheme of argument were carried over to the physical one. Mathematics
could also obfuscate as well as clarify, intentionally at times, and used to claim ter-
ritory rather than solve problems or answer questions. And as with any language,
the issues of precision have to be separated from those of certainty.
Eighteenth-Century Science
Earlier generations pursued their own problems with their own
instruments and their own canons of solution.
- Thomas S. Kuhn
The Structure of Scientific Revolutions, p. 141.
Chapter II
Vibrating Strings
and Eighteenth-Century Mechanics
In the late 1740s an acrimonious dispute broke out between Jean Ie Rond
d' AIembert and Leonhard Euler over the solution of the two-dimensional wave
equation. The issues were the nature of legitimate mathematical functions, the
definition of continuity, and differentiability. A description of this overheated
exchange illustrates the relations between mathematics, metaphysics and experi-
mental philosophy during the eighteenth century.l
D' Alembert was not the first mathematician to focus on establishing mathe-
matically, then solving, the equation of motion for a string under tension. In
examining the problem, d' Alembert exploited the newly developed partial differ-
ential equations. Apart from the general physical treatment he needed to establish
the equations, d' Alembert 's work was an exercise in the development of the calcu-
lus, an exploration of solutions to partial differential equations of a certain type. It
1 For details of the mathematical issues involved and the opinions of Euler and d' AIembert
see, Ivor Grattan-Guinness, From the Calculus to Set Theory (London: Duckworth, 1980),
chap. 1, Jerome R. Ravetz, "Vibrating Strings and Arbitrary Functions," in The Logic
of Personal Knowledge, (Glencoe II.: Free Press, 1961),71-88, and Clifford Truesdell,
"The Rational Mechanics of Flexible or Elastic Bodies, 1638-1788," in Euler, Opera
Omnia, series 2, vol. 11, part 2, (Bern: 1957),240-262, although Truesdell is too partial
to Euler. Henk Bas also noted this partiality in H. J. M. Bos, "Mathematics and Rational
Mechanics," in Ferment of Knowledge: Studies in the Historiography of the Eighteenth
Century, G. S. Rousseau and Roy Porter, eds. (Cambridge: Cambridge University press,
1980),327-355. For d'AIembert see Thomas Hankins, Jean d'Alembert: Science and
the Enlightenment (Oxford: The Clarendon Press, 1970). Michel Paty, d'Alembert
et son temps: Elements de biographie (Strasbourg: Universite Louis Pasteur, 1977)
characterized d' Alembert's previous work in mechanics more as "a branch of mathematics
than as an experimental science," in reference to d' Alembert, Traite de Mecanique (Paris,
1743). See also Hankins, "Introduction," to Traite. See also, G. F. Wheeler and W. P.
Crummet, "The Vibrating String Controversy," Amer. J. Phys. 55 (1987): 33-37.
was not the physical expression, in mathematical form, of the motion of a vibrat-
ing string. What d' Alembert proposed to demonstrate for a string under tension,
using Taylor's conditions, was that "there are infinitely many curves other than
the companion of the elongated cycloid [sine curve] that satisfy the problem in
question.,,2 Starting from Brooke Taylor's work, d' Alembert focussed on Taylor's
expression for the accelerating force on an element of the string, T{3, where {3
was the curvature and T the tension in the string. Using Newton's second law,
d' Alembert constructed solutions to expressions for the accelerating force on the
string that were equivalent to solving the wave equation,3
This expression included the "infinity of curves" d' Alembert promised. He pro-
ceeded to demonstrate their existence and explore the mathematical nature of the
functions, 1jf and D. a,nd their relationship for a string held taut and fixed at both
ends, for which, 5
y = 1jf(t + s) + 1jf(t - s).
2 Taylor's condition was that the acceleration of the oscillating body was proportional to
its distance from its equilibrium position, i.e., simple harmonic motion. Jean Ie Rond
d' Aiembert, "Recherches sur la courbe que forme une corde tendue mise en vibration,"
Mem. Acad. Sci. Berlin 3 (1747) [1749]: 214-219, p. 214, and "Suite des recherches sur
la courbe que forme une corde tendue, mise en vibration," same journal, 220-249.
3 D' Aiembert did not use modern notation but he manipulated his expressions as the partial
differential equation,
a
2y T 82 y
at
2 = p8s 2 ·
See S. S. Demidov, "Creation et developpement de la tMorie des equations differentielles
aux derivees partielles dans les travaux de J. d' Aiembert," Rev. Hist. Sci. 35 (1982): 3--42
and Steven B. Engelmann, "D' Alembert et les equations au derivees partielles," Dix-huit.
saxle 16 (1984): 7-203. Notation remained remarkably fluid until the late nineteenth
century. See Florian Cajori, A History ofMathematical Notations, (Chicago: Open Court
Press, 1929.)
4 For a discussion of d'Aiembert's derivation of the equation of motion and its problems,
see Hankins, d'Alembert, and Ravetz, "Vibrating Strings."
5 D' Aiembert obtained this solution through a change of coordinates, not the method
of separation of variables. D' Alembert called this "the method of multipliers." For
Eighteenth-Century Mechanics 33
Initially Leonhard Euler critiqued d' Alembert's assumptions that the vibrations
were infinitely small so that the length of the string stayed constant and that, as
it moved, the string formed one continuous curve. Simultaneously, Euler was in-
tent on establishing the equation of motion with his own method by examining
the balance of forces in an element of string under tension. He then turned to
d' Alembert's handling of the initial shape of the curve. For Euler, this shape deter-
mined all future motions and subsequent shapes of the curves that formed solutions
to the equation of motion. The first motions given the string were continued in-
definitely. By restricting this to analytical, i.e., geometrically continuous curves,
d' Alembert omitted curves. At this point Euler could not offer a reasonable, gen-
eral mathematical argument to counter d' Alembert's restriction. He could only
restate his assumption that at any time, "the state of vibrations following depend
on those preceding, and are determined through them; reciprocally by the state
of those following, one can conclude the disposition of those which preceded."
Therefore, if
y = f(x + t.Jb) + f(x - t.Jb),
then, in Euler's general solution, the form of the curve also "will represent the
figure given to the cord at the beginning of the motion."6
Euler rederived d' Alembert's solution but stated that it was defined only within
the interval 0 S x s e. He then explored the properties of his function f, finding
it periodic and odd, and extended the range of his solution to £ S x S 2£, and
so on. If the curve was continuous, "eel-like," it would cut the axis at an infinite
number of points. One such curve was
Jr X 2Jr X 3Jr X
= a sin- + f3sin-- + Y S l l l - - + ...
•
Y
£ £ £
where 2£ was the length of the cord. However, a trigonometric series was not the
most general solution to this partial differential equation since it could not duplicate
the initial shape of the plucked string. 7 In this statement Euler was addressing the
solution to the wave equation of a third protagonist, Daniel Bernoulli.
D' Alembert's paper stimulated Euler to reconsider his ideas on continuity and on
the nature of mathematical functions. For d' Alembert, an analytic representation
of the string may not always be possible. If it was possible, this "generating
curve" in his terminology was arbitrary, periodic, and odd, which meant that it
must represent some real mechanical case. The string was continuous. If it was
not analytically representable d' Alembert claimed "the problem was insoluble."
To be analyzable all the curves representing the string had to be differentiable.
Euler never questioned the form of the wave equation, only the necessity of
limiting the initial, arbitrary form of the curve and hence its final solution to such
"eel-like" functions. In this debate Euler explored discontinuous functions. In
this case he extended functions beyond their original intervals of validity. How-
ever, neither mathematician could definitively demonstrate the validity of their
approach.8 Within the language and concepts available to the calculus in 1750,
things were at an impasse. The polemic became a restatement of successful ex-
tensions of the notion of the function by Euler and the less successful ones on the
part of d' Alembert.
The dispute was complicated by the role d'Alembert was beginning to play in
the politics of the Berlin Academy of Sciences. Euler was acting, unpaid head of
the Academy, a position in which Frederick II would not confirm him. Frederick
the Great's francophilia was a disadvantage the Swiss Euler could never overcome.
While Euler's letters to bureaucrats, diplomats, and the politically powerful were
duly deferential and contain the leaven of Academy gossip, they are straightfor-
ward, if sometimes prejudiced. However, there was no lightness of touch, wit or
bite in his comments, virtues which d' Alembert had in abundance. Euler's posi-
tion in Berlin became more precarious with the Konig affair. When d' Alembert
was approached about becoming the next president of the Berlin Academy, Euler's
comments about him became more strident, and he began the negotiations that
would lead him back to St. Petersburg. The thought of d' Alembert descending on
Berlin, gaining the ear of Frederick the Great and disposing of the Academy at will
was too much to bear. From Euler's point of view, even though d' Alembert did
not become president, "some other Frenchman would.,,9
8 See Euler, "Sur la vibration." D' Alembert's reply to Euler's initial critique is in d' Alem-
bert, "Addition au memoire sur la courbe que forme une corde tendue, mise en vibra-
tion," Mem Acad. Sci. Berlin 6 (1750): 355-360, where he in turn critiqued Euler's
first paper. He reiterated his definition of the necessary geometric continuity for the
curves in d' Alembert, "Recherches sur les vibrations des cordes sonores," in Opuscules
Mathematiques (1761), vol. 1, 1-73. Euler's rejoinder appeared as Euler, "De motu
fili ftexilis, corpusculis quotcumque onusti,"Novi Comm. Acad. Sci. Petropolitanae 9
(1762-63): 215-245.
9 See Euler to Milller, April 1763, in Paul Heinrich von Fuss, compo Correspondance
mathematique et physique de quelques celebres geometres du xviii e siecle (New York;
Johnson Reprint of 1843 edition, 1968), vol. 2, p. 215. For Euler on d' Alembert, see vol.
Eighteenth-Century Mechanics 35
The importance of this dispute here is that the arguments were over mathematical
questions raised in solving a partial differential equation. The origins of the equa-
tion lay in a physical problem, but its solution did not. The focus of d' Alembert's
and Euler's attention, the fights over authority, and jostling for position were all
in the discipline of mathematics. The social locus of the solution lay within the
Berlin Academy and patronage of the Prussian state. From this time on most of
d' Alembert's and some of Euler's mathematical energies, apart from occasional
skirmishes over the vibrating string problem, were turned to exploiting partial
differential equations, the new area of the calculus so vividly illustrated by the
vibrating string problem. lO
Ignoring Physics
D' Alembert, Euler, and later Lagrange continued to argue over the mathematical
issues surrounding the wave equation. Unanimously they ignored an alternative
vision of the problem and its solution put forward by Daniel Bernoulli. Bernoulli
stood apart from the other mathematical members of his clan, and from other math-
ematicians and physicists of his generation. He also had a long, and complicated
relationship with Euler. Bernoulli's work encompassed experiments on vibrating
bodies of many kinds from which he teased their equations of motion. In this he
was more successful as an experimentalist than as a mathematician. As his ex-
periments developed, they provided him with solutions to the equations of motion
of some elastic bodies that were deeply at odds with those of his mathematical
peers. Despite these differences, Bernoulli's work on vibrating strings cannot be
separated from that of mathematicians concerned with figures of equilibrium and
the motion of flexible bodies in general, or from the work on the mathematical
analysis of musical instruments.
Early in their mathematical careers Bernoulli and Euler published and corre-
sponded with one another on the catenary problem. They both tried to find the
principle analogous to the isoperimetrical solution to the catenary problem for
other flexible bodies. Bernoulli suggested that for "elastic lamina," some power of
the radius of curvature must be a minimum for the lamina to hang in equilibrium.
2, p. 71. For his style, see also his letters to Miiller on candidates for the St. Petersburg
Academy, and his letters to the chaplain to Frederick, Prince of Wales in vol. 3.
10 D' Alembert's later mathematical work appeared in nine volumes, d' Alembert, Opus-
cules Mathematiques, (Paris, 1761-1780). See Demidov, "Creation," for d'Alembert
and partial differential equations. Euler's work on partial differential equations culmi-
nated in Euler, Institutiones Calculi Integralis, (St. Petersburg, 1770). See Demidov,
"The Study of Partial Differential Equations of the First Order in the Eighteenth and
Nineteenth Century," Arch. Hist. Exact Sci. 27 (1982): 325-350. A short account of the
history of partial differential equations is in J. Liitzen, "Partial Differential Equations," in
Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences,
Grattan-Guinness, ed. (London: Routledge, 1994),2 vols., vol. 1: 452-469.
36 Vibrating Strings
The discussion led him to propose that the vis viva of the body must be a minimum.
Not being able to demonstrate this, Bernoulli asked Euler for help.ll
From his experiments Bernoulli had already deduced the oscillations of a ver-
tically hung chain. Through a geometrical analysis of their oscillating forms, he
reached the idea of simple modes and proper frequencies. 12 His first published
account of this work, a mathematical paper, contained a number of theorems on
the oscillations of bodies hung along a horizontal, flexible thread. He assumed
Taylor's condition and expressed his results in the form of the length of a simple
pendulum oscillating with the same period as the body under examination. Taking
two bodies hung on a string, Bernoulli illustrated two possible types of motion.
This expanded to three with three bodies suspended along the string. He then
leaped from this case to a heavy chain, arguing that the chain moved in an infinity
of ways to execute the vibrations he observed in his experiments.
These and his other "theorems" were assertions, without proofs, along with
"Scholia" that were paragraphs on his experiments that illustrated his assertions.
This method of argument was a less rigorous form of Newton's techniques in the
Opticks. For the chain, Bernoulli characterized each motion by the number of
points of intersection of the chain with the vertical during its motion when the
excursion of the free end is at its greatest. He also drew an analogy between this
motion and that of a taut string, concluding that "experiment shows that in musical
strings there are intersections similar to those of vibrating chains."
Bernoulli's theorems were geometrical descriptions derived directly from exper-
imental data, although they were written as if they were affirmations of theoretical
results. In his second paper Bernoulli used these geometrical results and applied
them to the case of two, then three bodies. He assumed the system was constrained
and used the balance of forces to find their accelerations. From this and Taylor's
condition, Bernoulli deduced their displacement from the vertical. At any point
along the string, measured from the top down, the angle of contact was d 2 y / ds 2 ,
where y was the horizontal displacement. The accelerating force became
f dy dy
d ds - (i - s)d ds 2
which was set proportional to y. Changing variables from s to x = i - s, where
i was the length of the string, Bernoulli obtained a linear differential equation
n dydn +nx ddy = -y ddx,
----------------------------
11 This discussion began in correspondence after Daniel Bernoulli left St. Petersburg and
returned to Basel. See Fuss, Correspondance, vol. 2 and Enestrom, "Der Briefwech-
sel zwischen Leonhard Euler and Daniel Bernoulli," Bibliotecha Mathematica, series 3
(1906-1907): 126-156.
12 Daniel Bernoulli, "Theoremata de oscillationibus corporum filo ftexili connexorum et
catenae verticaliter suspensae," Comm. Acad. Sci. Imp. Petropolitanae, 6 (1732-33):
108-122, and, "Demonstrationes theorematum suorum de oscillationibus corporum filo
ftexili connexorum et catenae verticaliter suspensae," same journal, 7 (1740): 162-173.
Eighteenth-Century Mechanics 37
where n was a proportionality constant. This was the form of the curve. There
was no sense in his solution that the shape of the curve changed in time.
Even before the publication of Bernoulli's second paper, Euler became interested
in the problem and analyzed the case of a flexible, loaded string in the same way.
He reiterated Bernoulli's results and mathematically went beyond them. 13
Thus far, neither Bernoulli nor Euler caught the dynamical character of the
phenomenon. Although Euler's mathematical vision was clearer, Bernoulli un-
derstood that the principle of proper modes applied to all vibrating systems. He
was already concentrating on experiments with musical instruments. By Septem-
ber 1736 he claimed that, "I have quit mathematics almost completely, and if it
was not demanded by my relationship with the Academy [St. Petersburg] I would
abstain completely." Even allowing for hyperbole, this reflects the direction of
his work. In his letters to Euler he discussed experiments and mathematical ex-
pressions that emerged from them rather than purely mathematical issues. In the
eighteenth-century experimentalists, such as Bernoulli and their results established
the expressions from which emerged algebraic relationships that mathematicians
then manipulated into "equations of motion." These equations of motion offered
new opportunities to explore and extend the ca1culus. 14
Bernoulli continued to report his own mathematical findings on the oscillations
of strings, while Euler attacked the problem of elastic bodies through their equations
of motion. At each point Bernoulli asked Euler's opinions of his ideas and Euler's
solutions to problems of mutual interest. Initially, the tone of the letters was
informal and an easy continuation of their relationship in St. Petersburg. Gossip
was in French, with German as the main language that merged into Latin for
technical discussions. However, their interaction was more complex than one
of long-distance cooperation. Both were intensely ambitious and rivals for the
same prizes, literally and figuratively. The correspondence contained statements
of problems and results, no methods. The latter were matters for publication. IS
The increasingly acrimonious rivalry between Daniel Bernoulli and his father
13 Euler, "De oscillationibus fiJi ftexilis quotcumque ponduscuJis onusti," Comm. Acad. Sci.
Imp. Petropolitanae, 8 (1736): 30-47. Details of Euler's derivation are in Truesdell,
"Rational Mechanics," 162-165, and H. F. K. Burckhardt, "Entwicklungen nach oscil-
lirenden Functionen und Integration der differential Gleichungen der mathematischen
Physik," Jahresber. dtsch. Math. Verein, 10 pt. 2 (1901-1908).
14 Bernoulli to Euler, September 1736, in Fuss, Correspondance, vol. 2 p. 434. See also
Bernoulli to Euler, March 1739, Fuss, vol. 2, p. 456. In his November 1740 letter to
Euler Bernoulli discussed the fundamental frequencies of pipes of different lengths, Fuss,
vol. 2, p. 465. O. B. Sheynin also argues that Euler treated physical problems as purely
mathematical ones. See Sheynin, "Euler's Treatment of Observations," Arch. Hist. Exact
Sci. 9 (1972): 45-56.
15 This correspondence is important for establishing directions of research and when each
participant actually reached certain results. Publication of papers could be delayed up
to a decade depending on the sponsoring Academy. Their order of publication does not
38 Vibrating Strings
Johann complicated matters further, especially after Daniel learned through Euler
of some his father's hydrodynamic work. Father and son sought Euler's support
in their dealings with one another and Euler tried to treat them evenhandedly. Sci-
entifically, they were both useful to him. Although Daniel Bernoulli's treatise on
hydrodynamics had appeared years earlier, Euler sent his opinions on Daniel's Hy-
drodynamica to him within days of doing the same for Johann on his Hydraulica. 16
Daniel Bernoulli respected Euler and usually accepted his mathematical opinions
without demur. However, he considered Euler's work "abstract." It did not de-
scribe the real world. And, over the theory of the oscillations of vibrating systems
and musical instruments, there was a profound rupture between the two men.
Much of the Euler-Daniel Bernoulli correspondence was on the mutually inter-
esting problem of the vibrations of "elastic lamina" fixed horizontally at one end.
Bernoulli asked Euler whether the vis viva for elastic lamina must be a minimum or
would the body move spontaneously? Euler replied that such curves must follow
an extremum principle but what he wanted from Bernoulli was how to determine
"the quantity of potential forces that lie in the bendings" of the lamina. 17 It is clear
here, and elsewhere, that the two depended on one another, Bernoulli on Euler for
guidance with his mathematics, Euler on Bernoulli for his understanding of the
physical phenomenon from which he began his mathematical explorations. How-
ever, the two together did not produce a mechanical understanding of the vibrations
of "lamina," in which physical theory, expressed in the language of mathematics,
was joined to experiment, and where the direction and depth of the mathematical
development was bounded by the needs of the physical imagery. As their work
developed they saw each other as rivals and stood more and more in opposition to
each other. They offered alternative solutions, each of which excluded the other.
By 1740 Bernoulli experimentally had distinguished the simple motions of
already available mathematical solutions of Euler and d' Alembert. He was sending
this paper to the Berlin Academy.2o
There were other differences. Euler wanted to deduce the curve of a lamina
under load, a static problem, and trace its elasticity until its breaking point. Ber-
noulli was interested in the motions of laminae, not their equilibrium properties.
Mathematically he followed their motions using the balance of moments. The
goals of his analysis were ultimately to deduce the form of the curve they traced
as they moved isochronously, and hence obtain their frequencies of oscillation and
compare them with the frequencies of a simple pendulum. Bernoulli deduced the
form of the curve from the differential form,
where G was the force on the lamina, c was the amplitude of the oscillations at the
free end of the lamina, and m a constant determined by experiment. The y axis
ran along the lamina, with displacements in the x direction. Bernoulli gave two
solutions to this differential equation, one in series form,
where 1/[4 = G / m 4 c. Using special conditions, Bernoulli obtained values for the
coefficients in each term and from these deduced the result that the frequency of
the oscillations of the lamina varied inversely as the length of the lamina squared.
He then compared his mathematical results to the clear tones emitted by lamina
In his letter he explicitly asked whether this solution contained all possible "curves."
It is clear that Daniel Bernoulli thought so because through the use of arbitrary
constants the curves would pass through "any points we please." Here was a com-
bination of a physically determined, mathematically flexible solution. Inelegant,
clumsy but in this case it worked. 23
The virtues of Bernoulli's work were experimental and physical, while the math-
ematical analysis was unsatisfactory. One can sympathize with Euler's neglect of
Bernoulli's mathematical work. Bernoulli's papers were usually in the form of
theorems, whether the subject matter of the theorem was mathematical or experi-
mental. Proofs might or might not follow. On the theorems he then hung a series
of corollaries in which he explored the implications of the original theorem. The
scholia were descriptions, tantalizingly brief, of experiments where he demon-
strated the results of his theorems and their implications. Bernoulli gave no data
or the formulae that transformed that data into the results confirming the theorems.
He sketched a general method and the confirmation of the theoretical results. The
"geometrical" aspect, the "mathematical proof" of the theorems were in separate,
usually later papers and mathematically as incomplete as the descriptions of the
experiments. 24 His mathematical abstractions are geometrical descriptions of ex-
perimental situations. In all his proofs the concrete particulars of the experiment
are clearly before the reader. His mathematical methods are clumsy, limited, and
inadequate to the task, although suggestive and physically correct.
His father Johann, brother Nicholaus, and Euler clearly were in a different math-
ematicalleague. Time and again Daniel Bernoulli could present physical problems
to them in a form that highlighted the mathematical implications of the physical
experiments he was currently engaged in. At the same time he would describe
his experiments that impinged on the mathematical solutions to the equations of
motions that emerged from his earlier experiments that demonstrated the truth of
his mechanical ideas. Often he left the mathematics in mid course, as in the case
of a vibrating chain, and asked Euler to complete it. However, Bernoulli's usual
deference to Euler's authority broke down in the case of vibrating strings.
On his part, in the case of the vibrating string, Euler could not reconcile Ber-
noulli's physical insights with the promise of d' Alembert's mathematics. Euler
tried to retain all the abstract, general, yet physically arbitrary quality of his own and
d' Alembert's work, while admitting that it would be embarrassing to be "pulled
down by a simple physical consideration." To retain his own mathematical so-
lution, he argued that Bernoulli's solution, in terms of a trigonometrical series,
was incomplete. He also indicated that, when the number of terms in Bernoulli's
trigonometrical series became infinite, it seemed doubtful that the curve of the
string would consist of an infinity of sine waves. Each term clearly was indepen-
dent of the others. Euler also could not accept a physical limitation to the solution,
of what was to him, a mathematical problem. Bernoulli's infinite sine series,
nx 2nx 3nx
y = a sin-
a
+ f3sin--
a
+ y sin--
a
+"',
did not include all the possible curves. In addition, the initial curve of the string,
qui peuvent coexister dans un meme sysU:me de corps," same journal: 173-195.
24 For example see, Bernoulli, "Demonstrationes theorematum."
Eighteenth-Century Mechanics 43
as it was plucked, was quite arbitrary and the above equation might never reduce
to that initial form.
These remarks followed Bernoulli's papers in the same Academy journal where
Bernoulli set out his sine-series solution, his experiments and illustrations of su-
perposition. Bernoulli declared that a series made up of sine curves was the only
expression that represented simple, isochronous, regular vibrations of the kind Eu-
ler and d' Alembert claimed to establish in their general, mathematical solutions.
All vibrating strings, fixed at both ends, must move either in their fundamental
mode or in an overtone, of which there were an infinite number, or in a mixture of
these motions. As he understood them, Euler's and d' Alembert's solutions were
simply made up of mixtures of the type that all conform to the conditions set out by
Taylor. However, Bernoulli could not demonstrate these contentions mathemati-
cally and could only argue by analogy, using the evidence gleaned from musicians
and his own work on the behavior of musical instruments. 25
While claiming to admire the mathematical abilities of both d' Alembert and
Euler, Bernoulli described their analyses as "arbitrary" and apt to surprise rather
than to enlighten. They paid no attention to the simple vibrations of cords, or to
the actual motions of real cords. The latter die away quickly unless they move as
he, Bernoulli, had described. In particular he could not grasp what d' Alembert
"intended to say with his infinitely, isochronous vibrations and curvatures," par-
ticularly since he was always so abstract and never gave specific examples. While
finding in the two mathematicians' papers the "most profound analysis," they ap-
peared to Bernoulli arbitrary, "without synthetic examination of the question, and
they have not led to its clarification." The actual vibrations of strings confirmed
his trigonometric solution to the problem and by experiment he had confirmed the
"mixture of vibrations in one and the same sonorous body which are absolutely
independent of one another." He went on to compare this principle with that of the
"composition of motion."
For Bernoulli, mathematics had become an adjunct to experiment. In a letter
to Euler during this controversy, Bernoulli closed the matter by saying that he
"took his proofs from nature, not some principle of analysis." His understanding
of the phenomena restricted the possible mathematical solutions of the equation of
motion for the vibrating string.
Such restrictions never entered into the mathematical discussions of d' Alembert
or of Euler. Bernoulli could not make his assertions about the results of his ex-
periments, the motions of strings, and the sounds of musical instruments mathe-
matically sensible. He could not, therefore, change the terms in which the two
mathematicians carried on their debate. They could both safely ignore him. In
later papers Bernoulli could only reiterate his belief in his trigonometric series
solution, and that it was the general one since it contained "an infinity of arbitrary
quantities that make the curve pass through any given points that you wish, and
one can identify this curve to one proposed by them to any degree of precision as
one wishes.,,26
The different approaches of Euler and Daniel Bernoulli are illustrated further in
their exchanges over the science of music. Bernoulli thanked Euler for his treatise
on the subject but noted that his mathematical deductions were not recognized by
musicians or experimentalists, like himself, and did not represent the harmonies
actually in use. He then went on to give Euler a lesson on the actual tuning of
musical instruments. 27
For Euler and d' Alembert, the solution to the problem of the vibrating string re-
volved around the meanings of the terms "continuity" and "function." D' Alembert
required the function representing the curve of the string to be geometrically con-
tinuous and analytical in form. Only such a function could be a "general" solution
and contain all other possible solutions. Euler's criteria for the function was less
rigorous, but mathematically more imaginative. The functions might be contin-
uous only in certain intervals, or might be any arbitrary curve that might not
meet d' Alembert's criterion of differentiability.28 The debate was argued solely in
mathematical and metamathematical terms. They both abandoned further physical
considerations of the vibrating string as a guide to the solution of the equations
of motion. Reference to the physical problem that was the seed bed for this
mathematical challenge was abandoned. Daniel Bernoulli's suggestion of using
a trigonometric series solution to the wave equation already carried mathematical
26 Bernoulli, "Lettre de M. Daniel Bernoulli aM. Clairaut au sujet des nouvelles decouvertes
faites sur les vibrations des cordes tendues," J. des S~avans (March 1758): 157-166, p.
157. See also, Bernoulli, "Memoire sur les vibrations des cordes d'une epaisseur inegale,"
Mem. Acad. Sci. Berlin 21 (1765): 281-306, p. 283.
27 H. Floris Cohen, Quantifying Music: The Science ofMusic in the First Stage ofthe Scien-
tific Revolution (Boston MA: Reidel, 1984) notes that Euler's theory of consonance and
those of other mathematicians was "largely abstracted from physical and physiological
considerations and went back to operations with number," p. 237. There was no physical
theory of consonance until the work of Hermann von Helmholtz in the nineteenth cen-
tury. See also Albert Cohen, Music in the French Royal Academy of Sciences (Princeton
NJ: Princeton University Press, 1981) and Jamie C. Kassler, "The 'Science' of Music to
1830," Arch. Int. Hist. Sci. 30 (1990): 111-135.
28 Historians have traced mathematicians' understandings of functions in Jerome R. Ravetz,
"Vibrating Strings," Grattan-Guinness, Foundation, A. P. Iushevich, "The Concept of the
Function," Arch. Hist. Exact Sci. 16 (1976): 37-85, Pierre Dugac, "Des fonctions comme
expressions analytiques aux fonctions representables analytiquement," in Mathematical
Perspectives: Essays on Mathematics and Its Historical Development Joseph Dauben,
ed. (New York: Academic Press, 1981), 13-36.
Eighteenth-Century Mechanics 45
29 During the eighteenth century there was no general proof for the convergence of infinite
series of sines or cosines. Mathematicians sought functional equivalents to such se-
ries. See Grattan-Guinness, Foundations, Victor Katz, "The Calculus of Trigonometric
Functions," Hist. Math. 14 (1987): 311-324, and L. A. and R. W. Golland, "Euler's
Troublesome Series."
30 See Cohen, Music in the French Royal Academy.
31 Joseph Louis Lagrange, "Recherches sur la nature, et la propagation du son," Misc.
Taurin, 1 (1754): 1-112, and "Nouvelles recherches sur la nature et la propagation du
son," same journal, 2 (1760-1761): 11-172. Reprinted in Oeuvres de Lagrange (Paris:
46 Vibrating Strings
Lagrange's work on the problem was the culmination of over a century of math-
ematical attention to vibrating strings. Historically, however, investigations of
vibrating strings cannot be isolated from mathematicians' attention to the motions
of elastic bodies in general or the development of the calculus. 32 Lagrange's work
encompassed the motions of elastic bodies, one of the mathematically simplest of
which was that of the vibrating string. The roots of the modern problem lay in
Marin Mersenne's experiments and his deduction from those experiments of the
algebraic relationship between frequency of sound emitted by the string, its length,
and the tension in the string. Brook Taylor was the first mathematician to deduce
Mersenne's expression from the geometry of the string and the mechanics of con-
sidering it as a row of rigid particles. Arguing, reasonably, without reference to any
specific experiments, Taylor established that when the string's displacement from
the equilibrium position was small, the vibrations of the string were isochronous,
i.e., the acceleration of the string towards its equilibrium position was as its distance
from that position. From this he deduced the "time of vibration," v,
where 2£ was the length of the string, T the tension, and a a constant. He then
constructed the equation of motion for the midpoint of the string, treating it as a
simple pendulum of length U. Using fiuents and neglecting higher terms, Taylor
found the equation for the form of the string,33
. x
y =U Slll-.
a
As well as deducing a form for the curve of the string, Taylor laid down the
assumptions for subsequent mathematical derivations. However, there was no
search for a curve that changed in time. The derivation was static.34
While Taylor remarked that the shape of the curve of the string was a sine curve,
he did not use this physical result in any way to restrict his mathematics. Yet
35 See, John T. Cannon and Siglia Dostrovsky, The Evolution of Dynamics: Vibration
Theory from 1687 to 1742 (New York: Springer-Verlag, 1981).
36 Johann I Bernoulli, "Theoremata selecta pro conservatione virium vivarum demonstranda
et experimenta confirmanda excerpta ex epistolis datis ad filium Danielem," Comm.
Acad. Sci. Imp. Petropolitanae, 2 (1727): 200-207. Proofs appeared in the following
volume in Bernoulli, "Meditationes de chordis vibrantibus, cum pondusculis aequali
intervallo a se invicem dissitis, ubi mimirum ex principio virium vivarum quaeritur
numerus vibrantionum chordae pro una oscillatione penduli datae longitudinis," same
journal, 3 (1728): 13-28.
37 The history of the calculus up to the work of d' Alembert is in Florian Cajori, "The
Early History of Partial Differential Equations," Amer. Math. Monthly 35 (1928): 459-
467. John L. Greenberg, "Mathematical Physics in Eighteenth-Century France," Isis,
77 (1986): 59-78 discusses the early hostility then acceptance of the calculus in Paris,
and in Greenberg, "Alexis Fontaine's Integration of Ordinary Differential Equations and
the Origins of the Calculus of Several Variables," Ann. Sci. 39 (1982): 1-36 explores
the development of the calculus from Johann I Bernoulli and Maupertuis to Alexis-
Claude Clairaut. S. S. Demidov examines the history of partial differential equations in
Demidov, "La naissance de la theorie des equations differentielles aux derivees partielles,"
Proceedings of the xiv International Congress of the History of Science 2 (1974): 111-
113 and in Demidov, "The Study of partial differential Equations of the first Order in
the Eighteenth and Nineteenth Centuries;' Arch. Hist. Exact Sc. 26 (1982): 325-350.
For the overall development of the calculus see, Carl B. Boyer A History of the Calculus
(New York: Dover, 1959).
48 Vibrating Strings
38 See d' Alembert, "Recherches sur la courbe," "Suite des Recherches." For how these
papers fit into the development of the calculus see Demidov, "Creation et developpement
de la theorie des equations differentielles aux derivees partielles dans les travaux de J.
d' Alembert," Rev. Hist. Sci. 35 (1982): 3-42.
39 For details of Lagrange's goals, see Grattan-Guinness, Foundations. For Lagrange's
continuing interest in the foundations of the calculus see Judith Grabiner, The Origins of
Cauchy's Rigorous Calculus (Cambridge MA.: MIT Press, 1981). See also Craig Fraser,
"Joseph Louis Lagrange's algebraic Vision of the Calculus," Hist. Math. 14 (1984): 38-
53, "The Calculus as Algebraic Analysis," Arch. Hist. Exact Sci. 39 (1989): 317-335,
and "Lagrange's Analytical Mathematics," Studies Hist. Phil. Sci. 21 (1990): 243-256.
40 Lagrange's early interest in discontinuous functions also emerged in his correspondence
to Euler. See Lagrange, Oeuvres vol. 14, Correspondance, letters to Euler of August
1758 and October 1759. Mter Lagrange's paper on vibrating strings Euler's work on
discontinuous functions appeared in Euler, "De motu vibratorio fiJi fiexilis, corpusculis
quotcunque onusti," Novi Comm. Acad. Sci. Petropolitanae, 9 (1762-1763): 215-245,
"EcJairissemens sur Ie mouvement des cordes vibrantes," Misc. Taurin, 3 (1762-1763):
Eighteenth-Century Mechanics 49
of sound and in particular, the mathematics of the motions of vibrating strings. His
paper on the subject was thus very complicated as well as discursive. Lagrange
located in Turin, was young, and yet to make his mark in mathematics. He had to
demonstrate that he understood the history of the problem, the points of view of
his predecessors, and their limitations. That is, Lagrange needed to take care of
all his footnotes. To make his new approach to the calculus plausible, Lagrange
needed to transcend d' Alembert's restrictions on functional solutions to the wave
equation and go further than Euler in developing alternative, functional solutions.
Lagrange began, as did his predecessors, by considering a number of weights
strung along a weightless string which was then plucked. Starting with the case
of one body then two then n bodies and letting n ~ 00, he obtained the wave
equation. 41
With a new method of integration Lagrange reached a general solution which
he gave in the form of two integrals of a series of products of sines and cosines.
This result allowed Lagrange to reject d' Alembert's solution. His own ideas on
functions were less restrictive than d' Alembert's. However, Lagrange was also
convinced that series solutions were not legitimate. Therefore, he argued for
Euler's approach, on which he expanded at some length. He was also confident
that Euler had gone only part of the way in exploring discontinuous functions. He
needed to construct a functional solution that would include his series solutions.
In the process he hoped to confound Daniel Bernoulli. His ultimate functional
solutions in the form of exponentials were reached through exuberant displays of
algebraic manipulation.
Both Lagrange and Euler elaborated on their interest in discontinuous functions.
In both cases, pulses running along cords was the illustrative example that launched
these mathematical explorations. 42 In this relationship both gained a powerful ally.
For Lagrange, the support of Euler, the premier mathematician of Europe, was im-
portant at the beginning of his career. For Euler, the support from a mathematician
of such potential was welcome in his battles against the French in Berlin. Lagrange
27-59, "Sur Ie mouvement d'une corde, qui au commencement n'a ete ebranlee que
dans une partie," Mem. Acad. Sci. Berlin, 21 (1765): 307-334, "De chordis vibrationibus
disquisito ulterior," Novi Comm. Acad. Sci. Petropolitanae, 17 (1772): 381-409, "Con-
sideratio motus plane singularis qui in filo ftexili locum nature potest,"Novi Acta Acad.
Sci. Petropolitanae, 2 (1784): 103-120. See also, Lagrange, "Recherches sur la nature
et la propagation du son."
41 Lagrange was not just considering a vibrating string but the propagation of a pulse through
an elastic medium. The vibrating string was a simple example of this general problem.
42 Lagrange was already known to Euler through correspondence. In 1754 they began
corresponding on new methods of calculating maxima and minima as well as other issues
in the calculus. See Craig Fraser, "J. L. Lagrange's Early Contributions to the Principles
and Methods of Mechanics," Arch. Hist. Exact Sci. 28-29 (1983-84): 197-242. See
also, Jean hard, "Lagrange, Joseph Louis," in Diet. Sci. Bio. Charles C. Gillispie, ed.
(New York; Scribners, 1971), 14 vols., vol. 7,559-573.
50 Vibrating Strings
46 On Newton see Greenberg, Shape of the Earth, 119-120. On the more general issue of
the development of the calculus in this era, see chaps. 7, 8.
47 See Maupertuis, "Sur la figure de la terre et sur les moyens que l'astronomie, et la
geographie fournissent pour la determiner," Mem. Acad. Sci. Paris (1733): 153-164,"Sur
la figure de la terre," same journal, (1735): 98-105, and Examen desinteressee des
differents ouvrages qui ont he faits pour determiner fa figure de fa terre (Oldenberg,
1738).
48 Maupertuis, Discours sur fes differentes figures des astres (Paris, 1732), Maupertuis,
"De figure quas Fluida rotata induere possant," Phil. Trans. R. Soc. London (1733):
240-256, and "Loi du repos des corps," Mem. Rist. Acad. Sci. (1740): 170-176. For a
detailed discussion of Maupertuis' mathematics, see Greenberg, "Mathematical Physics
in Eighteenth-Century France," Isis, 77 (1986): 59-78, and Shape of the Earth, chap. 5,
and chap. 7, 243-258.
52 Vibrating Strings
also assumed that these mathematical ellipsoids represented the earth. 49 Clairaut
worked on the problem over a long period of time and his physical understanding
of how to establish his initial equations grew. While in this sense "mechanics
and mathematics interacted in his work," he treated the equations after they were
established solely as problems in the calculus. At the same time he developed
solutions to partial differential equations, opening up more of the possibilities of
the calculus.
That the results from the Lapland expedition did not agree with his mathematical
deductions did not interfere with Clairaut's assumptions or the exercise in math-
ematics that this problem presented to him. Clairaut could only compare the two
through various technical devices. In his opinion there were inevitably errors of
observation which, if properly attended to, would bring the empirical results closer
to his mathematical deductions. He seemed unperturbed by the discrepancies.
This hardly seems the reaction of physicists. Maupertuis saw the issue as meta-
physical and mathematical. Clairaut soft peddled the metaphysics, yet the problem
remained mathematically bounded. Neither tried to find a physical explanation for
the discrepancies their mathematical methods had uncovered. Neither wondered
how the their physical model, a rotating mass of fluid, could be made plausible
as well as mathematically possible. Greenberg sees the problem as physical and
Clairaut as solving a physical as well as a mathematical problem. However, he
notes that Clairaut's practices do not fit either Kuhn's description of belonging
to the mathematical or experimental traditions in physics. Nevertheless, as he
demonstrates, the mathematical solutions to the problem have no bearing on its
solution as physics. That was indeed observational and he does not follow the
observational history of the problem throughout the remainder of the eighteenth
century. Neither of the main actors behaved as we might expect from a physicist, if
we can impose the standards of twentieth century behavior on eighteenth-century
figures. 5o Points of dispute between the contestants here, along with others such
as d' Alembert, were mathematical.
Clairaut reverted to similar arguments in his later commentary on Newton's law
of gravitation. He criticized Buffon's defense of Newton as phenomenological and
dependent on observation. As a mathematician, he was free to explore several laws
There is, of course much more to the history of mechanics in the eighteenth cen-
tury than the vibrating string problem and its solution. As metaphysics, mechanics
gave both mathematicians and experimental philosophers the principles on which
to argue the validity of their solutions to mathematical and experimental problems.
Historians of the calculus agree that, in the eighteenth century, physical problems
were the starting point for a great deal of the construction of new differential and
partial differential equations. 52 The solution of these equations were the means by
which mathematicians developed the calculus. 53
51 Philip Chandler, "Clairaut's Critique of Newtonian Attraction: Some Insights into his
Philosophy of Science," Ann. Sci. 32 (1975): 369-378.
52 As well as Greenberg other historians of mathematics have stated this explicitly such as,
Henk Bos, "Mathematics and Rational mechanics," in Ferment of Knowledge Rousseau
and Porter, eds. 327-355, Grattan-Guinness, Foundations, and From Calculus to Set
Theory, Grabiner, Origins, and, Clifford Truesdell, "The Influence of Elasticity on Anal-
ysis."
53 This begins early. Newton's fluxions emerged from his mechanics. Questions of me-
chanics exercised Johann I Bernoulli and other early developers of the calculus. Of the
mathematicians mentioned in this chapter see, Euler, Mechanica sive motus scientia an-
alytica exposita (St. Petersburg, 1736), 2 vols. The calculus of variation began similarly
in the consideration of mechanical problems by Newton, Leibniz, Brook Taylor, Johann
I Bernoulli then Daniel Bernoulli amongst others. As a branch of the calculus Euler
systematized it in Euler, Methodus Inveniendi Lineas Curvas Maximi Minimive Propri-
etate Gaudentes (St. Petersburg, 1744). For a brief account see Craig Fraser, "Calculus
of Variations," Encyclopedia of Hist. Philos. Math. vol. 1, 342-350. See also H. H.
Goldstine, A History of the Calculus of Variations (New York: Springer Verlag, 1980).
For the technical developments in mechanics see, A. T. Grigorian, "On the Development
of the Variational Principles of Mechanics," Arch. Int. Rist. Sci. 18 (1965): 23-35.
54 Vzbrating Strings
54 Both Lagrange and Euler did both. Their work discussed here led them into new channels
of mathematics which they then explored as mathematics, not as outcomes of the prob-
lems of mechanics. See Fraser, "J. L. Lagrange's Changing Approach to the Foundations
of the Calculus of Variations," Arch. Hist. Exact Sci. 32 (1985): 151-191. Lagrange,
Mecanique Analytique (Paris: 1788) was a text on the the calculus and partial differ-
ential equations pertinent to mechanics. See also S. B. Engelmann, "Lagrange's Early
Contributions to the Theory of First Order Partial Differential Equations, " Hist. Math.
7 (1980): 7-23. Daniel Bernoulli's and Euler's work on the oscillations of lamina led
Euler to a general framework for the solution of n-th order differential equations. See
Demidov, "On the History of the Theory of Linear Differential Equations," Arch. Hist.
Exact Sci. 28 (1983): 369-387. See also Demidov, "The Study of Partial Differential
Equations of the First Order."
55 The first mathematician historians accept as working completely with problems set by
mathematics alone without reference to physical problems was Adrien Marie Legendre.
Eighteenth-Century Mechanics 55
principles were the legitimate starting point for understanding the motions under
discussion.
Because the disputes were metaphysical, their terms lay outside of the particular
problem under discussion, became bitter, and led nowhere. d' Alembert dismissed
the vis viva controversy as a war of words. However, they were words conveying
meanings beyond their supposed descriptions of physical reality.56 The idea of
the Principle of Least Action was based on characteristics of God's creation, not
on any analysis of how bodies move. In Pierre Maupertuis' words, "the quantity
of action necessary to cause any changes in nature are the smallest necessary."
And Maupertuis proceeded to apply this principle to the whole of animate and
inanimate nature. 57 In these metaphysical disputes the physical foundations for
establishing the mathematical problem connected the mathematical solution to
the real world and on this hinged the fundamental correctness of the solution.
Metaphysics mattered.
Once a particular form of an equation of motion was established, mathematicians
would reach that same equation despite differences in metaphysics. However, the
physical content of these deductions of the equations of motion was a hit and
miss affair. Physical arguments might-and at times were-beautifully succinct and
clear. Sometimes they were obscure and difficult to follow, other times they were
fudged. The main point of the exercise was attainment of the mathematical form
of the equations of motion.
The derivation of the equations of motion and their solution were contests over
technical issues as well. Confrontations erupted over the path a mathematician took
from experiment or observation to the equations of motion, rarely the form of the
resulting equations. The next step was to solve them, technically bettering one's
predecessors. And here, as we have seen with Euler, d' Alembert, and Lagrange,
metamathematical principles could enter into determining the completeness of
solutions. In both steps, metaphysical and technical considerations lay within
mathematics. No other kinds of considerations were needed in setting up or in
solving differential equations of all kinds. Above all, there was no return to the
56 For d' Alembert on the vis viva controversy, see Hankins, "Eighteenth Century Attempts
to solve the vis viva Controversy," Isis, 56 (1965): 281-297, and d'Alembert. See also
David Papineau, "The Vis Viva Controversy," Stud. Rist. Phil. Sci. 8 (1977): 111-
142, and J. Morton Briggs, "d'Alembert: Philosophy and Mechanics in the Eighteenth
Century," University of Colorado Stud. (1964): 38-56. For Euler's metaphysics, see
Jean Dhombres, "Les presupposes d'Euler," Rev. Rist. Sci. 40 (1987): 179-202, and
Stephen Gaukroger, " Euler's Concept of Force," Brit. 1. Hist. Sci. 15 (1982): 132-154.
57 For the principle of Least Action see Pierre Brunet, Etude historique sur Ie principe de
la moindre action (Paris: Hermann et Cie, 1938), p. 6, Philip Jourdain, "The Nature
and Validity of the Principle of Least Action," Monist, 23 (1913): 277-293. See also
A. Kneser, Das Prinzip der Kleinsten Wirkung von Leibniz his zur Gegenwart (Leipzig:
Teubner, 1928).
56 Vibrating Strings
mechanical problem that sparked the whole discussion. In generating their general
solutions, mathematicians were not required to interpret the physical implications
of any of their intermediate or final mathematical results. For mathematicians the
equations of motion were mute on what they contained in terms of how mechanical
systems moved. Their solutions contained nothing on the characteristics of physical
systems that they chose to explore.
Daniel Bernoulli's appeals to experiment to guide the solution to the problem
were simply irrelevant. The annexation of the problem and solution by mathemati-
cians is emphasized by their actions once they grasped the mathematical principles
behind linear differential and first order partial differential equations and variational
calculus. They reversed the order of the methods of solution and the problems that
gave rise to them both in the sense of their primacy and their place in their accounts.
Mathematicians produced texts that treated the equations that related to physical
problems as derivative of the mathematical techniques these problems had given
rise to initially. Lagrange's examination of vibrating strings ended debates on
some mathematical aspects of the legitimacy of discontinuous functions. Mathe-
maticians then focussed on discontinuous functions without reference to vibrating
strings except as an illustrative example.
Many mathematicians working on so-called physical problems had as their goal
was the reduction of physics to mathematics. The most well-known of these
was Lagrange, who wanted mechanics to become a new branch of analysis. 58 For
d' Alembert, mechanics was a completely deductive mathematical system untainted
by any hint of experiment. Euler was far more tolerant of experiment and more
attentive to the results of experimentation, but his work on physical, astronomical,
and engineering problems was mathematical and deductive. 59
Euler and other mathematicians used experiment to determine the initial con-
ditions of a problem from which the equations of motion would then be derived.
In the example chosen here, the experimental starting point was Mersenne's rela-
tionship between frequency of sound, tension in a plucked string, and its length.
However, the derivation of the equations of motion were matters of analysis, based
on metaphysical principles that were less, rather than more, connected to observa-
tions of the actual motions of bodies. The instance in which experiment had the
most effect on eighteenth-century mathematicians were the collision experiments
of Willem Jakob van 'sGravesande. Yet the mathematicians' discussions of how
to express the results of those experiments in mathematical form focussed on the
metaphysical issue of the notion of "force" and its mathematical expression. 60
matics, not physics. Thus, a whole series of considerations, tasks and skills taken
for granted as physical theory in the late nineteenth and twentieth centuries were
absent. They were clearly not part of the technical discourse of the eighteenth or
the early nineteenth-century physics.
In the eighteenth century, despite the excitement over electrostatic and other
phenomena, mechanics also had a place within experimental physics. 61 In the
experimental sciences, mechanics was the focus of some steady attention from
Daniel Bernoulli, Ernest Florens, Friederich Chladini, Jean TMophile Desaguliers,
van 'sGravesande, James Riccati and George Atwood, among others. However,
the purpose of these experimenters and lecturers on mechanics was not to produce
physical theory. Of these, Desaguliers, van 'sGravesande, and Atwood intended
to reach audiences beyond the community of other experimental practitioners.
Desaguliers and later Atwood worked to demonstrate the principles of Newton's
mechanics. Desaguliers went further. From demonstrating Newton's principles,
he progressed to mechanical devices and hence to manifesting the usefulness of
Newton's mechanics. 62
Atwood was not the first to try and develop a purely experimental approach to
mechanics, nor was he unique in Europe in offering university courses on New-
ton's mechanics featuring experiments demonstrating Newton's laws of motion.
His lectures in natural philosophy were well attended and leavened by experiments
in mechanics, optics, and electricity. However, there was no indication of linking
his experiments to physical theory in the modern sense. 63 Atwood's texts were not
mathematical except when analyzing practical problems. His was a mathematics
to display, not to explain, nature as were his experiments. With display the ex-
perimental and mathematical enterprise ceased. However, he did try to develop a
general theory of measurement and mathematics as an adjunct to experiment, not
65 See W. D. Hackmann, "The Relation between Concept and Instrument Design in Eight-
eenth-Century Experimental Science," Ann. Sci. 36 (1979): 205-224.
66 In France such experimental forms of mechanics were more likely to be met in courses
designed for engineers. See, Grattan-Guinness, "Varieties of Mechanics by 1800," Hist.
Math. 17 (1990); 313-338, 321-322, see also, C. C. Gillispie Lazare Carnot, Savant:
(Princeton NJ: Princeton University Press, 1971).
ics within its covers.68 Mathematics was the means only of displaying the results
of experiment more precisely. Mathematized theory is remarkable for its absence
rather than its presence in this or any other form.
Interpretations culled by experimentalists from the results of their work were un-
touched by higher mathematics. The language of explanation was the vernacular.
While using the same explanatory terms, force, vis viva etc., these explanations
could conflict with the constructions of mathematicians using the same terms.
Meanings, expressed in the vernacular and explanations ranged from the phe-
nomenological to the metaphysical were only tenuously connected to the results
of experiment through various rhetorical devices. Illustrative examples of how a
mechanical system might work in situations that through analogy mirrored those
of the experiment sufficed as the connection between the minute particles or the
various ethers that were the source of mechanical activity and the real bodies of the
laboratory. The two levels, one of the imagined world of theoretical entities and
the material world of the laboratory were tied together only tenuously. In addition
to illustrative example, metaphor and analogy served the function of connecting
what later would be bound together by the ties of mathematics. And the density
of experimental evidence drawn into this explanatory net could be very thin.
Daniel Bernoulli was rare because he was an experimentalist who could follow
through the maze set by the mathematicians and establish, using his continuing
work on vibrating bodies, whether or not, that mathematics spoke to the results
of his experiments. Most experimentalists could not. Their methods defined a
methodological and explanatory space that precluded such mathematics. When
used, mathematics was subordinate to the primacy of experiment and vernacular
explanation. There was, therefore, an experimental mechanics whose practitioners
worked independently of the mathematicians.
Here we have distinguished experiments and experimentalists in mechanics from
mechanical philosophy and philosophers which, in the eighteenth century, is well
nigh impossible. Metaphysical messages within the interpretations of experiments
were to the fore. This was especially true in lectures addressed to students and the
socially broader audiences of the experimentalists. Discourse within the world of
practitioners was about metaphysical and technical matters. The urgency of the
metaphysical truths over which they battled referred to the world beyond that of
the practitioners themselves. For some experimental craftsmen, the realm of ex-
planation of their metaphysics stayed within experimental philosophy. For others,
the implications of their ideas was far broader and the rightness of their ideas more
urgent. As has been argued elsewhere, it was through experimental demonstrations
that the new philosophy entered the reconstituted cultures of late seventeenth and
early eighteenth century Europe. However, the ideological content of these exper-
iments was not inherent in the metaphysics and experiment until those connections
were stated explicitly. Whether these connections were ever made depended on
both lecturer and audience. And, such connections led in opposite explanatory
directions ranging from the proof to the denial of God's existence as one example.
Nature and the natural had functioned before the seventeenth century, and still
did, to justify and even constitute a theory of the political, social, or economic
order. Whether those interpretations were imposed on nature depended upon the
expectations of both audience and speaker.
Therefore, mechanical interpretations aimed at other practitioners or broader au-
diences could range from descriptive, phenomenological discussions of experiment
to mechanical philosophies arranged independently of mathematics or experiment
for extra-scientific purposes. Many of these speculative metaphysical essays were
written to give voice to God's structure of his universe, to prove God's existence,
or to develop a metaphysics without calling on the necessity for a creator. The
"systems" ranged from the radical materialism of Julien Offray de la Mettrie, Denis
Diderot, or Baron d'Holbach to those of Christian Wolff and the Scottish Com-
mon Sense philosophers. There were also the excursions into natural theology
by British Newtonians and the continental materialists George Louis Buffon and
Albrecht von Haller. 69 Mathematicians could join in as well. Euler's letters on
natural philosophy used non-technical language free of mathematics. The little
mathematics that did enter, for example, the geometry of optical systems, was de-
scriptive. It was not used in interpretation of the nature of light. 70 In these works on
mechanical philosophy, mathematics had no place. A description of the operations
of nature mayor may not depend on evidence from experiments.
Mathematics could also be put to use for extra scientific purposes, although with
less success than that of experimental mechanics. 71 What therefore was meant by
69 This seems like a mixed bag but these authors fit the criterion of being more concerned
with developing a speculative system to meet extra scientific goals than bringing in the
details of experimental or observational data. The list could be extended. The literature
on such important thinkers for the eighteenth century is immense and in a study of this
size they are reduced to the status of name dropping.
70 Leonhard Euler, Letters on Different Subjects in Natural Philosophy, addressed to a
German Princess 2 vols. (New York: Arno reprint of 1833 edition). This was not a
popularization as depicted in Waiter D. Wetzels, "Popularization of the New Physics:
Euler's Letters," Stud. Voltaire Eighteenth Cent. 264 (1989): 796-800. Their metaphys-
ical and "antique" character is discussed in John Heilbron, Electricity in the Seventeenth
and Eighteenth Centuries: A Study of Early Modern Physics (Berkeley, CA: University
of California Press, 1979), p. 72-73. See also Casper Hakfoort, Optics in the Age of
Euler: Conceptions of the Nature of Light (Cambridge: Cambridge University Press,
1995).
71 See Tore Frangsmyr, "Mathematical Philosophy," in The Quantifying Spirit in the Eigh-
teenth Century, Frangsmyr, Heilbron and Robin Rider, eds., 27-44.
62 Vibrating Strings
mechanics depended on the author and his audience. If, as with Euler, the author
used mechanics for a variety of purposes, the ideas expressed to one audience
are not necessarily traceable in the ideas labeled mechanics addressed to another
audience. However much we might try to find unity in this situation, we have to
settle for fragmentation.
Despite the range of audiences and purposes to which it was put, mechanics
developed within two distinct communities of practitioners; experimentalists and
mathematicians. When they addressed other colleagues within their communities,
separate sets of criteria were used to define problem choice, methods, language
and terms that defined when a solution had been reached. Overlap between these
communities was minimal. Both disciplines were self-sufficient. If boundaries
were transgressed, the violations were either ignored or invalidated, the violator of
borders ridiculed, their work discounted.
This is just one example of the eighteenth-century division of labor within the
sciences. We now need to examine the depths of this partition, how it functioned
across other areas of experimental philosophy, and its social manifestations. We
can then examine the implications of this different disciplinary geography of the
study of nature for the development of modern theoretical physics.
Chapter III
Eighteenth-Century Physics
and Mathematics: A Reassessment
natural sciences to physiology.! Physics was the whole domain of research and
natural knowledge gained through experiment and observation. Experimental phi-
losophy, or physics, included, yet was distinguished from, physics "narrowly"
defined as the experimental investigation of light, sound, electricity, magnetism
and mechanics. This narrower definition of physics did not confine the inves-
tigator to the twentieth-century meaning of these terms. Research in light, for
example, could include the anatomy of the eye and the perception of color. For
Johann Heinrich Lambert, the study of light included the physiology of the eye
and the psychology of vision. The study of sound included music, the structure
of the ear, and the perception of sound. 2 Electricity was investigated in all its
manifestations, atmospheric and physiological, as well as those effects produced
in the laboratory. Henry Cavendish's research into electricity also included the
investigation of electric eels and "fishes.,,3 The study of lightning was central to
the developing understanding of quantity of electricity and then of potential in
the middle of the century. Luigi Galvani's focus on the reactions of frogs legs to
atmospheric electricity was not unique. 4
These interests and their place in physics are confirmed by later work in the nine-
teenth and early twentieth centuries. As uncovered by Susan Cannon, the study of
the earth became an organizing principle for some physicists in the 1830s. This
movement is a continuation of the earlier, broader understanding of physics. Her-
mann von Helmholtz's work on sound, accepted by physicists, was encompassed
within this broader sense of physics. In the Cavendish Laboratory, under J. J. Thom-
1 See Chapter I, note 25 for the changes in meaning of the term Physics in the seventeenth
and early eighteenth centuries.
2 Johann Heinrich Lambert, Photometria, 3 vols., E. Anding, trans. in Ostwald's Klassiker
series (Leipzig: 1892. See also the work of Thomas Young. See Chapter VI for more
details on these aspects of physics in the late eighteenth century.
3 Henry Cavendish, "Some Attempts to imitate the Effects of the Torpedo by Electricity,"
Phil. Trans. R. Soc. London, 66 (1776): 196-225, in Electrical Researches ofHon. Henry
Cavendish James Clerk Maxwell, ed. (Cambridge: Cambridge University Press, 1879),
194-215.
4 John Heilbron in his important study, Electricity, constrains this narrower meaning of
the term physics by including much of the study of meteorological, and physiological
electrical phenomena and electricity in animals and fish only as they were brought into
the laboratory and were necessary to narrate the conceptual developments in the field.
His physics is too modern. Luigi Galvani and his work on frogs appears from a narrative
vacuum. This has been noted also by Robert Palter, "Some Impressions of Recent Work
in Eighteenth-Century Science," Hist. Stud. Phys. Sci. 19 (1989): 349-401. As an
antidote see, Marcello Pera, The Ambiguous Frog: The Galvani-Volta Controversy on
Animal Electricity, trans. Jonathan Mandelbaum (Princeton NJ: Princeton University
Press, 1992) and Giuliano Pancaldi, "Electricity and Life: Volta's Path to the Battery,"
Hist. Stud. Phys. BioI. Sci. 21 (1990): 123-160.
Physics and Mathematics 65
5 See Susan Faye Cannon, Science in Culture: The Early Victorian Period (New York:
Science History Publications, 1978), and Peter Galison and Alexi Assmus, "Artificial
Clouds, Real Particles," in The Uses ofExperiment: Studies in theNatural Sciences David
Gooding, Trevor Pinch and Simon Schaffer, eds. (Cambridge: Cambridge University
Press, 1989),225-274.
6 The beginnings of experiment in physics as an explanatory form are in R. H. Naylor,
"Galileo's Experimental Discourse," in Uses of Experiment, Gooding, Pinch and Schaf-
fer, eds. 117-134. In the same volume Willem D. Hackmann, "Scientific Instruments:
Models of Brass and Aids to Discovery," 31-65, discusses the explanatory role of ex-
periment and the function of instruments in that role. See also Jan Golinski, "Precision
Instruments and the Demonstrative Order of Proof in Lavoisier's Chemistry," in Osiris 9
(1994): 30-47.
7 For the case of electrostatics, see Heilbron, Electricity, chap. XIX.
8 See Heilbron, Electricity, on van Marum's work on electrostatic machine.
9 The urge to quantify in the eighteenth century is examined in The Quantifying Spirit in
the Eighteenth Century Tore Frangsmyr, John Heilbron and Robin Rider, eds. (Berkeley:
University of California Press, 1990).
66 Eighteenth Century
10 For the simplicity of the mathematics used and what experimentalists actually did with
it in the eighteenth century see, Theodore S. Feldman, "Applied Mathematics and the
Quantification of Experiment: The Example of Barometric Hypsometry," Rist. Stud.
Phy. Sci. 15 (1985): 127-195, and John Heilbron Weighing Imponderables and other
quantitative Science about 1800 Supplement Rist. Stud. Phy. Sci. 24 (1993), chap. 2.
11 See Heilbron, Electricity chapter XIX.
12 See W. D. Hackmann, "The Relationship between Concept and Instrument Design in
Eighteenth Century Experimental Science." and "Instrumentation in the Theory and
Practice of Science: Scientific Instruments as Evidence and as a Aid to Discovery," and
Albert van HeIden and Thomas L. Hankins, "Introduction: Instruments in the History
of Science," Osiris, 9 (1994): 1-6. Allan Franklin, The Neglect of Experiment (New
York: Cambridge University Press, 1986). For a thorough examination of experimental
practices see Peter Galison, Row Experiments End (Chicago: University of Chicago
Press, 1987). Heilbron, Electricity assumes such prior theoretical structures always
exist. This issue touches upon the question of "tacit knowledge," or whether there are
other ways of investigating nature that yield rational order but are not subject to the
limited methodologies of philosophers. See, Edwin T. Layton, "Mirror Image Twins:
the Communities of Science and Technology," Techn. Cult. 12 (October 1971): 562-580
on different ways of "knowing."
13 Simon Schaffer, "Natural Philosophy and Public Spectacle in the Eighteenth Century,"
Rist. Sci. 21 (1983): 1-43.
14 See Schaffer, "Machine Philosophy."
15 Heilbron, Electricity, has shown the crucial role of electrostatics in the development of
the domain of physics. This is the reason for taking it as an example here. From the
masses of this material in the eighteenth century, we focus only on those examples usually
seen as steps in the development of mathematized, physical theory.
Physics and Mathematics 67
ical form, developing an algebraic expression for the repulsive force of a charged
plate on a particle of electric fluid. His mathematics remained largely as algebra
and appeared after his own vernacular argument explaining the meaning of the
phenomena under discussion. The mathematics illustrated his physics. It was not
the means for developing his physical argument.
Aepinius' attempt illustrates some of the missing steps that were needed to
join theoretical ideas, expressed in mathematical form, and to develop those ideas
through the mathematics into a physical theory whose deductions can be joined to
physical implication about the processes through which the physical system had
passed. In addition, this theoretical state must be related to measurable quantities. 16
He never took the algebraic expression for force, and through an argument ex-
pressed and developed by mathematical rules joined the mathematical expression
to the physical theory with which he began his mathematical exploration. How
he expected to connect his mathematical expression to his experimental results
was left moot. The solution of the mathematical problems posed by electrostatics
and magnetism occurred after 1800. A physical theory expressed in mathematical
form did not appear until the middle of the nineteenth century.
The difficulties of constructing such arguments are also seen in Henry Caven-
dish's attempt to argue, using geometry and fluxional calculus, that the particles
of the fluid of electricity repulsed every other electric particle and those of matter
with a force that was the nth power of the inverse of their distance apart, where
n < 3. His propositions were physical, his theorems geometrical or fluxional.
Cavendish took a cone of electric fluid, each particle of which repulsed the other
electric particles and acted on a particle at its vertex with a force that was l/rn.
He then considered the repulsion on the particle from the fluid up to a plane at a
distance from the vertex and that from the fluid beyond the plane. He argued that
if n :::: 3 the repulsion from beyond the plane was infinitely small compared with
that from the plane to the vertex. The mathematics at his disposal, geometry and
fluxions were unequal to his task. He could only demonstrate that n = 2 for a
series of particular cases. He was far more successful in demonstrating his case
for a repulsive force of n = 2 experimentally.17
as a military engineer see, Stewart Gilmor, Charles Augustin Coulomb and the Evolu-
tion of Physics and Engineering in Eighteenth-Century France (Princeton: Princeton
University Press, 1971).
21 See Coulomb, "Recherches theoriques et experimentales sur la force de torsion et sur
elasticite des fils de metal," Mem. Acad. Sci., Paris (1784): 229-269, Memoires, vol.
1, 64-103, and, "Sur la maniere dont Ie fluide electrique se partage entre deux corps
conducteurs mis en contact, et de la distribution de ce fluide sur les differentes parties
de la surface de ces corps," Mem. Acad. Sci., Paris (1787): 421-467, Memoires vol. 1,
183-229.
22 Coulomb, "Sur I' electricite et Ie magnetisme, quatrieme memoire, ou I' on demontre deux
principaux proprietes du fluide electrique;' Mem. Acad. Sci., Paris (1786) [1788]: 67-77.
See also Heilbron, Electricity, 495-498.
23 See Simon Schaffer, "Machine Philosophy,"and Jane Weiss,"Lecture Demonstrations
and the Real World: the Case of Cartwheels," Brit. J. Hist. Sci. 28 (1995): 79-90.
70 Eighteenth Century
28 For the case of ideology and Edinburgh, see Steven Shapin, "The Audience for Science
in Eighteenth Century Edinburgh," Hist. Sci. 12 (1974): 95-121. Modifying this view
by noting the care with which utility entered into the social language of the experimental
philosopher in Edinburgh, see Jan Golinski, Science as Public Culture: Chemistry and
Enlightenment in Britain, 1760-1820 (Cambridge: Cambridge University Press, 1992).
For broad claims for the mechanical philosophy and the ideological foundations of the
industrial revolution, see Margaret Jacob, The Cultural Meaning of the Scientific Revolu-
tion and Larry Stewart, The Rise ofPublic Science. For a specific example of mechanical
demonstration and economic problems, see Jane Weiss, "Lecture Demonstrations and
the Real World: The Case of Cart Wheels," Brit. J. Hist. Sci. 28 (1995): 79-90.
29 Arnold Thackary Atoms and Powers (Cambridge MA: Harvard University press, 1970)
liberated chemistry from Newton's grip. See Robert E. Schofield, Mechanism and Ma-
terialism: British Natural Philosophy in the Age of Reason (Princeton NJ.: Princeton
University Press, 1970). On natural philosophy in the eighteenth century, see also, Simon
Schaffer, "Natural Philosophy," in Rousseau and Porter, Ferment, 55-91. Clifford Trues-
dell, "Reactions of late Baroque Mechanics to Success, Conjecture, Error and Failure
in Newton's Principia," Texas Quart. 10 (1967): 238-258, undermines Newton's influ-
ence on eighteenth-century mechanics and mathematics, Henry Guerlac, "Newton on the
Continent: The Early Reception of his Physical Thought," in Newton on the Continent,
(Ithaca NY.: Cornell University Press, 1981),41-73, R. W. Home, "Out of a Newtonian
Straitjacket," in Studies in the Eighteenth Century, R. F. Baissenden and J. C. Eade, eds.
(Canberra: Australian National University Press, 1979), and Colin Russell, Science and
Social Change, chap. 3 "Alternatives to Newton." Peter H. Reill has argued that the
alternatives to Newtonianism attracted a large percentage of Europe's educated elite and
was a discourse "within" enlightenment scientific thought not against it. Reill, "Between
Mechanism and Hermeticism: Nature and Science in the Late Enlightenment," in Friihe
Neuzeit-Friihe Moderne? R. Vierhaus, ed. (G6ttingen: Vandenhoek and Ruprecht,
1992), 393-421.
30 On the reading of the Opticks see, Henry Guerlac, "Early Reception."
31 Although Newton's work on optics was patterned after his mechanics he never succeeded
in complete closure. For Newton's attempts to develop a mathematical theory of color,
see Alan Shapiro, "Experiment, and Mathematics in Newton's Theory of Color," Phys.
72 Eighteenth Century
Experiment was only one of the two methods that reoriented the study of nature
during the seventeenth century. The second was mathematics. 35 As in the case of
experimental philosophy, mathematics was a language of explanation, developed
Today, (September 1984): 32-42 and Fits, Passions and Paroxysms: Physics, Method,
and Chemistry and Newton's Theories of Colored Bodies and Fits of Easy Reflection
(Cambridge: Cambridge University Press, 1993).
32 Johann Karl Fischer Geschichte der Physik seit der Weiderherstellung der Kunste und
Wissenschaften his aufneuesten Zeiten (Gottingen, 1801-1808),8 vols. For a similar,
though condensed account, see F. A. C. Gren, "Geschichte der Naturwissenschaft," Ann.
Phy. 1 (1799): 167-204.
33 He was also involved in the construction and sale of equipment. See, Simon Schaffer,
"The Consuming Flame: Electrical Showmen and Tory Mystics in the World of Goods,"
in Consumption and the World of Goods, John Brewer and Roy Porter, eds. (London:
Routledge, 1993),489-526. See Heilbron, Electricity, for an alternative interpretation.
34 Joseph Priestley, History and Present State of Electricity with Original Experiments 2
vols. (New York: Johnson Reprint of the third edition, 1966) and Robert Schofield,
"Introduction".
35 These were not independent of one another. See Peter Dear, Discipline and Experience:
Physics and Mathematics 73
and improved through solutions to the problems posed by nature and addressed
through the languages of mathematics. At the same time mathematicians devel-
oped a new language of explanation, the calculus. As problems were tackled and
successfully solved, techniques and standards of solution changed and expanded
the range of this new language. Demonstrable success legitimated the techniques
and extensions and reenforced the power of the calculus to display the workings
of nature. 36
For mathematicians, the point of mathematics was to extend the domain of their
method into new fields, gaining access to those fields through the consideration of
particular problems. Nature both provided the problems that needed solution and
guaranteed that solutions existed. As problems were expressed in mathematical
language, they were annexed to mathematics and their solutions subject to the
techniques and standards of that developing discipline. Disputes arose over the
nature of functions and the legitimacy of certain solutions but seldom got further
than reiterations of preferences for certain definitions of functions. Importantly,
solutions were judged by criteria being developed within the calculus itself; they
were mathematical not physical. Experiments in mechanics posed the problems
for mathematicians to solve, but gave no clues to their solution. Those solutions lay
in the new, and developing, language of the calculus. The domain of mathematics
also grew larger as methods were improved through solutions being reworked.
The discipline also grew as solutions became more general as well as solutions to
different types of differential and partial differential equations succumbed to the
technical ingenuity of mathematicians. The invention of methods to accomplish
this work of expansion were as important as demonstrating that such solutions
existed.
In the course of addressing the problems of mechanics, the metaphysical princi-
ples on which the mathematical solutions were based came under close scrutiny and
new ones entered the mathematician's repertoire. Discussions on such hypotheses
were directed to mathematical ends, not those of producing a physical theory of
mechanics. The appearance of a concept within a mathematical context did not
signal, necessarily, a growing physical understanding of mechanics. At least these
physical insights did not impact the work of the small group of experimentalists in-
terested in mechanics. 37 Using a physical principle to set up an equation of motion
Detailed discussions were reserved for his methodological heritage. His Principia
was discussed as mathematics, or as metaphysics, not as theoretical physics. 49
Reactions to Newton's mechanics revolved around differing notions of a satisfac-
tory foundation for the calculus and acceptable solutions to a mathematical, not a
physical problem. 50
In ways that parallel experimental physicists and their treatment of the Opticks
as paradigmatic for their discipline, mathematicians accepted the Principia as
mathematics. What we take as the foundations of his physics, his three laws of
motion, his concept of force, and the idea of gravitation, were critiqued, accepted,
discussed as the work of some predecessor, or discarded for others. Newton's
physical problems became the basis for forays into mathematics using various
forms of metaphysics to establish the necessary physical imagery to enter into a
discussion of the mathematics. The result was always mathematics.
The geography of the disciplines in the sciences of the eighteenth was not that
of later centuries. Both mathematicians and experimental philosophers shared the
goal of increasing the explanatory range of their method by expanding the phenom-
ena covered by the methods they wielded and hence the reach of their disciplines.
For practitioners in both disciplines, there was no distinction between "pure" and
"applied." The broadening of the investigation of nature to areas of social impor-
tance, from theology to industrial production, was regarded as valuable, as notable
as research as any esoteric, abstract result of only intellectual interest. The ultili-
tarian potential of the sciences, moral, social, or material was part of its definition
as knowledge, and were important justifications for their pursuit. Chemistry's util-
ity became part of its definition. 51 Joseph Priestley was equally interested in the
chemistry of "airs" and in Josiah Wedgwood's problems with glazes and pottery
techniques. There were also his own attempts to exploit carbonated water. 52 Mem-
bers of the Academies of Sciences were involved in solving technical problems
49 For details of the reactions to Newton's Principia, see Clifford Truesdell, "Reactions of
late Baroque Mechanics." See also, The Annus Mirabilis of Sir Isaac Newton (1666-
1966), Robert Palter, ed. (Cambridge MA.: MIT Press, 1967), and Guerlac, "Newton on
the Continent."
50 See, Domenico Bertoloni Meli Equivalence and Priority: Newton versus Leibniz
(Oxford: Clarendon Press, 1993) for a discussion of these competing forms of the cal-
culus.
51 See, Arthur Donovan, "British Chemistry and the Concept of Science in the Eighteenth
Century," Albion 7 (1975): 131-144. See also the career of Joseph Black in Joseph
Black, A. D. C. Simpson, ed.(Edinburgh: Royal Scottish Museum, 1982).
52 See Robert E. Schofield, The Lunar Society ofBirmingham; A Social History ofProvincia I
Science in Eighteenth Century England (Oxford: Clarendon Press, 1963). Roger Hahn,
Physics and Mathematics 79
"Science and the Arts in France," Stud. Eight. Cult. 10 (1981): 77-93. For Swedish
science see, Tore Frangsmyr, "Swedish Science in the Eighteenth Century," Hist. Sci.
12 (1974): 29-42.
53 See Roger Hahn, The Anatomy of a Scientific Institution: The Paris Academy ofSciences,
1666-1803 (Berkeley CA.: University of California Press, 1965). Hahn notes that no
member of the Academie balked at the duties placed on him by the state. See also,
Charles C. Gillispie, Science and Polity at the End of the Old Regime (Princeton NJ:
Princeton University Press, 1981), chap. 4 and Robin Briggs, "The Academie Royale
and the Pursuit of Utility," Past Present, 131 (1991): 38-88. For the German states
see Henry E. Lowood, Patriotism, Profit, and the Promotion of Science in the German
Enlightenment: The Economic and Scientific Societies, 1760-1815 (New York: Garland,
1991). For the Berlin Academy of Science see, Hans Aarsleff, "The Berlin Academy
under Frederick the Great," Hist. Human Sci. 2 (1989): 193-206; Mary Terrall, "The
Culture of Science in Frederick the Great's Berlin," Hist. Sci. 28 (1990): 333-364, and
Roland S. Calinger, "Frederick the Great and the Berlin Academy," Ann. Sci. 24 (1968):
239-249.
54 See Michel Foucault The Order of Things: The Archaeology of the Human Sciences
(New York: Vintage Books, 1973). Foucault did not use his "archeology" to examine
science; that still existed in a separate category of knowledge.
80 Eighteenth Century
56 This division of astronomy into distinct disciplines, mathematical and physical is noted
in Rainier Baasner, Das Lob der Sternkunst: Astronomies in der deutschen Aufkliirung
(G6ttingen: Vandenhoek and Ruprecht, 1987).
57 For another example of this use of mathematics by an observational astronomer see, Eric
Forbes, "Mayer's Contribution to Observational Astronomy," J. Hist. Astron. 11 (1980):
28-49.
58 This decline might be partly explained by the positions at the Observatory being treated as
a family prerogative. Seymour Chapin, "The Academy of Sciences during the Eighteenth
Century: An Astronomical Appraisal," French Hist. Stud. 5 (1968): 371-404. Other
disputes within the Academy further weakened support for the Observatory.
82 Eighteenth Century
problem involving a very finite number of observations over a finite range of val-
ues. His problem was to determine the most likely value for a measurement after
all systematic errors were removed. The solution lay solely within the theory of
probability. Bernoulli was looking for the value with maximum likelihood which
he then assumed explicitly to be the true value. His distribution of values lay within
a circle whose diameter spanned observations on both sides of the "true" value. If
in the series of observations the smallest is A, the second A + a, the third A + b
and the most probable A + x, the highest probability for x was given by taking the
derivative of
Bernoulli deduced the value for x for n = 1,2,3 ... , noting the tedium in work-
ing out such calculations, and ended by illustrating his method with numerical
examples. 62
Taking examples of actual results from the available astronomical literature
Euler objected to Daniel Bernoulli's assumption that maximum likelihood was
the true reading but offered no improvements on his arguments. Indeed Euler's
mathematical excursions lead into territory where the error range was in danger of
becoming imaginary and he beat a hasty retreat. Euler had other problems. In his
example of deducing the shape of the earth from four measurements of the length of
an arc along a single meridian, he eliminated unknown parameters from his initial
equations. This left him with two equations in two unknowns, the corrections to
the lengths of the arc. However, the equations led to more than two solutions, one
of which seemed more "reasonable" than the other, although Euler did not indicate
why.63
While Laplace's work was important for the development of the mathematical
theory of probability, it was less useful for experimentalists and observational
1
astronomers. 64 However, Laplace investigated the "probability that an error should
Y
lie between ex and y by f(z) dz, where f(z) is a known function of the error
Z." He redefined the probability of error into a more general and more interesting
mathematical one. He examined the Gaussian distribution as well as the behavior of
other functions. However, Laplace offered no clues as to which function was useful
and under what conditions for dealing with the problems of errors of observation. 65
Karl Friederich Gauss was the exception. Together with Wilhelm Bessel, he
changed the relationship between mathematics and observational astronomy in the
crucial period after 1800. He defined the problem as an observational astronomer,
"to determine the orbit of a heavenly body, without any hypothetical assumptions,
from observations not embracing a great period of time, and not allowing a se-
lection with a view to the application of special methods." Using the method of
least squares, he worked out the principle characteristics of a new planet, Ceres,
first observed by Giuseppe Piazzi in 1801. Franz Xaver von Zach used these char-
acteristics to relocate the planet later that year. 66 Gauss' account of his methods
were directed to astronomers. He noted that "the motions of the heavenly bodies,
so far as they take place in conic sections, by no means demand a complete theory
of this class of curves;'67 His work was replete with specific examples taking into
account calculations with incomplete sets of observations and completed his text
with a set of log tables. 68
Most forays by mathematicians into the domain of experimental and obser-
vational natural philosophy did not stop those philosophers seeking their own
solutions within their terms of discourse. While mathematicians appeared to ap-
propriate the problem of the shape of the earth completely, even to making the
measurements in Peru and Lapland, they did not decide the issue. Their mathe-
matical explorations of the shape of the earth did not solve the problem. Geodesic
measurements continued throughout the eighteenth century. This measurement had
political and imperial implications as well as being of metaphysical use. These
efforts were, therefore, international and inconclusive. The incompatibilities of
these measurements seemed to indicate that the shape of the earth was not the sim-
ple oblate spheroid of the mathematicians. Observational closure came only with
the measurements of P. F. A. Mechain and 1. B. 1. Delambre during the revolution-
chemistry. What could physics offer chemistry in the late eighteenth century ex-
cept quantified experiment. However, the quantified experiments of Lavoisier and
Laplace on heat do not seem to change the terms of the theoretical debate about the
nature of heat. It remained in their study in the vernacular. Their elaborate theory
of heat as the vis viva of the particles of bodies was abandoned for the simpler as-
sumption that heat entered a body as it was heated and left as it cooled. From this
they developed their linear, algebraic expression for comparing specific heats. 72
Physics was still a loosely held bundle of practices whose theoretical conjectures
were as qualitative as those of chemists in the eighteenth century. 73
The practitioners of experimental physics were as divided as chemists over
foundational ideas, such as ether or caloric. Nor did practitioners in either discipline
structure theories in our sense of the term. Arthur Donovan's description of pre-
Lavoisian chemistry as a "dispersed and varied set of local didactic, experimental,
and explanatory practices" works for experimental physics in the same era and into
the next century.74 Physics in the eighteenth century shared the same characteristics
as eighteenth-century chemistry which differentiate them both from their modern
namesakes There is indeed more evidence that chemistry itself initiated the move
"into science" from natural philosophy.75 Lavoisier's new chemistry reorganized
that science around particular concepts rather than defining chemistry through
methods and phenomena. In trying to promote the new chemistry as a revolutionary
method, Lavoisier and the other anti-phlogistonists knew their colleagues well.
72 Lavoisier and Laplace, Memoir on Heat, Henry Guerlac, trans. (New York: Neale Watson
Pub., 1982), on vis viva 4-5. Their claims for precision were spurious. See also Heilbron,
Weighing Imponderables, 101-104.
73 On these points see the exchanges between Evan M. Melhado, "Chemistry, Physics
and the Chemical Revolution," Isis, 76 (1985): 195-211, C. E. Perrin, "Revolution or
Reform," Hist. Sci. 25 (1987): 395-423 and "Research Traditions, Lavoisier, and the
Chemical Revolution," Osiris 4 (1988): 53-81. Essentially, Perrin argues for a self-
contained tradition of research within chemistry, with chemistry as a disciplinary equal
of physics. Melhado and also Arthur Donovan in Donovan, "Lavoisier and the Origins
of Modern Chemistry," Osiris, 4 (1988): 214-231 argue that chemistry took as its model
the methods of physics, already assumed to be a modern scientific discipline. The debate
continued in Perrin, "Chemistry as Peer of Physics: A Response to Donovan and Melhado
on Lavoisier," Isis, 81 (1990): 259-270, Melhado, "On the Historiography of Science:
A Reply to Perrin," Isis, 81 (1990): 273-276, and Donovan, "Newton and Lavoisier:
Chemistry as a Branch of Natural Philosophy to Chemistry as Positive Science," in
Action and Reaction, P. Theerman and A. D. Seeff, eds. (Newark DEL: University of
Delaware Press, 1993). The issue begins in Guerlac, "Chemistry as a Branch of Physics:
Laplace's Collaboration with Lavoisier," Hist. Stud. Phys. Sci. 7 (1976): 183-276. For
the impact of the methods of physics on Lavoisier see, Donovan Antoine Lavoisier:
Science, Administration and Revolution (New York: Cambridge University Press, 1996),
chap. 3.
74 See Donovan, "Introduction," Osiris 4 (1988): 5-12, p. 11.
75 The phrase is taken from Donovan, "Lavoisier and modern Chemistry," p. 219.
Physics and Mathematics 87
The methodological division of the sciences was both intellectual and social. On
the continent, members of Academies of Science were divided into two sections,
mathematical and physical. These membership categories were further subdivided.
The publications of these societies reflected this same social division. In the Paris
Academie this division between the mathematical and deductive, versus the exper-
imental and observational and inductive, sciences was particularly long lived. In
the mathematics division was geometry, astronomy and mechanics. Under physics
was physics, anatomy, chemistry and botany. These particular subfields came un-
der revision as the methodological divisions of the disciplines became unwieldy.
Under Lavoisier's suggested reforms physics, narrowly defined, entered as a spe-
cific section within the Academie. This separation of physics from chemistry
meant that it no longer encompassed chemistry. Lavoisier reinforced chemists'
efforts to establish their discipline as separate and equal to the experimental field
of physics. 78
76 To understand that Lavoisier's chemistry was based upon new principles read Antoine
Laurent Lavoisier, Traite elementaire de chimie (Paris, 1789) trans. by R. Kerr as Elements
o/Chemistry (Edinburgh: 1790). The phrase is taken from Jan Golinski, "The Chemical
Revolution and the Politics of Language," Eighteenth Century: Theory and Interpre-
tation 33 (1992): 238-257. Bernadotte Besaude-Vincent in, "A View of the Chemical
Revolution Through Contemporary Textbooks: Lavoisier, Fourcroy and Chaptal," Brit.
1. Hist. Sci. 23 (1990): 435-460 discusses the differences between Lavoisier's text
and those of other chemists, particularly Chaptal who published texts before and after
becoming a Lavoisian.
77 The loud protests of chemists in the nineteenth century proclaiming their empiricism does
not alter the place of theory within their reorganized discipline. These protests occurred
after the acceptance of Lavoisier's revised system. His theory had become so ingrained
in practice as to be natural, not theoretical.
78 See Hahn Anatomy, for the divisions within the Academie, and Rhoda Rappaport, "The
88 Eighteenth Century
Liberties of the Paris Academy of Science, 1716-1785," in The Analytical Spirit: Essays
in the History of Science in Honor of Henry Guerlac, Harry Woolf, ed. (Ithaca NY.:
Cornell University Press, 1981),225-253, and James E. McClellan III, "The Academie
Royale des Sciences, 1699-1793: A Statistical Portrait," Isis, 72 (1981): 541-567,543-
544.
79 See Otto Sontag, "Albrecht von Haller on Academies and the Advancement of Science:
the Case of GOttingen," Ann. Sci. 32 (1975): 379-391.
80 David M. Griffith, "The Early Years of the Petersburg Academy of Science," Canadian-
American Slavic Studies, 14 (1980): 436-445.
81 This was also true in the early years of the Royal Society as later in local scientific
societies. See Marie Boas Hall, Promoting Experimental Learning: Experiment and
the Royal Society, /660-1727 (Cambridge: Cambridge University Press, 1991). This
changed as the eighteenth century wore on; see David P. Miller, "Into the Valley of
Darkness: Reflections on the History of the Royal Society in the Eighteenth Century,"
Hist. Sci. 27 (1989): 155-166.
Physics and Mathematics 89
82 The social range of the membership of these societies, their purposes and their fates are
the subject of a large literature. Most historians emphasize the provincial character of
these institutions and the local purposes they served. For provincial French scientific so-
cieties, see Daniel Roche, Le siecie des lumieres en province: Academies et academiciens
provinciaux, 1680-1789 (Paris: Mouton, 1978), 2 vols. For Holland, see Snelders, "Pro-
fessors, Amateurs and Learned Societies," in Dutch Republic, Jacob and Mijnhardt, eds.
For Scotland, see Donovan, Philosophical Chemistry, and R. L. Emerson, "The Philo-
sophical Society of Edinburgh (1737-1747)," Brit. J. Hist. Sci. 12 (1979): 154-171;
"(1747-1780)," same journal, 14 (1981): 133-176 and "(1768-1785)," same journal, 18
(1985): 255-303 and "Science, Origins and Consensus of the Scottish Enlightenment,"
Hist. Sci. 26 (1988): 333-336, and Kathleen Holcomb, "A Dance in the Mind: The
Provincial Scottish Philosophical Society," Stud. Eight. Cult. 21 (1991): 89-100. For
an early provincial English scientific society see Schofield, The Lunar Society. For the
German states see Lowood, Patriotism.
83 For arguments why twentieth-century sociological categories are useless for understand-
ing the social behavior of scientists even at the end of the eighteenth century, see Dorinda
Outram, "Politics and Vocation in French Science," Brit. J. Hist. Sci. 13 (1980): 27-43.
84 For the argument that eighteenth-century chemistry was a discipline but not a profession
see Karl Hufbauer, The Formation of the German Chemical Community, 1720-1795,
(Berkeley CA: University of California Press, 1982), and "The Social Support for Chem-
istry in Germany in the Eighteenth Century: How and Why did it change," Hist. Stud.
Phys. Sci. 3 (1971): 205-231. John Heilbron illustrates this for "electricians," although
he does not address the question directly, in Heilbron, Elements ofEarly Modern Physics,.
Roger Hahn also argues that even under the patronage of the French State scientists can-
not be defined as professionals. See, Hahn, "Scientific Research as an Occupation in
Eighteenth-Century Paris," Minerva, 13 (1975): 501-513, and "Scientific Careers in
Eighteenth-Century France," in The Emergence of Science in Western Europe Maurice
Crosland, ed. (New York: Science History Books, 1975). For an alternative view, see
Crosland, "The Development of a Professional Career in Science in France," Minerva,
13 (1975): 38-57, and Emergence Crosland ed., 139-159, although he claims the status
of profession in France only with the French Revolution. For a more general discussion,
see Professions and the French State, 1700-1900, Gerald L. Geison, ed. (Philadelphia
90 Eighteenth Century
its technical problems. Many did and this sometimes ended their ability to do
sustained pieces of research. 85 Making a living in science required entrepreneurial
skills to establish, consolidate, and sustain.
There was no established, systematic or regulated entry into the scientific dis-
ciplines. The study of nature was an integral aspect of polite, and not so polite,
cultures throughout the eighteenth century. At the universities of Europe the sci-
ences were taught as adjuncts to medicine, or some other profession, or as an
aspect of this male, educational rite of passage into elite, adult society. While not
on a par with dancing or fencing classes the sciences added a certain gloss to the
universities' offerings to the sons of the rich and the upper classes. For a student
the pursuit of science required aptitude, perseverance, and the opportunity to take
advantage of local resources. Training to become competent enough to publish
and enter the world of research was again a matter of personal commitment, social
place, opportunity, and patronage.
The arena of social and intellectual activity for most scientists in the eighteenth
century was the local scientific society, rather than the university or college. And
some of those studies accepted and lauded as research in the eighteenth century ap-
pear from the late twentieth century to be an unending recital of observations, and
some decidedly misplaced. Then as now, disciplinary colleagues were geographi-
cally scattered and might be addressed in technical terms. Yet the expectations of
this "imagined community" for the content of research reports were very different
from those of professionalized disciplines of the past century and a half. Exam-
ples of those were examined in the preceding chapter. This is not a matter of the
concepts of physics having changed, or of its problems being different. That is to
be expected. In the eighteenth century, experimentalists and mathematicians had
different notions of what constituted a valid research problem and the criteria for
their solution. Much that counted as explanation in the eighteenth, does not stand
up to scrutiny as science in this century. Similarly disciplinary boundaries are not
where we might expect them. What we find is that while names of broad disciplines
PA: University of Pennsylvania Press, 1984). Charles Coulston Gillispie locates the
professionalization of the sciences in France in the nineteenth century. See, Gillispie,
The Professionalization of Science in France, 1770-1850 (Tokyo: Kyoto Doshiha Uni-
versity Press, 1983). An enquiry into mathematicians and their status as "professionals"
based on an historically shifting definition of professions is in Ivo Schneider, "Forms of
Professionalization in Mathematics before the Nineteenth Century," in Social History of
Nineteenth Century Mathematics, Herbert Mehrtens, Henk Bos and Ivo Schneider, eds.
(Basel: Birkhiiuser, 1981), 89-110.
85 See Charles C. Gillispie, Science and Polity. Frederick the Great had the same expecta-
tions of his academicians in Berlin. See, Friederich II, "Discourse de I'utilite des sciences
et des arts dans un etat," (read by Therault Jan. 1772) Mem. Acad. Sci. Berlin (1772):
18. Whether he was as successful in mobilizing his academicians as the French state
is questionable. See also Calinger, "Frederick the Great and the Berlin Academy of
Science."
Physics and Mathematics 91
such as physics and mathematics have remained the same, the expectations and
practices of physicists and mathematicians have changed radically. Establishing a
theoretical point of view, then developing it to interpret phenomena, or as a valid
foundation for a mathematical argument, was a matter of metaphysics. The con-
duct of disputes, the curve of vocations and the distribution of honors and prizes
follow different intellectual as well as social rules.
Historians of the eighteenth-century sciences appreciate these differences and
take them as their starting point in analyzing their era. 86 We need to take the next
step and ask how then did the modern discipline of physics develop into a profession
whose practitioners were located in universities? How was it that the education
and practices of these professionals cut off most of the members of society from
participation in or even understanding of what those practitioners were doing?
How did the modern system of physics come into existence?
86 Symbolic of this realization are the essays in Ferment of Knowledge R. S. Rousseau and
Roy Porter, eds. This is continued in the review of Ferment ofKnowledge by G. N. Cantor,
"The Eighteenth Century Problem," Hist. Sci. 20 (1982): 44-63. See also J. F. Musser,
"The Perils of Relying on Kuhn," Eighteenth Cent. Stud. 18 (1984): 215-226, and E. M.
Melhado "Metzger, Kuhn and Eighteenth Century Disciplinary History," in Studies on
Helene Metzger, Gad Freudenthal, ed. Corpus, 8/9 (1988). For an earlier discussion of
the inherent difficulties of investigating science in the eighteenth century, see Crosland,
"Editor's Foreword," in Emergence of Science Crosland, ed. The pursuit of the place
of science in the growing consumer societies of the eighteenth century continues. See
Consumption Brewer and Porter, eds., part IV, as well as Golinski, Science as Public
Culture, and Stewart, Rise of Public Science.
Part II
Transitions, 1790-1830
Chapter IV
Well before 1790 Paris had become the social and intellectual center for scientific
life in Europe. It remained at the center until after 1830. Because this era in
the scientific life of France has been seen as the source of modern physics, we
need to examine the workings of Parisian scientific institutions and the practices
of mathematicians and experimental physicists. What, precisely, did this band
of intensely competitive men change in their mathematical and physical heritage
from the eighteenth century? After examining the social and political structures
of scientific Paris and their workings, we will turn to a series of problems and
prize-essay questions of the era. In France the solutions to these problems were
the occasions for fierce contests over the practices and future of both physics
and mathematics. The solutions also disclose what was accomplished in this era
in terms of changing the relationships between mathematics and experimental
physics. The mathematization of electrostatics by Poisson, and Fourier's work
on the conduction of heat through solids, will help us distinguish the technical
mathematician of the early nineteenth century from the theoretical physicist of a
later era. Similarly, the development of the wave theory of light and subsequent
work in France on elasticity will separate mathematicians from experimentalists
and reveal the changes in physics by 1830.
The place of science in French society and culture was strengthened in the era
between 1790 and 1815, despite the arbitrary rule of the radical period of the
French revolution, the instabilities of the Directory, and the manipulations of the
Napoleonic era. There were even signs of long-term social and intellectual conti-
nuity in the scientific community in the context of political change. 2 Institutions
1 Stephen Jay Gould coined this expression in, "The Stinkstones of Oeningen;' in Gould
Hens Teeth and Horses Toes (New York: Norton, 1983) 94-106,105 to describe Cuvier's
methodology in paleontology in his efforts to read the fossil record as it presented itself,
with no interpolations or theoretical leaps of the imagination.
2 This does not preclude the short-term dislocations, anxieties and even terror during the
changed, but as institutions they functioned in ways that were continuations of pre-
Revolutionary social practices within the sciences. Paris retained its international
leadership as the center for research and that leadership was even strengthened and
remained unchallenged until after 1830. 3 Scientists became even more important
to the State in this era. 4
1790s. See Dorinda Outram, "The Ordeal of Vocation: The Paris Academy of Sciences
and the Terror," Hist. Sci. 21 (1983): 251-274, and HahnAnatomy, chaps. 8 and 9.
3 The issue of the impact of the revolution and of Napoleon on science is yet to be de-
cided. See Henry Guerlac, "Some Aspects of Science in the French Revolution," Sci.
Monthly 80 (1955): 93-101, Rene Taton, "The French Revolution and the Progress of
Science," Centaurus, 3 (1953): 73-89, L. Pearce Williams, "The Politics of Science in
the French Revolution," in Critical Problems in the History ofScience, Marshall Clagett,
ed. (Madison WI.: University of Wisconsin Press, 1962),291-308, and Joachim Fischer,
Napoleon und die Naturwissenschaften (Stuttgart: Franz Steiner, 1988).
4 See Nicole Dhombres and Jean Dhombres, Naissance d'un pouvoir: sciences et savants
en France, 1793-1824 (Paris: Editions Payot, 1989) and Terry Shinn, Sa voir scien-
tifique et pouvoir social: l'Ecole Poly technique, 1794-1914 ( Paris: Presse Foundation
Nationale des Sciences Politiques, 1980).
5 See Charles Coulston Gillispie, "Science and Politics, with specific Reference to the
Revolution and Napoleonic France," Hist. Techn. 4 (1987): 213-223, and Gerald L.
Geison, Professions.
6 The epitome for this is Pierre Simon, Marquis de Laplace who flourished under the Old
Regime, survived the Revolution, Directory, Napoleon and the Restoration. Others were
less fortunate and suffered at certain periods, among them Augustin Fresnel, Joseph
Fourier, and Gaspard Monge. The fate of Antoine Laurent Lavoisier lies outside of our
story.
7 Gillispie, "The Encyclopedie and Jacobin Philosophy of Science," In Critical Problems,
Clagett, ed., 255-289.
8 Hahn discusses The fate of the Academie des Sciences and the academicians in Hahn,
Anatomy. Maurice Crosland The Society ofArcuei/. A View ofFrench Science at the Time
ofNapoleon I (London, 1967) recounts the attempts of some scientists to continue their
research careers during the chaos of the 1790s. However, this represents only a small
number of established scientists and their protegees.
in France, 1790-1830 97
9 See The Organization of Science and Technology in France, 1808-1914, Robert Fox
and George Weisz, eds. (New York: Cambridge University Press, 1980), and Geison
Professions. For a history of the Bcole Poly technique, see Ambroise Fourcy, Histoire de
I'Ecole Poly technique (Paris: Belin, reprint, 1987), and Terry Shinn, Sa voir scientifique
et pouvoir social, l'Ecole Poly technique, 1794-1914. For the later challenge to the
hegemony of the Bcole Poly technique, see Antoni Malet, "The Bcole Normale and the
Education of the Scientific Elite in Nineteenth-Century France;' Asclepio, 43 (1991):
163-187.
10 However, see Grattan-Guinness, "Grand-Bcoles, petits Universites: Some Puzzled Re-
marks on Higher Educations in Mathematics in France, 1795-1840," Hist. Univ. 7 (1988):
197-225.
11 The title and organization of the premier French scientific society supported by the state
changed with the political turbulence from 1790 to 1820. The Academie des Sciences in
Paris was disbanded in August, 1793 and replaced by the Institut at the end of 1795. The
Institut was organized into three classes, the first of which covered the sciences. These
included mathematics, physics, chemistry, botany and medicine. At the restoration of
the monarchy, the Institut reverted in March 1816 to the Academie des Sciences.
12 Cumul refers to the custom of accumulating positions in state teaching institutions that
exacerbated the competition for place within the sciences in early nineteenth-century
France.
98 Physics and Mathematics
reer in the sciences. No one was more adept and able at dispensing this patronage
than Laplace. Positions and preferment went wherever possible to young men
willing to use his ideas and approaches to the solutions of problems in physics
and mathematics. Biot's obsequiousness towards Laplace and his work in celestial
mechanics would be comical if it were not for the results. It worked. Biot's career
path was easier than that of other young colleagues. One of the latter was Augustin
Fresnel who offended the Laplacians and never invoked Laplace's intellectual au-
thority in his own work. However, Fresnel had the help of Fran<;ois Arago. Arago's
position at the Observatory, obtained originally through the patronage of Laplace,
and as editor of Annales de Chimie allowed him in turn to dispense patronage.
Poisson's early election to the physics section of the Institut was similarly a matter
of political patronage. His election was a further demonstration of the power of
a small group within the Institut to direct its affairs. The election of promising
scientists to sections not especially connected to their research specialty had been
used even before the Revolution. For Poisson, with no private income, election to
the Institut was crucial. 13
Patronage and political power within the institutions of science shaped the ca-
reers of individuals, the prizes they might, or might not be awarded, and the publica-
tion, or lack of publication of prize essays by the Institut. However, these political
forces did not prevent independent assessment of intellectual worth. While Joseph
Fourier's prize-winning essay on heat was not published by the Institut (or with
the restoration the Academie) for over a decade, its contents were well known to
mathematicians in Paris. He was also elected to the post of Executive Secretary in
1820, a sure indication of the decline of Laplacian influence. Fourier, being older
and a high-level government official, had the ability and connections to stay the
course. 14
13 While not delving into the politics ofInstitut elections, there is no evidence that Poisson's
election was other than another political victory for Laplace. His election may even have
been a payment on a political debt. Pierre Costabel, "Poisson, Simeon-Denis," Diet. Sci.
Bio. vol., 10, 480-490, 481. Neither Poisson nor anyone else regarded his papers on
electrostatics as other than mathematics, or marking a definitive change in the analysis
of physical problems. For an alternative interpretation, see R. W. Home, "Poisson's
Memoirs on Electricity: Academic Politics and A New Style in Physics," Brit. J. Hist.
Sci. 16 (1983): 239-259.
14 For the impact of Institut politics on the publication of Fourier's essay of 1811 see,
Grattan-Guinness and Jerome RavetzJoseph Fourier (1768-1830). A Survey of his Life
and Work (Cambridge MA.: MIT Press, 1972). For the impact of patronage in mathemat-
ics in general in this era see, Grattan-Guinness, Convolutions in French Mathematics,
1800-1840 (Boston: Birkhiiuser, 1990), 3 vols., vol. 1, chap. 2. For Biot, see Eu-
gene Frankel, "Career-making in post-revolutionary France: The Case of Jean-Baptiste
Biot," Brit. J. Hist. Sci. 11 (1978): 36-48, and "Corpuscular Optics vs. the Wave The-
ory of Light: The Science and Politics of a Revolution in Physics," Soc. Stud. Sci. 6
(1976): 141-184. See also, Crosland, Science under Control: The French Academy of
in France, 1790-1830 99
The Laplacians wielded great political clout within the scientific community for
a crucial period but this power should not be confused with intellectual authority.
In the long run, becoming a "Laplacian" did not garner those men secure reputa-
tions in French science. Poisson's scientific reputation peaked early in his career
and declined thereafter, even as his political power rose. The assessment of his
intellectual accomplishments sank to the point that, after his death Poisson, was
regarded as someone who used other's ideas to develop his own career.
The Laplacians were unsuccessful in stilling alternative visions of physical pro-
cesses, and of alternative foundations for the calculus. Laplace could and did
influence the choice and the wording of prize-essay questions in the mathematical
and physical sections of the Institut and later in the Academie. However, when
it came to awarding prizes, Laplace's influence was less monopolistic. Augustin
Fresnel's work was crowned even though his methods and ideas were not in accord
with those of Laplace, and despite Laplace's skepticism about the validity of his
mathematical methods so was Fourier's. The delayed publication of these works
became scandals that damaged the Academie and Laplace, not Fresnel or Fourier.
From 1790 on, in addition to Academie journals, others existed independent of
its influence. Fourier and Fresnel could make their ideas known and establish a
reputation beyond Laplace's influence. Indeed this multiple outlet for publications
in science made that tight centralized control impossible.
The name might change, and function of the Academie des Sciences might be
more restricted than in the old regime, yet entry into it capped a life in science.
Membership was the ultimate legitimation of a scientist's research and it was still
the most prestigious scientific institution in Paris. 15 The social structure of the
academy continued to reflect the changing disciplinary structure of the sciences.
While physics, narrowly defined, was already a section, others also appeared in
the Institut. This reflected the fragmentation of the experimental and observational
sciences already threatening the Academie's institutional monopoly of science in
the 1780s. However, the Academie no longer had a corner on the presentation and
publication of research. Important issues in chemistry were published elsewhere,
and the AnnaLes de Chimie and the Bulletin des Sciences par La Societe Philo-
matique de Paris offered alternative, and quick publication of important issues in
physics. 16 The ponderous pace of the full account of research that would appear
from the academic press bequeathed a polished presentation to posterity. Alter-
native journals were more attractive to address colleagues on important research
Sciences, 1795-1914 (Cambridge: Cambridge University Press, 1992),44-49. Patron-
age prevailed in other disciplines, see Dorinda Outram, Georges Cuvier (Manchester:
Manchester University Press, 1984).
15 The changes in the Academie are detailed in Hahn, Anatomy, chap. 9.
16 Crosland, In the Shadow of Lavoisier: The Annates de Chimie and the Establishment
of a New Science (Oxford: British Society for the History of Science, 1994), chap. 2,
discusses the changes in possible publication outlets for research in this era.
100 Physics and Mathematics
Experimental Physics
Physics as a term still carried both broad and narrower meanings. The work of
Arago and Alexander von Humboldt on geomagnetism kept the broader goals of
physics alive, although physics could no longer lay claim to the broad methodologi-
cal field of observation and experiment. Chemistry and the life sciences marked out
their own domains of competence. Physics in the narrower sense still included the
experimental exploration of sound and hearing, light and color. While experiment
still defined the discipline, the phenomena covered depended largely on individual
interpretation, or a journal editor, and the needs of the marketplace. 17 Experiments
were expected to be careful, quantitative laboratory experiments whose results were
often encapsulated in algebraic form. Speculations about the operation of nature
stayed close to the tabulated results of these quantitative experiments, and rarely
ventured beyond the phenomenological. Simultaneously, physics and chemistry
were seen as drawing closer together through electrochemistry. An instrument
developed in experimental physics had implications for some important problems
of experimental chemistry. While the goals of their research might diverge practi-
tioners in both disciplines shared a common methodological standard.
All of these developments bring early, nineteenth-century French physics closer
to that of the twentieth century. But whether this physics marks the definitive
breaking point between natural philosophy and physics in the modern sense still
requires closer examination. Relying on conceptual realignment to distinguish
between the two is clearly insufficient. 18 The concepts introduced between 1800
and 1850, while important are symptoms of change, not causes. Conceptual change
taken in isolation tells us nothing about practices. 19
17 For an alternative interpretation of the range of meaning of the term, see Crosland, Science
Under Control, 34-36. Crosland does not take into account the broader sense of the term
inherited in this era from earlier in the eighteenth century.
18 For example, see Pearce Williams, "The Physical Sciences in the First Half of the Nine-
teenth Century: Problems and Sources," Hist. Sci. 1 (1962): 1-15.
19 Robert Silliman, "Fresnel and the Emergence of Physics as a Discipline," Hist. Stud.
Phys. Sci. 4 (1974): 137-162, with a nod to methodological factors locates physics in
the introduction of the wave theory of light.
in France, 1790-1830 101
Clearly the era between 1790 and 1830 in France marked some kind of watershed
noted by more than one historian. Using textbooks in experimental physics Cannon
claimed that "physics itself was invented by the French around the years 1810-30."
Rene Hauy's Traite Elementaire de Physique of 1803 was definitely not physics
while Biot's Traite de physique experimentale et mathematique of 1816, "was
beginning to grasp at something like our concept of physics." Yet Cannon neither
detailed what physics was, or is, nor the differences between physics of Hauy and
Biot, nor those aspects of Biot that make him more "modern" contrasted with what
Hauy was doing. In short there is no detailed exploration of what the changes
were or the process of change. There is merely a sense that things progressed in
a direction that makes the product more familiar to us, although not completely.
Cannon named those men in the nineteenth century, Michael Faraday and John
Herschel who were not, and those of the mid-nineteenth century, James Clerk
Maxwell, who were physicists in our sense of the term without delineating what
differentiated the two groupS.20
John Heilbron also has located the beginnings of modern physics in the same
era and country through a detailed history of the practices of experimentalists and
the "theoretical work" of Laplace, Biot, and Poisson. Poisson's speculations about
electricity and its action were vague. His strengths lay in his mathematics, as "exact
description" over "a qualitative model deemed intelligible." Heilbron assumes here
an inherent clarity of mathematical over vernacular descriptions of phenomena and
ignores the history of mathematics. Vernacular descriptions of physical processes
are not inherently muddier than mathematical ones even though qualitative. And,
mathematics can be used to obfuscate and hide conceptual muddle while bringing
quantitative precision to physics. 21 He accepts the term "physique-mathematique"
as physics, not mathematics. 22
In this era, if mathematical physics was mathematics, we need to understand
what precisely was changing in physics to then decide where the origins of the
modern discipline might lie. We must judge whether the development of carefully
designed and executed quantitative experiments, whose results were analyzed for
error and then algebraically joined to the mathematical expression of those results,
is sufficient to define physics. If so, in what sense was the physics that these men
created "modern."
20 Susan Faye Cannon, "The Invention of Physics," in Cannon, Science in Culture: The
Early Victorian Period (New York: Neale Watson, 1978) 111-136, 115.
21 For an alternative view see Elizabeth Garber, "Simeon-Denis Poisson: Mathematics ver-
sus Physics in Early Nineteenth-Century France," in Beyond History of Science, Garber,
ed. 156-176.
22 For mathematical physics as mathematics in early nineteenth-century France, see Grattan-
Guinness, "Mathematical Physics in France, 1800-1835," in Epistemological and Social
Problems Jahnke and Otte eds., 349-370 and Convolutions, chap. 7.
102 Physics and Mathematics
servation, which were always subject to error, the mathematician must remain as
independent as possible of "every empirical process, and to complete the analysis,
so that it shall not be necessary to derive from observations any but indisputable
data."28 In addition Laplace believed that only empirical laws and the calculus were
necessary for the exploration of nature.
Laplace's interest in physical phenomena was as a source of problems within
the calculus. The physical problems he chose to examine could be directly related
to problems in astronomy, or those that could be reduced to the mathematics that
stemmed out of his work on celestial mechanics. The pattern of his approach was
to assume that the physical phenomena, refraction, capillarity etc., were caused
by central forces acting between the particles of matter at insensible distances.
Even while admitting that the effect might be macroscopic and there was no way
of knowing the force law that was operative, Laplace assumed that the force law
acted over sensible distances. Once this was accomplished, all the mathematical
techniques he had developed in celestial mechanics were brought to bear on the
problem at hand. And the amount of analysis brought to bear was prodigious.
Laplace approached these problems as he did the mathematics of celestial me-
chanics.
Laplace made his mathematical reputation by analyzing the complex interac-
tions of the planets, considering the planets themselves as finite bodies, taking into
account the subtle influence of their shape on their motions and the irregularities
of their paths. The key to his mathematical success in celestial mechanics was in
considering terms in series neglected by others, or integrating functions not inte-
gratable before, or by the invention of new mathematical techniques. To consider
the planets as finite mathematical bodies, Laplace needed a mechanics for finite,
masses. He established this mechanics by first deducing the law of universal grav-
itation "from observation." Kepler's laws were then used to establish the elliptical
paths of the planets and the parabolic paths of comets. Experiments on pendula
and astronomical observations verified this mathematical result for the earth from
which Laplace argued that every particle of matter on the earth must be such a
source of force or the centre of gravity of the earth would shift. Newton's name
did not appear anywhere in this discussion. 29 Whether his derivation was valid or
not, Laplace took Newton's law of gravitation not as an assumption about the oper-
ations of nature but as a statement deduced through mathematics from the results of
observation. No speculations about the operations of nature were necessary. The
law was operative everywhere, and became his foundation for the mathematical
field of physique-mathematique.
Physical principles did not exist for Laplace as independent sources of order.
There were only two such sources, experiment and analysis. Of these two, analysis
was clearly the more dependable. In the first book of Mecanique Celeste there was
no attempt at detailed physical reasoning to establish his general mechanics. 3o
Discussion of physical points was at a minimum. This led to less than satisfactory
explanations of physical circumstances. A particle moving on a sphere described a
great circle because "there is no reason why it should deviate to the right rather than
the left of this great circle." Not a word about forces acting on the particle on the
sphere. Having set up the most general laws of mechanics the rest of his celestial
mechanics was, as far as possible, a deductive system rooted in mathematics with
minimal reference to an empirical base. His deduction and then assumption of
the operation of gravitation between matter, macroscopically and microscopically,
only limited his mathematics by defining a relationship between the form of the
function that was the first derivative of the force and the function that represented
the force law. Mathematically, Laplace had introduced the potential function. Its
physical implications remained unexplored.
Laplace did not use his physical model to guide or limit the development of his
mathematics. Mathematical need was the criterion for deciding which terms, vari-
ables, or functions were eliminated or reduced to the status of constants or simply
dropped. There was no attempt at offering a physical justification for a mathemati-
cal necessity.31 He did not discuss the physical significance of eliminated variables,
and their disappearance is mathematically necessary but physically mysterious. In
short, all manipulations were mathematically, not physically, convenient.
Because of the overwhelming importance of analysis, Laplace deduced results
that were known empirically. His derivation of Snell's law never referred to its
empirical foundation nor did he mention that experiments existed that confirmed
his analytical result. He did not note that his deduction might provide evidence
30 Laplace, Celestial Mechanics opens with a general discussion of the laws of mechanics.
Laplace noted the conservation of vis viva and angular momentum and the Principle of
Least Action. This last principle did not, in Laplace's opinion, require a metaphysical
justification, it "is in fact nothing more than a remarkable result of the preceding differ-
ential equations." Of all the principles of mechanics, only the law of inertia and that of
force being proportional to the velocity depend on observation. See Laplace Celestial
Mechanics vol. 1, bk. I, chap. ii & viii.
31 This is particularly obvious in his theory of capillarity which is explicitly based on central
forces acting at insensible distances. Laplace set up an integral of a general function. The
form of the function remained undefined, although it could be limited if he introduced the
physical terms that he used to set up the problem. Also note the criteria he used to change
variables etc. See, Laplace, Celestial Mechanics, vol., IV, Supplement to bk., X. Jean
Dhombres, "La tMorie de la capillarite selon Laplace: mathematisation superficielle ou
etendue?" Rev. Hist. Sci. 43 (1990): 43-77, sees Laplace's work as a "separation of
physical inspiration from analytical calculation."
in France, 1790-1830 105
for his basic physical hypothesis. 32 For Laplace, the empirical evidence supplied
by experiments on capillarity were confirmations of his mathematics. There were
several instances in his discussions of analytically deduced results where he missed
the opportunity to comment on them physically, or see them as occasions for
experiment. They occur in the middle of an argument and are mathematically
uninteresting. The results finally deduced are physically incorrect or uninteresting
from an experimental point of view. 33 The physical fruits of all this complicated
calculus were very few. The results Laplace reproduced were known empirical
results. For all his analysis Laplace was unable to penetrate further into the structure
of matter or of the interaction of light and matter than contemporary, vernacular
theories.
Laplace did not integrate his physical model with his mathematical analysis.
There was no sense of what physical process was represented by the mathematics.
He did compare certain analytical results with experiment and the formulae were
interpreted mechanically, but not in terms of microscopic or macroscopic forces.
The explanations were vernacular and based on the changes of velocities of moving
particles. 34 The model and the analysis were decoupled. In complicated situations
he tended to add causes, to further complicate the physical description. This was
analogous to his work in celestial mechanics. Nor did he argue which of these
disturbing causes (friction in the case of capillarity, the attraction of particles of
heat to light in the case of refraction) could be put into analytical form, or how, or
indicate which terms they were represented by in his analytical relationships.
On balance Laplace does not seem to be doing modem mathematical physics.
He was adding a vernacular explanation to an eighteenth-century mathematical so-
lution to a mathematical problem defined in the context of a physical phenomena. 35
Physical phenomena were still serving mathematical purposes. Solutions to the
kinds of mathematical problems Laplace built out of the problem of the motions
of the planets required the utmost confidence in complicated analysis, along with
a technical brilliance in its manipulation to solve increasingly difficult differen-
32 The simplifications that he introduced to obtain this result are not strictly warranted by the
physical situation. Laplace Celestial Mechanics vol. 4, bk. X, 453, See also Bowditch's
note on p. 469 and his comment on p. 47l.
33 See his deduction of the limitation of the internal reflection of light, Laplace Celestial
Mechanics vol. 4, bk. X, 462. His attention was on the velocity of the particles of light
as they passed through several media. He argued that the velocity in the final medium
will be the same as if the light passed from the first to the final medium without passing
through all the intermediary ones.
34 See his explanation of total internal reflection, Laplace, "Memoires sur les mouvements
de la lumiere dans les milieux diaphanes," Mem. Institut (1808): 300-342.
35 This in contrast to Robert Fox, "The Rise and Fall of Laplacian Physics," Hist. Stud.
Phy. Sci. 3 (1971): 89-136, and Roger Hahn, Laplace as a Newtonian Scientist (Los
Angeles CA.: University of California Press, 1967).
106 Physics and Mathematics
tial equations with new methods or with the unexpected use of older ones. This,
together with the tenacity to work through such complexities with mathematical
imagination, produced methods that, if not elegant, were from an imagination that
delighted in complexity.
Poisson took much of Laplace's approach to problems, his goals of going beyond
colleagues' previous solutions through the construction of complex mathematical
problems, and the attachment to a particular, mathematically constructed model of
matter to annex electrostatics to "geometry." This annexation was the first success
in expanding the range of the calculus beyond mechanics. Beginning in Coulomb's
experimental results, Poisson literally transferred the methods of Laplace 's celestial
mechanics to the case of electrostatics. He noted in passing that, in the latter case,
there was attraction as well as repulsion, then focussed on attraction. Beginning
with Coulomb's results that the attractive force at a point within a closed conductor
was zero, Poisson commented that Laplace in his Mecanique Celeste had shown
that the "attraction of surfaces that were almost spherical to interior points was
zero." The physical cases of gravitation and electrostatics were mathematically
equivalent. Poisson then considered the mathematical problem of the depth of the
electric fluid over a spheroidal and almost spheroidal surface so that the attraction
at any interior point was zero. To do this, he had to look for the action of the electric
fluid on any point, within, on, or outside of the surface. Poisson then stated that
the result of his analysis was that the effect of the electric fluid was proportional
to its depth, and that while the problem appeared simple, it was actually tricky and
he had found a defect in previous analyses. Poisson considered the attraction of
spheroids covered with a thin layer made up of molecules between which central
forces act for which,
the components of attraction or repUlsion that a body exerts at a given
point, were expressed by the partial differentials of a certain function
of the coordinates of this point, namely, the function that represents the
sum of the molecules of the body divided by their respective distances
to the given point: therefore we designate the sum as V. 36
Having defined V mathematically, no more was said of its physical origins.
Poisson did not connect the results deduced using this definition of V to either
Coulomb's law or the results of Coulomb's experiments. While V = V(x, y, z),
Poisson did not use the functional relationship between the coordinates, given the
force-law he has defined above, to limit the kinds of solutions he sought. In using
the potential function Poisson deduced the equivalent of Gauss' law, but did not
connect it to the physical result that he seemed to be addressing. Immediately after
obtaining Gauss' law, Poisson then investigated the distribution of the electric fluid
to satisfy the condition that, in equilibrium, the force in the interior of the sphere
36 Poisson, "Memoire sur la distribution de l'electricite ala surface des corps conducteurs,"
Mem. Institut (1811): 1-92,14.
in France, 1790-1830 107
is null. His argument was complex and designed to show mathematically that the
force at the surface of the spheroid has no tangential component.
The mathematical thrust of Poisson's work was further emphasized in his second
paper on electrostatics. The main point of the paper was to draw mathematicians'
attention to his transformation of series that did not appear to converge into others
that did and that lead to a solution in finite form. These were important issues for
mathematicians of the time and so was the second issue on which he concentrated,
avoiding definite integrals. 37
The physics used in the introduction to the paper came directly from Coulomb's
works. And while he deduced important new physical results, Poisson did not
recognize them as such. They existed in a stream of analysis that climaxed in the
solution of a mathematical not a physical problem. Here, as in his later papers,
Poisson began with a specific physical model which was quickly translated into
a mathematical expression for a force. All the analytical apparatus of rational
mechanics could operate upon this expression and both model and physical problem
become irrelevant to the solution. The model of electricity as a fluid that spread
over the surface of the spheroid came from Coulomb, as did the idea that the depth
of the fluid was proportional to the intensity of the electric force. Poisson deduced
the depth of the fluid for many different particular cases of spheroids and spheres
acting on each other. He calculated V at some point due to spheres and spheroids
at very large, and at very close, distances in terms of the depth of the electric fluid.
The important issue was to reduce the expression for V into finite form and there
his interest in the function ceased. The electrical aspects of the case did not enter
into his solution, as he did not explain how the expressions containing V were
connected to the action of electricity. In his work were many clever mathematical
techniques without much indication of how all the analysis might be relevant to
the physics of electrified bodies.
Under these circumstances it is difficult to see Poisson's work on electrostatics
as "pivotal in the development of a new vision of physics.,,38 Poisson accomplished
what Laplace had hoped to do-extend the range of analysis beyond mechanics and
into experimental physics to create a physique-mathematique. The unification of
physics and mathematics was still in the future. 39 The problem Poisson solved
had been set as a prize problem by the mathematical section of the Institut. From
the reactions of his colleagues, Poisson had solved a mathematical not a physical
37 Poisson, "Second memoire sur la distribution de l'electricite a la surface des corps con-
ducteurs," Mem. Institut (1811): 163-274. These same patterns are followed also in his
later papers on magnetism.
38 Home "Poisson's Memoir in Electricity," 259. Heilbron, Electricity, shares the view that
Poisson's work was crucial for physics.
39 Garber, "Simeon-Denis Poisson," treats Poisson's work as mathematics and sets it in the
context of early nineteenth-century French mathematics.
108 Physics and Mathematics
problem. The problem was to express a phenomena in analytical form, and then
apply the calculus to solving the resulting equation, a partial differential equation
of the first order. Poisson did so by fixing on the force between the particles of the
fluid. The mathematics of these problems were already explored. Electrostatics
was opened to annexation by the calculus.
This is not to belittle Poisson's achievements but to recognize that if Poisson
did not discuss the physical significance of his analysis, neither did any of his
contemporaries. From the lack of hostile reactions to Poisson's papers on electro-
statics, we have to conclude that they were a satisfactory mathematical solution
to the problem in the terms acceptable to the French scientific community.4o He
had considerable talent for mathematics, which is evident even in his early papers,
but not for physics. He was inventive at solving mathematical problems not com-
pleted by others. Laplace's solution of the attraction of spheroids gave Poisson the
opportunity to solve it for more complex cases while encompassing a new domain
of experimental physics within the calculus. Even as the direction of mathemat-
ics changed, Poisson remained convinced that the proper direction of research in
mathematics lay in the solution of the partial differential equations emerging from
physical problems. The partial differential equations that were the proper focus of
mathematicians were those of the same form as the equations of mechanics. 41
While Poisson turned his attention to a field of experimental physics that had
resisted mathematization for decades, Andre Marie Ampere mathematized a com-
pletely new and unexpected phenomenon, the connection discovered by Hans
Christian Oersted between current electricity, magnetism, and mechanical force.
Ampere was initially a member of the mathematical division of the Institut and
had published in the growing controversy over the foundations of the calculus. In
1820, he turned to the experimental investigation of Oersted's results. 42 Ampere
was a mathematician blessed with manual dexterity and an interest in metaphys-
ical speculations as well as experiments. He assisted Arago in demonstrating
Oersted's experiment to the Academie and within the month reported on his own
experiments. 43
sur un memo ire lu a Academie Royale des Sciences, 4 Decembre 1820," [Sur l'action
mutuelle de deux elements de courans electriques,) 1. Phys. (1820): 226-230. Details
of Ampere's involvement and experiments are in Christine Blondel, A. -M. Ampere et la
creation de l' electrodynamique (Paris: Bibliotheque Nationale, 1982). However, see the
exchange between Blondel, "Ampere and the Programming of Research," Isis, 76 (1985):
559-561 and Pearce Williams' reply. Whether or not Ampere's work was programmatic
or, at a crucial stage, depended on chance does not affect the argument here. See also
Alfred Kastler, "Ampere et les lois de l'electrodynamique," Rev. Rist. Sci. 30 (1977):
143-157. See also J. R. Hoffmann Andre-Marie Ampere (Oxford: Blackwell, 1995).
44 Ampere, "Memoire sur la determination de la formule qui represente l'action mutuelle
de deux portions infiniment petits de conducteurs voitaiques," Ann. Chim. 20 (1822):
398-42l.
45 Ampere was not the first mathematician to perform his own experiments. Etienne Louis
Malus did his own experiments on refraction then polarization and developed his own
mathematical exploration of the phenomena. Jean Baptiste Biot first published papers on
the calculus and a companion volume to the mathematics of Laplace, Mecanique Celeste
before turning to experimental physics. Biot's first experimental work was in electricity,
sound and then in 1804, heat conduction.
46 For a discussion see Kastler, "Ampere et l'electrodynamique," 153-154.
110 Physics and Mathematics
space between the conductors. However Ampere did not connect these hypotheses
with the results of his continuing experiments. When he drew his experimental
results together to develop a mathematical theory of "electrodynamics," he used
none of these ruminations. Ampere's final statement on this stage in his electro-
dynamic work was presented in the mathematically elegant memoir of 1823.41 His
mathematics was based on a series of experimental results together with the results
encapsulated in his initial force law.
Ampere was at pains to separate his approach to the mathematization of electro-
dynamics from that ofBiot and Savart. The latter had their own mathematical forms
based on keeping intact the Laplacian approach to the mathematization of experi-
mental physics and retaining the distinction between magnetism and electricity.48
To obtain results for whole circuits, rather than circuit-elements Ampere needed
to use both line and surface integrals. This was the first time that such mathe-
matical techniques were required to bring a domain of experimental physics into
mathematical form. 49
In its final form the reader was presented with two papers, rather than one. The
first part of the memoir presented the phenomena and the non-mathematical phys-
ical model to explain the phenomena in terms of molecular forces and molecular
currents. 50 The second part was a generalized mathematical theory devoid of any
physical modelling or processes. The molecular models did not enter into either
the setting up of the problem nor in choosing how the mathematical development
of the problem might proceed. The two approaches existed fully developed in
their own spheres, both clear, both worked out in detail: Metaphysics and exper-
iment are contained in the first half, experimental results expressed algebraically
as the starting point for the mathematical second. The differences of Ampere's
from previous presentations in this tradition lay not in the simultaneous production
and presentation of a series of new experimental results, interpreted physically in
non-mathematical terms, together with their integration into an expanding domain
of mathematics. His uniqueness lay in his experiments and in the sophistication
of his mathematical techniques. The ultimate result does not read as theoretical
Heat
Both historians of mathematics and physics claim Joseph Fourier's work in the
conduction of heat for their disciplines. And, his work was indeed significant for
mathematics immediately, theoretical physics decades later. 53 Our concern here is
how Fourier's work on heat conduction fitted into the disciplines of physics and/or
mathematics in the early decades of nineteenth-century France. Its later absorption
into other disciplines will be considered elsewhere. Given the standards of practice
of French mathematicians in the first two decades of the nineteenth century and
the reactions of his colleagues, it makes more historical sense to place Fourier's
work in the history of mathematics. To judge how far Fourier was from being a
theoretical physicist, we need to consider the controversies surrounding Fourier
series when they first appeared and examine changes in the calculus that came to
fruition in the 1820s. We must, therefore explore the history of French mathematics
in two crucial decades and Fourier's work in development of the first, logically
defensible form of the calculus.
Historians of physics have retrospectively claimed Fourier for their discipline
from the importance of his mathematics for later developments within physics.
They have argued that his influence was crucial for the development of concepts
and as a means of expressing all manner of wave phenomena and arbitrary functions
that occur in physics and engineering. This later importance does not, however,
tell us how Fourier's work on heat or Fourier analysis was regarded and used in the
first decades of the nineteenth century. So important did Fourier's methods and the
concept of flux become for physicists that they and some historians have inverted
his purposes. The physics that was retrospectively seen within Fourier's work by
middle and late nineteenth-century physics must have been his intended object of
study in the first place.
Fourier followed the standard practice of early nineteenth-century mathemati-
cians by choosing a physical problem to explore a mathematical domain, arbitrary
functions. His initial attempt at transforming experimental results on the conduc-
tion of heat along a bar of metal into mathematical form was using a mechanical
model. He abandoned this approach, as it was only successful in some particular
mathematical cases and a mathematical failure in the general one. 54
Fourier's work also has been seen as creating a new theory of heat within physics,
conceptually beyond that of earlier caloric theories. 55 However, as most historians
of physics agree, none of the molecular or other processes Fourier used for the
conduction of heat ever entered into his mathematical analysis of the problem.
As Fourier noted, "if the mathematical laws which the effects of heat follows
are carefully examined, it is seen that the certainty of these laws does not rest
on any physical hypothesis."56 Fourier repeated the claim that the principles of
54 For an analysis of this initial attempt, its eighteenth-century mathematical roots and its
shortcomings, see Grattan-Guinness and Ravetz, Joseph Fourier, chap. 3, 36-81. See
also Amy Dahan, "J. Fourier: L'elaboration de la tMorie analytique de la chaleur," Sci.
Techn. Persp. 1 (1981): 7.1-7.41.
55 Robert Marc Friedman, "The Creation of a New Science: Joseph Fourier's Analytical
Theory of Heat," Hist. Stud. Phys. Sci. 8 (1977): 73-100.
56 Fourier, "Theorie du mouvement de la chaleur dans les corps solides," Mem. Acad. Sci.,
Paris 4 (1819-1820) [1824]: 185-555, 192. This appears also in Fourier, Analytical
Theory, 40-41. Fourier, "TMorie," listed here is the first part of Fourier's memoir
in France, 1790-1830 113
his work depended upon, "a very small number of primary facts, the causes of
which are not considered by geometers, but which they admit as the result of
common observations confirmed by all experiment.,,57 Immediately following this
disclaimer, Fourier gave a descriptive, molecular explanation of the conduction
process, one of radiation from molecule to molecule. But he did not put this
physical process into a generalizable mathematical form. Fourier abandoned the
approach. However, physical descriptions survived, scattered through the text
although they remained unrelated to the mathematical analysis of the problem.
Fourier published a separate account of the physical theory of heat conduction and
radiation. 58
On another level, historians of physics see Fourier as developing the important
concept of flux and giving it particular meaning in his theory of heat. 59 While flux
became important for later developments within physics, Fourier himself neither
named it, nor attached any physical significance to it. 6o Fourier established the
equation for the motion of heat on the basis of the principle of "the uniform, linear
movement of heat." He went to some lengths to demonstrate that the notion we
name flux emerged from experimental sources that supported his analysis of the
flow of heat into, then out of, a thin slab and the losses across its thickness. His
reasons for his detailed discussion were mathematical, "because the neglect of it has
been the first obstacle to the establishment of the mathematics." He argued further
that if we did not make a complete analysis of the elements of the problem, "we
should obtain an inhomogeneous equation, and, a fortiori, we should not be able to
form the equations which express the movement of heat in more complex cases.,,61
Within the context of his mathematics, Fourier was careful to avoid definitions
that were not phenomenological or deduced directly from experiment. All this
has been said before in the context of connecting Fourier to the later, intellectual
presented to the Institut in 1811. The second part was published as, Fourier, "Suite du
memoire intitule: Theorie du mouvement de la chaleur dans les corps solides," same
journal, (1821-1822) [1826]: 153-246. By the time these papers appeared their contents
were already published as Fourier, Theorie analytique de la chaleur, (Paris, 1822), trans.
by Alexander Freeman as Fourier, The Analytical Theory of Heat. This translation
follows the original closely and the translator's comments are carefully separated from
the author's original. Such is not the case in the version of this work in Fourier's collected
works.
57 Fourier, Analytical Theory of Heat, 6.
58 Fourier, "Questions sur la theorie-physique de la chaleur rayonnante," Ann. chim. phys.
6 (1817): 259-303.
59 John Herivel, Joseph Fourier: The Man and the Physicist (Oxford: Clarendon Press,
1975) chap. 9, emphasizes the importance of this as a contribution to physics. See also
Friedman, "Fourier."
60 The first time I have seen the name used was by Philip Kelland.
61 Fourier, Analytical Theory, 59.
114 Physics and Mathematics
62 The examples used are a row of disparate bodies, generalized into a line, a ring for the
two-dimensional equilibrium case, and various shaped bars for the three-dimensional
cases.
63 Fourier cited an experiment on a metal ring heated at different points and its temperature
taken at other points and claimed that these experiments "fully confirm" his mathematics.
Fourier, Analytical Theory, chap. 2, p. 90.
64 Fourier, Analytical Theory, 85.
65 AIl of Fourier's work on heat theory was completed before Cauchy began his lectures on
the calculus where he replaced the older calculus with new definitions of the derivative,
in France, 1790-1830 115
continuity, and the integral that he then used to prove the simple and not so simple
results of the calculus that others had taken for granted. For a full discussion of the
mathematical difficulties of Fourier's work, see Grattan-Guinness and Ravetz, Fourier,
and Grattan-Guinness, Convolutions, chap. 9, 597-602. Cauchy enters in chapter 10.
66 For particular examples See, Fourier, "Sur la temperature des habitations et sur Ie mou-
vement varie de la chaleur dans les prismes rectangulaires," Bull. Soc. Philo. (1818):
1-11, and "Extrait d'une memoire sur Ie refroidissement seculaire du globe terrestre,"
Ann. Chern. Phys. 13 (1820): 418-438. More papers by Fourier on the secular cooling
of the earth appeared in the 1820s after the publication of his Analytical Theory.
67 Fourier, "Extrait de theorie de la chaleur," Ann. Chirn. Phys. 3 (1817): 350--375.
68 Fourier, "Extrait," and Fourier, "Note sur la chaleur rayonnante," Ann. Chirn. Phys. 4
(1817): 259-303.
116 Physics and Mathematics
those of mechanics, he may well have had in mind Poisson's criticisms and the
latter's mechanical model for the flow of heat rather than rational mechanics that
he referred to repeatedly as a model. What he accomplished for heat had been
done already for mechanics. Even while the forms of the fundamental equations
were quite different both domains of physics were now encompassed by analysis. 69
Rational mechanics appeared in analogy to his work in several places and as an
exemplar throughout his preliminary discourse. Although the Laplacian model of
matter and the mathematical form for the equations of motion and other funda-
mental laws of mechanics might not apply specifically, rational mechanics was the
model Fourier tried to imitate.
As recent research in the history of mathematics amply demonstrates, the solu-
tion of problems defined mathematics as a discipline in the eighteenth and early
nineteenth centuries. The solutions of problems taken from mechanics were used
as arguments for the mathematical surety of the calculus before mathematicians
secured any rigorous foundation for the calculus. Problems grounded in reality
that the calculus could solve were guarantees of its mathematical validity. This
was the level of the usual response to criticisms of the calculus, especially that of
Bishop Berkeley.70 On this score Fourier echoed his contemporaries to reinforce
the validity of the shaky foundations of his own mathematics.
The profound study of nature is the most fertile source of mathematical
discoveries. Not only has this study, in offering a determinate object to
investigate, the advantage of excluding vague questions and calculations
without issue; it is besides a sure method of forming analysis itself, and
of discovering the elements which it concerns us to know, and which
natural source we ought always to preserve.?1
Fourier went on to observe that the analytical equations first introduced into math-
ematics by Descartes (the calculus), unknown to the ancient geometers, extend to
all natural phenomena, and,
There cannot be a language more universal and more simple, more free
from errors and from obscurities, that is to say more worthy to express
the invariable relations of natural things.
The reactions of Fourier's contemporaries to his work were to his mathematics,
complicated by the disputes already brewing over the proper foundations for the
calculus. Physical problems and the solutions to the equations they generated were
the battleground for the very soul of the calculus. By the time Fourier's work in
heat theory appeared, a rigorous foundation for the calculus was becoming urgent.
His work allowed other mathematicians to draw clearer battle lines. In his initial,
incomplete paper of 1807, Fourier challenged standard mathematical practice. He
used the separation of variables to solve partial differential equations of the second
order, as well as trigonometric series in mathematical situations where functional
alternatives were the more normal approach. 72 Before Fourier's paper could appear
in print, its mathematical lacunae were pointed out by Poisson. While noting the
incorrectness of the physical foundations of Fourier's physics, Poisson detailed the
problems with his mathematics. 73 Lagrange saw Fourier's mathematical approach
as simply unacceptable. This was hardly surprising. Lagrange had also developed
the only defensible formulation of the calculus whose canons of practice Fourier
so clearly violated.
Fourier responded to his critics both publicly and through Laplace. The issues
centered upon the ultimate legitimacy of Fourier's mathematics in its general form
and his competency as a mathematician. Fourier bested his major critic, Poisson,
even though Poisson persisted with his own version of a mathematics of heat pub-
lished years after Fourier's death.74 He refused to answer Biot in public. Fourier
had gained some crucial insight into the mathematical expression of the temper-
ature distribution along a bar from Biot's experiments and Biot's mathematical
expression of those results. Fourier did not acknowledge this debt. Ungracious as
this was, Biot's expression of his results was stated as the sum of two exponential
functions, with the remark that any mathematical theory of the phenomenon would
have to take the complexity of his data into account. Biot gave no details of how
he obtained this expression. This was the solution to the problem whose original
equation of motion was Fourier's quarry. Biot was not, in Fourier's eyes, a math-
ematician, only an experimentalist supplying grist for his mathematical mil1. 75 In
his first paper Biot described in the vernacular the equation for the conduction of
72 For details of Fourier's mathematical methods and how they differed from his colleagues,
see Grattan-Guinness and Ravetz, Fourier.
73 Poisson, "Memoire sur la propagation de la chaleur dans les corps solides," Bull. Soc.
Philo. (1807): 112-116. These arguments are repeated in Poisson, "Sur la distribu-
tion de la chaleur dans les corps solides," J. Phys. 80 (1815): 434-441. Both were
published before Fourier's text. For details of this criticism and Fourier's reaction, see
Grattan-Guinness and Ravetz, Fourier and Herivel and Costabel,Joseph Fourier face aux
objections contre sa theorie de la chaleur, lettres inedits, 1808-1816 (Paris: Bibliotheque
Nationale, 1980).
74 See Grattan-Guinness and Ravetz, Fourier, 463-471 for the derivative character of much
of Poisson's mathematics in this area.
75 Biot wrote both experimental and mathematical papers on the conduction of heat in 1804.
See, Biot, "Memoire sur la propagation de la chaleur, et sur un moyen simple et exact de
mesurer les hautes temperatures," J. des Mines 17 (1804): 203-224, and, Biot, "Sur la
loi mathematique de la propagation de la chaleur," Bull. Soc. Philo. 3 (1804): 215-216.
118 Physics and Mathematics
heat along, and the radiation of heat from, a bar in thermal equilibrium. The equa-
tion itself was not written down or solved until the second, separate mathematical
paper.76
Despite criticism and delays in the publication of his early work, Fourier ex-
tended his discussion of the expression of arbitrary functions in terms of trigono-
metric series, then developed their expression in integral form, that is, developed
Fourier analysis. All of this mathematics lay in the hands of the secretary of the
Institut, then of the Academie des Sciences, until the 1820s, and, as Fourier worked
his way into the inner circles of power within science. In 1822 the third version of
his Analytical Theory was published in the same year that he was elected secretaire
perpetuel of the Academie. This publication was his first opportunity to give an
extended account of his work, and to defend it publicly. 77
Given the practices of mathematicians in early nineteenth-century France, we
must place Fourier's work within the history of that discipline. Placing Fourier's
work on heat in the history of mathematics makes his use of his own experimental
work more understandable. This does not detract from the obvious value of the
work nor its position of influence in the later development of physics, mathematics,
and engineering. In examining reactions to Fourier's heat theory papers, we can see
patterns in the discussions that reveal what aspects of that theory were important
to his contemporaries. Fourier's mathematics were dissected. Equally important
are those issues passed over in silence, that is his experimental work.
The other aspect of early nineteenth-century French work on heat that became
crucial for the later development of physics was Sadi Carnot's examination of
the heat engine. This was passed over in silence by most of his contemporaries in
physics and mathematics for two reasons. The first was political. As a son of Lazare
Carnot, his work was without social grounding in the governmental institutions of
restoration France. It could be political folly simply to notice it. The other was
that he was an engineer and he addressed engineers and the problems they faced
in understanding steam engines rather than in producing a mathematical theory of
heat.
He followed the tradition of French engineering in reducing his steam engine
to an idealized, general form of a heat engine. Historians have seen Carnot's
understanding of the cyclic nature of the operation of the heat engine as derived
from his father's work in mechanics. However, his understanding of what was
76 Truesdell, The Tragicomical History, 51, suggests that Biot was not capable of thinking
on this level and that his equation was suggested by Laplace. There are indications that
in earlier mathematical problems, Biot was guided in his choice by Lacroix. His solution
of a problem in the partial differential equation of sound is pendantic. See Biot, "Sur
l'integration des equations differentielles partielles et sur les surfaces vibrantes," Mem.
Institut 4 (1802): 21-111. See also Eugene Frankel, "Career Making."
77 The publication of the 1811 prize paper was predictably slow. It oozed from the presses
of the Academie in two parts, appearing in 1824 and 1826.
in France, 1790-1830 119
physically happening along each of the four stages of the cycle was his own. And
his explanation of these stages was non-mathematical and in terms of caloric, and
its conservation in the complete cycle of the engine. The fall of caloric from a
higher to a lower temperature made the extraction of mechanical work possible.
While Carnot's description of the physical processes involved in completing a cycle
of his ideal heat engine was clear, his goals were those of an engineer. His aim was
an expression for the efficiency of such an idealized system, and understanding the
operating conditions for real steam engines to maximize their performance. Carnot
also needed to make plausible his assumption that the operation of his idealized
heat engine extended to all types of working substances. All his results on the
behavior of the specific heats of gases were deduced from known experiments.
The construction of his mathematical derivation for the temperature dependence
of the "motive power" of heat was relegated to a long footnote. This derivation came
directly from known gas laws and was completely independent of his assumptions
about the nature of heat. In the 1830s yet another engineer, Emile Clapyeron
constructed a mathematized theory based on Carnot's analysis of the heat-engine
cycle, and experiments on the specific heats of gases. Clapyeron specifically
based his mathematics only on well established hypotheses and distanced his work
from that of Lavoisier and Laplace. Neither work stirred the imagination of other
mathematicians or physicists until the late 1840s. 78
The last historical problem set we will consider is on the changes in ideas about
the nature of light that engendered a complex of mathematical theories in elasticity.
Some of these mathematical theories became important in the middle decades of
the nineteenth century as physicists adopted mathematics as the language of theory.
Until the early nineteenth century the phenomena of light were assumed as
empirically understood, although physical explanations of its nature and interaction
with matter remained problematical. In the eighteenth century, interpretations
of the nature of light remained as contested as those of caloric. The primary
assumptions were either that light was a stream of particles, or a disturbance
propagated through a substance that pervaded space. These hypotheses left the
details of the action of light to subsidiary assumptions, and all manner of tropes,
similes, and metaphors. 79 The principal area of difficulty for both sets of physical,
78 Sadi CarnotRejlexions sur la puissance motrice dufeu (Paris: Vrin reprint of 1826 edition,
1979). Emile Clapyeron, "Memoire sur la puissance motrice de la chaleur," J. Ecole Poly.
16 (1834): 153-190. Truesdell, Tragicomical discusses Clapyeron's mathematics. See
also Jean Dayantis, "Carnot, Clapyeron et la theorie calorique au dix-neuvieme siecle,"
Rev. Quest. Sci. 164 (1993): 105-130.
79 The ranges of such physical speculations have recently been explored by Casper Hakfoort,
Optics in the Age of Euler, and Geoffrey Cantor, Optics after Newton.
120 Physics and Mathematics
vernacular theories lay in the phenomena where light and matter interacted, namely
double refraction and to a lesser extent simple refraction. Laplace's political power
within the Institut in the first decade of the nineteenth century and his analytical
interest in light, as an offshoot of his work in celestial mechanics, guaranteed that
the problem areas of of light would become the subjects of prize essays. The
subject of the prize essay made public in December 1807 was double refraction. 8o
In 1810 the work of Etienne-Louis Malus was crowned. His prize essay included
the detection, experimental establishment, and analysis of a new phenomenon, po-
larization. This new phenomenon complicated the explanatory picture and guar-
anteed further prize questions on the same subject. Malus had presented his ex-
periments on the refractive power of opaque crystals to the Institut one month
before the prize problem was announced. In working further on Iceland spar, he
discovered the polarization of reflected light. 81 As Buchwald has amply demon-
strated, Malus' experiments changed experimental optics. Malus worked with
sophisticated instrumentation, reported, then analyzed his results carefully. 82
If we explore his long prize essay, we find that it falls into three sections, each
one separate from the others. The first section is mathematical on "Des questions
d' optiques qui dependent de la geometrie." This section consisted of the mathe-
matics of known phenomena dependent on light being propagated in straight lines.
When light is reflected or refracted from a surface, the equation for the system of
rays emerging from the encounter was
x - x' y - y' z- Z'
-----
m
= -----
n
= ----
o
where m, n, 0 were "arbitrary functions of x', y', Z'." SO far no physics. Malus
was working towards the general problem of considering,
a system of right lines emanating from all points of a curved surface,
that obey any analytical law whatsoever, this system being regarded as
the locus of the intersection of two systems of developed surfaces. 83
Malus proceeded to treat all the mathematical cases he could and expressed
them in general functional form. In the case of refraction, Malus used the notion
that at the surface light was turned through an angle by a force perpendicular to
that surface. This assumption reduced optics to the mathematics of mechanics. To
80 Events leading up to this prize problem and work done previously by William Hyde
Wollaston on double refraction are recounted in Jed Buchwald, The Rise of the Wave
Theory of Light (Chicago: University of Chicago Press, 1989) chaps., 1, 2.
81 The phenomenon was announced in 1809, Malus, "Sur une propriete de la lumiere
reflechie par les corps diaphanes," Bull. Soc. Philo. 1 (1809): 266-269.
82 Malus, like Coulomb who achieved the same transformation in the study of electricity,
was a military engineer. For the details of Malus' experiments, see Buchwald, Rise,
31-36.
83 Malus, "Traite d'Optiques," Mem. Institut, Paris 2 (1811): 214-302,221.
in France, 1790-1830 121
Malus it was irrelevant how the force operated on the light to cause it to deviate
from its straight-line path. It was only "necessary to calculate its effects."84 Malus
handled double refraction in the same way.85 The physical explanation of refraction
and polarization existed separately and escaped "all quantitative determination."
Malus relied on the speculative construction of short-range attractive and repulsive
forces acting close to the surface of the refracting body as physical, explanatory
devices. 86 He did not put these ideas into analytical form, nor could he in principle
subject them to experimental examination. 87
Experiments on polarization also became the ground on which Biot made his
professional mark. Initially Biot worked with Arago on the experimental deter-
mination of the indices of refraction of various gases. 88 He claimed to deduce
results only trivially different from those of Laplace in his Mecanique Celeste.
The mathematical part of the paper showed that Biot had developed his own ana-
lytical expression for the refracting power of gases and traced through Laplace's
work on the refraction of light for formulae corresponding to the conditions of his
experiments. However, the analytical derivation was in terms of rays of light and
their angular changes of path as they pass through different media. The vernacular
description of what occurred as the light passed through the media was in terms
of the forces causing these deviations. These forces were not connected to the
analytical derivations from his experiments. 89
Biot continued with a series of careful, quantitative experiments coupled with
long analytical deductions from his results. Yet, even while explicitly denying any
use of hypotheses, his papers on light are peppered with them. 9o In the mathe-
matical exploration of the results of his experiments on the polarization of light in
birefringent crystals, Biot expressed the equation of motion for the changes in the
84 Buchwald notes that this section contains no physics and is purely analytical, Buchwald,
Rise, 38.
85 Malus, "Theorie de la double refraction," in "Traite," 303-508.
86 See Malus, "Sur une propriete des forces repulsives qui agissent sur la lumiere," Mem.
Soc. Arcueil2 (1809): 143-158.
87 For details of Malus' work, see A. Chappert, Etienne-Louis Malus (1775-1812) et la
theorie corpusculaire de la lumiere (Paris: Vrin, 1977), p. 124.
88 Biot and Arago, "Sur les affinites des corps pour la lumiere, et particulierement sur les
forces refringentes des differents gaz," Mem. Institut 7 (1806): 301-387. The analytical
part of the paper is "Part II," 363-387.
89 For the cutthroat conditions in the competitive domain of experimental physics and in
particular the competition between Arago and Biot, see Buchwald, Rise, 79-88. Arago
did not follow the new trend toward experimental physics and quantification using com-
plicated instrumentation. Biot chose an aggressive prosecution of this new methodology
for experimental optics.
90 For example, see Biot, "Sur un nouveau genre d' oscillation que les molecules de la lumiere
eprouvent en traversant certains cristaux," Mem. Institut 12 (1812) [1816]: 1-371, p. 60.
122 Physics and Mathematics
dx 2
dt 2 =¢(i-X)-¢I(i-X),
where ¢ is the force and x the angle of the axis of polarization at time t. This
problem was worked out in its most general form only to be truncated into simple
harmonic motion to obtain numerical results that could be compared with experi-
ment.
In the introduction to this long paper, Biot claimed that his experiments showed
the successions of oscillations of the "luminous molecules." The equation of
motion of the planes of polarization were the equations of motion of these particles
of light. After assuming this he developed a non-mathematical, physical theory of
the changes undergone by the particles of light. This physical theory was devoid
of mathematical, although not of logical, reasoning or clarity of concept. While
his experiments on the polarisation of light in birefringent crystals were careful
and reported at length in the Memoires of the Institut, his theoretical account was
less than successful. The experiments recounted the changes in reflected and
transmitted light in birefringent crystals for all the colors of the rainbow. These
results and the differences in the colors of the reflected and two refracted rays were
important in his attempt to draw together the Laplacian theory of matter and light;
centre of force molecules acting on luminous particles that set up oscillations in
those luminous particles.91 Biot combined the particulate theory of matter and of
light that produced waves. However, this phenomenon was more easily explained
by waves than particles. His assumption was that the axis of polarization of the
light was gradually changed by the action of the molecular forces as the light
particles traveled through the medium.92
There were other conceptual problems. He implicitly assumed that the equation
of motion of the polarization axis was the same as the equation of motion of the
luminous molecules. Biot then investigated this simple harmonic motion for a
single molecule. However, his comparisons were between the mathematics of a
microscopic structure and experiments of macro-phenomena. No consideration
was given as to how this micro-motion led to the macroscopic effect. He assumed
they were the same. There were problems as well in the mathematical part of
the paper. The wave motions of the luminous molecules were reduced to the
simplest of motions without considering how this motion led to an ordinary and
extraordinary ray within the crystal. Again we have both a physical theory and a
91 See Biot, "Sur une nouvelle application de la tMorie des oscillations de la lumiere," Mem.
Institut (1812) [1816]: 1-38.
92 Biot continued to develop this vernacular, physical theory in Biot, Recherches experimen-
tales et mathematiques sur les mouvements des molecules de la lumiere (Paris: 1814), and,
"Nouvelles experiences sur Ie developpement des forces polarisantes par la compression
dans tous les sens des cristaux," Ann. Chim. Phys. 3 (1816): 386-394.
in France, 1790-1830 123
93 Buchwald details Fresnel's early life and experiments on light and the circumstances
under which they were performed in Buchwald Rise, chap. 5.
94 Fresnel, "Memoire sur la diffraction de la lumiere," Ann. Chim. Phys. 1 (1816): 239-
281. Some of these early memoirs were deposited at the Institut and later the Academie
des Sciences, and/or read by Arago at sessions of these institutions. All of his early
papers were published in Fresnel, Oeuvres completes H. de Senarmount, E. Verdet and
L. Fresnel eds., (Paris: Imprimerie Imperiale, 1866-1870), 3 vols. Fresnel's work on
light is in the first two volumes.
95 The first occurrence of these arguments is in Fresnel, "Premier memoire sur Ie diffraction
de la lumiere," Oeuvres vol., 1,9-34, 10-15. The best presentation is in his prize essay,
Fresnel, "Memoire sur la diffraction de la lumiere," Mem. Acad. Sci. Paris (1821-1822)
[1826]: 339-487, 341-349 where he builds a case that the emission theory contradicts
itself. Fresnel always referred, discretely, to the emission theory as "Newtonian."
96 Fresnel, "Premier memoire sur la diffraction de la lumiere," Oeuvres vol., 1, 9-34, 25-34.
124 Physics and Mathematics
reftection. 97 The above arguments were then repeated along with a group of im-
proved measuring techniques. 98
Fresnel was asked by the Academie committee considering the work that became
his prize essay, to add an analytical part to it. They deemed it incomplete. 99 In
his amended prize essay the mathematical work that accompanied his experiments
included an expression for the wave length of light that did not depend on any
particular model for the mode of action of light. The detailed development of
his ideas on the nature of light itself was accomplished through his experiments
and expressed clearly and argued logically in the vernacular. Fresnel did not
actually develop a mathematical theory of the nature of light and its action based
on his physical descriptions of its nature. The purpose of his mathematics was
not to explore the nature of light or its propagation or interaction with matter
but to replicate the expressions he had already deduced by geometry and algebra
from his experiments. Fresnel demonstrated graphically how the maxima and
minima of intensity represented by the fringes of his experiment occurred through
the superposition of waves. From the geometry of his experiment, he deduced
the algebraic relationship between the wavelength of the light, the distance of
the fringe maximum from the center of the pattern of fringes, and the distance
between the screen and the source of the diffraction pattern. The more sophisticated
mathematics of the calculus replicated these earlier results. 100
Analysis and the calculus did not bear much weight with Fresnel. In contrasting
the Newtonian and his own theory of light, Fresnel put forward his criteria for
choosing between the two. The choice must be made on the basis of the con-
ceptual simplicity of the hypotheses necessary to explain the phenomena, not on
the calculus, although the latter was more easily applied to the Newtonian than
the wave theory of light. In invoking the economy of nature, Fresnel noted that
"nature is not troubled by the difficulties of analysis," and appeared to produce the
maximum number of phenomena through the minimum number of causes. 101
And yet analysis had its uses. In the mathematical addition to his prize essay
Fresnel took over the available mathematics of fluid motion. He also accepted
the assumption that the velocity of propagation of the waves was the same for
all wavelengths and that the intensity of the wave motion was proportional to the
amplitudes of the motions of the particles. In addition to deducing the positions
of the maxima and minima for the fringes produced by diffraction at a slit, Fresnel
tried to find expressions for the intensity of the bright fringes. There was no attempt
at a complete theory of light, that is, to follow the particular motions of the ether
to produce the required intensities. Fresnel began with the total intensity of the
light at point P. This was defined as the sum of the elementary waves spreading
through P. The sum of all the "small motions at P," the actual intensity, was
(f dzcos(Jrz 2
(a+ b))2
abA) + (f' (a + b)
dzsm(Jrz 2 abA )
)2 ,
where the limits on the integration were zero and infinity. dz was any small distance
along the primary wave and z was the distance of P from the source. A was the
wavelength of the light and a and b were the distances of the point on the wave
front from the source and from the screen, respectively.102
Fresnel presented a confused mathematical argument. The goal of the mathe-
matical investigation was limited to the replication of his experimental results so
that he could undertake a direct comparison between the two. Biot and Malus had
done the same but from the basis of a more detailed analytical development of the
mathematical implications of their experiments. Much of the uniqueness attributed
to Fresnel by Robert Silliman is therefore undermined. Both Malus and Biot did
careful, numerate experiments, analyzed, and we could argue then developed their
ideas mathematically with far more confidence and mathematical skill than Fresnel.
However, we no longer accept the physical foundation for the analyses of those
experiments. Their efforts have been undervalued. Because we do still accept the
grounds for Fresnel's explanation, we overlook many of his shortcomings. 103 Fres-
nel did not have that kind of control or interest in the mathematical material. Both
Buchwald and Nahum Kipnis postulate much more mathematical and physical co-
herence in Fresnel's work than his papers suggest. Many of the physical concepts
made explicit in their analyses are implicit in Fresnel papers. His mathematics
is fragmented and needs frequent interpolations on their part. 104 While Fresnel's
102 Fresnel added all the waves that reach P from points that were ,X and 1/ 4,X and so on from
each other. See Fresnel, "Note sur la theorie de la diffraction," Oeuvres vol. 1, 171-181,
and "Memoire sur la diffraction de la lumiere," Mem. Acad. Sci. Paris (1819): 339-487,
383-407.
103 See Silliman, "Fresnel and the Emergence of Physics as a Discipline," Hist. Stud. Phys.
Sci. 4 (1973): 137-162.
104 Buchwald, Rise and Kipnis, History of the Principle ofInterference (Boston: Birkhauser,
1991).
126 Physics and Mathematics
105 See Humphrey Lloyd, "Report on the Progress and Present State of Physical Optics," Rep.
British Assoc. (1834): 295-413,387. See also Grattan-Guinness, "Review of Buchwald,
The Rise of the Wave Theory of Light," Ann. Sci. (1989): 185.
106 For Poisson and Cauchy on waves see Grattan-Guinness Convolutions, vol. 2, chap. 10.
For Sophie Germain see Louis L. Bucciarelli and Nancy Dworsky, Sophie Germain: An
Essay in the History of the Theory ofElasticity (Dordrecht: Reidel, 1980). Some aspects
of Poisson's work in the mathematics of elasticity is dealt with in Garber, "Poisson."
107 Fresnel, "Memoire sur la double retraction," Mem. Acad. Sci. Paris, 7 (1827) [1830]:
45-176,80.
in France, 1790-1830 127
the ether. lOS The problem of transforming physical images into a mathematically
expressed, physical theory of light was not solved easily. Cauchy developed the ra-
tional mechanics of the continuum that later was imbued with physical meaning. 109
By the second decade of the nineteenth century, there were two Parisian ap-
proaches to exploiting the mathematical opportunities offered by the deformation
of solids under external forces and constraints to produce internal motions. These
approaches were symbolized by the work of Lagrange versus that of Laplace. The
problem where these two methodologies clashed was in the analysis of Sophie
Germain and Denis Poisson on the vibration of elastic plates. 110 The goals of the
study of elasticity ranged from understanding engineering problems, as in the case
of Coulomb, to the explorations of the calculus afforded mathematicians by this
particularly difficult branch of rational mechanics.
Ernst Chladini's experiments on vibrating plates stimulated the Institut to offer,
as the prize essay for 1809, the mathematics of vibrating plates and to compare
the results with experiments. The question had to be reset twice before it was
awarded, with reservations, to Sophie Germain. Her flawed derivation of the
equations of motion for the elastic surface was based on Lagrangian mechanics.
In his Mecanique Analytique Lagrange reduced the physical properties of bodies
to geometry and used the principle of virtual velocities and variational calculus
to obtain equations expressing extremum conditions. He drew into one analytical
net the mathematical problems of both statics and dynamics. He also managed
to withdraw the mathematical consideration of both of these physical subjects
from the domination of hypotheses to the elegance of algebra. With analogical
reasoning, Germain argued that the sum of the curvatures of the plate would play
the same role in the theory of plates, as the curvature of the elastic central-line
did in the theory of rods. She used Euler's work on vibrating rods and extended
his reasoning to vibrating plates. Germain argued that the action of the forces
on the plate was proportional to the sum of the inverse of the change in the radii
of curvature of the plate. Using the principle of virtual work, she obtained an
equation for the vibration of the elastic surface. Her basically incorrect equation
was reworked by Lagrange into the form Germain adopted and then solved for
particular casesyl
Denis Poisson was hot on Germain's mathematical trail. He criticized her work
on technical and conceptual grounds. 112 Poisson argued that the only foundatiop for
the mathematical analysis of physical problems was the consideration of the forces
between the molecules making up bodies.The problem with Germain's analysis was
that it was geometrical and lacked any appropriate hypotheses for the mathematical
study of physical phenomenaY3 Germain was a stand-in for Lagrange whom
Poisson also criticized on the same grounds. He had also recently published his
own text in mechanics, a rival to that of Lagrange. He then rederived Germain's
revised equation of motion on what he took to be more appropriate foundations.
In his first paper on the equilibrium of elastic surfaces, Poisson looked at an
isolated molecule and assumed that deformation of the surface changed the distance
between molecules and sought the force that would return the surface to its original
form. 114 Poisson also derived the equation of motion for a vibrating plate that
required the expansion and then truncation of series. All relationship to either the
original problem, or the model of matter used to set up the equations, were lost.
His paper was derivative at best and a demonstration of mathematical acrobatics
in search of a known goal. 115 Navier later questioned Poisson's actual use of his
111 Her solution was published as Sophie Germain, Recherches sur la tMorie des surfaces
elastiques, (Paris 1821). For details of Germain's derivation of her equation of motion
and solutions, see Bucciarella and Dworsky, Sophie Germain. While they do not spare
Germain's essay from criticism, they make the importance of being male and having
powerful, committed patrons in the mathematical world of early nineteenth-century Paris
all too evident.
112 As soon as the professional mathematicians entered into the fray Germain was marginal-
ized. Her isolation from the discipline and inability to gain the necessary technical
training meant that her work could not be technically as sophisticated or regarded with
anything but condescension.
113 Poisson, "Memoire sur les surfaces elastiques," Mem. Institut (1812) [1814]: 167-226.
Bucciarella and Dworsky, Sophie Germain argue that the prize essay was initiated by
Laplace to further Poisson's career, although they offer only circumstantial evidence.
However, Poisson's paper on the prize topic was published by the Institut even as he
withdrew from the competition on his election to that institution.
114 Poisson had introduced this model in 1811 in Poisson, Traitede Mecanique (Paris: 1811),
1833 edition translated by H. H. Harte asA Treatise on Mechanics (London, 1842).
115 Even Todhunter notes this might be a display of analytical skill that did not add to the
physical discussion of the problem. See Todhunter, History of Elasticity vol. 1, 212.
in France, 1790-1830 129
Bucciarella and Dworsky, Sophie Germain, 75-76, note the mathematical character of
this paper.
116 Poisson, "Memoire sur l'integration de quelques equations lineaires aux differences par-
tielles, et particulierement de l'equation generale du mouvement des ftuides elastiques,"
Mem. Acad. Sci. Paris 3 (1818) [1819]: 121-176, and "Sur les integrales definies," J.
Ecole Poly. 11 (1820): 295-341. For a discussion of Poisson's mathematical concerns
see, D. H. Arnold, "The Mecanique Physique of Simeon-Denis Poisson: The Evolu-
tion and Isolation in France of his Approach to Physical Theory (1800-1840)," Arch.
Rist. Exact Sci. 27 (1984): 248-367, vol. 28 (1984-1985): 27-266, 287-307, and
Grattan-Guinness Convolutions.
117 This argument is in contrast to Arnold, "Poisson," who treats Poisson's molecular model
not as a necessary start for his mathematics but as a physical theory.
130 Physics and Mathematics
17= 1
a
00 2rr
-p4F(p)dp
15
and F (p) is the force between two points at distance p apart. To eliminate F (p) was
a mathematical exercise in Lagrangian mechanics. Essentially Navier constructed
the work done by all the forces that he assumed acted upon a single molecule from
all the molecules in the solid. Using the calculus of variations for the equilibrium
case, Navier deduced the differential equations above and the boundary conditions
at the surface of the body.
As Navier recognized, this was an exercise in mathematics.u s He had taken the
simplest case for the external forces. Cauchy weighed in with the more general case
of forces acting at any angle to the surfaces of the solid. Cauchy also considered
the solid as a continuum. The paper was not published by Cauchy in full until
1828. 119
Among the stream of papers on analysis Cauchy published during the 1820s
and into the 1830s were several on the mathematics of elastic bodies. In these
papers he clearly represented the internal physics of bodies under various forces.
While setting up the equations of equilibrium and motion within the elastic solid,
the physical imagery of stresses and strains could be vivid and explicit. However,
they were not consistent, and the physics of elasticity was not developed from
the back and forth between Poisson, Cauchy, and Navier in these mathematical
papers of the 1820s. All three derived similar, if not identical, equations from
different physical starting points. They all quarreled over the validity of their
118 Navier, "Memoire sur les lois de l'equilibre et du mouvement des corps elastiques,"
Mem. Acad. Sci. Paris 7 (1821) [1827]: 375-393, an abstract appeared in Bull. Soc.
Philo. (1823): 177-183.
119 An abstract appeared earlier as Cauchy, "Recherches sur I'equilihre et Ie mouvement
interieur des corps solides, ou ftuides elastiques," Bull. Soc. Philo. (1823): 9-13. The
full paper was published in three installments, "De la pression ou tension dans un corps
solide," "Sur la condensation at la dilation des corps soli des," and "Sur les equations
qui expriment les conditions d'equilibre ou les lois de mouvement interieur d'un corps
solide," in Cauchy Exercises de mathematiques vols. 2 and 31827 and 1828 respectively.
in France, 1790-1830 131
various mathematical results. l2O The point was to best one's competitors in being
able to solve the partial differential equations of the motions of elastic bodies for
cases that one's competitor had failed to solve. Or, to add complexity to a solution
in some other case, turning a factor assumed to be a constant into a function. These
were exercises in mathematical proficiency not physical insight. Between Poisson
and Cauchy there was also the serious mathematical issue of how, or if, summations
should replace integrations. They both used integrations to obtain solutions to the
particular equations that they could in principle solve. Poisson mathematically
deduced a ratio for lateral contraction to longitudinal elongation for a thin bar that
was not valid in general. 121 Cauchy also extracted other such seemingly measurable
ratios from his mathematics. Experiments that were difficult did not decide the
issue of the validity of either mathematical approach.I22
These strenuous mathematical exercises were given added urgency with the
development of Fresnel's theory of light. Mathematical exploration of the prop-
agation of waves through solids and liquids was given renewed impetus. The
mathematical theories of elastic solids were elaborated along the lines of Cauchy's
approach during the 1830s and 1840s by Gabriel Lame. The physical muddle that
resulted was sorted out later in the century by George Gabriel Stokes and others
when mathematical theories of the ether became an urgent issue in the developing
discipline of physics in Britain.
To assess the changes in both French mathematics and physics between 1790
and 1830, we need to keep in mind the differences between these two disciplines in
terms understood in that era. Even as the calculus was redefined through the work
of Cauchy in the 1820s, mathematicians continued to appropriate the expanding
domains of experimental physics. The algebraic expressions of experimental re-
sults remained the starting points for excursions into the mathematics of light, heat,
and elasticity. Cauchy's analysis opened up new avenues of mathematical explo-
ration and expression. The display of mathematical virtuosity through problem
solving was still an important route to a reputation and career in French mathe-
matics. Joseph Liouville used Cauchy's analysis to explore generalizations of the
mathematics of heat conduction. He promoted the coefficient of conductivity from
123 See Jesper Liitzen, Joseph Liouville (1809-1882): Master of Pure and Applied Mathe-
matics (New York: Springer-Verlag, 1990), 22.
in France, 1790-1830 133
124 However, in his mathematics the function representing that summation of forces bore no
relationship to the law of force between the molecules through which Poisson established
his equations of motion. Poisson, "Memoire sur l'equilibre et Ie mouvement des corps
elastiques," 366. See Garber, "Poisson."
125 Poisson published papers on purely mathematical issues throughout his life. The ones
of concern here are Poisson, "Sur les integrales definies," J. Ecole Poly. 9 (1813):
215-246. "Sur l'integration de quelques equations lineaires aux differences partielles,
et particulierement de l'equation generale du mouvement des fluides elastiques," Mem.
Acad. Sci. Paris 3 (1818) [1819]: 121-176 and "Sur la maniere d'exprimer les fonctions
par des series de quantites periodiques, et sur l'usage de cette transformation dans la
resolution de diffCrents problemes," J. Ecole Poly. 11 (1820): 417-489.
126 Representing a sum by an integral whenever convenient did not appear to bother him in
his earlier papers.
127 Reworking Fourier's methods on heat, to reproduce both his equations of motion for heat
and his solutions to those equations under various boundary conditions, took Poisson over
two decades. Poisson, Theorie mathematique de la chaleur (Paris: Bachelier, 1835).
134 Physics and Mathematics
using problems drawn from physics were rich in physical implications that were
deciphered by others. However, the physics from which they drew their problems,
while still centered on experiment, had changed radically. Emblematic of those
changes were the textbooks by Rene Hauy and Jean-Baptiste Biot examined by
Cannon. 128 Both textbooks were popular, yet they addressed different audiences.
Hauy described physics as the science of familiar phenomena. The instruments
described and the objects of study were deliberately chosen to explain the com-
monplace. This was a text for the general public, or a text on an elementary level.
In Hauy's text the relationships used to express experimental results were in terms
of simple geometry. At its center were the descriptions of phenomena, not the dis-
cussion of the instrumentation, methods of taking measurements, or their expected
accuracy. However, Hauy text was up-to-date with references to current research
and researchers and was liberal with descriptions of their work.
Biot's audience were clearly pre-professionals, those planning careers as engi-
neers or physicists for whom discussions of the commonplace would be digres-
sions. The laboratory and quantitative experiments were the focus of Biot's text.
In it Biot devoted large amounts of space to instrumentation, and the results of
experiments and their analysis. Yet the only mathematics in the text was con-
tained in the algebraic relationships Biot drew up between measured quantities.
And these became unduly complicated because Biot insisted on the primacy of the
data. No mathematical development of theoretical ideas intruded. What theory
existed was expressed non-mathematically, if technically. Biot's modernity lies in
his focus upon the technical details of experiments and their results. These were
clinically dissected and analyzed. Mathematics was subordinated to the needs
of the experimentalist and this was Biot's meaning of "physics experimental and
mathematical.,,129 Biot was a Laplacian, but on the other side of the disciplinary
divide, an experimentalist for whom theory was still expressed in the vernacular.
His objections to Fresnel were matters of physics, not mathematics, and expressed
in the non-mathematical terms used by physicists. In his papers as a mathematical
Laplacian, particular solutions of equations could well match specific, idealized
experimental situations. This was an important innovation.
By 1830 the practices of experimentalists had redefined the discipline but not
its relationship to mathematics. Experiment was still the heart of the discipline,
but it was quantitative experiment conducted with elaborate instrumentation so
that the measurements taken and their accuracy could be understood and analyzed.
This self-conscious understanding and analysis of methods was a function of the
increasing competition for priority in the development of a reshaped field. As
128 Rene Hauy, Traite Elementaire de Physique, and Jean-Baptiste Biot Traite de physique
experimentaie et mathematique.
129 Frankel's argument cannot be sustained by either Biot's textbook or his research. See
Frankel, "BioI."
in France, 1790-1830 135
Prologue
Paris was not the only site for the practice and development of mathematics and
experimental philosophy around the turn of the nineteenth century.l Throughout
the eighteenth century, in Britain and the German States, experimental philoso-
phers and mathematicians built their own traditions, interacting with, but not over-
whelmed by, the research of the French. After 1800 the achievements of French
experimentalists and mathematicians intruded into those traditions and began to
change them. These intrusions reoriented research problems and the terms of their
solution by both experimentalists and mathematicians. At the same time and on
a broad scale, British and German societies went through metamorphoses. In the
German states, the invasion and occupation of the Rhineland by the French accel-
erated these changes. Experiences in the Napoleonic wars added to the structural,
economic translocations already affecting Britain.
In these countries the changes wrought by war and economic transformation re-
cast the educational systems and the place within those societies of both the study
of nature and mathematics. New opportunities and social forms for the practice of
experimental philosophy and mathematics emerged within either new or renewed
educational institutions. While the details of the patterns of these developments
1 Although this narrative focuses on events and changes in Britain and the German States,
other sites in Europe continued their own traditions in the study of nature and interacted
with those states. Among the more important in this period are those of Sweden and
Denmark. Physics in the United States does not enter this account until physics became a
profession and only in the person of Josiah Willard Gibbs. American physics had its own
distinct social history. However, the pervasive "baconianism" that Robert Bruce, The
Launching of Modern American Science, 1846-1876 (New York: Alfred Knopf, 1987)
Prologue, found so dismal was not so different from the standards of Europe of similar
eras.
were unique to each society within the German States and Britain, the general trend
and end results were the same for experimental philosophy and mathematics. Ex-
perimental philosophers and mathematicians became professionals, mainly within
institutions of higher learning. The actual practices of mathematicians and exper-
imental philosophers within these newly professionalized disciplines were quite
different from those of earlier generations. Alterations in these disciplines were
shaped by changes internal to their particular societies. Socially, these re-formed
disciplines became the modern scientific professions of physics and mathematics.
The social function of research became the advancement of careers rather than the
affirmation of social place within a general literate culture. The social justifications
for such research quickly became moral, economic, and political to secure then
consolidate these new professions within the emerging industrial societies of the
nineteenth century.
Intellectually members of these disciplines also reacted to, and then tried to
emulate, the achievements of early nineteenth century French mathematicians and
experimentalists. The intellectual reorganization of the practices of both British and
German mathematicians and experimentalists manifested these reactions. How-
ever, emulation did not reach replication. British and German mathematicians
and experimentalists interpreted French mathematics and experimental physics in
light of practices already in place. In both societies the French example and native
traditions led to reconstitutions of mathematics and physics. These reconstitutions
required periods of students hip, followed by reappraisal, then refashioning the
complex heritage of methods, problems and standards of solution available from
the French after 1800.
The periods of social and cognitive reappraisal set mathematics and physics
into new channels that led to the redefinitions of both research problems and their
solutions. One outcome of these reorientations was that France was replaced by
Britain, and then Germany as the center of scientific life in Europe.
The ultimate effects of these processes take us beyond 1830. In the era, 1790-
1830, France remained the center of the scientific life of Europe and Britain and
Germany were still on the margins. The period of emulation of the French by
the British and the Germans began in the 1820s. The cultural matrices of the
educated elites of the German states and Britain led natural philosophers and
experimentalists, in their interpretations of the French experimental physics and
mathematical physics, to struggle with problems the French no longer worried
about. These problems emerged from the interaction of experimental philosophy,
natural philosophy, and mathematics. Both mathematicians and experimentalists
in these societies were drawn into a series of philosophical dilemmas because
members of both these disciplines believed that examinations of the appearances
of nature could reveal its underlying operations.
In the German States and Britain, one of the more important issues that was
German States, 1790-1830 139
debated was the relationship between empirical and deductive knowledge. Other
more specific metaphysical topics exercised experimental physicists and philoso-
phers. As in earlier metaphysical disputes, general guiding principles upon which
theories might be established could be a node of consensus. The actual working
out of those speculations about nature led in their details to a myriad of different
theories. General principles had to be supplemented by other hypotheses which
directed theories in wildly different directions. This was especially true in an
era when analogy and illustrative example still filled the place that mathematical
exposition served later.
The diverse social and cultural settings of the German States and Britain led
to quite different interpretations of the French heritage they all sought to emu-
late. The particular institutional forms that emerged to house British and German
mathematicians and experimentalists who reshaped this French heritage were also
quite different. This was partly due to the forms already available and the differ-
ing opportunities that opened up within these societies between 1790 and 1830.
Therefore we will examine them separately. However, after 1830 this more open
access began to close. Entering into the disciplines of mathematics and experi-
mental philosophy began to be mediated by new formal educational requirements.
Formal education became a prerequisite to joining the research community. These
educational requirements also intruded into merely understanding the content of
mathematics and experimental philosophy. Intermediaries between the research
communities and the broader, educated public became necessary. The general au-
dience was disengaged from the practice of the investigation of nature and became
more passive observers of the feats of scientific professionals.
Thus the era from 1790 to 1830 was pregnant with possibilities of which only a
few were brought to term and safely delivered.
intellectual settings haunted physics in Germany long after 1830, as later changes
transformed the discipline into its modern social forms.
Physics in this era was still experimental philosophy and a part of the shared
culture of intellectuals. In German-speaking areas of Europe, natural philosophy
and philosophy could not be separated. This was partly because of the belief that the
study of experimental phenomena could uncover the actual workings and structures
of nature. The findings of experimental philosophers and interpretations of the
meanings of their results contained within them metaphysical and philosophical
implications no philosopher could ignore. Debates within the sciences were of
vital interest to philosophers. Yet within this vital, contentious culture there was no
one metaphysics, experimental or explanatory approach available to experimental
physicists. 2
Historians trace this fragmentation ultimately to the fragmented character of the
German-speaking world. Yet there existed a semblance of social cohesion in the
group interested in the relationship between philosophy, metaphysics, and experi-
mental philosophy. The community consisted of the faculty at state universities, of
private secondary or higher educational institutions, state civil servants, members
of the professions of law or medicine, members of the state academies of science
or letters, and finally private scholars. This corps saw itself and self-consciously
acted as a distinct group in the varied German societies. Members of this com-
munity were self-recruited from the full range of social ranks from aristocracy to
burgers to skilled artisans. 3 They no longer acted as members of the social rank
into which they were born. As a group they exhibited no attachment to a particular
place or state. Careers came before social or political identity. To enhance their
individual careers, they moved from one state to another, and from the service of
one state to that of another.
Throughout the disruptions of the Napoleonic wars, small gatherings of such
like-minded individuals would discuss, sometimes passionately, the ties of experi-
mental philosophy to metaphysics. Historians have traced the common metaphys-
ical ground shared by philosophers and experimentalists through the eighteenth
2 The fragmented intellectual and social character of this era has been detailed by David
Knight, "German Science in the Romantic Period, 1781-1831," in The Emergence,
Crosland, ed. 161-178, and Barry S. Gower, "Speculation in Physics: The History
and Practice of Naturphilosophie," Studies Hist. Phil. Sci. 3 (1973): 301-356. For recent
studies on Romanticism and science, see Romanticism and the Sciences, Cunningham
and Jardine, eds. (Cambridge: Cambridge University Press, 1990) and Romanticism
and Science in Europe (1790-1840), Stefano Possi and Mauritz Bossi, eds. (Dordrecht:
Kluwer Academic, 1994). For a useful survey of the literature, see Trevor Levere, " Ro-
manticism, Natural Philosophy, and the Sciences: A Review and Bibliographical Essay,"
Persp. Sci. 4 (1996): 463-488.
3 For a full discussion on these points, see C. E. McClelland, State, Society and University
in Germany, 1700-1914 (New York: Cambridge University Press, 1980).
German States, 1790-1830 141
Goethe also opposed both Naturphilosophie and Newtonianism. Yet his work
followed patterns of problems and of solutions that fit into the Naturphilosophie
approach to the study of nature. 14
In these specific examples and those of many others, the philosophical issues
central to physical theories and general speculations about the operations of na-
ture were shared with other experimental philosophers of differing philosophical
persuasions. This shared, general understanding of the important problems to be
solved, and the terms in which the solutions to those problems should be couched,
continued throughout the nineteenth century in German physics. Explanations of
the operation of nature in terms of bipolar forces continued through several decades
and generations of physicists. Specific adherence to Naturphilosophie might later
be denied, but investigation of change and the explanation of change in terms of
forces remained. Also shared was the belief that the investigation of the appear-
ances of nature, the phenomena, would uncover the real structures and processes
of nature. Experimental philosophy was not merely descriptive but penetrated the
actual workings of nature.
Whatever divisions there were within the German physics community, physics
as a discipline was still defined broadly, in the eighteenth-century sense of the
term. It covered all the experimental sciences and included the important areas of
mineralogy, meteorology, and climatology. This range of concerns was reflected
in the pages of the first journal addressed to German experimental philosophers,
Journal der Physik. In 1790 the manifesto of its editor, F. A. C. Gren proclaimed
that the journal was addressed to a narrow audience, not those who "merely lecture
as a pastime," or, "write diverting essays." However, this was not a research
journal. It was one of news in experimental philosophy. He reprinted material
from foreign and domestic journals and used its pages to argue for phlogiston
and other issues in chemistry. IS The next editor, L. W. Gilbert inherited Gren's
chair in chemistry and physics, and his journal. Gilbert's own research was in the
domain of what he called "physical chemistry" or "chemical physics." While he
sketched Lavoisier's chemical ideas, he also used explanatory components taken
14 For Goethe's work on light and opposition to Naturphilosophie see Keld Nielsen, "An-
other Kind of Light: The Work ofT. J. Seebeck and his Collaboration with Goethe," Hist.
Stud. Phys. Sci. 20 (1989): 107-178,21 (1991): 317-397. See also H. D. Irmscher,
"Goethe und Herder im Wechselspiel von Attraction und Repulsion," Goethe Jahrbuch,
106 (1989): 22-52.
15 Gren, "Vorrede," J. der Phys. 1 (1790): 137. See also Dieter B. Hermann, Die Entste-
hung der Astronomischen Fachschriften in Deutschland, 1798-1821 (Berlin: Archen-
hold-Sternwarte, 1972), 137. Thomas H. Broman, "J. C. Reil and the "Journalization"
of Physiology," in The Literary Structure of Scientific Argument, Dear, ed. 13-42, estab-
lishes the same pattern of publication, news and reprinted materials, in the early years of
the Archiv for die Physiologie, although in other ways it represented a break with earlier
medical journals.
German States, 1790-1830 145
from atomism, together with the language and imagery of dynamics and forces. He
was also highly critical of the theories of both Schelling and Hegel as speculative.
He continued Gren's policy of reprinting articles. However, the articles were
explicitly limited to articles from journals of physics, not mathematics. It was
through this journal that a great deal of the new, French, quantitative experimental
physics became known in Germany. 16
Some historians have taken the reprinting of articles from foreign journals as
evidence of the low quality of the sciences in the German states in this era. How-
ever, Gren and Gilbert's journal was not backed by state funds, as were those of
academies of science. We must see the journal as performing a different function
for a different audience than that of the research journal for a profession. The con-
tent of this journal followed the same pattern as other eighteenth-century journals
for a general, elite audience. In this case, the audience sought was not quite so
broad and the content of the publication narrower than usual, yet the eighteenth-
century form continued. Gilbert and Gren's venture was private. To be successful
they had to appeal to a broad audience, but did not have enough active contributors
for this new medium of publication-the journalP In the early 1800s the presen-
tation of ideas in texts that served student needs was more rewarding, financially
and professionally for the German professoriate.
Johann Karl Fischer's history of physics also reflected the broad and inclusive
meaning of the term in the Journal. Fischer distinguished physics, broadly and
narrowly, and decried the current fashion of trying "to derive natural phenomena
and their laws as mere conclusions a priori." For Fischer, physics was a discipline
grounded in experiment and the goals of its practitioners were to understand the
processes of nature. His account of physics ranged over experiments and obser-
vations in physics and chemistry, observational astronomy etc. This wide range
and experimental bias is also present in J. D. Reuss' organization of physics in his
Repertorium. 18
Physics as reflected in these pages, Gren's, and then Gilbert's journal, and else-
where was an empirical science, not necessarily quantitative, but done with care
and precision. Theory was presented in non-mathematical terms understood by a
broad general educated audience and theorizing meant launching oneself into meta-
physics. Vague ideas and experiments whose goals were obscure were present in
the literature. Yet carefully delineated, qualitative experiments and descriptions
16 Hans Schimank, "Ludwig Wilhelm Gilbert und die Anfange der 'Annalen der Physik',"
Sudhoff's Archive, 47 (1963): 360-372.
17 Journals in natural philosophy published in Britain and France in the same era as com-
mercial ventures shared the same mix of reportage, replication and articles.
18 Johann Karl Fischer, Geschichte der Physik, and J. D. Reuss, Repertorium Commen-
tationum et Societatibus Litterariis Editarum (New York: Burt Franklin reprint of the
G6ttingen edition of 1805) 6 vols., vol. 4 Physics.
146 Physics and Mathematics
with clear explanations of the meanings of the results also existed in the same
journals. As in the work of Oersted and Ohm, and the later work of Michael
Faraday, the absence of quantification did not necessarily preclude clarity either of
experimental purpose and accomplishment, or of physical interpretation. Clarity
was more than quantity. And numbers can only convey meaning if they measure
something that is predetermined on other grounds as meaningful.
In such a discipline, seen as related to chemistry with interpretations of ph en om-
ena tied closely to metaphysics and philosophy, mathematics held only a marginal
place. At its most useful it served the purpose, for experimentalists doing quanti-
tative experiments, of reducing their data and expressing those results in algebraic
form, if they took the analysis of their work that far. Most experimentalists were
suspicious of the use of mathematics within physics. For some it was impossible to
believe that any knowledge of nature could be gained through mathematics. This
was the opinion of Seebeck and a loudly voiced, majority opinion. Analysis (the
calculus), it was argued, was devoid of the imagery necessary to develop ideas
about the real structure of the external world and its processes. In the case of
electromagnetism,
The mathematician conceives the phenomena of electromagnetism, in the
first instance only as diverse modifications of motions; if he succeeds in
setting up a fundamental equation into which all the factors that influence
the kind and magnitude of the motion enter as elements, whose particular
values, exactly determinable by the formula itself, exactly specify the
motion itself, then he has incontestably satisfied the demands placed on
the so-called mathematical physics. 19
In mathematical versions of electromagnetism, attractive and repulsive forces
appear merely as positive or negative quantities. A mathematical investigation
could proceed no further. The source of such effects was of no concern to mathe-
matics, as the latter gives only a quantitative account of the phenomena. A physical
explanation, however, demanded such an exploration into the sources of the posi-
tives and negatives. Only physical theories were "capable of capturing the essence
of the phenomena." This criticism was mild compared with those launched by ad-
herents to a metaphysics in which mathematics was taken to be a purely deductive
form, a product of the mind, that in principle could not relate to phenomena of the
real world. 2o
Not all experimentalists with such philosophical commitments were necessar-
ily hostile to mathematics. In Oersted's opinion, mathematical abstractions were
different from physical quantities and hence distinct from them. The disciplines
that formed around these different entities were complementary. On philosophic
grounds, Jakob Friederich Fries rejected Goethe's color theory because there was
no mathematical development to complete it. Fries represented an important, mi-
nority, opinion. 21 Pfaff also placed himself in this minority camp when he added
that, at the same time, physical explanations must also be mathematical. While
he did not explain exactly what he meant, there were experimentalists prepared
to concede that both a physical explanation of phenomena and a mathematical
description of those phenomena might be equally valid. Yet these two kinds of
examinations of nature remained in separate disciplines. Johann Tobias Mayer
went so far as to acknowledge that when branches of physics became subject to
mathematical treatment, they were handed over to mathematicians to become part
of mathematics. Mathematical physics of the French variety was the domain not
of physicists but of mathematicians. He regarded the work of Euler and Lagrange
on sound and Fourier on heat as mathematics, not physics. 22 Opinion on the rela-
tionship of the experimental investigation of nature and mathematics ranged over
the map, and was expressed only in general terms.
While the relationship between the empirical results of experiment and the anal-
ysis of those results by mathematicians had to be addressed in the philosophically
charged atmosphere of early nineteenth-century German science, no experimen-
talist did it very systematically. With the triumphs of French mathematical and
experimental physics, a more systematic handling of this issue was in order. In ad-
dition to Fries, Ernest Gottfried Fischer tried to develop a generalized, systematic
study of this relationship. For Ernest Fischer, knowledge of nature came from both
investigations into the nature of forces and their effects on bodies. Working from
Kant's ideas of the necessity of deductive thought in the construction of knowledge,
he argued that a philosophy of nature was only possible when expressed in the most
general fashion, and that was impossible without mathematics. Physics provided
the observations, the foundation upon which mathematicians could build. Fischer
used Kant's idea that most general knowledge of bodies lay in their motions, and
that necessarily introduced mathematics. Yet mathematics only entered into the
picture to attain a complete synthesis once all knowledge of the phenomena was
In the third decade of the nineteenth century, the practices of physicists and
mathematicians changed again, driven by institutional opportunities and cogni-
tive realignments. The cumulative impacts of the first generation of appointments
made to reform the Prussian universities become manifest in the 1820s. This re-
forming impulse brought with it a new philosophical foundation for the study of
mathematics. This new philosophy would indirectly affect the practice of physics
and mathematics in later decades. At the same time experimentalists and math-
ematicians faced the issue of the relationship between mathematics and physics
more systematically. In addition to this philosophical topic was the more prag-
matic one of developing research problems and solutions that brought recognition.
The obvious model was that of French mathematics and quantitative, experimental
physics. Firstly, mathematicians needed to argue that mathematics was a study
that on philosophical grounds belonged legitimately within the university.
The philosophical debates to justify the study of mathematics within the newly
constituted universities and attempts to domesticate French methods did not lead
Wilhelm von Humboldt, lasted less than one year in his civil-service post. 27 His
successors, who actually forged the institutional reforms within the universities
were compromisers, bureaucratic realists, and ambitious for careers within the
Prussian civil service. Their ambitions were also hedged by the limitations of their
budgets and of available personne1. 28
The rhetoric surrounding the philological seminar spread, along with the method
of the seminar, as the teaching mode in other university disciplines. However, these
did not help to define research problems and methods specific to those other disci-
plines such as mathematics and physics. Acceptance of the rhetoric of description
from philology and the use of research as the teaching model meant that mathemat-
ics could shed the stigma of "bread study." The choice of problems discussed in
seminars and the standards of solution passed to students came from other sources.
However, it followed that only those with specialized training could claim a place
in that discipline and that training was only available at the university, inculcated
in the context of the new teaching mode of the seminar. When this model became
the norm in both mathematics and physics, access to even experimental physics
was narrowed and proscribed for many.29
However, no one has denied the opportunities for the pursuit of careers in the
sciences that opened up in these revitalized state institutions of higher education.
Yet we cannot see the career opportunities as determining, in and of themselves,
either the success, or the directions of development, of research in either the disci-
plines of mathematics or of physics. Social change did not necessarily determine
cognitive realignments. 30
Interpretations of cognitive changes in the sciences, specifically physics that
are solely based on sociological criteria, cannot explain why the disciplines in
the natural sciences were cognitively transformed in the nineteenth century.31 The
problems shared by Marxist scholars, ethnomethodologists, sociologists of knowl-
edge, and cultural contextualists is that no explicit, general connections can be
made between social structure, ideology, or cultural context and the details of the
27 Paul R. Sweet, Wilhelm von Humboldt: A Biography 2 vols., (Columbus OH.: Ohio State
University Press, 1980), gives ample evidence that Humboldt was only happy as a private
citizen. He shed public offices as soon as possible.
28 For details of these and other limitations, see McClelland, State, Society and University
in Germany, 122-132.
29 An example of this was Seebeck's attempts to obtain a university post and the reasons
for his difficulties in doing so. See Nielson, "A Different Kind of Light, Part II." Ohm
suffered similar difficulties.
30 We shall examine France in the nineteenth century in chapter IX. See also Laudan, "Ideas
and Organization," for a case of active organization without intellectual redirection.
31 For a discussion of the problems of social constructivism, see Stephen Cole, Making
Science: Between Nature and Society (Cambridge MA.: Harvard University Press, 1992).
German States, 1790-1830 151
results of physical arguments. 32 The theory that the university formed the vanguard
of bourgeois hegemony with the privat docent as its proletariat must explain why
the particular form of the intellectual changes that occurred in the sciences can
be linked to the bourgeoisie and no other class. 33 It has been argued that the uni-
versities gave unprecedented career opportunities for the sons of the bourgeoisie.
Ideology and social change did the work of economic interest. 34 Yet it is not ap-
parent that through much of the nineteenth century, the German bourgeoisie held
the universities in high regard. Many of the men discussed in this and subse-
quent chapters were the sons of gymnasia teachers, the clergy, civil servants, or
minor aristocracy. In another highly regarded sociological approach, Stichweh's
exploration of the development of the modern scientific disciplines is descriptive
rather than interpretive. None of the details of the actual contents of the research
problems that implicitly define his notion of physics enter into his analysis. 35
Recent sociological examinations of the nineteenth-century German university
systems emphasize the role of the state in extending its power over the previously
autonomous corporations ofthe universities. 36 The problem with this interpretation
is that it is not at all clear why the Prussian state would support reform within the
universities when its goal was control. Yet the state was largely successful in this
enterprise. Or was it? The expansion of the role of the state in the reform suggests
that the state bureaucracy reclaimed its elite position in Prussian society by captur-
ing the universities through the creation and wielding of Wissenschaftideologie. 37
32 Elizabeth Garber and Fred Weinstein, "History of Science as Social History," in Advances
in Psychoanalytic Sociology, Rabow and Platt, eds. (Malabar FL.: Krieger Pub., 1987),
279-298.
33 Alexander Busch, Geschichte des Privatdocenten (G6ttingen: Abhandlungen zur Sozi-
ologie, 5 (1959» for this Marxist interpretation. This and other interpretations of the
development of the universities in the German states are outlined in McClelland, State,
Society and the University, chap. 1.
34 One has to question just how many young men from the bourgeoisie, or any other class,
could be absorbed by the university system. The rigors of the course of study, then
the shear physical stamina to attain the status of Ordinarius even in the middle of the
nineteenth century must have defeated most aspirants. There are indications that the
education ministry of the Prussian state discouraged university attendance when careers
that required a university education were, in their opinion, overcrowded. McClelland,
State, Society, University, chap. 3.
35 Rudolph Stichweh, Zur Entstehung des modernen Systems wissenschaftlicher Diszi-
plinen: Physik in Deutschland, 1740-1890 (Frankfurt am Main: Suhrkamp, 1984.)
36 R. Steven Turner, The Prussian University and the Research Imperative, 1806-1848,
unpublished PhD dissertation, 1973.
37 By Wissenschaftideologie is meant here "the active pursuit of integrated, meaningful and
pure knowledge" as "the highest calling of man," McClelland, State, Society, University,
p. 24. The social conservatism of Wissenschaftideologie is discussed in Turner, "The
Prussian Professorate and the Research Imperative, 1790-1840," in Epistemological and
152 Physics and Mathematics
Social Problems, Jahnke and Otte, eds. 109-122, although he does not link it with the
aristocracy.
38 McClelland State, Society, University. Kees Gispen supports this view in his explanation
of the place of engineers in German society in the nineteenth century. Kees Gispen, New
Profession, Old Order: Engineers in German Society, 1815-1914 (Cambridge: Cam-
bridge University Press, 1989) chap. 1. One can challenge his ideas on the commitments
of all academics later in the century to the notions of social hierarchy he claims they
accepted. Hermann von Helmholtz's son Richard became an engineer, a career he fol-
lowed with the encouragement of his father. Helmholtz also understood the economic
need for research institutions for industry, part of the justification for the establishment
of the Physikalisch-Technische-Reichanstalt. See David Cahan, An Institute for an Em-
pire: The Physikalisch-Technische-Reichanstalt, 1871-1918 (Cambridge: Cambridge
University Press, 1989).
39 McClellandState, Society, University, p. 112, who finds Wilhelm von Humboldt's rhetoric
a nostalgic recollection of his years of freedom as a student at Halle then Gottingen. See
also Robert S. Leventhal, "The Emergence of Philological Discourse in the German
States, 1790-1810," Isis, 77 (1986): 243-260.
40 See William Clarke, "On the Dialectical Origins of the Research Seminar," Hist. Sci. 27
(1989): 111-139, offers a more complex description of the origins of the nineteenth-
century seminar over a long time period.
41 See R. Steven Turner, "The Growth of Professorial Research in Prussia, 1818 to 1848-
German States, 1790-1830 153
pline was the model of "Wissenschaft" pure, intellectual, universa1. 45 Yet Felix
Klein had another program for mathematics which he propagated and expanded
upon through the authors of the series he edited. Mathematics was indispensable
for and actually subsumed the other sciences, most notably physics. 46 There was
a power struggle within the mathematics profession in the late nineteenth century
that had its roots and manifestations in the ways in which both the philosophy
and the history of mathematics were portrayed, as well as in the problems and
methods of mathematics itself. Recent historiography of mathematics developed
by professional historians includes the intellectual history of the discipline with a
consciousness of its social and cultural context. 47
Yet the changes that redefined what problems mathematicians worked on as re-
search, and the training offered by the research seminar, could only effect changes
in the profession if the trainees from the seminars could be placed. This in turn
depended on gaining enough social support to continue training the next genera-
tion. This was not assured, and it took time. The new mathematics spread slowly.
Indeed the two processes, social and intellectual occurred simultaneously. During
the 1820s mathematicians elaborated a philosophy of mathematics that claimed
purity from their redefinition of mathematics. 48 In their research German mathe-
maticians annexed the problems and methods of the French before creating their
unique research traditions in mathematics based on their understanding of "pure"
in mathematical terms.
Experimental physicists were less burdened than mathematicians with the need
to justify their work as "pure." However, it took longer for physicists to change
their discipline structurally. They too took their models from France and through
learning, then practicing these methods refashioned their discipline. In the devel-
opment of both disciplines we see the same patterns of emulation, then reworking
of French methods and problems into new channels.
During the decade of the 1820s physicists, especially those of the first generation
trained in the reformed Prussian universities, turned away from Naturphilosophie
and towards the French for models of investigating nature. This did not change
the sense that physics was an empirical study, or, removed the problem that Ger-
man physicists saw in the relationship between such empirical knowledge and the
deductive knowledge of mathematics. 49
Perhaps the first German experimental physicist whose research revealed the
direct influence of the French was Georg Ohm. However, the subject of his research,
galvanic electricity, was not one in which the French had shown any interest.
Using galvanic electricity, the French explored the relationship between electric
currents, and between electricity and magnetism. Ohm examined the properties of
current electricity itself. He was also explicitly influenced by Fourier and based
his own research patterns on Fourier's work.50 His 1827 mathematical paper on
galvanism was preceded by a series of experimental papers on many aspects of the
behavior of the galvanic circuit and its components. Through these experiments
Ohm developed laws of conduction that took into account all of the components of
the circuit, and included measurements on the conductivity of the different metals
in the circuit. In this context Ohm developed the notion of "equivalent length" as
a measure of the resistance of a component under study compared to a standard
wire. 51 None of these experimental papers included any mathematics other than
49 This interpretation clearly diverges from that of Jungnickel and McCormmach Intellectual
Mastery, vol. 1, who accept as physics much of what was mathematics in the early
nineteenth century. While they recognize the importance of French mathematical physics,
they do not explore what that discipline was, or, how the Germans understood that
discipline. They also do not define what they mean by the discipline whose history they
are narrating, that is, theoretical physics. On this last point, see also Cahan, "Pride and
Prejudice in the History of Physics: The German Speaking World, 1740-1945," Hist.
Stud. Phys. Sci. 19 (1988): 173-191, and Pearce Williams, "Review of Intellectual
Mastery of Nature," Hist. Math. 15 (1988): 389-392.
50 There are numerous references to Fourier in Ohm's mathematical paper on galvanic
electricity. See Kenneth Caneva, "Ohm, Georg Simon," Diet. Sci. Bio. vol. 10, 186-194,
188.
51 See John L. McKnight, "Laboratory Notebooks of G. S. Ohm: A Case Study in Experi-
mental Method," Amer. 1. Phys. 35 (1967): 110-114,111-112. The first paper in which
this comparative measure of resistance appears is Ohm, "Vorlaufige Anzeige des Geset-
zes, nach welchem Metalle die Contact-Electricitat leiten," 1. fUr Chem. Phy. 44 (1825):
110-118, reprinted in Ann. Phy. 4 (1825): 79-88, and Ohm, GesammelteAbhandlungen,
E. Lommel ed., (Leipzig: Barth, 1892), 1-8. Ohm described experiments comparing the
conductivity of several metals and ordering them with respect to their conductivity in
Ohm, "Uber Leitungsfiihigkeit der Metalle fUr Elektricitat," 1. Chem. Phy. 44 (1825):
245-247, reprinted Ohm,Abh., 9-10. He argued with the work of Becquerel and Barlow
on conductivity in Ohm, "Uber Electricitatsleiter," same journal and volume, 370-373,
156 Physics and Mathematics
the statement of the algebraic laws deduced from his data relating "loss of force"
to various "equivalent lengths of the conductors" in his circuit and one simple
differentiation and integration to reach a more general form of his empiricallaw. 52
The development of the mathematical implications of his law were published
separately. In the first half of this paper, Ohm used a geometrical analogy to
illustrate the meaning of "electroscopic force" and "tension" for the overall circuit
excluding the electrochemical cell, and reconstructed his experimental law through
this analogy.53 Physically Ohm was trying to develop a language for the potential
in the galvanic case, much as Priestley and Cavendish had in the electrostatic one. 54
Because Ohm used accepted explanatory terms, most importantly force, and did
not clearly differentiate between force and potential he complicated his physical
explanations.
Ohm then investigated the circuit by considering the "flow" of electricity at
individual points of the circuit, excluding the electrochemical cell. His language
in this section directly reflected Fourier's in his theory of heat, for example his
observations on the "diffusion" of electricity. Taking the annulus as an appropriate
analogy for a circuit, Ohm constructed a partial differential equation for the diffu-
sion of electricity in a closed annulus, which is precisely the form given by Fourier
for the diffusion of heat in the same geometrically shaped thermal conductor. 55
du d 2u bc
y-=k---u.
dt dx 2 w
--------------
reprinted Ohm, Abh. 11-13.
52 The mathematical manipulation appeared in Ohm, "Voriaufige." The linear form of
his law appeared in Ohm, "Bestimmung des Gesetzes, nach we1chem Metalle die Con-
tacte1ektricitat lei ten, nebst einem Entwurfe zu einer Theorie des Voltaischen Apparate
und des Schweiggerischen Mutliplicators," J. Chem. Phy. 46 (1826): 137-166, reprinted
Ohm,Abh., 14-36,25.
53 Ohm, Die galvanische Kette, mathematisch bearbeitet (Berlin, 1827), reprinted, Ohm,
Abh., 61-186, translated as Ohm, "The Galvanic Circuit, Investigated Mathematically,"
in Scientific Memoirs, selected from the Transactions ofForeign Academies and Learned
Societies and from Foreign Journals, Richard Taylor, ed. vol. 2 (New York: Johnson
Reprint of 1841 edition, 1966),401-506, that closely follows the German edition. The
geometrical analogy appears on pps., 405-416.
54 It is best to keep in mind that in the 1820s, while one can connect electrostatic and galvanic
phenomena, they were kept separate, even seen as two kinds of electricity. See Thomas
Archibald, "Tension and Potential, Ohm to Kirchhoff," Centaurus, 31 (1988): 141-163
on the development of the concept. Gustav Robert Kirchhoff, "Uber eine Ableitung der
Ohm'sche Gesetze, we1che sich an die Theorie der Elektrostatik anschliesst," Ann. Phy.
76 (1849): 50&-513, translated as, Kirchhoff, "Ohm's law and Electrostatics,"Phil. Mag.
37 (1850): 463-468, drew Ohm's work and electrostatics together.
55 Ohm, "Galvanic Circuit," Scientific Memoirs, 451. This equation appeared in Fourier
Analytical Theory ofHeat, 88. The differentials are partial differentials. See also Bernard
L. Pourprix, "La mathematisation des phenomenes galvaniques par G. S. Ohm (1815-
1817)," Rev. Hist. Sci. 42 (1989): 139-154, on the mathematical aspects of Ohm's work.
German States, 1790-1830 157
In setting up his equation Ohm applied Fourier's arguments in the thermal cases
directly to the galvanic circuit. 56 Leading up to the above derivation was a long dis-
cussion on whether such inhomogeneous partial differential equations were proper
in mathematics. Ohm traced this mathematical quarrel back to Fourier and Poisson
on the one hand and Laplace on the other. He argued against Laplace because, in
transforming physical phenomena into differential form, Ohm assumed that the ef-
fects of forces between infinitely small bodies could only extend a certain distance.
Laplace did not limit his microscopic forces to microscopic distances. However,
Ohm's actual mathematics depended neither on microscopic forces between par-
ticles, nor on the assumption that matter is made up of particles. The derivation
was appropriated from Fourier. 57
There was no explanation in Ohm's construction of his partial differential equa-
tion of physical processes. Nor did Ohm attach any physical significance to the
constants in the functions of integration. The special case Ohm explored, where
b = 0, was the steady state case, although this was not stated as such by Ohm. 58
The final expression that Ohm obtained for u, the "electroscopic force" was,
To reach this solution Ohm used the functional methods of Laplace and Poisson
as much as the mathematically more radical ones of Fourier. There were actually
two exponential terms, the functional equivalent of the Fourier series, which were
reduced to one by taking a mathematical condition so that only the negative ex-
ponential was left. Ohm asserted that, as t ~ 00, the expression would become
linear with only the first term remaining. In this linear, rump equation, x was the
direction in which the electricity flowed, and a was left undefined physically.
Ohm was not merely influenced by Fourier but annexed pages of Fourier to
solve the problem of reducing the galvanic circuit to mathematics. He then solved
the mathematical equation following the methods of Fourier and other French
mathematicians. In hewing to the French model, Ohm examined a mathemati-
cally special case to replicate the empirical results of his experiments. This would
56 Heidelberger, "Change in the Baconian Sciences," argues that in accepting Fourier Ohm
abandoned ontology. This meant that Ohm went against the prevailing notion in physics
that knowledge of nature could not be gained using mathematics. He was, therefore,
judged to be doing bad physics. Ohm suffered outraged criticism from Georg Friedrich
Pohl, a Hegelian, and his experiments were also criticized by Gustav Theodor Fechner.
However, the evidence is that in the 1820s a commitment to physical explanation did not
preclude investigating the mathematical implications of empirically deduced relations
given the opinions current amongst mathematicians and physicists.
57 This was noted also by Caneva, "Ohm,"190, although his language is more muted than
my own.
58 This was also noted by Caneva, "Ohm," 191.
158 Physics and Mathematics
validate his mathematics. He did not consider whether the special, mathemati-
cal case mirrored the conditions of his experimental one. An omission seen also
amongst French mathematicians. 59 Ohm was working closest to French mathe-
matical physics. While he did not create the mathematics, he explored it in some
depth. And, this was hardly surprising, Ohm was a mathematician and a teacher
of mathematics at the Koln gymnasium. 6o Ohm's subsequent mathematical papers
on the same subject were further manipulations of the mathematical equations of
his 1827 paper. They were also published separately from his physical, vernacular
explanation of the difference between force and tension. 61
In his practices Ohm was replicating the French. He presented the complete
solution to a problem, experimental in physics, and the mathematical implications,
in mathematics. And in the mathematical papers Ohm discussed only mathemat-
ically significant cases. In his answers to criticisms of his empirical law, Ohm's
defense of his work was confined to his experiments and the physical models of
his rivals. 62 This pattern of considering the mathematics of physical phenomena as
mathematics reappeared in his subsequent work on sound and light. Again Ohm
took all kinds of mathematically developed special cases and connected them to
empirically known results without any specific assumptions about light or sound
other than their wave nature. They were exercises in the mathematics of waves
moving through homogeneous substances. 63
The later, physical model and physical interpretative weight placed on this 1827
paper did not appear in the original. To label Ohm as a physicist and his math-
ematical work as physics was to miss the point of what he was actually trying
to accomplish. At the beginning of his 1827 paper, Ohm echoed Fourier in his
belief that his investigation would "secure incontrovertibly to mathematics the
59 Yet physical information lies within the mathematics. The rate at which the exponential
term approaches zero depends on k'7r 2 i 2 t / £2 where e is the length of path, k' is the
conductivity, and i = 1,2, etc. Ohm did not investigate this physically or mathematically.
60 Jungnickel and McCormmach, Intellectual Mastery vol. 1, see Ohm's purposes as phys-
ical because Ohm did not test for the convergence of the series he was using. His model,
Fourier, did not either. By contemporary standards both were doing mathematics.
61 Ohm, "Nachtrage zu seiner mathematischen Bearbeitung der galvanischen Kette," Ar-
chive for die gesammte Naturlehre 14 (1828): 475-493. The physical explanation ap-
peared in Ohm, "Nachweisung eines Uberganges von dem Gesetze der Elektricitatsver-
breitung zu dem Spannung," same journal 17 (1829): 1-25.
62 See Ohm, "Zur Theorie der galvanischen Kette," J. Chem. Phy. 67 (1833): 341-354,
reprinted in Ohm, Abh., 560-572.
63 See Ohm, "Uber die Definition des Tons, nebst daran geknupfter Theorie der Sirene und
ahnlicher tonbildender Vorrichtungen," Ann. Phy. 59 (1843): 513-565, reprinted Ohm,
Abh., 587-633, and Ohm, "Erklarungen in einaxigen Krystallplatten zwischen geradlinig
polarischtem Lichte wahrnehmbaren Interferenz-Erscheinungen in mathematischer Form
mitgetheilt," Abh. der Math.-Phy. Ct. Konig. Bayerische Akad. Sci. 7 (1853): 43-149,
267-370, reprinted Ohm, Abh., 665-855.
German States, 1790-1830 159
possession of a new field of physics, from which it had hitherto remained almost
totally excluded."64 Ohm's experimental results were physically important but his
goal was not to produce a physical interpretation of those results in mathematical
form. 65
Neither can we straighforwardly interpret Franz Neumann's research as physics.
In the same decade as Ohm, Neumann emulated the French in producing exper-
imental results and separate, mathematical explorations arising from his experi-
ments. Neumann's early research followed a pattern he exploited all his life. In
all his work, experimental and mathematical, he had predecessors on which he
modeled his own work. The focus of much of his early experimental research was
the exploration of the physical properties of crystals, initially extending the work of
Gustav Rose. In the mathematical analysis of the properties of crystals, Neumann
used Joseph Fourier for heat, Augustin Fresnel, Navier, Cauchy, and Poisson for
light. In his work in electromagnetism he followed Ampere.
As in all mathematical physics, Neumann tried to push the mathematical analysis
beyond the solutions of his predecessors into more general mathematical territory.
However, much of this mathematical work was limited by physical considerations.
In his mathematical work on heat conduction in crystals, he went beyond Fourier's
work on homogeneous solids, where thermal conductivity was a constant, into
solids in which conductivity became a function of the symmetry of the crystal.
In his work on light, the elasticity of the ether also became a function of the
symmetry of the crystal, not a constant of its motion as in homogeneous solids. In
electromagnetism, Neumann included more geometrical cases than those explored
already in the experimental results of Heinrich Friederich Emil Lenz and Faraday.
In this last example Neumann was able to develop a general mathematical approach
from which all the physical occurrences of electromagnetic induction could, in
principle, be deduced. In the case of his work on crystals there was no general
mathematical expression or function from which he could deduce the physical
cases that led to the experimental results which were the starting point of his
mathematical work.
Neumann's position was difficult. He had examples of completed solutions
to physical problems, both the experimental and the mathematical part, but he
did not share with his French predecessors the same level of formal training in
mathematics. He did understand the mathematical game enough to know that in
mathematical physics points were scored through the display of technical prowess.
The mathematical problem usually emerged as a partial differential equation to be
64 Ohm, "The Galvanic Circuit," 404.
65 Initial reactions to Ohm's law were to his experiments. In 1831 Fechner carried out
an extensive series of experiments to test Ohm's experimental results that were in turn
criticized by Ohm. Fechner, Maassbestimmungen iiber die galvanische Kette (Leipzig:
F. A. Brockhaus, 1831). Fechner was followed over the century by many others. For the
fate of Ohm's Law see Caneva, "From Galvanism to Electrodynamics."
160 Physics and Mathematics
solved either for the first time, or in a more general form than before. Thus if the
density, in the case of elasticity, had already been considered as a constant, the
next technical step was to assume the density p = p(x, y, z). After reconstructing
the equations of motion for such a material, the solution might be pushed further
by particular cases by assuming a given functional form for p, or, for the general
case. The only limitation on the mathematical exercise was that the mathematician
had to begin in known experimental results and replicate other known results. This
was usually done by imposing restrictions on the mathematical solution obtained,
without actually noting whether the restrictions were physically plausible.
To complement his mathematical work, Neumann extended previous experimen-
tal work into new domains. He began with heat conduction in crystals. Neumann
also added the new German passion for precision. He applied the example set
by Bessel and his analysis of astronomical measurement to all his experiments on
crystals. 66 Neumann's experimental research began as an extension of the quanti-
tative methods of French physics into mineralogy. His purely geometrical exam-
ination of crystal symmetry became his dissertation in which he developed new
methods of stereographic projection. 67
His subsequent experimental work on crystals was published separately and kept
distinct from his mathematical work on crystals. In doing both he was following
his model in the exploration of heat, Joseph Fourier. Fourier was an early and
crucial influence upon Neumann and his research methods. Neumann copied
from Fourier's theory of heat at least enough to remember his results, if not his
derivations. He also copied Fourier's justification for his theory of heat,
The differential equations of the propagation of heat express the most
general condition and reduce the physical questions to problems of pure
analysis and this is the proper object of theory.68
66 See Kathryn M. Olesko, Physics as a Calling: Discipline and Practice in the Konigsberg
Seminar for Physics (Ithaca NY: Cornell University Press, 1991), chap. 2 for a discussion
of Bessel's analysis of the second-pendulum, and Neumann's use of Bessel's approach
in the analysis of experimental data.
67 The experimental methods were developed in Franz Ernst Neumann, Beitrage zur Krys-
tallonomie (Berlin: 1823). Neumann published one paper on crystal symmetry before
his dissertation, Neumann, "Uber das Crystallsystem des Axinits," Ann. Phy. 4 (1825):
63-76. Neumann's dissertation was published as, "De tactionibus atque intersectionibus
circulorum et in plano et in sphaera sitorum, sphaerarum atque conorum ex eodem vertice
pergentium commentatio geometrica,"/sis, (1826): cols., 349-369, 468-489, see Franz
Ernst Neumann, Gesammelte Werke, M. Krafft, E. R. Neumann, H. Steinmatz and A.
Wangerin, eds. 3 vols. (Leipzig: B. G. Teubner, 1906-1928), vol. 1. In these papers
Neumann displayed his ability to visualize relationships in space that is so apparent in
his work in optics and electromagnetic induction.
68 Olesko, Physics as a Calling, 123. Olesko is the latest to note this. See Woldemann
Voigt, "Gedachtnissrede," in Neumann, Gesammelte Werke vol. 1,3-19. The quotation
is from Fourier, Analytical Theory, 6.
German States, 1790-1830 161
This remained Neumann's goal. Mathematical physics was distinct from physics
itself which was quantitative experiment. Mathematical physics was, and remained
for Neumann, part of mathematics. His research commitment was to solving
problems completely, both the experimental and the mathematical part.
As a potential new faculty member within the reformed Prussian University
system, Neumann asked that his teaching assignment match his research com-
mitments. He wanted a post where he might teach the mathematics that would
complement his experimental research, namely, mathematical physics. He there-
fore requested a position where he could teach those aspects of physics which
had "received a higher mathematical development or those which are capable of
being so treated.,,69 He also set about teaching experimental physics, which he
accomplished quickly. To reach the goal of teaching mathematical physics took
longer. He also spent the next twenty years expanding his research into a complete,
experimental and mathematical understanding of crystals.
His later experimental work on the specific heats of minerals was new and distin-
guished by the ways in which Neumann treated his data, not in his stance towards
"mathematical" developments. Here he used Fourier's expression for the loss of
heat from the surface of a sphere to correct an expression in the reduction of data
from the method of mixtures. Neumann was using analysis, not a theory, to ex-
amine data. What was new was his ability to appropriate Fourier's analysis into a
novel experimental situation. Fourier allowed him to improve, in ways parallel to
Bessel's work in observational astronomy, his understanding of his data. This was
not the experimental confirmation of deductions from a generalized mathematical
solution for an equation expressing physical processes. His models were geomet-
rical and algebraic. As an experimentalist, Neumann's considerable innovations
were directed to improving physics as experiment and grafting mathematics onto
that. In these early papers there was no new image of mathematical physics.
We have in the 1820s two important specific examples of German physicists
emulating the French in the pursuit of experimental and mathematical physics.
Their mathematical sophistication was decidedly below that of their model. Given
the quality of the training available in mathematics for them as students, it was re-
markable that Ohm and Neumann accomplished so much in this decade. However,
available mathematical training for would-be mathematical physicists in Germany
was about to change.
69 Luise Neumann, Franz Neumann Erinnerungs bliitter von seiner Tochter (Leipzig: F. S.
B. Mohr, 1904),226. Also quoted in Olesko, Physics as a Calling, 129.
162 Physics and Mathematics
philosophy of the Prussian university system. 70 With this new philosophical jus-
tification, mathematicians could claim the ultimate value of mathematics-training
the mind. The mathematician most successful in articulating this philosophy was
August Crelle. He also embodied that new philosophy in the title and contents of
a new journal, Journal fUr Reine und Angewandte Mathematik. 71 To develop his
philosophy of mathematics, Crelle reached back to the ideas of Immanuel Kant.
Among other aspects of Kant's philosophy he adopted was the separation of math-
ematics into "pure" and "applied." "Pure" mathematics was the mathematics of
quantity, pure number-pure because it was the product of human intellect alone.72
Geometry was not such a pure product of the human intellect, being both depen-
dent on reason and experience and hence "applied." Crelle made this philosophy
manifest in the contents of his journal. Within its covers were papers on both the
algebraic and geometrical parts of mathematics.
Crelle and others carried the banner of this new philosophy and content of Ger-
man mathematics into the institutional forms made available through the reformed
university systems. Mathematicians quickly remade their discipline into an aca-
demic profession that became the pattern sought by organizing members in other
scientific disciplines.
These values were successful within the university but led to less happy out-
comes for other institutions. One such struggle was over the curriculum for the
proposed Berlin Polytechnic. 73 The curriculum both defined the social place of the
institution and its graduates. 74 For Crelle, mathematics was the foundation of all
technical education. The crucial issue here was what Crelle would have included
in "mathematics" both pure and applied. Mathematics also embraced "the mathe-
70 We will only consider those aspects of German mathematics that affected physics in this
era.
71 See Wolfgang Eccarius, "Der Techniker und Mathematiker August Leopold Crelle (1780-
1855) und sein Beitrag zur Forderung und Entwicklung der Mathematik im Deutschland
des 19 Jahrhunderts," NTM, 12 (1975): 38-49, and, "Zur Griindungsgeschichte des
Journals fi.ir reine und angewandte Mathematik," NTM, 14 (1977): 8-28.
72 See Gert Schubring, Die Entstehung des Mathematiklehrerberufs im 191ahrhunderts
(Basel: Beltz Verlag, 1983), and Jahnke and OUe, "Origins of the Program of the
'Arithmetization of Mathematics'," and Gert Schubring, "The Conception of Pure Math-
ematics," in Social History, Mehrtens, Bos and Schneider, eds. 21-49 and 111-134
respectivel y.
73 Many hands and opinions tried to shape this institution. Alexander von Humboldt tried
to add chemistry to its curriculum and make the institution a second Ecole Poly technique.
74 Kees Gispen, New Profession, Old Order, chap. 1, 15-43 sketches the problems of en-
gineers in German society against the general social and cultural background and the
role of education in the defining of engineers in German society in the early nineteenth
century.
German States, 1790-1830 163
mati cal parts of physics.,,75 Mathematics was to train the mind and number, space,
and force were its subjects. 76
This was neohumanism at its most forceful but not necessarily at its most suc-
cessful. When it was finally established, the Berlin Polytechnic did not have a
curriculum based on Crelle's ideas. 77 However, Crelle was successful, along with
Jacobi, in establishing this definition of mathematics in universities and of remold-
ing mathematics there. Jacobi claimed the high ground for mathematics because
of its esoteric nature. In a letter to Alexander von Humboldt, Jacobi voiced the
opinion that the most "lofty of sciences were the most impractical." He considered
his work in astronomy (the subject under discussion was Neptune) as in the proper
sense mathematics. In this work he had never considered its application to actual
astronomical problems.78 Jacobi delighted in the consternation he caused at the
1841 meeting of the British Association when expressing similar opinions. 79
Crelle's definition of the content of mathematics also emulated the content of
French mathematics. Mathematical physics and the mathematical parts of physics
remained mathematics. Although the problem with the French, in Crelle's opinion,
was that they emphasized application too much.
Crelle's assessment of what the discipline of mathematics covered, number,
space and force, was commonly adopted amongst mathematicians and younger
academics. Jacobi and the other young faculty would create this new academic
profession and discipline of mathematics and outstrip the French. Not that there
emerged from these efforts one exclusive set of practices that defined mathematics.
However, by the last third of the nineteenth century, the approach to mathematics
through the solution of physical problems was relegated to the level of a secondary
75 Schubring, "Mathematics and Teacher Training: Plans for a Polytechnic in Berlin," Hist.
Stud. Phys. Sci. 12 (1981): 161-194,174.
76 Crelle defended the "purity" of mechanics in Crelle, Encyklopiidie deutsche Darstellung
der Theorie der Zahlen (Berlin: 1845), vol. 1, iii-iv. See also Schubring, "Mathematics
and Teacher Training," 178.
77 Gispen, New Profession, Old Order, notes that the Polytechnic and its curriculum re-
flected the now lowly place given "bread study." The mathematics in the curriculum
was low-level and the Berlin Polytechnic became the training ground for engineers. The
Polytechnic was overseen by the Trade Ministry rather than the Kultus Ministerium which
had important political outcomes as well as effects on the curriculum.
78 K.-R. Biermann, "Uber die F6rderung deutscher Mathematiker durch Alexander von
Humboldt," in Alexander von Humboldt: Gedenkschrift zur 100 Wiederkehr seines
Todestages (Berlin: Akademie Verlag, 1959) 83-160, p. 88. See also Biermann,"Der
Briefwechsel zwischen Alexander von Humboldt und G. J. Jacobi tiber die Entdeckung
des Neptun," NTM, 6 (1969): 61-67.
79 See Briefwechsel zwischen C. G. J. Jacobi und M. H. Jacobi, W. Ahrens, ed. (Leipzig:
B. G. Teubner, 1907) 22.
164 Physics and Mathematics
80 In the 1820s we detect similar divisions of the intellectual territory between mathematics
and physics in the new philosophical context in the short-lived ZeitschriJt for Physik und
Mathematik edited by Andreas von Baumgartner and Andreas von Ettingshausen.
81 Gurt Konig, "Einleitung zur Abteilung," to Jakob Friedrich Fries, Siimtliche SchriJten vol.
13, Schriften zur Angewandten Philosophie II Naturphilosophie und Naturwissenschaften
(Darmstadt: Sci entia Verlag reprint, 1979),7-15,12-14.
82 Muncke, "Physik," in Physikalisches Worterbuch 11 vols (Leipzig: Schweikert, 1825-
1845) vol. 7 (1833): 493-573, 510-51l. His effort to distinguish theoretical and exper-
imental physics appear on pages 503-504.
German States, 1790-1830 165
claim that "as long as we neglect to use mathematical methods" physics was in-
complete. He could only point to the example of the French and their role in
developing mathematics and to state simply that, for the further study of phenom-
ena, the calculus and geometry were as important as experiment.
Many statements reinforcing this view of mathematics and physics occurred in
the entry on "Mathematics" by Heinrich Wilhelm Brandes. Pure mathematics was
about quantity and had no bearing upon the empirical realm. He did not include
geometry in pure mathematics because it referred to experience, only quantity
qualified for inclusion. While mathematics was independent of experience, it
did develop the consequences of hypotheses in the sciences through the rules of
arithmetic and geometry. In its development there were no contradictions, unless
from human frailty. From the rules of arithmetic, those of algebra and the calculus
followed "in a natural sense." Reducing natural phenomena to mathematical form
had much to recommend it, as mathematics could easily examine the correctness
of hypotheses. Brandes gave some general, loose justifications for a relationship
between physics and mathematics but no systematic understanding of how this
might be accomplished in practice. Also Brandes did not seem to understand the
difficulties of examining the correctness of hypotheses with mathematics. 83
Other definitions of physics existed that did not generate such tensions. Poggen-
dorff published a manifesto, as he began his long career as editor of the Annalen
der Physik und Chemie. Only purely scientific matter was to appear. The drift
of physics towards chemistry was overwhelming but his volumes would include
work in meteorology and physical geography. Pure mathematics lay outside its
coverage. Mathematics was included only in so far as it "made experiments more
precise or where a series of data can be brought together into an essential rela-
tionship through a theory created from the principles of mechanics." Poggendorff
assumed, along with his colleagues, that mechanics was a branch of mathematics.
Overuse of formulae would be avoided and mathematics would find a place in his
journal only "when it reflects the true interests of physicists.,,84
In his first decade as editor, Poggendorff reprinted many foreign articles in exper-
imental physics, but afterwards domestic research papers predominated. It was not
until the 1840s that mathematical physics entered the journal as a steady stream
in papers on the wave theory of light, and electricity. Many of them were also
translations from foreign journals. 85 In the 1840s most of the papers were experi-
particular reason for their existence except for their usefulness at the time, and
the personal preferences of the historical actors. It also leads to some complex
problems of accounting for certain forms of physics such as Neumann's. And it
does not explain why mathematical physics continued to be considered different
from theoretical physics and treated as such. It also ignores how and why mathe-
maticians could still publish in areas of "physics" and loudly claim the domain of
mathematical physics as their own.
What this historical approach does require is that we accept any mathematical
expression of a physical problem and its mathematical development as physics.
This belies the contention that theoretical physics was created in the reformed
universities of Germany in the nineteenth century. Such mathematical solutions
to physical problems were available from the time of Newton. We need to pay
attention to the descriptive language of these two intellectual disciplines. No one
in Germany used the terms mathematical and theoretical physics interchangeably,
or simultaneously. The first was an inheritance from the French, the second a
creation of the nineteenth century.
Changes occurred in both the professions and disciplines of mathematics and
physics that were interconnected and form a continuum of interests through time
and across changing disciplinary boundaries. Both disciplines later laid claim
to the mathematical development of physical problems. German mathematicians
continued to use emerging areas of physics as sources for research problems in
mathematics. At the same time academics who became physicists or who held
appointments as physicists began to publish mathematical investigations of the
same physical phenomena. Initially the investigations within either discipline were
mathematics. In the 1840s these mathematical investigations began to diverge as
the aims of physicists became more closely defined and included the conscious
exploitation of hypotheses. Mathematicians continued to develop their version
of mathematical physics using the mathematics of physics problems, solved in
more general ways. Simultaneously, physicists were creating a new discipline,
theoretical physics. Physicists annexed parts of that domain, mathematical physics,
that had belonged exclusively to mathematicians, then reconfigured it to match their
changing disciplinary needs through the consideration of a few crucial problems
in electrodynamics, light, and heat.
Chapter VI
German case, native traditions mutated this newly acquired heritage into unique
forms.
All of these processes began in the thirty years that are the subject of this chapter.
They were not completed until the last third of the nineteenth century.
One of the social costs of these changes was the isolation of physicists from
the general public and the need for mediators between research practitioners and
that public. In the nineteenth century these mediators were often professionals
who understood the different audiences that they needed to address, research peers,
student-apprentices, students destined for other careers, an informed public and the
more general, interested public. These various listeners were looking for different
kinds of understanding through physics and each required different languages of
explanation, depth of coverage, and the kind of argument necessary to draw that
audience into the subject matter.
Many of these changes were completed after 1830. Yet by 1830 the possibilities
so apparent in the early nineteenth century for the practice of experimental philos-
ophy had narrowed in form, access, and content. While the final outcomes of these
transformations can be described in general outline in language close to the de-
scription of developments in the German states, the passage to professionalization
and theoretical physics was unique, much more diffuse, and driven by economic
as well as ideological and political forces. The net results were quite different in
detail from those of Germany. Within the three decades described here, only a
few of the social and intellectual forms explored became lasting possibilities. The
universities of Oxford and Cambridge retained, for the most part, their function
of a general education for the elite. By the middle third of the nineteenth century,
with the economic development of southern Scotland, Scottish universities offered
an alternative more open to the educational needs of the industrial economy. At
the end of our period, symbolically, if not practically, London University was be-
ginning to do much the same. Higher education was opening up for more men
further down the social scale. The education offered was for a society, economy,
and political order of a different kind from that assumed and still operative in the
educational philosophy of Oxbridge.
Social Institutions
The particular dates chosen for this section emphasize this era's essential so-
cial and intellectual continuity of the study of nature with the eighteenth century.
However, the social and intellectual forms of the eighteenth century came under
increasing strain during the 1820s and then dissolved to re-form during the 1830s
and 1840s.
In the decades around 1800 experimental philosophy was still an avocation for
most of its practitioners who formed a loosely connected network. Experimental
philosophy still defined a set of practices rather than a specific series of research
Britain, 1790-1830 171
problems or a theoretical stance. Physics still covered, for example, every aspect
of electricity from static electricity to the physiology of electric fishes. The study
of sound and optics included the anatomy of the ear and eye and the perception,
as well as the nature of, sound and light and their propagation through space. The
study of nature was an increasingly popular aspect of a general culture shared
by many varied groups in society. Institutions to inform, demonstrate, and even
develop the study of nature multiplied during this era. They catered to all groups
and classes under a broad range of economic, cultural, and ideological hats.
Many of the institutions that supported the study of nature were volunteerist and
provincial. Their titles, purposes and functioning often included much more than
experimental philosophy. Their success depended upon meeting the aspirations
and interests of a broad, local, audience. I While these institutions were local, their
members were not isolated or scientifically unsophisticated. 2 The societies might
function for social or cultural self-improvement, as a center for sharing information,
research results amongst local practitioners, friends of science and local "worthies"
of science. 3 Some institution offered the enterprising an opportunity to transmute
1 For the diversity represented in such organizations, see Metropolis and Province: Science
in British Culture, 1780-1850, Ian Inkster and Jack Morrell, eds. (London: Hutchinson,
1983). See also Ian Inkster, "Cultural Enterprise: Science, Steam, Intellectual and
Social Class in Rochdale, circa 1833-1900," Soc. Stud. Sci. 18 (1988): 291-330; Jack
Morrell, "Early Yorkshire Geological and Polytechnic Society," Ann. Sci. 45 (1988):
153-167, and "Bradford Science, 1800-1850," Brit. J. Rist. Sci. 18 (1985): 1-23; J.
N. Hays, "Science in the City: The London Institution 1819-1840," BritishJ. Hist. Sci.
7 (1974): 146--162, and "London Institution," Ann. Sci. 39 (1982): 229-274. The
localism and attempts to preserve it in Edinburgh as natural philosophy became science,
and national rather than local, are detailed in Steve Shapin, "The Audience of Science in
Eighteenth-Century Edinburgh." D. S. L. Cardwell Organization of Science in England
in the Nineteenth Century second edition (1973), makes little mention of these institutions
while focussing on mechanics institutes and the redbrick universities. The narrative of
the mechanics institutes is well known, its meaning still contested. See Roy Heyden,
"The Glasgow Mechanics Institution," Phil. J. 10 (1973): 107-120; Steve Shapin and
Barry Barnes, "Science, Nature and Control: Interpreting Mechanics Institutes," Soc.
Stud. Sci. 7 (1977): 31-74; Ian Inkster, "Science and Mechanics Institutes, 1820-1850:
The Case of Sheffield," Ann. Sci. 32 (1975): 451-474, and, "The Social Context of
the Educational Movement: A Revisionist Approach to English Mechanic's Institutes,
1820-1850," Oxford Rev. Educ. 2 (1976): 277-307. See also Gordon W. Roderick and
Michael D. Stephens, Scientific and Technical Education in Nineteenth-Century England
(New York: Barnes and Noble, 1972) chaps., 8 and 9.
2 For a sense of the networks these institutions formed see, Jack Morrell and Arnold Thack-
ray, Gentlemen of Science: Early Years of the British Association for the Advancement
of Science (Oxford: Clarendon Press, 1981), chap. 2 and Appendix II, 544.
3 Nathan Reingold's classification of the supporters and practitioners of science, while
developed for the American context, seems apt for this period in early nineteenth-century
Britain.
172 Physics and Mathematics
4 Arnold Thackray,John Dalton: A Critical Assessment ofhis Life and Science (Cambridge
MA: Harvard University Press, 1972) chaps. 4 and 5, details the ways John Dalton
fashioned his career at the Manchester Literary and Philosophical Society. Dalton was not
the first to do this through the Manchester institution, see Frank Greenaway, John Dalton
and the Atom (Ithaca NY.: Cornell University Press, 1966), 91-95 on Thomas Henry.
There were also Humphry Davy and Michael Faraday at the Royal Institution. For John
Phillips at the Yorkshire Philosophical Society see Morrell and Thackray, Gentlemen
of Science and Martin Rudwick, The Great Devonian Controversy: The Shaping of
Scientific Knowledge among Gentlemanly Specialists (Chicago: University of Chicago
Press, 1985.) On Charles Lyell's use of London institutions in his early career see, Jack
Morrell, "London Institutions and Lyell's Career, 1820-41," Brit. J. Hist. Sci. 9 (1976):
132-146.
5 The Royal Institution and many Mechanics institutes were caught in this situation. For the
links between architectural form and social values see Sophie Forgan, "The Architecture
of Science and the Idea of a University," Studies Hist. Phil. Sci. 20 (1989): 405-443.
6 For a case study of such political purposes see, Steve Shapin, " 'Nibbling at the Teats of
Science': Edinburgh and the Diffusion of Science in the 1830s," and Michael Neve, "Sci-
ence and Commercial Utility: Bristol, 1820-1860," in Metropolis and Province, Inkster
and Morrell, eds. 151-178 and 179-204, respectively. See also Shapin, "Mechanics
Institutes," and Thackray, Science in Manchester. The ideological purposes of the Bristol
Pneumatic Institution should also be noted here.
7 See Dorinda Outram, "Science and Political Ideology, 1790-1848," in Companion to the
History of Modern Science R. C. Olby, G. Cantor, J. R. R. Christie, and C. Hodge, eds.
(New York: Routledge, 1990), 1008-1023.
Britain, 1790-1830 173
8 For such goals in the foundation of some societies see, Arnold Thackray, Science in
Manchester and Robert E. Schofield, The Lunar Society.
9 Physicians were a significant percentage of its membership. It was still important for
ambitious physicians to be seen as learned advisers as well as mere healers, and as having
social contact with the wealthy and powerful. See Harold Cook, "The New Philosophy
and Medicine in Seventeenth-Century England," in Reappraisals of the Scientific Revo-
lution, David Lindberg and Robert S. Westman, eds. (Cambridge: Cambridge University
Press, 1990) 397-436. For the Royal Society'S social function and political ties, see Marie
Boas Hall, All Scientists Now: The Royal Society in the Nineteenth Century (Cambridge:
Cambridge University Press, 1984), chap. 1.
10 The diversity, intensity, productivity and independence of the research done by John
Dalton in Manchester, Humphry Davy, Michael Faraday, and Thomas Young at the
Royal Institution argues for this social function of the Royal Society.
11 Thomas Young,A Course in Natural Philosophy (London: Joseph Johnson, 1807) 2 vols.
174 Physics and Mathematics
The institution was shaped by Davy and his career and was reshaped to attract
the fashionable London audience. 12 By 1807 the syllabus of the Royal Institution
included lectures on moral as well as natural philosophy, drawing, engraving,
music, and poetry. Count Rumford's vision of an institution for the diffusion of
useful knowledge was gone. Yet the Royal Institution flourished even as its appeal
for funds on a national level floundered. It became one of the most successful local
cultural institutions of the early nineteenth century.
The multiplication of these institutions and their popularity in the early decades
of the nineteenth century meant that a career in experimental philosophy was not
consumed by constant, countrywide, itinerant lecturing or teaching. A base in
London or some institution in a large city could support a career. Dalton did some
lecturing, usually in the summer in the Lake District for the tourist trade. 13 Such
professionals also brought tensions into the institutions that housed them. Their
needs sometimes clashed with the survival of the institution. Research brought
recognition but not income. The lecture series such professionals had to deliver
were for large audiences and directed to entertainment as well as enlightenment.
It was rare that they could repeat Davy's success and combine the two. Michael
Faraday separated the two functions, instituting the Friday night lecture series at
the Royal Institution.
For a fee, these institutions offered the public opportunities for an education in
natural philosophy unavailable except for the privileged few who attended univer-
sity. How many besides Faraday transformed such opportunities into careers is
still obscure. 14 As obscure is the role that these institutions played in educating
the working class in those scientific principles that were commonly seen as the
foundation for the arts they practiced. According to this philosophy of progress,
knowledge of such sciences should improve their economic future. It was also a
means of cultural and moral uplift. However, in the institutions controlled by the
working class, indications are that, in the interest of survival, systematic lecture
series in the sciences were sacrificed for less demanding and more entertaining
fare. Only those institutions catering to the upper and middle classes could afford
to offer lecture series in natural philosophy. And these were only delivered in large
cities where a significant percentage of the population could afford the expense.
Lectures in experimental philosophy delivered over several weeks were more likely
12 On the uses that Humphry Davy made of the Royal Institution in shaping his career see,
Jan Golinski, "Davy and the 'Lever of Experiment,''' in ExperimentalInquiries, Homer
LeGrand, ed., 99-136.
13 For Dalton and his lecturing see Thackray, Dalton, and Greenaway, Dalton and the Atom,
chap. 5. For London see J. N. Hays, "The London Lecturing Empire, 1800-1850," in
Metropolis and Province, Inkster and Morrell, eds. 91-119.
14 Jan Golinski has traced chemists active in London in the early nineteenth century, their
education and their professions before they specialized in chemistry. See Jan Golinski,
Science as Public Culture, chap. 8.
Britain, 1790-1830 175
15 Evidence for the most systematic education in chemistry available in Britain was at
Edinburgh University, the locus of the best medical training.
16 See Metropolis and Province, Inkster and Morrell, eds.
17 The empirical foundation of the sciences assumed in Britain in this period has been
emphasized more than once. See Michael Shortland, "A Mind for the Facts, some
Antimonies of Scientific Culture in early nineteenth century Britain," Arch. Int. Hist. Sci.
36 (1986): 294-324.
18 However, authors of texts in natural theology were already feeling the pressures of ex-
perimental philosophy and natural history. They had to adjust their arguments to accom-
modate new phenomena and current interpretations of a mechanical world. On William
Paley see, Neil Gillispie, "Divine Design and the Industrial Revolution: William Paley's
176 Physics and Mathematics
the student from novice to professional competence. Nor was there any need for
certification as an end to such an educational enterprise.
There was one great difference between the lectures given by university faculty
in England and the lecturers such as Young, Davy, and later Faraday. These three
men constructed their careers through their lecture courses and drew their audiences
into their research. Their audiences were at the very creation of the knowledge,
not just exposed to a demonstration of nature's readily repeated characteristics.
In those audiences were their research peers as well as the general public. 19 Such
professional lecturers were freer to question the prevailing, broadly Newtonian,
explanations of the phenomena revealed in their lectures. They could offer very
successful alternatives to long-standing research issues in experimental philosophy
and chemistry.
The same imperative to introduce students to research results did not motivate
most university faculty. They were primarily expected to transmit to their students
the rudiments for entry into gentlemanly culture. A faculty member who developed
a reputation as a good teacher would enhance his income. Assuming that he did
research, he was not under the same pressure to present his research results to his
audience. Some faculty did present their research within their lectures to students,
the most obvious being Joseph Black at Edinburgh. The pressures to do so could not
match the imperatives of survival in the newly created scientific institutions such
as the Manchester Literary and Philosophical Society and the Royal Institution.
At a university, research and teaching need not be integrated. In some settings, the
problem of losing status existed if the university lecturer was seen as pushing his
ideas upon the public, as a projector pushed his schemes for making money.20
The function of the faculty, to pass to the next generation the accepted cultural
foundations for a gentlemanly existence, actually worked against the intrusion of
research results into lectures. At Cambridge, as the Tripos became more important,
students were reluctant to attend courses that did not "pay." Unless the material in
the course was related to the subject of likely examination questions, classrooms
were mostly empty.
University texts for students offered a blander, more acceptable form of experi-
mental philosophy for their students than the texts of independent lecturers. These
texts share some common features. Courses on experimental philosophy were sur-
veys dominated by experimental demonstrations. The broadest terms were laid out
by John Playfair as, "the knowledge of the general laws obeyed by the phenomena
of nature, whether in the intellectual or the material world." Playfair's description
abortive Reform of Natural Theology," Isis, (1990): 214-229. Gillispie points out that
Paley's audience included the urban middle class as well as Cambridge students.
19 See Jan Golinski, Science as Public Cu/tllre, chap. 7.
20 See Jan Golinski, Science as Public Culture, chap. 2 and William Cullen's problems with
"speculation" at Edinburgh University.
Britain, 1790-1830 177
betrayed the presence of Common Sense and the educational philosophies of Scot-
tish universities. Other texts do not go so far as to include psychology in the net
of natural philosophy.21
University lecturers shared the same Newtonian, non-mathematical and non-
technical explanations for the phenomena that were the heart of their lectures. 22
In this vernacular Newtonianism, as at Oxford, the "mathematical approach is ap-
parently deliberately avoided."23 The subject matter of the lectures joined natural
theology to natural philosophy that elevated the mind and, with demonstration ex-
periments, proved God's power. 24 Utility was popular, although proclaimed rather
than systematically explored. 25
The only mathematics in these unrelenting factual narratives punctuated with
descriptions of experiments and physical explanations were algebra and geometry.
Many experiments were qualitative, very few quantitative as in Biot's text of the
same era. 26 The sample experiments were chosen to demonstrate as simply as
possible an explanation of the operation of mills, machines, etc.
This utilitarianism reaches into other sections of the lectures as well. Playfair's
section on physical astronomy was descriptive. His discussion of the regularities
21 For Cambridge see the textbooks discussed in chapter II. In the Scottish universities the
same pattern prevailed. See, John Playfair, Outline of Natural Philosophy (Edinburgh:
A. Constable, 1812-1814), 2 vols., as 1. The volumes are clearly from notes for lectures.
They are too cryptic for delivery to students. See also John Robison, Elements ofMechan-
ical Philosophy, being the Substance of a Course ofLectures on that Science (Edinburgh:
Constable and Co., 1804). The broad sweep of the meaning of the term "physics" in this
era is in Michael Short land, "On the Connexion of the Physical Sciences: Classification
and Organization in Early Nineteenth-Century Science," Hist. Scientiarium 41 (1991):
17-36.
22 For a discussion of mechanical philosophy and its inclusive character, see Crosbie Smith,
" 'Mechanical Philosophy' and the Emergence of Physics in Britain: 1800-1850," Ann.
Sci. 33 (1976): 3-29, 6-14. Smith locates the beginnings of this tradition in Robison's
lectures and sees the same patterns in William Meikleham's lectures in natural philosophy
at Glasgow University from William Thomson's notebooks on those lectures.
23 Gerald L. E. Turner, "Experimental Science in Early Nineteenth-Century Oxford," in
Hist. Univ. 8 (1989): 117-135, 123.
24 The ends of science, as expressed by natural philosophers themselves in this era, are dis-
cussed by G. A. Foote, "Science and its Function in Early Nineteenth-Century England,"
Osiris 11 (1954): 438-454.
25 Predictably, Davy saw the ends of science as Truth. The relationship between early
nineteenth-century science in Britain and the ideas and values of Romanticism are tenuous
at best. Very few natural philosophers expressed even Davy's early commitments. Utility
was far more compelling within the changing economic environment of northern England
and Scotland.
26 A need was however seen for such a text because Biot was translated. See Jean-Baptiste
Biot, Traite de physique experimentale et mathematique 4 vols., (Paris: Dateville, 1816),
translated by John Farrar.
178 Physics and Mathematics
of the planets was in terms of Kepler's laws put into geometrical form. Fluxions
were mentioned in his definition of velocity and acceleration, then dropped. He
noted Laplace's Mecanique Celeste but did not explore it. On the basis of the
rest of the course students would have a hard time making any sense of Laplace.
Algebra and geometry were used to express empirical laws, such as those of optics.
From these empirical laws, other, equally empirical results were deduced, some of
which had already been demonstrated. Beyond a statement of Newton's laws of
motion and gravitation, the Principia did not enter into experimental philosophy.
There were no problem sets.
Initially in the lectures, the definition of experimental philosophy was broad.
The actual topics covered were narrowly defined by mechanics with some optical
phenomena thrown in. The patterns discussed here for lectures were replicated in
all universities including the later ones of Dionysius Lardner at London.
In short the lectures read as protracted encyclopedia articles, many of which
were indeed written by the same lecturers. They were both lectures and articles
intended for the same audience to serve similar cultural purposes. John Robison's
article on "Physics" in the Encyclopedia Brittanica remained unchanged from the
third through the seventh editions. In his article on mechanics Robison separated
mechanical philosophy from mathematics. D' Alembert and Lagrange were,
merely employing the reader in algebraic operations, each of which
he perfectly understands in its quality of an algebraic or arithmetical
operation, and where he may have the fullest conviction of the justness
of his procedure. Well all this may be (and, in the hands of an expert
algebraicist, it generally is,) without any notions, distinct or indistinct,
of the things, or the processes that are represented by the symbols made
use of.27
Study of the natural world and the manipulations of mathematical entities were
distinct activities. Similarly, astronomy was based on accurate observations, nec-
essary for "philosophical inference." Mathematics did not enter into Robison's
discussion. 28
John Play fair made much the same distinctions. However, Playfair set the sci-
ences in a hierarchy with mathematics in first place. Its "progress" has been
"one principal instrument applied by the moderns to the advancement of natural
knowledge." The other instrument of progress was experiment and the method of
induction. Bacon's philosophy and experiment received far more attention than
did mathematics. Even in this systematic account of the sciences mathematics
entered only by deducing results logically from principles established "by experi-
ence." The example used was Galileo's relationship between the distance fallen by
a body and the square of the time taken for the descent. There was a decided gap
between Playfair's claims for the importance of mathematics and his demonstrated
use of it in both experimental philosophy and astronomy.29 Playfair discussed the
development of mathematics only so far as Descartes and the invention of loga-
rithms, ending on a Scottish note. There was no mention of fluxions or the calculus
with which Playfair was very familiar. He divided experimental philosophy into
divisions, mechanics, astronomy, optics, then the imponderables, heat, electricity
and magnetism. His narrative was organized to illustrate the central function of
induction in the "progress" of the sciences.
John Leslie's article on the eighteenth century made even stronger claims for
mathematics in the development of natural philosophy but had no better illustration
of this claim than Playfair's. Leslie stated that in the eighteenth century mathemat-
ics, when introduced into physics and the practical arts, had brought great results.
He classified all of the sciences into two great classes, the "pure or speculative"
and the "applied or practical." The latter included optics, electricity, magnetism
and theories of heat and their application in the mechanical arts. "Pure physics"
was now limited to magnetism and electricity, neither of which contained much
"geometry." Since Leslie only stated the basic principles behind each of his sci-
ences, how their application worked on the mechanical arts was no clearer than
how mathematics was applied in the other sciences. These essays also give us
an idea of the history of natural philosophy from the Edinburgh point of view in
the early nineteenth century. It was unrelentingly empirical, inductive, and experi-
mental. Mathematics entered only after experiment, to confirm, with its generality,
the results of those experiments. 3D
Published from Edinburgh, the Encyclopedia Brittanica displayed, wherever
possible, the concerns of Scottish academics for philosophical consistency. The
Encyclopedia Metropolitana had quite different goals and a different set of authors.
The editors and contributors were either educated or taught at the University of
Cambridge which heavily influenced the classification and content of the entries
on the sciences. The philosophical categories of Samuel Taylor Coleridge dictated
the organization of the text. In their turn his categories reflected the concerns of
contemporary German academics, for whom the pure sciences were those of the
29 John Playfair, A General View of the Mathematical and Physical Sciences since the
Revival of Letters in Europe in Stewart, Mackintosh, Playfair and Leslie, Dissertations
on the History of Metaphysical and Ethical and of Mathematical and Physical Sciences
(Edinburgh: Adam and Charles Black, 1824-1835), (Edinburgh: 1824), 2 vols., as
one, 433-572. Richard Yeo, "Reading Encyclopedias: Science and the Organization of
Knowledge in Dictionaries of Arts and Sciences, 1730--1850," Isis, 82 (1991): 24-49
discusses the function of these discourses.
30 John Leslie, A General View of the Progress of Mathematical and Physical Science,
chiefly during the Eighteenth Century in Stewart et ai, Dissertations.
180 Physics and Mathematics
mind, separated from the "mixed sciences" such as mechanics, optics, and astron-
omy. In Coleridge's hierarchy there was also a third level, the applied sciences that
depended on changes in bodies. These sciences included electricity, magnetism,
and chemistry. These applied sciences did not rest on any general, purely intel-
lectual source for the knowledge they generated and hence were on a decidedly
lower intellectual plane. This philosophical hierarchy was reinforced by the order
of publication of the volumes, starting from the top down. Articles only referred to
material already published displaying the intellectual dependency in Coleridge's
scheme. 3 !
The content of the individual articles reflected the content and organization of
current teaching at Cambridge. The mathematics included geometry, algebra, and
fluxional calculus. Mechanics was included in the volumes on mixed mathematics,
yet treated as a subfield of mathematics. 32 John Herschel's article on physical as-
tronomy was also in the volume on the "mixed" sciences. Neither of the articles on
mechanics and physical astronomy could be read with any understanding without
intimate knowledge of fluxions or the French calculus respectively. Audiences
for this level of treatment was small and the enterprise failed. The volumes also
reflected some of the crucial changes occurring in sciences in the 1820s. Articles
were narrowly focused and written by specialists. Herschel on astronomy detailed
the new standards of measurement. He also showed how to use the mathematical
results of celestial mechanics and turn them to solving problems of observational
astronomy, both uses of mathematics that were recently imported from Germany.
By the time the Encyclopedia Metropo!itana began publication, the older, long
accepted intellectual geography of the sciences was under considerable strain, as
was their social organization. Simultaneously, in the late 1820s agitation began
for an education to meet the expectations of the middle class and a generation that
needed technical training for the new economy.
of the powers of nature, properties of natural bodies, and their interactions, the
objects of study of physics, restricted its methods to those of experiment. And,
experiment was defined as the trials made to uncover these attributes of nature.
Natural philosophy encompassed ideas and theories about those powers, proper-
ties and interactions and was coextensive with experimental philosophy. After a
statement of Newton's laws of motion, Newtonian philosophy was discussed at
some length non-technically as a gloss on the books of the Principia. Charles
Hutton included mechanics, however, as a "mixed mathematical science." Mixed
mathematics itself was the effort to,
reason mathematically upon physical subjects, such just definitions can-
not be given as in geometry: we must therefore be content with de-
scriptions; which will be of the same use as definitions, provided we be
consistent with ourselves, and always mean the same by those terms we
have once explained. 33
Geometry belonged to abstract or pure mathematics, the "science of quantity." Pure
mathematics was speculative but also had the advantage of leading more surely to
truth than experiment. To end a dispute in pure mathematics, all one needed to
do was to show that an opponent had not stuck to his definitions, or had argued
incorrectly.
From 1790 to 1820 research in experimental philosophy followed these same
patterns. The encyclopedia literature only occasionally reflected the new principles
introduced into natural philosophy. These new principles were not introduced by
university professors and dons, the contributors to those articles. They came mainly
from outsiders, constructing new ways of doing natural philosophy both socially
and cognitively.
While John Dalton worked within a Newtonian tradition, his concept of the
atom was only tenuously connected to Newton's commitments. In Dalton's work
the ultimate physical and chemical particles of matter became one. His theory,
developed within the net of questions of meteorology, poached upon ground that
chemists had regarded as theirs. The sometimes fierce opposition from chemists
to Dalton's ideas was only partly due to the incompleteness of Dalton's evidence
and the arbitrary nature with which Dalton was forced to construct his molecular
formulae. 34 Specialist boundaries were breached and the newly established claims
of chemists to empiricism were called into question. 35
Dalton's opponents used hypotheses as freely as he, yet chemists were not able
to acknowledge a place for "speculation" for many decades. Experimental philoso-
phers already had recognized the necessity for hypotheses. 36 Natural philosophy
was both a method of exploring nature and, properly separated and reported, a
series of speculative ideas about nature. The same consciousness of both method
and hypotheses are in the lectures of Robison and other university lecturers on
mechanical philosophy. Disputes between experimental philosophers were not
over the use of hypotheses as such but whether particular hypotheses met current
criteria for legitimacy.
In the early decades of the nineteenth century, Dalton was forced to defend his
ideas on many fronts. So was Thomas Young. Claiming also to be a Newto-
nian, Young fractured that heritage beyond his contemporaries' recognition of it. 37
Young's lectures and research on light conformed to inherited patterns of natural
philosophy, but with a new emphasis on certain aspects of experimental philosophy
emerging from France. His work in optics emerged from his medical interest in
vision, and that followed his earlier work in acoustics. 38 The latter encompassed
both the phenomena of sound and their explanation, the functioning of the ear as
the organ of hearing, and the sense of hearing itself. Optics, therefore, included
the phenomena of light and their explanation, as well as consideration of the eye
and vision.
Young's contemporaries made no clear distinction between the phenomena of
sound and hearing, light and vision. The means of detection and observation of light
and sound were direct and depended upon the acuity of the observer's hearing and
sight. Even in an age in which musical ability was an important social skill, there
were disputes about the phenomena of sound, that is, what could be heard. The
all-encompassing aspects of these studies, which mirrored the breadth of natural
philosophy itself, led to many misunderstandings of Young's intentions and the
meanings that underlay his analogy between sound and light as wave phenomena.
see Thackray, John Dalton. For the arguments surrounding his theory and chemists'
claims of empiricism, see Allen J. Rocke, Chemical Atomism in the Nineteenth Century
from Dalton to Cannizzaro (Columbus OH.: Ohio State University Press, 1984).
36 For the reality behind the rhetoric of the chemists, see Rocke, "Methodology and its
Rhetoric in Nineteenth-Century Chemistry: Induction versus Hypothesis," in Beyond
History of Science, Garber ed., 137-155. Geologists were also less able to see any legiti-
mate function for hypotheses during this era. See Rachel Laudan, "Ideas and Organization
in British Geology."
37 As much research has recently demonstrated, Newton's work was sufficiently complex as
to admit of a broad range of theoretical opinions being attached to his name. For example,
see Thackray, Atoms and Powers, and Conceptions of Ether, Cantor and Hodge, eds.
38 Young's work in vision begins with Thomas Young, "Outlines of Experiments and En-
quiries respecting Sound and Light," Phil. Trans. R. Soc. London, (1800): 106-150, and
"Mechanism of the Eye," same journal (1801): 23-88. He retained this interest in his
lectures at the Royal Institution. See Young, Lectures on Natural Philosophy.
Britain, 1790-1830 183
One phenomenon over which there was much dispute was that of beats in sound.
Some experimentalists claimed that they existed, others that they did not. 39 Young's
wave theory of light was dismissed on grounds that no longer apply in the study
of sound and light. Young differentiated sound and light from hearing and vision,
but did not explicitly express this to his contemporaries. Many of them, besides
Henry Brougham, had difficulty understanding Young's experiments. These men
included John Robison and Robert Woodhouse who needed repeated exposure to
Young's work to finally extract from it Young's principle of interference. 4o
At the same time Young had to invent a language in which to express physi-
cal ideas that he was uncovering piecemeal. The idea of wavelength is absent,
although Young used the ambiguous term "breadth" of an undulation without fur-
ther explanation. Similarly, amplitude was absent, although Young wrote of the
height and depth of a wave. 41 Young inherited the term frequency from acoustics
but did not relate this to any of the other characteristics of his waves. 42 He also did
not use the kind of geometrical diagrams developed by Fresnel that eased the task
of his audience with visual representation. Young's work was within experimental
philosophy and his explanations were verbal.
And finally, Young's hypothesis that light was a wave motion was at odds with
contemporary ideas about light. He also embedded that hypothesis within his
narrative account of his experiments, not as a climatic statement at the end of the
series. He drew his new concept of light out of analogies between well-known
phenomena in acoustics with known results in experiments on light. Explaining
Newton's rings, Young noted that rings of the same color occurred at distances
from the center of the pattern where the distance between the two glass plates
were in an arithmetic progression, that is, at d, 2d, 3d, and so on. This was the
same relationship that occurred with the production of the same note in "organ
pipes which are different multiples of the same length." If light was a continuous
impulse of ether, "it may be conceived to act on the plates as a blast of air does on
organ pipes, and to produce vibrations regulated in frequency by the length of the
lines that are terminated by the two refracting surfaces.,,43 Young also contrasted
39 Nahum Kipnis, Principle of Interference, chaps. II, and III, notes this in his account of
the criticisms of Young's early work on acoustics and vision.
40 See Kipnis, Principle of Interference, for Robison, 56 and Woodhouse, 147-148.
41 The notion of wavelength did not exist mathematically, although the idea of amplitude
was defined mathematically.
42 Young was further hampered by an inability to communicate his own work and ideas,
especially to a general audience. See George Peacock, Life of Thomas Young (London:
John Murray, 1855), 135 and Alexander Wood, Thomas Young, Natural Philosopher,
1773-1829 (London: Cambridge University Press, 1954), 137.
43 Young, "Outline of Experiments." The phenomena are, however, different. The colors
from thin plates are the result of refraction, then interference. The organ pipe phenomena
are from standing waves.
184 Physics and Mathematics
Huygens' and Newton's theories of light and the difficulties with the latter in
explaining refraction.
The acceptance of Young's suggestion that light was, like sound, an undulation,
in the ether rather than in the air depended upon his audience accepting his interpre-
tation of Newton, a series of his own experiments, and the reinterpretation of other
still controversial phenomena. He used the principle of superposition to explain
beats as well as interference, yet the annihilation of sounds from different sources
contradicted experience. 44 Young was exposed on various grounds. In his replies
Young leaned towards the undulatory theory because of the phenomena of colors,
and it was here that he focussed his own research, presenting his experiments, and
the new phenomenon of interference, in his two Bakerian Lectures. 45
To emphasize his Newtonian roots in his first Bakerian lecture, Young presented
his theory in the form of Propositions and Scholia. The only demonstrations offered
were analogies to the behavior of fluids and sound. In his second Bakerian lecture,
and in his lectures at the Royal Institution, he intermixed experiments and theories.
Young therefore lost the dramatic climax of usual accounts of empirical research
where piling up empirical evidence appeared to force the researcher by induction
into a particular theoretical position. As his ideas and experiments developed
and he reacted to criticism, Young also changed details of his explanation of the
phenomena. 46
Most of his readers and audience did not dispute the quality of his experiments.
Many of those experiments included measurements, thicknesses of glass plates, the
distance of the diffracting object from the eye and the screen. He also included other
factors. In his algebraic relationships were sines, cosines and tangents of measured
angles. 47 In other cases his data were presented raw to the reader with no further
explanations. Neither Young's use of measurement, nor of geometry and algebra to
deduce a general relationship from his data, were points of comment in the barrage
44 Technically the most serious criticisms were from John Robison, "Temperament of the
Scale of Music," Supplement to Encyclopedia Brittanica 3rd., ed. This was a reply to
Young's earlier work on sound. The more damaging critique from Brougham was yet to
come.
45 Young, "The Bakerian Lecture [1801]. On the Theory of Light and Colors," Phil. Trans.
R. Soc. London (1802): 12-48, and, "The Bakerian Lecture [1803]. Experiments and
Calculations relative to Physical Optics," same journal, (1804): 1-16. Between these
two lectures Young had presented a brief description of his two-slit experiment in his
lecture series at the Royal Institution.
46 His papers give a more than usually intimate account of the evolution of his ideas,
complete with imprecisions, muddles, and backtracking in the research process. This
interpretation is in contrast to that of Kipnis, Interference ofLight who sees Young's work
as a linear progression moving towards a modern, generalized theory of wave motion.
47 See Young, Lectures, figure 442 for his two-slit experiment; "Bakerian Lecture [1803],"
171 and "[1801 ]," 160.
Britain, 1790-1830 185
of criticism that ensued. Yet his work hardly constituted a mathematical theory
of light, or even of interference. In his lectures and papers on natural philosophy
what mathematics he did develop was presented separately from his physical work,
experiment and hypotheses. 48 Despite Young's radical stance in denying Newton's
theory of light and in his presentation of his own ideas, he did not challenge
contemporary demarcations between physics as experiment and hypothesis, and
mathematics, generated from the results of physical experiments. 49
Young's presentation of his ideas on light were complicated by his multiplication
of hypotheses about the ether and its relationship with matter, and effect of this
interaction on light. In 1801 he explained the production of fringes inside and
outside the shadow of a small object as depending upon the refraction of light in
the ether atmospheres surrounding the particles of matter. By 1803 he explained
interference by the difference in the length of path traveled by two portions of light
reflected from two parts of the body. Young admitted that ether atmospheres were
unnecessary. 50
Between his two Bakerian lectures Young delivered a long series of lectures on
natural philosophy at the Royal Institution, the last of which were on light. Like
Davy, Young drew his audience into his research, to enhance the credibility of
his ideas, and answer his critics which, by this time included Henry Brougham.
Brougham had performed experiments on light himself and had his own strongly
held views on its nature. In his opinion, Young's experiments were not new and the
phenomena well known, irrelevant, or irreproducible, and his interpretations of all
his experiments were incorrect. Nothing Young had done merited the name philo-
sophical. Young's work challenged the methodology of Brougham's one domain of
direct research experience. Since method defined the field of experimental philos-
ophy, Young's challenge was fundamental. Brougham was defending a philosoph-
ical tradition, and the carefully constructed rhetoric of experimental philosophy
where empirical evidence led to, but was not intermingled with, hypotheses. 51
Young's answer to Brougham came in his lectures at the Royal Institution. He
48 Kipnis argues that Young "must" have had a mathematical theory of the interference of
light although there is no evidence for it. He even argues that Young also understood the
modern concept of wavelength and its relationship to the frequency of a wave motion.
49 See Young, Lectures, vol. 2. In this volume is a catalogue of some 2,000 published items,
many of which are followed by Young's remarks on their contents along with those of
other authors. Many items are from continental mathematicians.
50 See G. Cantor, "The Changing Role of Young 's Ether," Brit. J. Hist. Sci. 5 (1970): 44-62,
for a discussion of Young's ether.
51 Henry Brougham, "Bakerian Lecture on Light and Colors," Edinburgh Rev. 1 (1803):
450-456. Brougham was writing for the same kind of audience that attended Young's lec-
tures. See also G. Cantor, "Henry Brougham and the Scottish Methodological Tradition,"
Stud. Hist. Phi/os. Sci. 2 (1971): 69-89. I would put more weight on the significance of
this challenge because of the central place of method in defining experimental philosophy.
186 Physics and Mathematics
presented an overview of optics, from physical optics to vision and the anatomy
of the eye. Young repeated much of his previous work, including his criticism of
Newton's theory of light, but the tone had changed. The emphasis was on denying
the validity of the particle theory of light, rather than compiling evidence for the
undulatory alternative. Beginning with the assumption of light as an undulation in
an elastic Medium, Young stated that light must display the phenomena of superpo-
sition, as do sound and water waves. Here Young needed to define superposition,
then describe clearly its visual effects. He described the phenomenon by asserting
that waves travelling along different paths could destroy or enhance each other
at certain points. This led to the production of dark fringes when the difference
in path length was some multiple of an odd number of half "undulations." The
bright fringes were formed from path differences of whole undulations. 52 From his
experiments, Young estimated the wavelengths of the various colors. He discussed
the effect of varying slit-width and of removing the barrier between the slits. After
a survey of the production of colors from thin plates and soap bubbles, together
with interpretations in terms of diffraction, interference, refraction and reflection,
colored bodies, and their lack of fringes. The meteorological production of color
rounded out his survey. Clearly Young was the master of the subject. Brougham
was reduced to ad hominum attacks.
Young's lectures on optics, like Davy's on chemistry, were important in the
presentation of his research to the public within whose culture his results would
be judged. However, rather than focussing simply on his research Young plunged
into a protracted survey of the whole of optics, describing many phenomena, in-
struments, and experiments. This avalanche of empiricism buried the significance
of his own research. Young violated the general principles of instruction that
Davy clearly exploited, especially those particular to the cultural context of his
audience. Cruickshank's satirical cartoons of Davy's performance at the Royal
Institution catch one important feature missing in Young's lectures, the need to
entertain as well as inform. 53 Young had that opportunity. While experiments on
light were not as dramatic as those of Davy's chemistry, they had a beauty and
even a romantic appeal he seemed unable to exploit.
On several grounds Young needed the support of this group. His ideas evolved
rapidly and his idiosyncratic presentation of hypotheses intermingled with ex-
periments worked against acceptance of those hypotheses. Within the group of
specialists wedded to particular practices that Young transgressed, his experiments
were admired, his ideas ignored. Young, therefore failed, in this unique cultural
enterprise. Davy prevailed and established his research as knowledge in front ofthe
audience at the Royal Institution before his research colleagues accepted his exper-
52 Young, Lectures, vol. 1, 464, for the definition of superposition, 464-464, for his de-
scription of fringes.
53 See Golinski, Science as Public Culture, for science as theater.
Britain, 1790-1830 187
iments and results as valid. Despite some early disasters, Davy built a successful,
very public career in science. He drew into both chemistry and experimental phi-
losophy ideas alien to the prevalent Newtonian tradition. For natural philosophy,
these innovations lay in his focus upon the agents of change, or power, and his
concept of the unity of nature. 54 Davy's dramatic experiments in electrochemistry
gave currency to his emphasis on power and on his chemical theory of galvanic
electricity. While Davy's speciality was chemistry, his research had an impact on
ideas on the nature of matter and helped to reorient experimental philosophers to
the agents of physical change.
Such lectures and the production of research in this era pinpoints that education
and practice in experimental philosophy was very dependent upon the individual.
While an education in experimental philosophy could on the individual level be
systematic, there were no institutional forms for such training. 55 The product of
such self-education was a diversity of interpretation and variety of approaches
to the study of nature that has to be understood individual by individual. The
labels with which practicing natural philosophers covered such discrepant ideas
imply that the labels point to subtexts rather than the scientific questions at hand.
Newton's name still added respectability to ideas and his work was still a rich
source for hypotheses, most of which Newton would not have recognized as his
own. 56 Possibilities in terms of hypotheses multiplied partly because there were
no tight disciplinary matrices into which individual experimental philosophers had
to fit. Only a series of loose confederations existed, made up of others of similar
philosophical persuasion.
About the only consensus available was the common understanding that natural
philosophy was first and foremost experiment. Opposition sprang up whenever
the primacy of experiment was undermined as in the case of Young, Davy, and
Dalton. This trio of outsiders to the gentlemanly vocation of natural philosophy
made natural philosophy a means of earning a living and forced new ways of using
experiment onto their colleagues. They gave a new emphasis to hypotheses as a
necessary aspect of natural philosophy. We can hardly see these three men as con-
stituting any coherent group, other than in their social climbing and abilities to use
opportunities to secure careers in science. While we can argue that Dalton's work
lay in the broad tradition of Newtonianism, he did not share the all encompassing
mechanism of many other natural philosophers. Young's rejection of Newton's
54 See Golinski, Science as Public Culture, chap. 7, and "Humphry Davy and 'the Lever of
Experiment' ," in Experimental Inquiries, Legrand ed., 99-136.
55 As an illustration the example of Michael Faraday points to the possibilities of such
an education. That his training was largely through the Royal Institution lectures and
reading points to social disorganization.
56 The subtexts could be ideological or social-simply using a name to enhance a fledgling
career.
188 Physics and Mathematics
work was on entirely different grounds than those of Davy, but as radical within
experimental philosophy as Davy's rejection was in chemistry. One could hardly
connect Young to Davy's romantic vision of both the cognitive and the cultural
aims of the study of natureY The intellectual spread among these three men might
be broader than among their academic brethren but the differences, in their visions
of the cultural place and uses of the study of nature were a mirror of the breadth
of possibilities for that study in the early nineteenth century.
However, the wide ranging possibilities of such plurality enhanced by these
early entrepreneurs began to narrow during the 1820s, as the standards of French
experimental physics and of mathematics began to change the practices of British
natural philosophers. Economic changes were also forcing the study of nature
into new, narrower paths as attempts to regulate education in, and the practices of,
experimental philosophy and mathematics came to a head in the 1830s.58
57 Recent biographers see Davy's commitment to certain key ideals of romanticism one of
the few aspects of his life that gave it any coherence. David Knight, Humphry Davy:
Science and Power (Oxford: Blackwell, 1992). See also Trevor H. Levere, "Humphry
Davy, "The Sons of Genius" and the Idea of Glory," in Science and the "Sons ofGenius ":
Studies on Humphry Davy, Sophie Forgan, ed. (London: Science Reviews, 1980),
33-58, and Christopher Lawrence, "The Power and the Glory: Humphry Davy and
Romanticism," in Romanticism and the Sciences, Cunningham and Jardine, eds., 213-
227.
58 The nature of this study does not allow me to explore the ways in which the working
class constructed their own versions of experimental philosophy for their own purposes.
59 For example, see the correspondence between Colin MacLaurin and Clairaut in Green-
berg, The Shape of the Earth, 412-425.
Britain, 1790-1830 189
mathematicians. They neither knew the principles nor the methods that continental
mathematicians could take for granted in their readers. Playfair blamed the reward
system in science, the value placed on utility, and the hostility of the Royal Society
of London towards mathematics. The reference to the Royal Society was code for
Sir Joseph Banks its president. By 1800 his hostility to mathematics and mathe-
maticians was legendary. A near rebellion in the ranks of the Society's fellows in
the 1780s, many of whom were mathematicians, began with the removal of Charles
Hutton as foreign secretary of the society. The discontent was against what was
seen as Banks' arbitrary rule and hostility to mathematics and the mathematicians
within the society. Order and Banks were restored. 63
James Ivory and William Wallace, both employed as teachers at military col-
leges, made significant steps towards understanding continental calculus. In the
1790s Wallace abandoned fluxions for the French calculus and was anticipated by
Legendre in some methods in perturbation theory.64 In 1819 Wallace succeeded
Leslie as professor of mathematics at the University of Edinburgh.65 Together with
Charles Hutton and Peter Barlow, Wallace introduced the French calculus to British
mathematicians. In the first decade of the nineteenth century, other mathemati-
cians, such as Ivory, abandoned fluxions altogether. By 1814 Toplis had translated
Laplace and Ivory was following some points in the Mecanique Celeste on the at-
traction of ellipsoids using Euler's notation for partial differentials and developing
aspects of Legendre's mathematics. 66
Long before the agitation of the Cambridge "Analysts" there was a network
of mathematicians, many outside of Cambridge, working to bring attention to the
new continental calculus. 67 Analysis in the sense used by the Cambridge group was
broad and meant the use of algebraic rather than geometric methods to solve math-
ematical problems. It also included the use of algebraic definitions of functions,
284, and "Review of Laplace's System of the World," same journal, 15 (1808): 396-417.
Playfair was not alone in this lament. See, John Toplis, "On the Decline of Mathematical
Studies, and the Sciences dependent upon Them," Phil. Mag. 20 (1805): 25-31.
63 For an account of this disturbance, see Russell McCormmach, "Henry Cavendish and the
proper Method of Rectifying Abuses," in Beyond History of Science, Garber ed. 35-50,
37-38. See also David P. Miller, "Into the Valley of Darkness."
64 Wallace translated Legendre as well as Lagrange into English in the Mathematical
Repository .
65 For Wallace's role in the mathematical importations from the continent see M. Pantiki,
"William Wallace and the Introduction of Continental Calculus to Britain: A Letter to
George Peacock," Hist. Math. 14 (1987): 119-132.
66 See James Ivory, "On Attraction of Homogeneous Ellipsoids," Phil. Trans. R. Soc.
London, 99 (1809): 345-372, and same journal 102 (1812): 1-45.
67 For a reassessment of the role of the Cambridge group see Phillip Enros, "The Analytical
Society, 1812-1813: Precursor to the Renewal of Cambridge Mathematics," Hist. Math.
16 (1983): 24-47.
Britain, 1790-1830 191
and the acceptance of the new French calculus and their notation. The brash young
members of the Analytical Society emphasized the use of notation, but that was
not the heart of the matter. In ways parallel to Lavoisier's new chemical notation,
French mathematical notation carried with it different notions of differentiation,
integration, and the nature of functions, as well as their representation.
Much of the rhetoric of the Cambridge group was exaggerated and self-serving.
While Robert Woodhouse might have been the first to introduce French mathemat-
ical notation in his textbook, he was not as isolated as the Analysts pictured him.
Although he used differential notation in his textbook, Woodhouse's attempt made
little impression on his colleagues and students at Cambridge. 68 Woodhouse's fail-
ure was probably because French methods did not "pay" in terms of the Cambridge
examination system that was already driving the curriculum. 69 Similarly the under-
graduates of the Analytical Society were ineffective until 1817 when they began to
infiltrate the Senate House Examination system.7° By 1800 the lowest denomina-
tion of a "pass" degree was well established and might meet the criteria for a liberal
training of the mind. What constituted the upper limit, the best, was open ended.
This became even more crucial as honors in mathematics, doing well in the Senate
House examinations, could lead to a position at one of the Cambridge Colleges.
By 1800 the examination was so competitive that the curriculum was designed to
meet the expectations of the examiners.71 Teaching might be in the hands of the
Colleges, but the dons had to cover the topics that would arise in the examinations.
These subjects included Euclid, algebra, conic sections, trigonometry, fluxions,
fluents, Book I of Newton's Principia, the mathematics of astronomy, mechanics,
and hydrostatics. The transformation of mathematics in Britain accelerated with
the capture of the examination system at Cambridge.
Examination questions and textbooks reveal the meaning of mathematics at Cam-
72 An easily available example of this is in Ball, Origin, in the examination questions for
1801,30-33, and in Cambridge Problems: Being a Collection of the Printed Questions
proposed to the Candidates for the Degree ofBachelor ofArts at the General Examination,
1801-1810 (Cambridge: J. Deighton, 1810).
73 James Wood and Samuel Vince, The Principles of Mathematics and Natural Philosophy
(Cambridge: J. Burges, 1795-1799) 4 vols.
74 Woodhouse An Elementary Treatise on Astronomy (Cambridge: J. Smith, 1812) was
largely descriptive. The equation of time and method of computing eclipses derived from
the tradition of observational astronomers. Woodhouse explained that unlike geometry
astronomy did not spring from simple principles followed by logically deductible results.
Everything was connected and accuracy achieved through successive approximations.
Woodhouse, Treatise on Astronomy, Theoretical and Practical (Cambridge: J. Smith,
1821) separated the physical from the mathematical treatment of the subject. He included
the mathematical techniques of Laplace but no discussion of the physical implications of
all these manipulations.
Britain, 1790-1830 193
During the 1820s the general public began to lose its role in scientific institutions
and as actors in the production and dissemination of research that had been so
important for the first professional scientists in the early years of the nineteenth
century. This, paradoxically, was because of increasing opportunities to engage in
research, and to obtain a more systematic education in natural philosophy.
At the same time that the number of opportunities to engage in research opened
up, positions in the older university were filled with younger men committed to
research as well as teaching. Nationally, the number of men practicing within
certain subfields of experimental philosophy expanded. Increasingly, the numbers
engaged in closely allied research problems reached a critical mass that could sup-
port a society dedicated to the narrowly defined needs of this research community.
The membership, provincial in its intellectual interests, became geographically
national, inverting the intent and the geographical reach of the older, culturally
broad, geographically provincial societies. 76 These specialist societies launched
their own journals, and restricted membership in the societies to individuals ac-
tively engaged in the research the membership regarded as legitimate. The journals,
meetings, papers, discussions, and the management of the society focussed upon
the needs of this group. The tensions inherent within the earlier, philosophical
societies that sought cultural support by appealing to local educated groups were
77 See Rudwick, The Great Devonian Controversy, for an account of these new tensions,
complicated by the search for the correct social tone that would not undermine the
gentlemanly pretensions of its practitioners.
78 The social divisions that rent the Geological Society in a later decade are detailed in
Rudwick, The Great Devonian Controversy. The care with which the British Association
orchestrated its social and cultural place in British society throughout the 1830s is detailed
in Morrell and Thackray, Gentlemen of Science.
79 For the struggles of the geologists see, Rudwick, "The Foundation of the Geological Soci-
ety of London: Its Scheme for Cooperative Research and its Struggle for Independence,"
Brit. J. Hist. Sci. 1 (1962): 325-355.
80 For example, see the humiliations that Michael Faraday was prepared to endure to be-
come elected to the Royal Society in 1824, June Z. Fullmer and Melvyn C. Usselman,
"Faraday's Election to the Royal Society: A Reputation in Jeopardy," Bull. Hist. Chern.
11 (1991): 17-28.
81 For example see John Cawood, "The Magnetic Crusade: Science and Politics in Early
Victorian Britain," Isis, 70 (1979): 493-518.
Britain, 1790-1830 195
these activities as part of their specialist purview. 82 The Royal Society yielded to
reform slowly. In the 1820s Humphry Davy was able to institute improvements
in running the actual meetings of the Royal Society, the quality of the papers
delivered and published by the society, and began to run the library as a more
scholarly institution. He even opened up the social intermingling of fellows, that
is those elected for their social and political connections and the scientific fellows.
However, he managed to alienate politically powerful groups within the Society,
including those that had helped to elect him. He resigned the presidency in 1827.
His reforms did not address the systemic problems of the society, its relationships
with the government and its function as a social ladder for London physicians
that was so offensive to the growing group of scientific professionals. In 1828
any structural reform was precluded with the election of the Duke of Sussex as
president. Jostling for power within the society and hence for control over its future
continued until the 1840s. 83
By the time that the Royal Society was controlled by professional scientists its
function as the major site for the display of science and its intellectual develop-
ment had been usurped by the British Association for the Advancement of Science.
However, membership in the Royal Society never lost its place as the crowning
social achievement in a career. The overlapping membership of the British Associ-
ation and the Royal Society helped the former to consolidate its place in Victorian
society and helped to divert government largesse to the the Association and the
Society.84
Specialist societies, dedicated to the needs of the research community, also
locked out the general public from participation in the process of the creation,
display, and legitimation of research that were the structural support of the careers
of Dalton, Davy, and Young. The audience that was so important for science to
acquire the correct cultural tone in the early nineteenth century were increasingly
only allowed into the conversaziones or the Friday evening general lectures of the
Royal Institution and other scientific societies. Social and intellectual mediators
became necessary between those engaged in the sciences and the general public.
82 See William J. Ashworth, "The Calculating Eye: Baily, Herschel, Babbage and the
Business of Astronomy," Brit. 1. Hist. Sci. 27 (1994): 409-441.
83 For a narrative ofthese events, see Marie Boas Hall,All Scientists Now. For a sociological
analysis, see Roy Macleod, "Whigs and Savants: Reflections on the Reform Movement
in the Royal Society, 1830-1848," in Metropolis and Province, Inkster and Morrell, eds.
55-90. On Davy's tenure as president, see David P. Miller, "Between Hostile Camps:
Sir Humphry Davy's Presidency of the Royal Society, 1820-1827," Brit. 1. Hist. Sci. 16
(1983): 1-47.
84 For the patronage of the sciences in this era, see, Cawood, "The Magnetic Crusade," and
J. B. Morrell, "The Patronage of Mid-Victorian Science in the University of Edinburgh,"
in The Patronage of Science in the Nineteenth Century, G. L' E. Turner, ed. (Leiden:
Noordhoff, 1976),53-93.
196 Physics and Mathematics
Not insignificantly many of those intermediaries between the world of science and
of general culture were women. They were still essential for the enterprise, even
when pushed to its margins. 85
During the same decade criticism, educational, social, and political, began to
be directed against Oxford and Cambridge. Part of this critical barrage claimed
that their curricula were antiquated. It was no longer enough to train the mind
the man must be trained in a particular branch of the arts or the sciences. The
professional was defined. Simultaneously, this pressure to reform was felt acutely
by the Scottish Universities. Their curriculum, management, and parochial focus
was under the scrutiny of a central government ready to economize at any oppor-
tunity. While the reforms reached into parts of the fiscal management of these
universities, neither the curriculum nor the distinct function that the universities
saw that they filled within Scottish society seemed to have been affected. 86 The
opening of University College, London introduced other educational alternatives
that claimed to offer a more modern curriculum. In large cities philosophical
societies also began to offer such courses of study in the 1820s. These courses
tended to be systematically delivered on narrowly defined subjects and replaced the
broadly conceived surveys in natural philosophy. The rhetorical appeals to moral
and intellectual improvement no longer appealed to an audience bent on specific
knowledge for economic purposes rather than for entertainment or cultural and
social uplift. They were also taught by men recognized for their more narrowly
focussed technical abilities and accomplishments. 87 Such courses attracted smaller
numbers of students, that is men bent on using the course work in their economic,
not their cultural, lives. Appeals to aesthetic ideals and cultural uplift were largely
replaced by strict utility as justifications for studying nature. The latter was usually
emphasized by mechanical philosophy that pointed in general to the connections
of mechanics to mechanisms. Truth was replaced by narrow economic aims. 88
85 The preeminent examples are Jane Marcet in chemistry and Mary Somerville in astron-
omy. See Somerville, Mechanism ofthe Heavens (London: J. Murray, 1831), a translation
and commentary on Laplace's Mecanique Celeste, and Preliminary Dissertation on the
Mechanism of the Heavens (London: 1832). See Elizabeth Patterson, Mary Somerville
and the Cultivation of the Sciences, 1815-1840 (The Hague: Nijhoff, 1983). Faraday
became his own mediator, establishing Friday evening lectures at the Royal Institution.
86 See J. B. MorreIl, "Science and Scottish University Reform: Edinburgh in 1826," Brit.
J. Hist. Sci. 6 (1972): 39-56.
87 This can be seen in the courses offered by Michael Faraday at the Royal Institution in
the 1820s. For changes in lecturing content and styles in London during the 1820s see,
1. N. Hays, "The London Lecturing Empire," in Metropolis and Province, Inkster and
MorreIl, eds. 91-119.
88 See George A. Foote, "Mechanism, Materialism, and Science in England, 1800-1850,"
Ann. Sci. 8 (1953): 152-161, and "Science and Its Function," Osiris, 11 (1954): 438-
454.
Britain, 1790-1830 197
None of these criticisms, agitations and changes in the ways in which natural
philosophy was taught appeared to change the intellectual boundaries of exper-
imental philosophy as a research discipline. With the new phenomena of light
and electromagnetism, British experimentalists became more sensitive to French
research. Yet experimentalists in Britain took note only of French experimental
results, not mathematical physics. David Brewster was actively involved in the
early research into polarization and fought a losing battle against the hypothesis of
undulations. As such, he reacted to the experimental work of Fresnel and even of
Thomas Young but not the mathematics of Cauchy or any other mathematician.
In Brewster and Faraday we can see the interaction of British experimental
philosophers with French experimental physics. Neither succumbed to precise
quantitative experiments or the seduction of French ideas about light in Brewster's
case, nor electricity, magnetism and their interaction in that of Faraday. Faraday's
experiments in electromagnetism were precise and developed in a sequential series
that allowed him to explore the specific consequences of his own ideas on electricity
and magnetism in great detail. The clarity of the expression of his ideas defies
the notion that mathematics was a necessary development for the intelligibility
of theories in physics. Later Faraday regretted his ignorance of mathematics,
yet in the I820s he did not feel compelled to address Ampere's mathematical
electromagnetism. He confined himself to answering the physical ideas expressed
in Ampere's experimental papers.89
In the I820s the mathematical side of the disciplinary divide was kept intact in
George Green's work on electricity and magnetism. Scientific life in Nottingham
in the early nineteenth century centered upon the Bromley House subscription li-
brary. There Green had access to Laplace in Toplis' translation, Charles Hutton's
Course in Mathematics and other resources in the new French mathematics. He
also had the encouragement of Sir Edward Ffrench Bromhead, one of the original
founders of the Analytical Society at Cambridge. 9o While Green's work in elec-
tricity and magnetism was classified as "mixed" mathematics, it was mathematics.
Green explored the mathematical properties of the "potential function" and the
a= Jp d; - JXdX+YdY+ZdZ.
In this case p was the density of electricity on the surface of a conducting sphere of
radius r, and da was an element on the surface. The components of the electrical
force from an external charge at the surface were X, Y, Z. The first integral
ranged over the whole surface, and the expression X dx + Y d y + Z dz was an exact
differential. Poisson had performed the integrations in particular cases and the
f
solutions "must be looked upon as an effect of chance rather than of any regular
and scientific character.,,93 To attack this problem Green wrote V = pd a / r
where V satisfied,
a 2v a 2v a 2v
0=-+-+-.
ax2 ay2 az 2
Green systematized the mathematics implicit in Poisson's work and rederived
some of the latter's results far more directly, noting at the same time that many
were also available in Laplace. He developed all the mathematical examples for
which he could reach explicit solutions. He also noted in the case of the Leiden jar
that the sum total of electricity on all surfaces was zero. He remarked that such a
result, while surprising, "would not be difficult to verify" by experiment. Clearly
he was concerned with the mathematical, not the experimental literature. Green
then proceeded to reach many of Poisson's results in magnetism using the same
elegant mathematical approach. 94
In the 1820s the boundaries between mathematics and physics remained in-
tact. While mathematical methods were transformed by the importation of French
mathematics they only reinforced traditional boundary lines between experimental
philosophy and mathematics. The starting point for mathematics remained the
same. Yet some of the critical changes in the development of theoretical physics in
Britain were effected by the graduates and faculty at the university of Cambridge
to which we must, from now on, give increasing amounts of attention.
Cambridge was changing, being forced in new directions by its teachers, its
students, and the world beyond the Fens. These changes were symbolized by the
establishment of the Cambridge Philosophical Society in 1819, and the improved
pulse of research at Cambridge and the kind of research done by its faculty. On
the student's side was the Apostle's Club in the 1820s. Affecting both students
and faculty was the formalization of Cambridge examination practices with the
establishment of the Mathematical Tripos in 1824.95
Even as their educational philosophy denied this as their goal, the establishment
of the Tripos guaranteed that Cambridge would graduate professional mathemati-
cians. With the Tripos, mathematical proficiency defined intellectual excellence.
Since honors in the Tripos led to college and university appointments, college
tutors, fellows, and university professors became professional mathematicians.
With the introduction of analysis into the university examination system, research
in mathematics developed in several different directions. 96
The selection system for faculty and teaching, now driven by the formalized
system for honors in mathematics, negated the official educational philosophy of
Cambridge. By 1830, despite Whewell's efforts, the Tripos no longer served as a
means for a liberal education but as the technical training ground for mathemati-
cians. If the Analytical Society was a symptom of this change, the Apostle's Club
was its confirmation. The initial purpose of the Apostles Club was to give to a se-
lect few undergraduates interested in literature much that was intellectually absent
97 See Peter Allen, The Cambridge Apostles: The Early Years (Cambridge: Cambridge
University Press, 1978).
98 See A Don [Leslie Stephen] Sketches from Cambridge (London: Macmillan, 1865),
32-47. Frances M. Brookfield, The Cambridge Apostles (London: Pittman and Sons,
1906), chap. 1. Arthur Gray, Cambridge University, an Empirical History (New York:
Houghton Mifflin, 1927), p. 276, on Tennyson's view of his tutor, William Whewell,
"Billy Whistle." Thackray got revenge in The Book of Snobs where Crump, Master of St.
Boniface was based on William Whewell, as was Dr. Sargent in Lowe the Widower. John
Clive, Macaulay: The Shaping of the Historian (New York: Knopf, 1973), p. 21 recounts
the disappointment of both Macaulay and his family in the 1830s at his Cambridge record.
Macaulay saw himself as an academic failure because he was not a mathematician. See
also the agonies of Darwin trying to understand mathematics and struggles to achieve
a pass degree. See Desmond and Moore Darwin, and Browne Darwin. On students
see Sheldon Rothblatt, "The Student Sub-Culture and the Examination System in early
Nineteenth-Century Oxbridge," in The University in Society, Stone, ed. (Princeton NJ:
Princeton University Press, 1973) 2 vols., vol. 1,247-303.
99 For the establishment and the initial aims of the Cambridge Philosophical Society, see A.
Rupert Hall, The Cambridge Philosophical Society: A History, 1819-1969 (Cambridge:
Philosophical Society, 1969).
100 The same process occurred at Oxford in the same period. See Sheldon Rothblatt, The
Revolution of the Dons (New York: Basic Books, 1968) and Arthur Engel, "Emerg-
ing Concepts of the Academic Profession at Oxford, 1800-1854," in The University in
Society, Stone, ed. vol. 1,305-351.
101 See Crosbie Smith, "Geologists and Mathematicians: The Rise of Physical Geology," in
Wranglers and Physicists, Harman ed., 49-83, 52.
102 William Whewell, "Report on Recent Progress in Mineralogy," Rep. British Assoc.
Britain, 1790-1830 201
Students who did attend the reactivated lectures by university professors were
treated to a different form of knowledge than earlier generations. The lectures were
more narrowly defined and closely linked to the research interest of the faculty. 103
During this decade the generation that began the transformation of the Cambridge
examination system scattered geographically and intellectually. As their careers
developed, their attitudes to mathematics and its function in the other sciences and
in the Cambridge curriculum spread across a broad spectrum of opinion. These
changes were driven by their experiences beyond Cambridge, even for those like
Herschel whose earliest publications were in the mathematics so recently imported
from France. 104
John Herschel lost direction after he left Cambridge until he entered the family
trade of observational astronomy. By 1820 he already had learned the craft of
grinding mirrors and the techniques of astronomical observation. Simultaneously
he reported to the Cambridge Philosophical Society on his own experiments on
double refraction and polarization. 105 These were not his first forays into experi-
mentation. He was already exploring chemistry as well as the optical properties of
various substances. Herschel did not invite mathematics into experimental philos-
ophy, yet he appreciated the new concern with quantification and error in experi-
ment. He also began to establish, then develop an epistemology along with ideas
concerning the place of mathematics within experimental philosophy. The disci-
plines of physics and mathematics were distinct and experiment the surer method
of exploring nature. While hypotheses were a legitimate aspect of experimental
(1831-32): 322-365, is a convenient place to see how Whewell separates the hypo-
thetical and physical from the geometrical and mathematical.
103 This is best seen in the geology lectures by Adam Sedgewick, as well as those of E. D.
Clark, followed by the lectures of Henslow and Whewell.
104 John Herschel and Charles Babbage published on functional equations and operational
methods of the calculus. Herschel's first papers appeared in Memoirs of the Analytical
Society (1813) and continued into the 1820s in the Philosophical Transactions ofthe Royal
Society of London and the Transactions of the Cambridge Philosophical Society. They
continued to appear even as his research began to focus on chemistry and observational
astronomy. See Gunther Buttman, The Shadow of the Telescope: A Biography of John
Herschel, Bernard Pagel, trans. (New York: Scribner's Sons, 1970), and Aspects of the
Life and Thought of Sir J. F. Herschel, S. S. Schweber ed. (New York: Arno Press,
1981), 2 vols. Babbage's first paper on functional analysis appeared in 1815. See
J. M. Dubbey, The Mathematical Work of Charles Babbage (New York: Cambridge
University Press, 1978). His assessment of Babbage's achievement must be modified in
light of contemporary French work in the same field. See Grattan-Guinness, "Babbage's
Mathematics in its Time," Brit. J. Hist. Sci. 12 (1979): 82.
105 See, Buttman Shadow. His work on double refraction was published as Herschel, "Double
Refraction as a Deviation from Newton's Scale," Trans. Cambridge Phil. Soc. 1 (1822):
21-42, and that on polarization as, Herschel, "Polarization," Trans. Cambridge Phil. Soc.
(1823): 1-52.
202 Physics and Mathematics
106 See Herschel, Admiralty Handbook of Scientific Enquiry (London: Dawson reprint,
1974). His ideas also appeared in his various encyclopedia articles that are examined
below.
107 Systematic discussion of Herschel's ideas on the place of hypotheses in "science" appear
in Richard Yeo, "Reviewing Herschel's Discourse," Studies Hist. Phil. Sci. 20 (1989):
541-542, "Reading Encyclopaedias," and in Defining Science: William Whewell, Natu-
ral Knowledge, and Public Debate in early Victorian Britain (Cambridge: Cambridge
University Press, 1993), 92-99.
108 See Herschel, Treatise on Physical Astronomy, Light, and Sound (London: Richard
Griffin and Co., n. d.). These were reprints, with the original pagination, of articles
from the Encyclopedia Metropolitana of, "Physical Astronomy," vol. 3, 647-729, 647;
"Light," vol. 4, 314-586; "Sound," vol. 4, 747-824, 747.
Britain, 1790-1830 203
In his own research he had used Bessel's methods and could extract just the
right amount of mathematics to make visible the physical reasons for its use. Yet
in his general discussion of the relationship between the experimental and the
mathematical sciences nothing had changed. "Pure" mathematics was a higher
form of attainment than "mixed" mathematics. The former depended only on
the "intuitive perception of abstract truth" and hence led to absolutely correct
conclusions. Mixed mathematics, although a lesser art, was definitely mathematics,
as he demonstrated in his section on mechanics. Clairaut was the first to derive
"the correct and general laws which regulate the equilibrium of a fluid mass acted
on by any force, and to point out their connexion with the equations of condition
which render a function an exact differential.,,109 He began in physics and ended in
mathematics. The mathematics of mechanics culminated in the works of Laplace
and Lagrange. Lagrange's MecaniqueAnalytique "is a compendium of the general
formulae and analytical artifices necessary for the treatment of every problem which
can be proposed in the equation of motion of matter." Mathematics, not physics. lID
As Herschel's research turned to observational astronomy, mathematics became
a tool for experimentalists and observers. Anything more was superfluous and for
the most part irrelevant, an opinion shared and expressed more vehemently and less
eloquently by George Biddell Airy. Airy had not taken part in the initial debates
over the new analytical methods. He was one of the first students to go through
the examination system at Cambridge after these methods were in place. lll Very
quickly Airy developed his own ideas about mathematics that particularly suited his
own talents and from which he developed a singular career. In his autobiography
Airy claimed to dislike "mere theoretical problems." This was written after a
lifetime of work making mathematics practical at Greenwich Observatory "at any
cost oflabor." He viewed skeptically any mathematical result that were not based in
the physical entities of space, time, and matter. While expressing what seems like
a traditional understanding of the relationship between mathematics and physics
Airy narrowed this view. He could not see any value in mathematics not used in
solving the problems posed by nature. Airy's earliest texts and papers bear out
these values, made more explicit and general later. His Mathematical Tracts were
purely utilitarian treatments of those parts of Newton's Principia required for the
Cambridge examinations, translating physical problems into analytical form and
109 On the term "mixed mathematics" and its changing meanings, see Gary I. Brown, "The
Evolution of the Term "Mixed Mathematics"," J. Hist. Ideas (1991): 81-102. Brown
points out that during the nineteenth century the term was dropped and replaced by
applied mathematics but does not discuss any reasons for this change.
110 Herschel, "Mathematics," Edinburgh Encyclopedia, David Brewster, ed. (Edinburgh: W.
Blackwood, 1830) vol. 13, pt. 1,359-383. On Clairaut see p. 381. See also p. 382 where
Herschel noted that Clairaut was a mathematician not an astronomer. For Lagrange see
p.383.
111 He was Senior Wrangler in 1823.
204 Physics and Mathematics
developing only that much mathematics necessary to solve those problems. He also
took the student through the problems as if they were observational astronomers. 112
Airy's mathematical tract on the undulatory theory of light was not so obviously
a training manual. Airy differentiated between the "geometrical" part of the theory
that depended only on assuming that light was a transverse wave that traveled at
different velocities in different media and the "mechanical" part of the theory
that depended upon hypotheses, "far from certain" on the internal behavior of
the ether. In developing mathematical methods, he used only those that were
necessary to solve specific problems. The general solution to the wave equation
was discarded for various particular ones that were related to actually observed
phenomena. Physical interpretations of mathematical results were explicit and
unnecessary hypotheses were avoided. Mathematics was focussed here on the
needs of physics.
Airy left Cambridge for Greenwich in 1835 and was replaced by James Challis
as Plumian Professor. Challis therefore took over the training of students as astro-
nomicalobservers. He interpreted his responsibilities as having to teach astronomy
as natural philosophy. His astronomy lectures dealt with the instrumentation and
observational techniques of astronomy; diagrams and drawings were used as illus-
trations. After the construction of the university observatory, Challis drew students
into the observation process and the reduction of data and developed a fine teaching
and research facility, a result that surprised the French. l13 Similarly, in his demon-
stration lectures on natural philosophy, mathematics only entered as the expression
of the laws extracted from his experiments. However, in his research Challis de-
veloped his own mathematical theories of physical phenomena on a grand scale,
closer to French mathematical physics than theoretical physics.11 4
George Peacock, one of the original members of the Analytical Society, was
at the other end of the mathematical spectrum. Peacock stayed at Cambridge
long enough to become embroiled in the debates over the content of the Cam-
bridge curriculum. He defended algebra that was as mathematics legitimate as
112 George Biddell Airy, Mathematical Tracts on Physical Astronomy, The Figure of the
Earth, and the Calculus of Variations designed for the Use of Students in the University
(Cambridge: Deighton, 1826).
113 See Harvey Becher, "Voluntary Science in Nineteenth-Century Cambridge to 1850," Brit.
1. Hist. Sci. 19 (1986): 57-87,68-70.
114 James Challis,Notes on the Principles of Pure and Applied Calculation: and the Appli-
cation ofMathematical Principles to Theories ofPhysical Force (Cambridge: Deighton,
Bell and Co., 1869). Challis reduced gravity, heat and other forces of nature to mechanical
pressure in a fluid ether. While analytically defensible, James Clerk Maxwell described
Challis' work as self-consistent mathematically, but physically indefensible. Challis'
fluids could not behave as ordinary fluids. See Maxwell,"Challis' 'Mathematical Prin-
ciples of Physics' ," Nature, 8 (1873): 279-280, reprinted in Maxwell on Molecules and
Gases, Garber, Brush and Everitt, eds. (Cambridge MA.: MIT Press, 1986), 126-132.
Britain, 1790-1830 205
geometry for training the mind. Geometry had this primary function in the Cam-
bridge liberal education philosophy and teaching at the university. Continuing the
eighteenth-century tradition geometry was a mathematics rooted in the physical
world. Philosophically geometry was well grounded and based on axioms and
definitions and connected through theorems to known results. It could also be
taught that way as well and justified as training for the mind. Algebra could claim
no such philosophical or pedagogical structure. This criticism threatened the very
methods ofthe analysis so recently introduced into Cambridge. Peacock countered
these arguments by putting algebra on the same foundations and creating for it a
logical structure. He thus met head-on Whewell's growing opposition to analysis
in the curriculum. lI5 This exploration of the foundations of algebra did not lead
Peacock to Cauchy's calculus but to the development of symbolic algebra-algebra
as logic. For Peacock, arithmetic was a restricted form of algebra. Algebra was
a generalized form of arithmetic but non-commutative algebra, for example, was
impossible.lI6
During the 1830s and 1840s Whewell appeared to win the battle over the cur-
riculum. However, the examinations system worked against his proclaimed goal
of using mathematics as the foundation of a liberal education. The competition for
place in the Tripos ensured that the examination, rather than the curriculum, mat-
tered among ambitious students. This in turn guaranteed that Cambridge trained
the narrowly educated, yet technically accomplished, mathematician Whewell de-
115 Whewell's opinions on analysis and its place in the curriculum changed over time and in
the context in which he was placed and the subject discussed. As his career at Cambridge
developed his defense of geometry against the inroads of French analysis grew. For the
arguments over the Cambridge curriculum and the meaning of a liberal education see,
Martha McMackin Garland, Cambridge before Darwin: The Ideal ofaLiberal Education,
1800-1860 (Cambridge: Cambridge University Press, 1980). M. V. Wilkes has explored
the roles of Peacock, and Herschel in shaping the Cambridge curriculum in the middle
of the nineteenth century in Wilkes, "Herschel, Peacock, Babbage and the Development
of the Cambridge Curriculum," Notes Rec. Roy. Soc. London 44 (1990): 205-219, that
of Whewell in Harvey Becher, "William Whewell and Cambridge Mathematics," Hist.
Stud. Phys. Sci. 11 (1980): 1-48.
116 Peacock, Treatise on Algebra (Cambridge: Deighton, 1830). For a survey of the his-
toriography and the development of symbolic algebra, see Menachem Fisch, "'The
Emergency which has arrived': the Problematic History of Nineteenth-Century British
Algebra-a Programmatic Outline," Brit. J. Rist. Sci. 27 (1994): 247-276. For the de-
velopment of British explorations of "foundational" issues in mathematics see, Elaine
Koppelman, "The Calculus of Operations and Rise of Abstract Algebra," Arch. Hist. Ex-
act Sci. 8 (1971): 155-242, Joan Richards, "The Art and Science of British Algebra: A
Study in the Perception of Mathematical Truth," Rist. Math. 7 (1980): 343-365, Helen
Pycior, "George Peacock and the British Origins of Symbolical Algebra," same journal
8 (1981): 23-45, "Early Criticism of the Symbolic Approach to Algebra," same journal
9 (1982): 392-412, Joan Richards, "Rigor and Clarity: Foundations of Mathematics in
France and England, 1800-1840," Sci. Context 4 (1991): 297-319.
206 Physics and Mathematics
plored. This was the French model of the mathematician he began to criticize in the
1820s. While the Analytical Society began at the tail end of the process of introduc-
ing the new French mathematics into Britain, the Analysts were largely successful
in introducing it into the Cambridge examination system. Initially, Cambridge
mathematicians took the forms of the calculus of Lagrange, Laplace, and Poisson,
while ignoring the debates that had divided these mathematicians earlier in the
nineteenth century. They also largely ignored or criticized the physical models
that informed the mathematics of Laplace and Poisson. The Cambridge Tripos
carried on the tradition of French mathematical physics, using the mathematics en-
capsulated within Lacroix's texts and now largely abandoned by the French. Thus
it was unnoticed by the French. Their French models also gave them the physical
subjects upon which the Cambridge mathematicians built. The mathematics they
developed initially was generated by mechanics and celestial mechanics, which
formed the basis of examination questions and research problems.
After 1830, later forms of the calculus developed by Cauchy and Fourier became
a source of problems and of methods for English mathematicians none of whom
explored the incompatibilities between their earlier sources and these later French
mathematicians. In the late 1830s, the sources for their mathematics had spread
into light, electricity, and magnetism.
What this education and research did not guarantee was a turn towards natural
philosophy or theoretical physics. At Cambridge as in France, physical problems
continued to be the inspiration for forays into mathematics. Theoretical physics
did not appear at Cambridge until after 1850. In the 1820s we already have the
development of extremes with respect to mathematics, with Herschel and Airy,
proclaiming it as a tool, and Peacock defending what became a uniquely British
form of pure mathematics. We also have both Airy and Herschel trying to connect
mathematics and its manipulation to physical understanding of their observational
results. However, both were suspicious of the use of hypotheses within what they
regarded as the purely empirical and inductive sciences.
Yet out of this mix of battles over the curriculum and uncertainty over the place
of hypotheses in the experimental sciences, theoretical physics emerged in some
of its incarnations during the late 1840s and into the 1850s in the work of George
Gabriel Stokes, William Thomson and James Clerk Maxwell.
Part III
Transformations, 1830-1870
Mathematics constitutes the language through which alone
we can adequately express the great facts of the natural world. *
• Ada, Lady Lovelace, "Sketch of the Analytical Engine invented by Charles Babbage," in
Taylor's Scientific Memoirs, (1843): 696.
Chapter VII
1 This was made explicit by Thomas Kuhn, Revolutions, and L. Pearce Williams, "The
Physical Sciences in the First Half of the Nineteenth Century: Problems and Sources,"
Hist. Sci. 1 (1962): 1-15. For Williams the history of nineteenth-century physics after
the discovery of the conservation of energy was a footnote to what had gone before.
2 Williams," The Historiography of Victorian Science," Viet. Stud. 9 (1966): 1977-204.
3 The clearest documented example of this thus far is Darwin's network of colleagues and
acquaintances from which he gleaned support, information, and opinions. On a much
smaller scale this can be seen in the mutual support system of Maxwell, Thomson, and
Tait.
4 The historiography, difficulties, and advantages of biographies in science are discussed in
Telling Lives in Science: Essays on Scientific Biography, Michael Shortland and Richard
Yeo, eds. (Cambridge: Cambridge University Press, 1996).
Keywords
During the early decades of this period, a number of keywords were introduced
into the general discourse of science or changed their meanings significantly. Doc-
trine became theory. Theory connoted a logically developed sequence of conse-
quences from a set of defined ideas. Metaphor and analogy alone no longer served
to connect experimental results to hypotheses about nature. Hypothesis replaced
Britain 1830-1870 211
5 For his remarks on science, scientist, and physicist, see William Whewell, "Review of
Mary Somerville, Connexion of the Sciences," Quarterly Rev. 51 (1834): 58-68,59-60.
See also S. Ross, "Scientist: The Story of a Word," Ann. Sci. 18 (1962): 65-85.
6 See Richard Yeo, Defining Science. Of his contemporaries John Herschel admired his
philosophical work yet disagreed with him, as did Airy and Brewster.
7 Geoffrey Cantor has investigated the juxtaposition of natural philosopher versus scientist
on more philosophical grounds in Cantor, "The Reception of the Wave Theory of Light
in Britain: A Case Study illustrating the Role of Methodology in Scientific Debate," Hist.
Stud. Phys. Sci. 6 (1975): 109-132. However, Cantor assumes that all papers on wave
212 From Natural Philosophy to Physics
The British Association for the Advancement of Science gave scientists a na-
tional institutional unity that was previously lacking. Some historians have seen
the rich diversity of pre-Association scientific institutions as dividing the scientific
community into intellectual, methodological, religious, and ideologically warring
factions. Others have tied the lack of a unified social structure for science to pre-
theory lie within physics without looking at the role of physical imagery in some papers
where it is marginal at best.
8 British observational astronomers then physicists went beyond the quantitative French
experimentalists in their investigations of the limitations of these new methods. For the de-
velopment of these methods, see Zeno G. Swijtink, "The Objectification of Observation:
Measurement and Statistical Method in the Nineteenth Century," in The Probabilistic
Revolution, Lorenz KrUger, Lorraine J. Daston and Michael Heidelberger, eds. (Cam-
bridge MA.: MIT Press, 1990),2 vols., vol. 1,261-285, Simon Schaffer, "Astronomers
Mark Time: Discipline and the Personal Equation," Sci. Context 2 (1988): 115-145. See
also John Cawood, "The Magnetic Crusade." Cawood narrates the advent of accurate
geomagnetic measurement as well as a major lobbying effort to gain funds from the
government for this worldwide, long-term effort.
9 This was not the only function of the Section. See Lawrence Goldman, "The Origins of
British 'Social Science': Political Economy, Natural Science, and Statistics, 1830-1835,"
Hist. J. 26 (1983): 587-616.
Britain 1830-1870 213
10 The differences are emphasized in Parliament a/Science, Roy Macleod and Peter Collins,
eds.(Norwood: Science Reviews, 1981) and glossed over in Morrell and Thackray, Gen-
tlemen a/Science. For Individualism, see J. B. Morrell, "Individualism and the Structure
of British Science in 1830," Hist. Stud. Phys. Sci. 3 (1971): 183-204.
11 For a recent discussion on the historiography of the Reform Act, see John A. Phillips and
Charles Wetherell, "The Great Reform Act and the Political Modernization of England,"
Amer. Hist. Rev. 100 (1995): 411-436. The authors argue that despite recent scholarship
1832 marked a watershed in the political life of England.
214 From Natural Philosophy to Physics
suspicions were exacerbated by the battles within the Royal Society and the Decline
of Science debates. Political damage to the founding of a national organization
became very possible unless handled carefully because of the Royal Society's
connections with the government and the criticisms of those connections. As well
as fostering external skepticism and suspicion, the debates forced divisions within
the ranks of the scientists making the practical problem of drawing them together
that much more difficult. 12
William Whewell wanted to retain the monopoly of Cambridge University over
mathematics, the science regarded as a key element of the change from natural
philosophy to science. Any changes in the status quo, including governmental
funding, threatened this primacy.13 He quickly moved to help frame the British
Association when convinced that the movement would be out of his grasp if he
did not join. John Herschel was not against government funding, although he
considered himself above the political fray because of his wealth. This public
persona gave him immense credibility as a spokesman for science. While Charles
Babbage, Herschel, and Whewell could join David Brewster in arguing for the
utility of science, that utility was tied to the "higher" aims of national government.
Herschel and Whewell did not see utility in terms of commerce, industry or the
lives of ordinary citizens, closer to Brewster's concerns.
Personal disputes and ambitions could become other reasons to wreck the fledg-
ling organization unless kept within reason. If the British Association was to
act as a national organization, scientists with reputations to match had to be at
its center. If government was to be convinced that science was important to the
nation, scientists with access to the government were necessary to lead that same
organization. This new national institution had to be suitably framed and tamed to
pose no social, cultural, or ideological threat in such a tense political atmosphere. 14
Internally the purposes of the Association meshed with those of emerging pro-
fessionals. Reports and papers delivered in the sections were geared to the research
community of the various disciplines. Cultivators of science were welcomed, their
numbers helped to demonstrate and add to the claims for government attention.
They had no say in the shaping of the institution. The scientific business of the
Association was in the hands of specialists who arranged the meetings, chose the
programs, and vetted submitted papers. They also chose who addressed the gen-
eral public in the open sessions, and hence represented science to the laity. In the
1830s the clerisy could define and control this image and even the doing of sci-
ence within the Association. However, there were already signs that such control
12 These conflicts and the solutions to them are detailed in Morrell and Thackray, Gentlemen
of Science, chap. 2.
13 However, Whewell was not above accepting government monies later in the 1830s.
14 The British Association, its social and cultural framing and the maintenance of this stance
are detailed in Morrell and Thackray, Gentlemen of Science, chap. 3.
Britain 1830-1870 215
was impossible beyond the annual meetings and even within some sections of the
Association as well.
The very size and public display of the annual meeting did not meet with univer-
sal acceptance. The theatrical aspects of the meetings, the sightseeing expeditions,
the social spectacle all guaranteed that the message that the study of nature was
being done in new ways was not lost on the public. 15 Yet this new enterprise still
functioned within the confines of particular cultural norms. Science remained a
gentlemanly pursuit while proclaiming that it was culturally, economically, intel-
lectually, and morally necessary to the nation. In cutting its annual swath through
British society, members of the British Association separated themselves visibly
from the values expressed in the Royal Society with the latter's ties to outmoded
political values and forms. The Association managed to do this without visible
signs of political entanglements.
Within a decade the Association was indispensable for the practitioners of the
sciences. The Association met many needs of the growing professions within
science whether or not the rank and file of the sciences shared the political ideology
and cultural pretensions of its leaders. Careers were launched with papers at the
sections. New work presented in its meetings was important for physicists and
mathematicians who formed no national, disciplinary societies of their own. It
united individuals through the research they shared at the annual meetings. If this
was not enough, there was a steady stream of commissioned review articles on
various branches of the sciences. In the 1830s the subject matter, content, and
viewpoints expressed within these reports helped to define what science was.
During the 1830s the papers of Section A helped to define the practices of math-
ematicians and physicists. The reports in physics and mathematics were narrowly
focussed on the tides, electricity and magnetism, conical refraction, algebra, etc.
Their authors were all centrally placed in the research community of those subjects.
They did not necessarily alter every aspect of those disciplines. The influence of
the French was heavy in terms of the problems chosen as subjects of reports. The
French also provided the results and methodologies in attacking research prob-
lems whose solutions were the subject of annual progress reports. Exploitation of
French practices only reinforced the traditional geography and boundaries between
the experimental science of physics and mathematics. In detail the reports often
reflect the practices and prejudices of the individual making the reports rather than
present a coherent sense of either experimental physics or mathematics. They also
served other purposes. Both Whewell's and Peacock's reports were intellectual
salvos in the ongoing battle over the Cambridge curriculum.
15 See A. D. Orange, "Idols of the theater: The British Association and its Early Critics,"
Ann. Sci. 32 (1975): 277-294. For the growing unease of some religious leaders over
the developing cultural power of science, see Frank M. Turner, "The Victorian Conflict
between Science and Religion: A Professional Dimension," Isis 69 (1978): 356-376.
216 From Natural Philosophy to Physics
16 David Brewster, "Report on Recent Progress in Optics," Rep. British Assoc. (1832):
308-322; Baden Powell, "Report on Our Present Knowledge of the Science of Radiant
Heat," Rep. BritishAssoc. (1833): 259-301; S. Hunter Christie, "State of Our Knowledge
Respecting the Magnetism of the Earth," Rep. British Assoc. (1834): 105-130. See also
James David Forbes, "Report on recent Progress and present State of Meteorology," Rep.
British Assoc. 1 (1831): 196-258.
17 George Peacock, "Report on the Recent Progress and Present State of Certain Branches of
Analysis," Rep. British Assoc. (1833): 185-352, his discussion of Fourier, 248-259. For
Peacock's mathematics, see Koppelman, "The Calculus of Operation," Arch. Hist. Exact
Sci. 8 (1972): 155-242; Richards, "The Art and Science of British Algebra,"; Helene
Britain 1830-1870 217
The critical issue of the relationship between mathematics and observation arose
in the reports on experimental physics, which also had been the subject of sustained
mathematical development. Humphry Lloyd gave voice to a new relation between
experiment and mathematics. Vague physical speculations were no longer enough
when mathematicians compared experimental results to those of their mathemat-
ics. However attractive an hypothesis, "it is only when it admits of mathematical
expression, and when its mathematical consequences can be numerically compared
with established facts, that its truth can be fully and finally ascertained." This was
important for Lloyd to state because the point of his report was not to investigate
"mathematical optics" in detail but to establish the validity of the hypothesis that
light was a transverse wave in the ether. He presented the mathematics of William
Rowan Hamilton, Augustin Cauchy, James MacCullagh and Augustin Fresnel
as mathematical theories. 18 Lloyd pointed out Cauchy's work as, "an interesting
department of analysis" but not strictly a physical theory. This was because in
Cauchy,
the phenomena of light are not connected directly with any given physical
hypothesis; but are shown to be comprehended in the results of the
general theory, in virtue of certain assumed relations among the constant
which that theory involves. 19
Cauchy chose the coefficients in his equations for the wave fronts in a crystal so
that they were compatible with Fresnel's experimental results. The actual behavior
of light in a crystal remained a mystery.
Physically detailed theories were not part of mathematics. Experiment only
tested the foundations of the mathematics, the hypotheses on which they were
based. Physical explication remained in non-mathematical language and belonged
to physics proper. In Lloyd's case, he gave Cauchy's mathematics physical meaning
(wave fronts) but realized that arbitrary constants added nothing to the physical
explanation of the behavior of light. What he did not tell his listeners was how he
deduced the physical meaning he had found in Cauchy's analysis.
In a similar fashion Whewell used geometry to treat mineralogy as a branch of
mathematics. The physical foundations for mineralogy lay in the optical, and other
measured, properties of minerals. 2o James Challis shared the common view that
as Whewell restricted his remarks to those cases where Poisson compared his
analytical results directly with experiment. Whewell chose to quibble with the
physical model from which Poisson extracted his mathematics. Yet Whewell took
Poisson's comparisons at face value. He assumed that the analytical conditions
reflected those of the experiments. When he dealt with the differences between
mathematical and experimental results, inaccuracy belonged to the observations.
Yet he concluded that mathematics and experiment "coincide as near as could be
expected."
Whewell understood Poisson's mathematics as its form was that of Laplace's
celestial mechanics and grounded in problems familiar from the Cambridge cur-
riculum. He was not so charmed by Laplace's mechanism for the conduction of
heat in a body. Nor for that matter did he like Fourier's. He judged Fourier's physi-
cal reasoning incorrect. However Whewell had to accept Fourier's equation for the
conduction of heat. Hammering away at Fourier, Whewell insisted that he could
not dispense with molecular reasoning in his account of the cooling of the earth.
He echoed Poisson's criticisms as well. Fourier's solution for the equilibrium
conduction of heat in a lamina,
such as T = e- mx cos my, were only particular solutions. For a complete solution
to this equation boundary conditions for the temperature, T, must be introduced
in the form of some "prescribed law." This led him to the consideration of discon-
tinuous functions, a "curious and perplexing part of analysis."
He focused on one issue, an example of the equilibrium radiation of heat from a
sphere. He had to consider the work of French mathematicians but regretted that
the mathematics of Fourier, Laplace, Poisson and Libri,
has not been in all respects favorable to the progress of the subject as
a branch of experimental and inductive science. The great beauty and
curiosity of meaning of the mathematical investigations which offered
themselves to our analytical discoverers, have led them to wander in that
deep and charmed labyrinth much longer and farther than the demands
of physical science required. 25
Whewell assumed that mathematics contained physical significance and he fo-
cussed on unearthing those significances by directly comparing mathematical de-
ductions and experimental results. However, he accepted the methods of math-
ematicians as sufficient to define the interactions with experiment. He assumed
that mathematical theories were necessary but that the focus of the French was
mathematics, not the investigation of the physical properties of heat, electricity,
25 Whewell, "Report on the Recent Progress and Present Condition of the Mathematical
Theories of Electricity, Magnetism, and Heat," Rep. British Assoc. (1835): 1-34,29.
220 From Natural Philosophy to Physics
or magnetism. However, he did not indicate here how to proceed to reorient this
kind of study. He could only complain that the French left the majority of mathe-
maticians behind them. Some simpler form of mathematics might suffice, that of
Newton perhaps, although Whewell conceded this would mean some sacrifice of
rigor and generality.
During the 1830s Whewell's was not the only report on heat given before the
British Association. In 1837 Phillip Kelland wrote a text on the subject partially
encompassing Fourier's work. In both text and report, Kelland tried to develop a
physical model for heat. He rejected caloric theory because it could not explain
radiation and turned to a vernacular version of Poisson's molecular model. When
he developed his mathematical theory of heat, he used Fourier. The text remained
in two distinct parts. This was not a copy of French mathematics since Kelland de-
veloped special cases that led to real physical circumstances. These circumstances
were reflected in experiments whose results could be directly compared to the math-
ematics. He focussed on this aspect of his work in his British Association report.
He was hard put to do this given the relationship of physical model to mathematics
in his own work and the absence of physical process in Fourier's mathematics. He
worked out specific examples that might be tried experimentally. The four mathe-
matical theories did not allow him to do this. Kelland noted that mathematicians,
Poisson in particular, had "not presented their results in a form sufficiently tangible
to direct or suggest the application of experiment to them." Experiments in and
of themselves could not decide among the various mathematical interpretations.
Available experiments also were not consistent enough to lead to anyone empirical
law of conduction. Kelland went on to suggest some experiments that might do
that and the difficulties they presented to the experimenter. 26
Kelland's work on heat was characterized later as mathematically ingenious but
physically flawed. Supposedly Kelland confused heat flow and temperature and
wrote of "temperature flow." In his research Kelland sought a physical expres-
sion for the mathematical generalization of Fourier's relation that heat flow was
proportional to temperature difference. Kelland changed this linear relationship
into the more general one where heat flow was dependent on some function of the
temperature. 27 Kelland was a mathematician who was trying to find an empirical
foundation for the mathematical work of Fourier. He then tried to to validate this
mathematics as his French models did by comparing experiment and mathemat-
ics. However, the experimental results at his disposal were inadequate. He was
a mathematician and generalized the conditions of Fourier's work mathematically
26 Kelland, Theory a/Heat (Cambridge: J. J. Deighton, 1837) and Kelland, "On the Present
State of our Theoretical and Experimental Knowledge of the Laws of Conduction of
Heat," Rep. British Assoc. (1841): 1-25,25.
27 George Chrystal and Peter Guthrie Tait, "The Reverend Professor Phillip Kelland," Proc.
R. Soc. Edinburgh 10 (1879): 319-321.
Britain 1830-1870 221
28 Airy, "Report on Recent Progress in Astronomy in the present Century," Rep. British
Assoc. 1 (1831): 125-188, p. 172. Airy's extreme utilitarian streak appalled William
Rowan Hamilton who reported that Airy stated "the Liverpool and Manchester Railway,
he said, playfully perhaps, but, I think sincerely, he considered as the highest achievement
of man." Hamilton's hesitation may well mean Airy also had a sense of humor. See Robert
Percival Graves, Life of William Rowan Hamilton (New York: Arno Press reprint of 1882
edition, 1973), 3. vols., vol., 1, 444.
29 We should, however, state that with Peacock's work in algebra the British developed their
own sense of "pure" mathematics distinct from those of either France or Germany. See,
Joan Richards, "Rigor and Clarity."
222 From Natural Philosophy to Physics
In the middle third of the nineteenth century all British universities, including
Cambridge, came under pressure to update their curricula and broaden the social
spectrum of its students. In reaction Cambridge reinforced the narrow foundations
of its curriculum and only grudgingly acknowledged the existence of the natural
sciences in its examination system in the 1850s. Cambridge seemed to retreat from
the new continental mathematics.
In mathematics and physics, other universities responded to the French leader-
ship by reforming the content of their courses and systematizing their teaching of
them. Natural philosophy remained a subject taught separately from mathematical
physics. However, the content of the natural philosophy courses shifted. Term, or
even year-long courses on particular fields in physics, were available to students.
Faculty made concerted efforts to include courses on heat, light, and other fields of
research in physics while much of the material was encompassed within a mechan-
ical explanatory net,30 Such focussed courses were empty of the mandatory ties to
natural theology of the first decades of the century. The universities of Edinburgh
30 See David Wilson, "The Educational Matrix: Physics Education at Early Victorian Cam-
bridge, Edinburgh and Glasgow Universities," in Wranglers and Physicists, Harman, ed.
12-45, for the courses at Glasgow and Edinburgh. For James David Forbes' work in this
Britain 1830-1870 223
and Glasgow further adapted the teaching of mathematics and natural philosophy
to the needs of engineers and other future professionals.
New French mathematics and experimental physics had a more immediate im-
pact in Dublin in terms of changes in teaching and in research. 31 The most important
research to emerge during the early decades of this influence was that of William
Rowan Hamilton. His research in optics and dynamics were in the French math-
ematical tradition. His optics began as an investigation into the mathematical
properties of systems of rays and the surfaces light formed in passing through
optical systems. He then turned to the work of other mathematicians on double
refraction and the surfaces formed by the ordinary and extraordinary rays in bi-
axial crystals. Although he preferred the wave theory, Hamilton understood that
his mathematical results were independent of any assumptions about the nature of
light. His work was,
not to discover new phenomena, nor to improve the construction of
optical instruments, but with the help of the differential or fluxional
calculus to remold the geometry of light, by establishing one uniform
method for the solution of all problems in that science, deduced from
the contemplation of one central, or characteristic relation. 32
Hamilton's goal was to reduce optics to analysis as Lagrange had reduced
mechanics. 33 While only "a secondary result," this deductive method had lead
to "some unexpected conclusions." Out of his consideration of the mathematics of
Fresnel's work on biaxial crystals Hamilton deduced that the surface of the wave
front within the crystal,
1st has four cusps (at the ends of the optic axes) at each of which the
tangent planes are (not, as he [Fresnel] thought, two but) infinite in
number; and 2nd , four circles of plane contact, along each of which the
ray is touched, in the whole extant of the circle, by a plane (parallel to
direction see also, John Campbell Shairp, Peter Guthrie Tait and A. Adams-Reilly, Life
and Letters ofJames David Forbes (London: Macmillan, 1873). Forbes also introduced
written examinations into Edinburgh.
31 See Grattan-Guinness, "Mathematical Research and Instruction in Ireland, 1782-1840,"
in Science in Ireland 1800-1930, J. R. Nudds et. aI., eds. and the account of Hamilton's
education at Trinity College, Dublin in Thomas L. Hankins, Sir William Rowan Hamilton
(Baltimore: The Johns Hopkins University Press, 1980),22-23.
32 Hamilton to Samuel Taylor Coleridge, October 1832, in Graves, Hamilton, vol. 1,592.
Graves notes that this letter was actually never sent. See also Hankins, Hamilton, 61-62.
Hankins gives a detailed discussion of Hamilton's work on optics, chaps. 4 and 5, and
conical refraction in chap. 6. Hamilton identified the characteristic function with the
principle of least action, or "least time."
33 Hamilton made this comparison explicit in Hamilton, "An Account of a Theory of a
System of Rays," Trans. R. Irish Acad. 15 (1824) [1828]: 69-174; 16 (1830): 4-62; 16
(1831): 85-92; 17 (1837): 1-144. In this paper Hamilton was careful to refer to both
fluxions and the calculus, although he used continental methods.
224 From Natural Philosophy to Physics
34 Hamilton to Herschel, December 18, 1832, Graves, Hamilton, vol. 1, 627. See also
James G. O'Hara, "The Prediction and Discovery of Conical Refraction by William
Rowan Hamilton and Humphry Lloyd," Proc. R. Irish Acad. 82A (1982): 231-257.
Emphasis in the original.
35 Lloyd, "On the Phenomena presented by Light in its Passage along the Axes of Biaxial
Crystals," Phil. Mag. 2 (1833): 112-120,207-210,116--117, his report to the British
Association, Lloyd, "Conical Refraction," Rep. British Assoc. (1833): 370, reprinted in
Lloyd, Miscellaneous Papers Connected with the Physical Sciences (London: Longmans
Green, 1877), 1-18. Hamilton's report to the British Association on his optical work
appeared as Hamilton, "On Some results of the View of a Characteristic Function in
Optics," Rep. British Assoc. (1833): 360-370.
36 Hankins in Hamilton, sees his real strengths in analysis which, in the formal sense of
the papers Hamilton produced and his importance within mathematics, is true. However,
what is striking in his account is Hamilton's ability to translate analytical results into
geometrical imagery where the reader can visualize the result.
37 Hamilton, "On a General Method in Dynamics," Phil. Trans. R. Soc. London, pt. II
(1834): 247-308, reprinted in Hamilton, The Mathematical Papers ofSir William Rowan
Hamilton, A. W. Conway andJ. L. Synge, eds. (Cambridge: Cambridge University Press,
1931) 3 vols., vol. 2, Dynamics, 103-211, 105. Hamilton had previously reported on his
Britain 1830-1870 225
However, the first mathematical results of Hamilton's work was not a new branch
of analysis but a new method for the integration of partial differential equations.
These were also published as a series of lectures on dynamics. 38
Hamilton drew no new physical conclusions about the motions of point centers
of force and while Hamilton put Boscovich's ideas into mathematical form the
physical implications of any of his results were seemingly of no interest to him. In
general he accepted the idea of light as a wave motion and the idea of immaterial
matter. He did not explore either to develop physical theories of the behavior of
light or of matter. As with French mathematicians specific physical issues set
Hamilton exploring new mathematical possibilities. 39
While Hamilton was the most prominent, James MacCullagh also took a math-
ematical approach to the theory of light. In his papers on double refraction Mac-
Cullagh noted that the phenomena were just so many isolated facts. He supplied
the connective tissue of mathematics by explaining known experimental laws "hy-
pothetically, by introducing a differential coefficient of the third order into the
equations of vibratory motion"-a mathematical fix, but with no imagery to catch
a physical process. 40 MacCullagh required that his mathematical relationship lead
to known experimental laws but his system was still deductive. While his image
of the ether was of a particulate medium, MacCullagh did not use its properties
to derive his equation of motion. He began with an examination of the geometri-
cal properties of ellipsoids and concluded from the results of Fresnel's work that,
since the wave front in a biaxial crystal was an ellipsoid, the particles of the ether
could only move in certain directions with respect to the wave fronts within the
crystal. From these deductions, he derived Biot's Law and argued by analogy that
his results agreed with Brewster's law as well. Important for his future work were
MacCullagh's deductions of the mathematical form for the elastic force and the
work in dynamics and its roots in optics in Hamilton, "On the Application to Dynamics
of a general mathematical method previously applied to Optics," Rep. British Assoc.
(1834): 513-518.
38 See Hankins, Hamilton, 196-197.
39 On Boscovichean atomism in Hamilton, see Robert Kargon, "William Rowan Hamilton
and the Revival of Boscovichean Atomism," l. Hist. Ideas 26 (1965): 137-140, and,
"William Rowan Hamilton, Michael Faraday and Boscovichean Atomism," Amer. l.
Phys. 32 (1964): 792-795.
40 James MacCullagh, "On the Laws of Double Refraction of Quartz," Proc. R. Irish A cad.
1 (1836-40): 385-386. Reprinted in MacCullagh, The Collected Works oflames Mac-
Cullagh, John H. Jellett and Samuel Houghton, eds. (Dublin: Hodges, Figgis and
Co., 1880), 63-74, 63. MacCullagh published a series of papers on double refraction
and reflection, and refraction throughout the 1830s and 18405. See MacCullagh, "On
the Double Refraction of Light in a crystallized Medium, according to the Principles of
Fresnel," Trans. R. Irish Acad. (1830-32): 79-84; "On the Properties of Surfaces of the
second Order," Trans. R. Irish Acad. (1836-40): 89-90 and "On the Dynamical Theory
of crystalline Reflection and Refraction," Trans. R. Irish Acad. 21 (1848): 17-50.
226 From Natural Philosophy to Physics
41 MacCullagh, "A Short Account of Some Recent Investigations concerning the Laws of
Reflection and Refraction at the Surface of Crystals," Rep. British Assoc. (1835): 7-8.
Reprinted in MacCullagh, Collected Papers, 55-57, 56.
Britain 1830-1870 227
tal and mathematical research. British mathematicians seemed blind to the quarrels
that separated the French into different mathematical camps. Hamilton did not see
that embracing a form of metaphysics favored by Poisson and Laplace forced
him to abandon the geometric elegance of Lagrange's mathematics. MacCullagh
used whatever mathematical results and methods suited his immediate needs. One
mathematician that the Irish did not seem to know or to use was Fourier.
The same could no longer be said of Cambridge mathematicians. By the late
1830s, despite the campaign of Whewell to purify the curriculum and rid the uni-
versity of the French threat, Fourier and the mathematical promise of his methods
had infiltrated Cambridge. However, any account of Cambridge mathematics in
this era must include Whewell's struggle for control ofthe curriculum and the ways
in which he ultimately lost that struwe.
Whewell and Peacock were both on the side of "reform," although they differed
over just what aspects of change they were prepared to support. Peacock, along with
other teachers at Cambridge, wanted to retain the French mathematics introduced in
the early 1820s in the examination system and in the official curriculum. Whewell
would have none of this and he had a good pedagogical point. A Cambridge
education trained the mind and the next generation of clergy for the Church of
England. A few highly-trained mathematicians schooled in the esoteric arts of
analysis were not an advertisement for the claims he and other reformers were
making about the Cambridge curriculum. Peacock's work on the foundations
of algebra undermined Whewell's contention of the philosophical barrenness of
analysis. In their battle over the curriculum at Cambridge, Peacock and Whewell
needed to reach two audiences. The first consisted of mathematicians, a minority
community within Cambridge but with allies outside Cambridge. They also had
to reach the tutors and other faculty within Cambridge, and this is where the
philosophical voice was so important. In this struggle Peacock won over the
mathematicians. Together with the long term effects of the examination system at
Cambridge, this annulled Whewell's short term gains over the curriculum.
Whewell's behavior in the struggles over the curriculum mirrored his actions
within the British Association. After his initial offense at Brewster's slurs on Cam-
bridge intellectual life, Whewell wholeheartedly supported the new organization.
He also worked to ensure that control of this new venture remained in safe social
and political hands. The Association never strayed into radical territory. It was re-
liable in the sense that the power relationships of the status quo were not upset. At
Cambridge Whewell also sought to keep the power relationships intact. 42 This was
42 There is ample evidence that national politics and the internal upheavals in Cambridge
were closely linked. For the involvement of both see Joseph Romilly, Romilly's Cam-
bridge Diary, annotated and introduced by J. T. Bury (Cambridge: Cambridge University
Press, 1967). Romilly notes those occasions at which politics did not enter the conver-
sations during 1831 and 1832. For the impact of political reform on Cambridge, see
Garland, Cambridge Before Darwin, chap. 2.
228 From Natural Philosophy to Physics
precisely what Whewell set out to accomplish within Cambridge. Power relation-
ships between College and Universities would remain the same. The educational
function of the university would not change. This commitment to conservation
crystallized with the development of Whewell's career which was nurtured by
those same power relationships. And to reiterate, as Master of Trinity he had to
concern himself with the education of the whole body of students not merely a
handful of wranglers. During the 1830s and 1840s, career, experience in teaching
and research, and developing philosophical interests mixed with ambitions were
channelled into his growing hostility against French mathematics.
By 1840 Whewell had lost the war within Cambridge and growing pressure from
Westminster was to force reform on Cambridge in the early 1850s. Whewell's was
a holding action. The subjects Whewell did not want introduced into the Natural
Science Tripos appeared the year after he died. 43
As Menachem Fisch has pointed out, Whewell approached the issues of re-
form in practical ways as well as philosophical ones. He wrote textbooks. It is
through those textbooks and developing philosophical ideas that Whewell argued
against the new mathematics, not simply on pedagogical grounds but as mathe-
matics. However, his attempt to provide a philosophically defensible alternative
to continental analysis was incomplete. 44 These textbooks were only part of his
motivation and only one aspect of his published assault. In his Bridgewater Trea-
tise, Whewell argued that those who were truly great inductive scientists were
drawn to God through their scientific work, "the very imperfection of the light in
which he works his way, suggests to him that there must be a source of clearer
illumination at a distance from him." Among those in this group were Robert
Boyle, Nicholas Copernicus, Galileo Galilei, and Johannes Kepler. In the group of
lesser scientists who gained no religious sense through their science he included
d' AIembert, Clairaut, Euler, Laplace, and even Lagrange. This rogue's gallery
of mathematicians, whom Whewell knew to be mathematicians not inductive sci-
entists, were the very men whose work he was arguing should be left out of the
Cambridge curriculum. Their work formed the foundation for the mathematics
that would displace Newton's formulation of mechanics, fluxions, and geometry
as the educational mainstay of the university.45
43 See Lewis Campbell and William Garnett, The Life ofJames Clerk Maxwell (New York:
Johnson reprint of 1882 edition, 1969), 325, chap. 12.
44 See, Fisch, "The Emergency that has arrived', " 266-276.
45 Whewell, Astronomy and General Physics considered with Reference to Natural The-
ology, Third Bridgewater Treatise, 1833 (London: W. Pickering, 1834). This treatise
went through many editions during his lifetime. The title alone indicated that Whewell
was categorizing astronomy and physics as observational and experimental sciences,
not mathematical ones. See also Richard Yeo, "William Whewell, Natural Theology,
and Philosophy of Science in Mid-Nineteenth Century Britain," Ann. Sci. 36 (1979):
493-516.
Britain 1830-1870 229
46 Some of Whewell's own research in mathematics have pedagogical goals, see Whewell,
"Rotary Motion of Bodies," Trans. Cambridge Phil. Soc. 2 (1827): 11-20.
47 For Whewell's remarks on Fourier, see Whewell,"Report on the Progress and Present
Condition of Electricity, Magnetism," 24-28. His work on the foundations of mechanics
is in Whewell, "On the Principles of Dynamics as stated by French Writers," Edinburgh
J. Sci. 8 (1827-28): 27-39, and, "On the Nature of the Truth of the Laws of Motion,"
Trans. Cambridge Phil. Soc. 5 (1834): 149-172. Whewell does not always differentiate
the various meanings of the term force used by the authors under discussion. See also
Whewell, On the Free Motion of Points and on Universal Gravitation including the
Principal Propositions of Books I and II of the Principia. First Part of a Treatise on
Dynamics (Cambridge: Deighton, 1833) for a later version of his mechanics.
48 Todhunter, William Whewell (New York: Johnson reprint of 1876 edition, 1970) p. 25.
He also discusses the problems of Whewell's understanding of the physics.
49 The question of the empirical status of the various forms of mechanics in Newton, Leibniz,
Euler and later French authors such as Laplace and Poisson can also be traced through
the various editions of his texts. See especially Whewell, An Elementary Treatise on
Mechanics designed for the Use of Students in the University (Cambridge; Deighton,
1819), second edition 1824, further editions appeared in 1828, 1833, 1836 and 184l.
The last edition appeared in 1847. The physical content of each of these editions is
detailed in Todhunter, Whewell, chap. 2. Whewell fiddled with the contents of his texts
in other ways. In 1833 he separated Statics from the fourth edition of his mechanics as
Whewell, Analytical Statics (Cambridge: Deighton, 1833) as a supplement to the fourth
edition of his elementary text on mechanics.
230 From Natural Philosophy to Physics
integral and differential calculus. Yet Whewell never completed a smooth transition
from physical example to increasingly difficult mathematics that culminated in the
calculus. Because of his growing hostility to analysis, at best he managed only to
graft the calculus intuitively on to specific problems. 50
Whewell consistently narrowed the mathematics addressed to the majority of
Cambridge students. He could thus refuse a place for the study of the new mathe-
matical domains of light, electricity and magnetism. These subjects represented the
locus of the new analysis. However, he could not control the choice of examiners for
the university examinations. Nor could he stop college tutors or university profes-
sors pursuing French mathematics. Because the university examinations seemed
beyond his grasp, the actions of Whewell and other conservatives to stave off the
French mathematical menace only aggravated the schizoid situation at Cambridge-
a situation that students understood by the 1820s. 51 The college curriculum was so
inadequate that any student striving for honors needed a private tutor. There was
an official and an unofficial teaching stream at Cambridge. The training required
for students to excel in university examinations demanded "very extensive and sin-
gularly accurate knowledge, in a wide range of mathematical subjects, combined
with perspicuity of thought and language in answering the questions proposed in
the examinations." The teaching they received from their college was inadequate
"to instruct and discipline the student so as to enable him to attain to that degree
of excellence in these points to which he is capable of attaining.,,52
Two systems existed at Cambridge, the official curriculum, which if followed
would lead to a pass degree. Distinction of any kind required students and tutors
to acknowledge this and supply the necessary training. The teaching members of
this underground system were integrated socially into university society. Their
teaching function was clandestine and subversive. Private tutors coached students
in the aspects of French analysis demanded by the last half of the Tripos examina-
tion. William Hopkins was preeminent among these tutors, training more senior
wranglers than anyone else of his generation. Important here is the training he gave
to three key Cambridge students, James Clerk Maxwell, George Gabriel Stokes
and William Thomson. His lectures were grounded in Lagrangian mechanics and
mathematics, elegant and an example of French mathematical physics. Rather
out of date in terms of its mathematics by the 1850s but required for the Tripos.
50 Harvey Becher, "William Whewell and Cambridge Mathematics," argues for the mathe-
matical goals of Whewell's texts. However, the unity he sees in the physics and mathe-
matics is in question.
51 Peter Allen, The Cambridge Apostles, chap. 1.
52 William Hopkins, "Remarks on the Mathematical Teaching of the University of Cam-
bridge," Trinity College Library. These are notes for Hopkins presentation before the
1851 Parliamentary Commission. As a private tutor Hopkins needed to understand the
problems of the system. His livelihood depended upon it and discerning how to bridge
the gap.
Britain 1830-1870 231
53 Hopkins never published a text. These remarks are derived from the lecture notes of
Maxwell, Thomson and Stokes in the Cambridge University Library. It is interesting to
note that all three preserved these notes.
54 Hopkins, "Researches in Physical Geology," Trans. Cambridge Phil. Soc. 6 (1835)
[1838]: 1-84. Crosbie Smith, "William Hopkins and the Shaping of Dynamical Geology,
1830-1860," Brit. J. Hist. Sci. 22 (1989): 27-52, sees Hopkins work as geology, while
not understanding that during the 1830s and 1840s Hopkins' goal was literally to make
mathematics from the empirical foundations of geology. In this case the mathematics
was the geometry of geology.
55 See Stephen G. Brush, "Nineteenth-Century Debates about the Inside of the Earth, Solid,
Liquid or Gas," Ann. Sci. 36 (1979): 225-254.
56 Duncan Gregory, "Preface," Cambridge Math. J. 1 (1837-1839): 1-2.
57 In 1850 the journal became Quarterly Journal of Mathematics. In 1845 it had already
232 From Natural Philosophy to Physics
In its early years students were able to brush up on the physical foundations
of Tripos questions through discussions of physical principles of mechanics and
astronomy. In the first five volumes there were a string of articles on new ways of
solving problems, useful methods, and reminders of important points sure to come
up on examination questions. There were also articles amounting to short courses
for students on aspects of the calculus and others reworking Tripos topics in ways
that were clearer than those usually encountered in textbooks and lectures. The
articles directed to students were usually short, and most of them were on strictly
mathematical subjects. They ranged from subjects encountered in the first year to
more esoteric methods, such as the solution to linear equations of finite and mixed
differences to Jacobi's methods of solving partial differential equations. In fact
the journal contained a great deal on the solutions to partial differential equations,
analytical geometry, and so on.
Yet even in these volumes many authors were outside of Cambridge. 58 Many of
the papers were on specific problems, many of which began in physics. In the early
1840s there was also a great deal of discussion of mathematical theories of light.
The physical theory of Fresnel was discussed as well as his mathematical work.
Archibald Smith noted that Fresnel usually used "mixed geometry" which was the
best method of establishing theorems even though clumsy and tedious. He planned
to establish the same theorems in the more general form of algebraic geometry.
In a later paper Smith considered Fresnel's work on crystals, made reference to
Hamilton's work, and included his mathematical deduction of conical refraction
from the properties of the surfaces of the wave fronts. Physics led to mathematics,
then returned to phenomena directly from the mathematics. 59
Papers also addressed mathematical issues directly from French mathematical
physics. The subjects incorporated the work of Joseph Liouville and Gabriel Lame.
And more and more articles represented research, done by mathematicians and
written for an increasingly professional audience. Within five years the number of
articles addressed to students fell, although William Thomson as editor from 1845
to 1850, hoped to revive them. As if to emphasize the journal's commitment to the
new calculus, Gregory wrote the first article on Fourier analysis. It was an attempt
to rewrite Fourier's results in functional form because Fourier's proofs were unsat-
isfactory. The improvement of Fourier's proofs was the subject of Thomson's first
paper in 1841. 60 Gregory returned to Fourier in his attempts to develop a differential
expanded into the Cambridge and Dublin Mathematical Journal signalling that its audi-
ence was mathematicians. The needs of students were becoming less and less the concern
of the editors.
58 These include Arthur Cayley, George Boole and others.
59 Archibald Smith, "The Wave Theory of Light," Cambridge J. Math. 1 (1839): 3-10,
and, Smith, "Notes on the Undulatory Theory of Light," 84-95.
60 Gregory, "Notes on Fourier's Heat," Cambridge J. Math. 1 (1837-1839): 104-107.
Britain 1830-1870 233
William Thomson
64 Thomson, "On the uniform Motion of Heat in Homogeneous Solid Bodies and its Connec-
tion with the Mathematical Theory of Electricity," Cambridge J. Math. 3 (1841-1843):
71-84. Gravitation is introduced on p. 83. Helmholtz, "Sir William Thomson's Math-
ematical and Physical Papers," Nature 32 (1885): 25-27 also noted the mathematical
character of Thomson's early work. See also Cross, "Integral Theorems," p. 35.
65 Thomson, "On the Equilibrium of the Motion of Heat referred to Curvilinear Coordi-
nates," Cambridge J. Math. 4 (1843-1845): 33-42, and "On the Lines of Curvature of
Surfaces of the second Order," same journal 4 (1843): 279-286.
66 Thomson, "Demonstration of a Fundamental Proposition in the Mechanical Theory of
Electricity," CamhridgeJ. Math. 4 (1843-1845): 223-226. He was referring to Gauss,
"Allgemeine Lehrsatze," in which Gauss stated that gravitation, electrostatics and mag-
netism are "special cases of the particular mathematical solutions being sought." p. 241.
Britain 1830-1870 235
infiltrating the natural philosophy courses at all Scottish universities that set them
apart from their English counterparts. Most historians of physics do not worry
about what constituted mathematics in the middle of the nineteenth century, nor
the common elements shared by Scottish and English scientists through that math-
ematics. If we do consider mathematics, then the disciplinary boundaries derived
from twentieth century practice dissolve and reform into a different geography.
The inheritance of historians of physics included categories created in the mid-
dle of the nineteenth century partly by the rewriting of the history of mechanics
by, among others, William Thomson. The early disciplinary geography and the
changes in boundaries and relationships between physics and mathematics became
invisible. 67
At Glasgow University, even if mathematical and experimental natural philos-
ophy were taught by the same professor, they were kept distinct. The first was
mathematics, the second the phenomena and vernacular explanations of those
phenomena. Experimental results were the starting points for the mathematics de-
veloped in the mathematical physics course. The evidence of textbooks indicates
that the courses lived alongside, yet largely unconnected, to each other. 68 Meik-
leham's course in natural philosophy was phenomenologically grounded with no
mathematical development of the principles drawn from experiment.
Thomson was, therefore, familiar with the frame of reference of Cambridge
teaching. He was also trained in mathematics by using physical problems to
generate mathematical conclusions and even new branches of analysis. He quickly
became a Cambridge mathematician in the radical camp of mathematical research at
the university. Continuing his work on the mathematics of electrostatics, Thomson
argued that both Coulomb's and Faraday's results were true. What was needed
was a mathematics that brought them together. In at least three places he pointed
out that his was a mathematical theory, "independent of physical hypotheses."
67 Crosbie Smith and Norton Wise, Energy and Empire: A Biographical Study of Lord
Kelvin (Cambridge: Cambridge University Press, 1989), have done precisely this. They
also treat Thomson as a physicist not as someone who helped to create this new discipline.
While Norton Wise has argued that Thomson was led to energy conservation through
mathematics he treats all the methods that Thomson used as aspects of physics, thus
reading twentieth-century givens back into the 1840s. See Wise, "William Thomson's
mathematical Route to Energy Conservation: A Case Study in the Role of Mathematics
in Concept Formation," Hist. Stud. Phys. Sci. 10 (1979): 49-83, and Harold Sharlin,
"William Thomson's Dynamical Theory: An Insight into a Scientist's Thinking," Ann.
Sci. 32 (1975): 133-147.
68 This is reinforced in Thomson's case as his father was professor of mathematics at Glas-
gow and wrote textbooks for his own courses covering some of the new analysis. See
Smith and Wise, Energy and Empire, chap. 1, where they claim that the elder Thomson
in his texts minimized abstraction. This is an indication that he followed the mathemat-
ical path of the French closely in outline, physical problems leading to mathematical
excursions, if not in the details of techniques.
236 From Natural Philosophy to Physics
From this he created the absolute scale of temperature. 72 This was reinforced by his
brother James' equally physical deduction from mathematics-that the freezing
point of a substance decreased with an increase in pressure. William Thomson
devised and performed the experiments that confirmed the mathematics. 73
However, James Prescott Joule's experiments indicated the conversion of heat
into work and work into heat. These experiments posed a direct challenge to the
principle of the conservation of heat upon which Thomson and Clapyeron based
their mathematics. 74 Already doubting Carnot's physical theory of heat, Thomson
nevertheless published his mathematical generalization of Clapyeron's work. 75
Thomson then refereed William J. C. Macquorne Rankine's paper on the me-
chanical theory of heat. In Rankine's work, the idea of heat as work was given
explicit mathematical expression as the vis viva of the motion of the particles of
bodies. Rankine then assumed that the forces between the particles of bodies were
some function of the density of the body. With this assumption, Rankine deduced
Joule's relationship between heat and work, and other known gas laws. Rankine
limited his mathematics to physically plausible cases, ignoring the mathematically
generalized ones. Yet he deduced his physical results from his mathematics without
72 Heat and its measurement, along with temperature, were important in marine engineering.
73 William Thomson, "On an Absolute Scale of Temperature founded upon Carnot's Theory
of the Motive Power of Heat, and Calculated from the Results of Regnault's Experiments
on the Pressure and Latent Heat of Steam," Phil. Mag. 33 (1848): 313-317. James
Thomson, "Theoretical Considerations on the Effects of Pressure in Lowering the Freez-
ing Point of Water," Trans. R. Soc. Edinburgh 16 (1849): 575-580, Cambrdge Dublin
Math. J. 5 (1850): 248-255. The experiments were reported in William Thomson, "On
the Effect of Pressure on the Freezing Point of Water, experimentally demonstrated,"
Proc. R. Soc. Edinburgh 2 (1850): 267-271, Phil. Mag. 37 (1850): 123-127. Neither
the absolute scale of temperature, nor the lowering of the freezing point, depended on any
assumptions about the nature of heat itself. It can be deduced directly from the form of
the equations, giving the symbols their initial physical meanings. See Clifford Truesdell,
The Tragicomical History of Thermodynamics.
74 In 1847 Joule read an account to the British Association of improved experiments that he
was convinced demonstrated the conversion of mechanical work into heat and suggested
the reverse transformation should also take place. See James Prescott Joule, "On the
Existence of an equivalent Relation between Heat and Ordinary Forms of Mechanical
Power," Rep. British Assoc. (1847): 55, in full in Phil. Mag. 31 (1847): 173. For
accounts of Joule's experiments and the origins of his work in heat, see John Steffens,
James Prescott Joule (New York: Science History Pub., 1979), D. S. L. Cardwell,James
Joule: A Biography (Manchester: Manchester University Press, 1989), and William
Cropper, "James Joule's Work in Electrochemistry and the Emergence of the First law
of Thermodynamics," Studies Hist. Phil. Sci. 19 (1988): 1-15.
75 Thomson, "An Account of Carnot's Theory of the Motive Power of Heat with Numerical
Results deduced from Regnault's Experiments on Steam," Proc. R. Soc. Edinburgh 24
(1849): 198-204 and in full in Trans. R. Soc. Edinburgh 16 (1849): 541-574.
238 From Natural Philosophy to Physics
ative time values were meaningless. Mathematical expressions for a system at zero
time now carried cosmic as well as physical significance. 81 Thomson stated his
new found physical understanding of irreversibility and changed his interpretation
of his own earlier work on Fourier, "when heat is diffused by conduction there is a
dissipation of mechanical energy, and full restoration of it to its primitive condition
is impossible." From this proposition Thomson deduced the mechanical work ex-
tractable from an unequally heated body by equalizing the temperature of that body.
This was the mechanical equivalent of the heat "put out of existence." Although he
never published a mathematical derivation of this result, Thomson understood that
in every cycle of a heat engine, some energy becomes "unavailable" for work and
was cast out as heat. This loss was inevitable and irreversible. Thomson followed
this statement with an integral expression for the final temperature in the cycle of
a heat engine coupled with his conclusion of the heat death of the universe. For
detailed mathematical arguments for the cosmical consequences of irreversibility
Thomson relied on Fourier and the cooling of the earth as the concrete physical
example. 82
Within a short period of time Thomson's research had changed radically. His
experiences with heat convinced him of the necessity for theories of physical
processes. Within those mathematical theories the mathematics was limited by
requirements of the mechanical model, under the guidance of the principle of
conservation of energy. Like many of his contemporaries, Thomson explored the
mathematical implications of the principle of conservation of energy, but in his case
there was now a real integration of mechanical principles, with mathematics con-
fined to physically plausible outcomes along with the integration of experimental
work to exemplify this tight connection. 83
Thomson became bold in predicting the physical meanings encoded in mathe-
mati cal language that went beyond known experimental results. His early daring is
best exemplified by his work on the Atlantic Telegraph and his explanation of signal
theory. See Thomson, "On the Universal Tendency in Nature to the Dissipation of
Mechanical Energy," Phil. Mag. 4 (1852): 304--306, and Proc. R. Soc. Edinburgh 3
(1852) [1857]: 139-142.
81 In the papers he published on Fourier analysis in the 1840s Thomson dealt with negative
time as a mathematical quantity. It had no physical significance. See Thomson, "Note
on Some Points in the Theory of Heat," CambridgeJ. Math. 4 (1843-1845): 67-72.
82 See Joe Burchfield, Lord Kelvin and the Age of the Earth (New York: Science History
Publications,1975). See also Smith and Wise, Energy and Empire, chaps. 4, 5,16.
83 See Thomson's papers on thermoelectricity. He enlisted his students in the experimental
half of the labor. Thomson first mentions thermo-electric phenomena in Thomson, "On
the Dynamical Theory of Heat," (1851): 15-16. Other papers followed, see, Thomson,
"Account of Researches in thermo-electricity," Proc. R. Soc. London 7 (1854--1855):
49-58; "On the Dynamical Theory of Heat. Part VI Thermoelectric Currents," Trans. R.
Soc. Edinburgh (1854): 123-172, Phil. Mag. 11 (1856: 214--225,281-297,379-388,
433-446.
240 From Natural Philosophy to Physics
attenuation. His physical grasp of the problem was encompassed in that expla-
nation, and in his recommendations for countermeasures. 84 His physical theories
became dependent on ever more intricate mechanical molecular models. They
were specific, mechanical systems that on the micro-level were miniature systems
that obeyed the same mechanical laws as macro-phenomena. Directly from the
mechanical properties and behavior of these molecules, Thomson shaped the func-
tioning and operation of the macroscopic physical system. The most graphic of
his later physical approach to phenomena were in his models of the ether. Me-
chanical models became Thomson's way of visualizing the operations of nature.
If he could not make a mechanical model of a theory, he could not understand it.
His mechanical models were models whose motion could be grasped visually as
well as expressed mathematically. Whether these models were compatible with
one another to make a coherent picture was a lesser problem. Thomson invented
them to solve one physical problem at a time. 85
Thomson's approach to the solutions of physical problems was encapsulated in
his textbook on mechanics authored with Peter Guthrie Tait. This text was not just
a reworking of mechanics using energy conservation as its conceptual foundation.
It redefined the subject pedagogically. Force, not a concept either Thomson or
Tait used in their research, physically tied the fields of mechanics together. This
replaced the analysis that mathematicians used to hold mechanics together. Math-
ematicians moved from statics to dynamics through virtual displacement. 86 Here
the text began in kinematics. Through the balance of forces, statics became a
special case of dynamics. Dynamics supplied the explanations for the motions of
bodies described kinetically. Thomson and Tait devoted a lot of space to statics
84 For details on Thomson's work on the Atlantic telegraph, see Smith and Wise, Energy
and Empire, chap. 19,661-684. Usually seen as evidence of the marriage of research
and engineering the experiments done by Thomson and his students on the electrical and
chemical properties of the copper in the cables are another indication of his commitment
to the experimental investigation of nature.
85 See Kelvin (Thomson), Baltimore Lectures and Modern Theoretical Physics: Historical
and Philosophical Perspectives, Robert Kargon and Peter Achinstein, eds. (Cambridge
MA.: MIT Press, 1987). This is a reprint of the notes of the lectures as they were given,
rather than the longer version published later, along with historical and philosophical
essays. The models that appear in almost every lecture are made up of various combina-
tions of spring systems or vortices. See, 9-10, 48-52, 77-81, 82-94, 108-111, 120-124,
125-128,135-144,145-150,152-157. The focus of the lectures was on the difficulties
besetting the mechanical, molecular theories of dispersion, refraction, and fluorescence.
For the power and limitation of Thomson's models see Smith and Wise, Energy and
Empire, chaps. 12, 13. For Thomson on vortex atoms see Robert Silliman, "William
Thomson: Smoke Rings and Nineteenth-Century Atomism," Isis, 54 (1963): 461-474.
For mechanical atomic models and their limitations see, Garber, "Molecular Science in
Late Nineteenth-Century Britain," Hist. Stud. Phys. Sci. 9 (1978): 265-297.
86 The other approach was to separate the two subjects, motion and statics altogether.
Britain 1830-1870 241
and elasticity, both important subjects for their engineering students. Elasticity
was also becoming important in Thomson's own research into the ether and its
interactions with matter.
Thomson and Tait also rewrote the history of mechanics. 87 Theorems were taken
from Euler, Green, Gauss, Legendre, and others in mathematical isolation. The
only criteria applied was the theorem's relevance to the solution of the physical
problem at hand. The theorems were stated mathematically, then explained phys-
ically and applied to solve specific mechanical problems. Many problems sprang
from Tripos questions. Yet they were exercises in statics, kinematic, and dynamics
not mathematics. Physical explanations of the results of the mathematics followed
during and at the end of the mathematical solutions. Mechanics became a sub-
ject of physics, not a launch pad into mathematics. The authors developed the
mathematics they needed to solve physical problems. Physics remained the focus
of their attention. 88 And they conceptually refocussed mechanics by interpreting
many results in terms of potential and kinetic energy.89
Thomson and Tait had annexed mechanics for physics and it became a source
for colleagues as well as students. Their approach meant that they could reinter-
pret the mathematics of Euler, Green, Gauss, Legendre, and others in terms of
the physics implicit in their mathematics. Therefore, "Euler discovered that the
kinetic energy acquired from rest by a rigid body in virtue of a impulse fulfills a
maximum-minimum condition.,,9o By attributing to Euler a concept only recently
developed, the foundation of their physics was given a respectable ancestry al-
though it falsified Euler's understanding of the problem. They also contended that
Lagrange extended this to a connected system of bodies struck with any impulse.91
They forced the past of mechanics, which had been part of mathematics, into
their version of what it must have really been physically. They made the past of
mechanics over into physics.92 All of this is understandable given the audacious
87 Smith and Wise, Energy and Empire, chap. 11, recount the writing of the text and its
historical importance in terms of the new conceptual foundation of energy conservation.
However, they also accept much of Thomson and Tait's interpretation of the work of their
predecessors without comment or investigating the reasons for their interpretations.
88 This explains, in part, the popularity of the text and its rapid translation into German.
89 The clearest examples of this are in their treatment of Green's potential. See Thomson
and Tait, Principles, vol. 2, article 482, 28-29. This physical interpretation is in marked
contrast to Thomson's mathematical uses of the potential in the 1840s. See Thomas
Archibald, "Physics as a Constraint on Mathematical Research,"
90 Thomson and Tait, Principles, vol. 1, article 311, 285.
91 Thomson and Tait, Principles, part 2, article 37.
92 Their reworking of the history of mechanics and its reshaping into the concept of energy
was not always successful. Horace Lamb and George Darwin made additions to later
editions and noted that Thomson and Tait's "attempt to deduce the principle of virtual
velocities from the equation of energy alone can hardly be regarded as satisfactory."
242 From Natural Philosophy to Physics
character of their text, and their need to establish credentials that would give to
them a respectable pedigree. They were successful and erased the mathematical
context of the development of mechanics.
While Thomson and Tait accepted the use of hypotheses within natural philoso-
phy, they never addressed this issue formally, although they discussed the grounds
for believing in the hypotheses they did use. That ground was experimental evi-
dence. Mathematical theories of planetary motion were well grounded, those of
geometric optics were carried "far beyond the limits of experiment." The hypothe-
sis that heat was a form of energy came from experiment although many formulae
were still "obscure and uninterpretable," as the mechanics of the motions of the
particles of matter were unknown. Only mathematical analysis existed in those
physical fields of the lowest tier of this hierarchy of surety, electricity, magnetism,
heat and light.
The contingencies of building careers in the new professions opened up by the
sciences led Thomson and Tait into investigations of nature and away from their
common starting point in Cambridge mathematics. Yet that transition was never
quite complete. For Thomson, mechanical models were means of solving indi-
vidual problems. The consistency with which he used mathematical methods was
absent from his mechanical models. The details of the latter depend entirely on
the specific problem at hand and could change radically even when dealing with
the same physical body, such as the ether. Models were heuristic and necessary for
Thomson to grasp the mathematics and its physical meaning. Consistent with this
were the ways in which Thomson taught. Experimental and mathematical physics
were separate courses. He limited use of his mechanics text to the mathematical
physics course for honors students for the MA.93 Comprehensive physical theories
expressed in the language of mathematics did not emerge from his work, although
his research contained streams of ingenious solutions to particular problems based
on the mathematical analysis of mechanical models. Tait was even less interested
in physical, rather than mathematical, consistency. His quarrel with Josiah Willard
Gibbs in the 1880s over vectors revolved around Gibbs' desecration of the mathe-
matical integrity of quartemions. The issue of the usefulness of vectors for physics
did not enter into his argument.
Thomson and Tait published a model on how to teach this new discipline; others
had already done this in their teaching. Although he left no textual monument with
which to bedevil historians, the most important of these men was George Gabriel
Stokes. Stokes was quite clear on where mathematical argument was appropriate
and where physical hypotheses began and the extent of their legitimacy. And it is
with Stokes, we see the first conscious and consistent separation of mathematical
issues on the one hand and the needs of physical theory on the other.
William Thomson was not alone in recognizing and transcending the limitations
of Cambridge mathematics in the exploration of nature. George Gabriel Stokes
had begun earlier, yet his work followed a different line of development from
Thomson's. While committed to a mechanical view of nature, Stokes was a good
deal more discerning in its use. Like Thomson, Stokes was trained in the Cam-
bridge mathematical tradition and was able to extend that mathematics through the
consideration of physical problems. However, from the beginning of his career
Stokes's research papers were of three types. He worked within Cambridge math-
ematical tradition by using physical problems to extend other mathematicians'
work. His first papers on hydrodynamics were improvements upon the theorems
of Cauchy and Poisson and the mathematics of Laplace. What he was after were
better solutions to certain partial differential equations. 94
Secondly, Stokes clarified and examined key issues about the physical hypothe-
ses being actively pursued by his contemporaries. At the time he graduated as first
Wrangler and Smith's Prizeman, Stokes was performing experiments and specu-
lating about the nature of the ether. His early physical papers on the ether contain
a minimum of mathematics. Stokes claimed results without going through the
analytical details and speculated on whether the ether was at rest, or was dragged
along with the earth. His focus was on the physical implications of Fresnel's
and later authors' mathematical work. Stokes concluded that the laws of reflec-
tion and refraction were unaffected by any motion of the ether and there were no
experimental tests available to choose between the two hypotheses. 95 Published
separately from his mathematical work Stokes dealt here with a subject entirely
within natural philosophy. The interaction of the ether and matter was speculative
and hypothetical. 96
94 See Stokes, "Steady Motion of Incompressible Fluids," Trans. Cambridge Phil. Soc. 7
(1842): 439--454,465; "Some Cases of Fluid Motion," same journal 8 (1849): 105-137,
409-414, abstract in Phil. Mag. 31 (1847): 136-137, and, "On the Theories of the
Internal Friction of Fluids in Motion, and of the Equation and Motion of Elastic Solids,"
Trans. Cambridge Phil. Soc. 8 (1849): 287-319, abstract in Phil. Mag. 29 (1846):
60-62. Stokes' mathematical work is discussed in Cross, "Integral Theorems," 136-137,
144-145.
95 Stokes, "On the Aberration of Light," Phil. Mag. 27 (1845): 9-15, and, "On Fresnel's
Theory of the Aberration of Light," same journal (1846): 76-8l.
96 Stokes, "On the Constitution of the Luminiferous Ether, viewed with reference to the
Phenomenon of the Aberration of Light," Phil. Mag. 29 (1846): 6-10.
244 From Natural Philosophy to Physics
dp au au au au a 2u a2u a2u
-dx = p(x - -
at
- u- - v - - w-)
ax ay az
+ f1(-
ax 2
+ -ay2 +-)
az 2
f1 d du dv dw
+"3 dx(dx + dy + dz)'
with similar equations for dpjdy and dpjdz, where u, v, w were the components
of the velocity of the fluid along the x, y, z axes, p the pressure and t the time,
p the density of the fluid and f1 "a certain constant dependent on the nature of the
fluid." Stokes then confined himself to a series of special cases of these equations
that were dictated by physical conditions. The motions of the fluid were small so
that terms in the squares of the velocities could be neglected and the density could
97 The experiments he referred to were those performed by Bessel in the 1820s and by
Sabine and others on the seconds pendulum.
Britain 1830-1870 245
. . . au + -av + -aw = O.
and the equatIOn of contmUlty, -
. .
Workmg systematIcally
ax ay az
through the mathematics, Stokes eliminated cases by imposing physical conditions
until he could consider pendulums that were cylinders and spheres performing small
oscillations in spaces restricted by other cylinders and spheres. These conditions
meant that the number of arbitrary constants introduced into his solutions was
restricted to one. Stokes called this the "index of friction" of the fluid which could
be determined by experiment. Taking a qualitative demonstration of its existence,
he developed a theory based on the general mathematized properties of fluids,
expanded to include this new phenomena, then reduced the mathematics to a state
of direct comparison with several different experiments. 98
In his report to the British Association on hydrodynamics, Stokes interpreted
the results of all mathematicians physically. He only dealt at length with those
results that had physical content. He mentioned Ostrogradsky's paper in passing,
although it was on the motion of a fluid in a cylindrical basin. However, "the
interest of the memoir, however, depends almost exclusively on the mathematical
processes employed, for the result is very complicated, and has not been discussed
by the author.,,99 In another case Stokes suggested that a mathematical investigation
was characterized as "one of great complexity and very little interest," that is, of
physical interest. lOO Here Stokes also gives explicit physical meaning to the terms
in the mathematics of fluids. 101
Quite explicitly Stokes separated his physical and mathematical understanding
of the same piece of work. 102 In his discussion of Fourier series, he was at pains to
show the mathematical advantages of Fourier analysis over functional solutions to
the same partial differential equations. Here the point was mathematical, to extend
98 Stokes, "On the Effect of the Internal friction of Fluids on the Motion of Pendulums,"
Trans. Cambridge Phil. Soc. (1851): 8-106.
99 Stokes, "Report of Recent Researches in Hydrodynamics;' Rep. British Assoc. Part
I (1846): 1-20, reprinted in Stokes, Mathematical and Physical Papers (Cambridge:
Cambridge University Press, 1880) vol. 1, 157-187, 162.
100 Stokes, "Report," 168.
101 See Stokes, "Report," 183-184 where he explains physically what terms St. Venant used
to describe the motion of fluids where the pressure was not equal in all directions, and
the physical results of these suppositions.
102 In the case of fluorescence Stokes could not complete the transition from experiment to
mathematics and physics. The physical foundation of his mathematics were insufficient
to analyze even his own experiments.
246 From Natural Philosophy to Physics
Fourier series to cases beyond their usual range. Temperatures become functions
of coordinates, and the results of these mathematical explorations were not referred
back to measurable temperatures or other physical conditions. He left the results
in terms of functions and arbitrary constants. The results illustrated mathematical
methods, but did not elucidate any physics. 103
Stokes was quite conscious of his separation and treatment of mathematics from
physics. He was also clearheaded about his use of physics to generate mathematics.
He put it to Cayley that
Thomson and I are at present writing to each other about potentials.
I think that potentials may throw light on the interpretation of f (x +
Hy). How horrible you would think it to prove, even in one's own
mind, a proposition in pure mathematics by means of physics. 104
Whether Cayley was horrified or not, his report on dynamics of 1857 served to
show the distance between mathematicians and physicists in the middle decades
of the nineteenth century. Cayley's report traced "the investigations of geometers
in relation to the subject of analytical dynamics." He recounted the successive
development of mathematical methods. In conclusion Cayley reminded his au-
dience that the differential equations of dynamics "are only one of the classes of
differential equation which have occupied geometers." He then noted the work
of Jacobi and Pfaff in the theory of the solution of partial differential equations.
Mathematicians could still claim mechanics. 105
Stokes' ability to differentiate mathematical nicety from physical meanings was
put to the test when he was appointed as Lucasian professor of mathematics at
Cambridge. In this capacity he took over the lecture demonstrations on hydrostatics
and optics from Challis. As one student later put it, before Stokes,
we had to get up natural philosophy by a painful exercise of the imagi-
nation on diagrams and descriptions, and the abstractions formulated by
mathematicians to make calculations possible which presented Nature
as a lifeless statue. 106
103 Stokes, "Critical Values of the Sums of Periodic Series," Trans. Cambridge Phil. Soc. 8
(1849): 533-583, abstract in Phil. Mag. 33 (1848): 309-31l.
104 Stokes to Cayley, 29th Oct., 1849 in David B. Wilson, The Correspondence between Sir
George Gabriel Stokes and Sir William Thomson, Baron Kelvin of Largs (Cambridge:
Cambridge University Press, 1990),2 vols. vol. 1,81 footnote. This runs contrary to the
usual interpretation of Stokes' work which is seen as dominated by physics. See David B.
Wilson, Kelvin and Stokes, and E. M. Parkinson, "George Gabriel Stokes, 1819-1903,"
Dict. Sci. Bio., vol. 13,76-79.
105 Arthur Cayley, "Report on the Recent Progress of Theoretical Dynamics," Rep. British
Assoc. (1857): 1-42.
106 G. D. Liveing, in Memoir and Scientific Correspondence of the Late George Gabriel
Stokes, 1819-1902, Joseph Larmor, ed. 2 vols. (Cambridge: Cambridge University
Britain 1830-1870 247
Stokes "at once set the study on a new footing" with his experiments. He demon-
strated conical refraction, a result that was well known and liable to be on the
Tripos.
Stokes was well aware of his audience. Professional audiences were now becom-
ing large enough that papers on the different aspects of one topic, mathematical,
experimental and hypothetical were placed in different publications. He used these
publication alternatives to address specific issues. By the middle of the 1840s with
the Cambridge Mathematical Journal, mathematicians had an outlet that catered
to their interests. 107 Physical speculations were of no interest to this group unless
they led to new mathematical puzzles. Stokes explained the physical content of
mathematical theories in articles published in Philosophical Magazine that catered
to experimental physicists. Stokes' physical interpretations appeared here without
losing his audience in a sea of impenetrable analysis. Stokes' papers, therefore,
had clearly delineated purposes that he followed through in their structure. Those
directed towards mathematical investigations and those of physical interpretation
and investigation were quite distinct.
The clarity and shift of priorities were a break with mathematical physics at
Cambridge where any physics emerged from the particulars of a generally devel-
oped mathematics. In Stokes' work mathematical physics became a branch of
physics as well as one of mathematics. In the mathematical physics addressed to
physicists, Stokes focussed only on the amount of mathematics necessary to make
the physical points clear. In addressing mathematicians, the physical meanings of
the mathematics was either absent or a minor point of the paper.
Stokes' and Thomson's interests and talents complemented one another. In their
early correspondence Stokes and Thomson explored the mathematics and physics
of their separately favored subjects, fluids for Stokes, electrostatics for Thom-
son. In all cases Stokes was the more perceptive mathematician. 108 He also helped
Thomson to define his physical cases from which the mathematics developed more
precisely. In trying to understand Gregory's recent work on differential and in-
tegral calculus, Thomson used examples from physical cases. Physics generated
and made mathematics intelligible. While Stokes did not follow Thomson into ex-
periments in electricity he could, through mathematics, advise Thomson on both
his physics and mathematics. Stokes chose not to speculate physically in the same
way as Thomson or even Maxwell. However, both of the latter referred to him and
Press, 1907), vol. 1, 91-97, 96. Liveing points out the costs to Stokes of all this work.
Stokes had to set up the demonstration experiments on his own, at great expense in time
and trouble.
107 The Cambridge Philosophical Society together with the Royal Societies of London and
Edinburgh as before published papers over a broad range of topics to their members.
108 See Stokes' comments on Thomson's paper on orthogonal surfaces. See Stokes to Thom-
son, 10th April, 1847, in David B. Wilson, Correspondence vol. 1,23-25,25.
248 From Natural Philosophy to Physics
109 This was captured by Tait in "George Gabriel Stokes," Trans. Cambridge Phil. Soc. 18
(1904) : 303-304 and in Scientific Correspondence of Stokes, Larmor ed., vol. 1. It was
echoed in Thomson's remarks on the same occasion, 277.
Britain 1830-1870 249
hydrodynamics, as well as his work in light and on the ether that was an extension
of his work in fluids.
Maxwell shared with Stokes and others an interest in the relationship between
images of nature and reality, and the particular example of Stokes' clear criticism
of mechanical models of the ether. Much has been made of the roots of Maxwell's
philosophy in the unique aspects of his Scottish education. However, in looking at
the group of natural philosophers and physicists in Scotland and England as well
as Ireland, the same philosophical issues and range of responses to the problems
posed by the relationship between images of nature and its reality are found across
cultural and social boundaries. The assumption of the uniqueness of the Scottish
philosophical heritage of Thomson and Maxwell cannot explain the baroque na-
ture of Thomson's later detailed molecular models of the ether, magnetism, and
electricity. Within these accounts Stokes' distance from these same models also re-
mains somewhat mysterious given his educational background. Stokes' research
matched the "geometrical" descriptions given to the kind of natural philosophy
that supposedly emerged from Scottish natural philosophy. Maxwell's use then
discarding of the same kind of mechanical models as Thomson's speaks for a
continuing search for heuristic methods to explore nature, rather than a lifelong
commitment to any philosophical program.
The argument over the special place of geometry within this Scottish philo-
sophical tradition also fails, if we take into account the research interests as well
as the textbooks of many professors of natural philosophy and mathematics in
ScotlandYo What these physicists had in common was intensive mathematical
training at Cambridge. What they did with that training depended on the contin-
gencies of their careers. Stokes remained at Cambridge. The research problems
of interest there and the requirements of his chair led him to consider some physi-
cal problems rather than others. The opportunities of Glasgow and study in Paris
changed the direction of Thomson's work. Maxwell was able to exploit several
different approaches to the exploration of nature. Unlike the other two, he had an
independent income that distanced him for some years from the immediate needs
and the social context of a profession. It allowed him to pursue theoretical physics
along several different fronts simultaneously.
What also joined these and other physicists of the mid-nineteenth century was
the conviction that they were searching out the true structure and functioning of
nature. Maxwell simultaneously explored the philosophical grounds on which
to develop theories of nature while constructing them in mathematical form and
interpreting them physically. Early in his career Maxwell explored the physical
110 On this heritage, see Richard Olson, "Scottish Philosophy and Mathematics, 1750-1830,"
J. Hist. Ideas 32 (1971): 29-44, and Scottish Philosophy and British Physics (Princeton
NJ.: Princeton University Press, 1975). See also Peter Harman, "Edinburgh Philosophy
and Cambridge Physics: the Natural Philosophy of James Clerk Maxwell," in Wranglers
and Physicists, Harman, ed., 202-224.
250 From Natural Philosophy to Physics
111 Maxwell, "Faraday's Lines of Force," Trans. Cambridge Phil. Soc. 10 (1856): 27-83,
reprinted in Maxwell, Scientific Papers vol. 1, 155-229, 188-189,209.
112 Maxwell's commitment to the specific mechanism of vortices has been reaffirmed in
Daniel M. Siegel, Innovation in Maxwell's Electromagnetic Theory: Molecular Vortices,
Displacement Current and Light (Cambridge: Cambridge University Press, 1991), and,
"Mechanical Image and Reality in Maxwell's Electromagnetic Theory," in Wranglers
and Physicists, Harman, ed. 180-201, and "Thomson, Maxwell and the Universal Ether
in Victorian Physics," in Conceptions of Ether, Cantor and Hodge, eds. 239-268.
Maxwell's reformulations of his electromagnetic theory are discussed in C. W. F. Everitt,
James Clerk Maxwell Physicist and Natural Philosopher (New York: Charles Scribner's
Sons, 1975),80-111.
113 Other historians and philosophers also maintain Maxwell's skepticism towards models
and specific mechanisms. See Peter Harman, "Edinburgh Philosophy and Cambridge
Physics," in Wranglers and Physicists, Harman, ed. 202-224, and "Maxwell and Modes
of Consistent Representation," Arch. Hist. Exact Sci. 6 (1970): 171-213. Others have
argued that aesthetic principles guided at least Maxwell's theory of the electromagnetic
field. See, Alfred M. Bork, "Maxwell's Displacement Current and Symmetry," Amer. J.
Britain 1830-1870 251
Phy. 31 (1963): 854-859, and Joan Bromberg, "Maxwell's Displacement Current and
his Theory of Light," Arch. Hist. Exact Sci. 4 (1967): 218-234.
114 See Daniel Siegel, "The Origins of Maxwell's Displacement Current," Hist. Stud. Phys.
Sci. 17 (1986): 19-146.
115 M. Norton Wise, "The Maxwell Literature and British Dynamical Theory," Hist. Stud.
Phys. Sci. 13 (1982): 175-205, noted this impasse.
116 For the importance of philosophical issues throughout his life see, Lewis Campbell and
William Garnett, The Life ofJames Clerk Maxwell (New York: Johnson Reprint of 1882
edition, 1969), passim.
117 For Maxwell on physical analogies, see Joseph Turner, "Maxwell on the Method of
Physical Analogy," Brit. J. Hist. Sci. 6 (1955): 226-238, and Robert Kargon, "Model
and Analogy in Victorian Science: Maxwell's Critique of the French Physicists," J. Hist.
Ideas 30 (1969): 423-436.
118 Campbell and Garnett, Life, p. 261.
252 From Natural Philosophy to Physics
119 Maxwell, "Analogies. Are there Real Analogies in Nature?" in Campbell and Garnett,
Life, 235-244, 243.
120 Maxwell Elementary Treatise on Electricity William Garnett ed. (Oxford: Clarendon,
1881), section 64. For a more extended discussion see, Maxwell, "Faraday'S Lines of
Force," in Scientific Papers, vol. 1, 155-229, 156-158.
Britain 1830-1870 253
physics from a physical point of view. To demonstrate the plausibility of his rings
of small satellites, Maxwell transformed his mathematical model into mechanical
reality. He had models constructed to demonstrate the motions of these satellites
as a disturbance travelled as a wave around a circle of the particles that made up a
ring.121
By 1860 Maxwell had moved beyond Stokes on two fronts. He was construct-
ing two theories, one of gases, the other of magnetic phenomena based upon the
particular physical behavior of specific mechanical models, pursued mathemati-
cally and interpreted physically. The first was on the interaction of the molecules
of gases, the other his vortex model of the action of magnets and of electromag-
netism. The first focussed on matter, the other was a material representation of
physical change. Both of these mechanical models led to predictions that Maxwell
followed up himself. Taken from Clausius, his kinetic model of a gas made up
of billiard ball molecules randomly colliding with one another led to predictions
about the thermal behavior of the transport properties of gases. The viscosity of
such a gas was independent of its pressure, a very unexpected result, and varied
as the square root of its absolute temperature. 122 While his experiments seemed to
confirm his model for a gas, Maxwell took them as only confirming the rightness
of his mathematical conclusions. They did not tell him anything of the validity of
his molecular model. 123
The steps from his vortex model of electromagnetic interactions to the labo-
ratory were less direct than those from kinetic theory to the measurement of the
transport properties of gases. 124 When Maxwell first began to study electricity and
magnetism, the range of phenomena to be covered by any mathematical theory was
orders of magnitude greater than in gas theory. Mathematical theories abounded to
121 See Maxwell, On the Stability of the Motion of Saturn's Rings (Cambridge: Macmillan
and Co., 1859), reprinted with commentary in Maxwell on Saturn's Rings, Stephen G.
Brush, C. w. F. Everitt and Elizabeth Garber, eds. (Cambridge MA.: MIT Press, 1983).
122 These earl y kinetic theory papers are reprinted with commentary in Maxwell on Molecules
and Gases, Elizabeth Garber, Stephen G. Brush and C. W. F. Everitt, eds. (Cambridge
MA.: MIT Press, 1986). His experimental work on viscosity became the Bakerian
Lecture of the Royal Society in 1866.
123 Maxwell, "Illustrations of the Dynamical Theory of Gases," Phil. Mag. 19 (1860): 19-
32,20 (1861): 21-37, and, "On the Viscosity or Internal Friction of Air and other gases,"
Phil. Trans. R. Soc. London, 156 (1866): 249-268.
124 Details of Maxwell's model and its mathematical expression are in Siegel, Innovation,
chap. 3. Maxwell, "On Physical Lines of Force. Part I The Theory of Molecular Vortices
Applied to Magnetic Phenomena," Phil. Mag. 21 (1861): 161-175, "Part II The Theory
of Molecular Vortices Applied to Electric Currents," same journal (1861): 281-291,
"Part III The Theory of Molecular Vortices Applied to Statical Electricity,"same journal
23 (1862): 12-24, "Part IV The Theory of Molecular Vortices Applied to the Action of
Magnetism on Polarized Light," same journal (1862): 85-95. Reprinted as Maxwell,
"On Physical Lines of Force," in Maxwell, Scientific Papers, vol. 1,451-513.
254 From Natural Philosophy to Physics
cover aspects of these phenomena but no one theory unified all of them. Michael
Faraday had shown that galvanic and static electricity were the same and extended
the work of Oersted and Ampere to show how mechanical force, current elec-
tricity and magnetism were inextricably linked. Maxwell began his mathematical
constructions in magnetism as he had with his fluid analogy. In working on the
mathematical expression of the mechanics of vortices, Maxwell found that the
mathematics representing the centrifugal force of the rotating vortices acted as the
magnetic force. The mathematical form representing the motions of the particles
acting as idler wheels between the vortices behaved as the electric current. The
mathematics that represented the changes in the velocity of the vortices was the
same as that for electromagnetic induction.
To complete the mathematical description of the full range of phenomena, Max-
well needed a mathematics of some mechanical property of his vortices that could
represent the phenomena of electrostatics and then link those to the equations
representing current electricity.12s To do this Maxwell endowed his vortices with
elasticity. With a medium endowed with elasticity, Maxwell could bring to bear
all the mathematics of the ether and capture light in his mechanical net by tracing
the propagation of elastic waves through his mechanical medium. His mechanical
model from which he could deduce the mathematics of known electromagnetic,
electrical, and electrostatic phenomena led him to predict the velocity of propaga-
tion of these elastic waves (the ratio of electrostatic to electromagnetic units) as
the velocity of light. He extended his mechanical model in other directions that
led to a relationship between the refractive index of a dielectric with its dielectric
constant. Neither prediction was brought to the experimental fruition Maxwell had
hoped for. 126
The mechanical model from which Maxwell developed a unified mathematical
theory of electricity, magnetism, and light depended on Stokes' understanding that
solids and fluids differed only in the degree of their mechanical properties, not in
kind. Simultaneously Maxwell used fluid-theory mathematics in his parallel devel-
opment of electromagnetism and a second version of his theory of gases. In his gas
theory he avoided the specifics of a molecular model until it was necessary to deduce
expressions for the transport coefficients. He then used a centers-of-force model
that allowed him to integrate a crucial equation and bring about mathematical and
physical closure. 127 While uniting a broad range of phenomena mathematically it
is unclear that Maxwell accepted these molecular models as representing physical
125 This was a crucial development as Wilhelm Weber's theory, the only serious rival to
Maxwell's, had accomplished just that.
126 Siegel documents Maxwell's responses to the relevant experiments in Siegel, Innovation,
155-158.
127 On the importance of the mechanics of fluids in Maxwell's physics see Maxwell on
Molecules and Gases, Garber, Brush and Everitt, 23-26.
Britain 1830-1870 255
reality. It was heuristic. In the manipulations of his equations, the mechanics of the
model allowed him to follow physical processes that were mechanically defensible
and consistent. Only mathematical manipulations with such mechanical counter-
parts were recounted and followed through in his papers. This was a search for an
understanding of the processes of nature, not a coherent mathematical description
of a broad, experimentally connected set of phenomena. Mathematical extensions
were through the development of the mechanical properties of the medium, not
the manipulation of equations. Mathematics was subordinated to the extensive
exploration of a mechanism. 128
Maxwell did not take these specific mechanical models as the image of nature.
He later reformulated both his gas theory and his electromagnetic theory of light.
He reworked them so that all his results depended on the general principles of me-
chanics, not on the particular characteristics of anyone model. 129 He also explored
the shortcomings of theories based on mechanical models. Maxwell preferred a
more Stokesian approach, basing his theories on general hypotheses that simply
assumed the phenomena only depended upon the configuration and motion of a
material system. 130
He used models to generate mathematical relations and guide the development
of the mathematics along physically defensible lines. The mathematics and model
together also allowed Maxwell to reach beyond known experimental evidence to
create new encounters between mathematics and experiments. Only mathemat-
ics guided by possible mechanical processes could lead to plausible outcomes.
Mechanical models were necessary for him to structure his physical theories. As
an image of nature and her operations, he found mechanical models less than
satisfactory.
Maxwell was both a theoretical and experimental physicist who made few con-
tributions to pure mathematics. Those he did make were in his Treatise and his
final paper on gas theory. In the Treatise, pure mathematics was confined to two
128 For his mechanical models in electromagnetism see Maxwell, "Physical Lines of Force."
For gases see Maxwell, "Illustrations," and, "On the Dynamical Theory of Gases," Phil.
Trans. R. Soc. London 157 (1867): 49-88, reprinted in Scientific Papers, vol. 2, 26-78,
and Maxwell on Molecules and Gases, 419-470.
129 For Maxwell's reformulation of his electromagnetic theory see Maxwell, "A Dynamical
Theory of the Electromagnetic Field," Phil. Trans. R. Soc. London, 155 (1865): 459-
512, and for the papers leading to his form of statistical mechanics, with commentary
see Maxwell on Heat and Statistical Mechanics: On "Avoiding all Personal Enquiries of
Molecules" Garber, Brush and Everitt, eds. (Bethlehem PA.: Lehigh University Press,
1995).
130 For Maxwell on mechanical models in general see Maxwell, Treatise on Electricity and
Magnetism (New York: Dover reprint of third edition, 1945), vol. 2, chap. 5. Maxwell
added a chapter, "On the Equations of Motion of a Connected System," in the second
edition of his treatise. For Maxwell's criticisms of specific molecular models including
vortices, see Garber, "Molecular Science in Late Nineteenth-Century Britain," 275-279.
256 From Natural Philosophy to Physics
131 Maxwell Treatise, vol. 1, chap. 5, "General Theorems," where there is a generalized
discussion of Green's theorem, and chap. 9, "Spherical Harmonics." Spherical harmonics
are also the subject of an addition to Maxwell, "On Stresses in Rarified Gases from
Inequalities in Temperature," Trans. R. Soc. London, 170 (1880): 231-256. Reprinted
in Maxwell, Scientific Papers, vol. 2, 681-712. See Maxwell on Heat and Statistical
Mechanics, Garber, Brush and Everitt, 77-78 for a discussion. Maxwell also published
on reciprocal statics, curvilinear coordinates, and the calculus of variations.
132 Maxwell "The Elements of Natural Philosophy by Professors William Thomson and
Peter Guthrie Tait (Macmillan and Co., 1873)," Nature, 7 (1873): 324-328, reprinted in
Maxwell, Scientific Papers, vol. 2, 324-328.
Britain 1830-1870 257
analogs were quickly built.133 He based his proposed classification of these analo-
gies on the physical effects he saw signified by the mathematical quantity. Curl
thus signified a rotation, convergence, a focussing at a point.
By the time of Maxwell's death in 1879, several approaches to and goals for the
study of nature using mathematics had emerged, several of which he had defined,
shaped and made credible. Mathematicians whose forays into analysis began in
problems of physics continued to use the new domains of physics for their own
purposes. They frequently inverted the direction of the development of these fields
by using physical problems to illustrate the uses of new mathematical domains.
This constituted "mathematical physics" as practiced by mathematicians. This
type of mathematics was quickly being reclassified as a branch of "applied math-
ematics," and relegated to a secondary place in a changing hierarchy of research
in mathematics. The newer pursuit of research into foundations constituted the
highest rung of the research ladder in "pure" mathematics. Mathematicians also
developed mathematical theories based in physical hypotheses which they claimed
were theories of physical phenomena that bore no relationship or points of contact
through experiment with any aspect of physical reality. These were not confined to
James Challis; it was even a characteristic British exercise in mathematics. Some
German mathematicians indulged in the same kind of research that Maxwell found
irksome and fundamentally useless.
Not all explorations of the purely mathematical kind that began in physical prob-
lems were without interest for those using mathematics to interpret the functioning
of nature. Mathematical conclusions, especially using particular cases, were ex-
amined for their physical content. These examinations were no longer done with
the casual assumptions of Poisson et aI., but required that the behavior of the orig-
inal physical entities encoded in mathematical symbolism be traced through the
mathematical manipulations. This was to make sure that those manipulations con-
sistently represented plausible physical processes and the conclusions were a truly
idealized version of real experimental circumstances. Such examinations began
with Herschel and Whewell and culminated in the work of Stokes who used such
a generalized approach masterfully.
To go further and generate physical meanings, not merely seeing them retrospec-
tively in mathematics already developed, required that each step in the evolution
of the mathematical operations, going from one equation to the next, carried phys-
ically consistent meanings. A certain mathematical operation always represented
a rotation, etc. In addition, this operation was only of interest if and when the
physical results of it were consistent with the general physical characteristics of
133 He extended this idea of mathematical analogy from algebra to geometry and formally
began the transition from quarternions to vector analysis. See Maxwell, "On the Math-
ematical Classification of Physical Quantities," Proc. London Math. Soc. 31 (1869-71):
224-232, reprinted in Maxwell, Scientific Papers, vol. 2, 241-266.
258 From Natural Philosophy to Physics
the system under study. Conservation laws for example and other accepted prin-
ciples had to be obeyed. Even if only such generalities were followed, the results
of mathematical manipulations might be unexpected. If these results were physi-
cally plausible, they were allowable. The paths taken by the mathematician were
hedged about by the needs of physical plausibility and consistency in the meanings
attached to mathematical operations. The meanings of mathematical operations,
as well as its symbols, were becoming infused with physical meanings. Every
mathematical result was reached to make a physical point, or bypassed. Or such
results might be passed over in silence and explored later before a mathematical
audience.
Further sUbjugation of the languages of mathematics to the purposes of physical
interpretation came by using the specifics of mechanical models. The range of
both mathematical and physical validity was more limited, although they might
be physically suggestive and insightful. Thomson's growing preference for this
approach meant that his work in physics was piecemeal, while the mathematics
that he used was consistent. He produced insights into physical problems without
developing any broad theories of any type of physical phenomena. 134
There were also cases where both physical imagery and the logical outcomes of
mathematical manipulation joined together more equably. The logical structure of
the mathematical manipulation suggested directly how a physical system ought to
behave. This was true in the development of thermodynamics and is most easily
seen in English in Maxwell's Theory of Heat. 135
Practitioners of all these different methods shared the assumption that mathe-
matics was central to understanding nature. All three approaches led to predictions
of new phenomena through the extension of mathematics guided by physical prin-
ciples, or models, beyond the confines of known behavior and experimental results.
Maxwell understood the power of this new use of mathematics that was indepen-
dent of the older experimental methods of natural philosophy. He also exploited
the potential of these predictions to justify this new discipline. He also understood
that the failure of mathematical results to meet those of experiment doomed the
theoretical enterprise and was a forceful argument against the use of theory in the
study of nature. It was however this new combination of mathematics, detailed
physical imagery, and experiment that characterized the new discipline. Experi-
mental results were integrated into the justification of using mathematics in one
way rather than in another through the behavior of postulated physical systems.
134 For the popularity of Thomson's vortex atom see Robert Silliman, "William Thomson:
Smoke Rings and Nineteenth-Century Atomism." For other atomic and molecular models
see, Garber, "Molecular Models."
135 Maxwell, Theory of Heat (London: Longmans Green, 1871). This text went through
several editions in Maxwell's lifetime. See the section in the later editions on Gibbs's
thermodynamic surface. See also Maxwell on Heat and Statistical Mechanics, Garber,
Brush and Everitt, eds. 48-51,232-247.
Britain 1830-1870 259
For Maxwell and his contemporaries, predictions did not necessarily mean that
the physical imagery that led to the experimental result represented the actual work-
ings of nature. Such philosophical subtleties were lost on following generations
who took for granted the power of models and hypotheses to mimic and predict the
operations of nature. This heady combination of mathematics, physical imagery,
and experiment characterized a new discipline. Yet the mix and actual usage of
these elements depended on the individual and even the specific problems under
study. In Britain this range remained within the practices of theoretical physicists
throughout the nineteenth and into the twentieth century. Mechanical models of
the ether were not the only characteristic of British physical thought as Pierre
Duhem would have us accept.136 The centrality of mathematics to this new dis-
cipline meant that new developments within mathematics might have immediate
implications for theoretical physics. And withina generation, vector analysis had
such an effect in the reformulation of electromagnetic theory by Oliver Heaviside
and Josiah Willard Gibbs. 137
In Britain the inheritance of French mathematical physics was transformed
through the traditions of natural philosophy. The descriptive function of French
mathematical physics was inadequate. Philosophical issues were fundamental to
and necessary for the development of theoretical physics, as was training in the
mathematical languages. All practitioners of this first generation of theoretical
physicists shared training in French mathematical physics and natural philosophy.
For those educated at Cambridge, especially at Trinity, the philosophical interests
of Whewell were never far from the surface of their experience and their letters.
This reenforced the philosophical discussions within the demonstration courses
in natural philosophy that all were exposed to both before and during their math-
ematical training. These factors, together with the contingencies of developing
careers in the middle of the nineteenth century in the new professions within the
sciences, converged to give a few extraordinary individuals opportunities for the
136 For example, in the 1890s George Hartley Bryan argued from general principles that no
mechanical model could represent the behavior of the second law of thermodynamics.
In Britain, at least, mechanical mimicry of thermal systems ended. Through the same
papers efforts to propose mechanical models to avoid the implications of the equipartition
theorem came to a similar end. See Brush, The Kind ofMotion we Call Heat (New York:
North-Holland, 1976),2 vols., vol. 2, chap. 10.
137 On Oliver Heaviside see Bruce Hunt, The Maxwellians, chap. 3. For Gibbs on vectors
see Gibbs, Elements of Vector Analysis, arranged for the Use of Students of Physics
(New Haven, 1881). Reprinted in Gibbs, The Scientific Papers ofJ. Willard Gibbs Henry
Andrews Bumstead and Ralph Gibbs Van Name, eds., ( New York: Dover reprint of
1906 edition) 2 vols., vol. 2, 17-90. While printed privately Gibbs' text on vectors fell
into the hands of Tait and Gibbs defended vectors in a series of papers in Nature in 1891
and 1893. Reprinted in his collected papers. On the mathematical history of vectors see
Michael J. Crowe, A History of Vector Analysis: The Evolution of the Idea of a Vectorial
System (Notre Dame: University of Notre Dame Press, 1967).
260 From Natural Philosophy to Physics
By 1830 in the German States the new opportunities to make careers within the
sciences and mathematics were secure within the reformed universities. Teaching
was no longer the only expected activity of faculty. Research had become a defining
quality for appointment to and promotion within universities across the German
states. This ideal was complicated by the justification for continued state support
for these renewed institutions, the training and certification of future teachers in
the lower echelons of the education systems within each state. l
Entry into the research community in the sciences required a student to navigate a
series of formal steps. He had to be trained and certified as competent in a specialty
by a university. 2 The specialist training was offered in demonstration lectures and
seminars, and the first laboratory courses in the experimental sciences. Seminars
and laboratories were apprenticeships in the practices of the disciplines. Ultimate
certification was through the acceptance of a PhD dissertation. The dissertation
certified the ability of the author to understand a body of material, identify a
problem, then bring to bear and manipulate the methods of that specialty in the
search for its solution. 3
These general remarks outline the formalized academic training at any Ger-
man university in any scientific discipline. What redefined both mathematics and
1 Jungnickel and McCormmach, Intellectual Mastery of Nature give the details of the
formalized, German academic system as it affected the members of the discipline of
physics, and then the problems of physicists in obtaining adequate support. The first
volume covers the period of this chapter.
2 It was no longer possible for someone like August Crelle for example, who studied math-
ematics privately, to obtain their PhD. Nor were the cases of Friedrich Wilhelm Bessel
and Joseph Fraunhofer repeated, where two men who began their lives as craftsmen over-
came social and intellectual barriers to gain appointments as professors at universities.
Hermann von Helmholtz was trained as a physician, i.e., in a learned profession, so that
his transition to physics was possible.
3 Certification in teaching in a university required a further dissertation, the Habilitation.
physics lay in the details of the practices developed by mathematicians and physi-
cists as they confronted specific research problems and passed the skills of solutions
on to their students. Precisely what the practices of mathematics and physics were
to become depended upon the subject matter of the lectures, the methodologies
taught in the seminars and laboratories of these same universities. There was no
consensus about these crucial, defining elements. What specifically was taught
and how depended on the occupants of the chairs, the heads of seminars, and the
directors of the research institutes. They introduced their experiences and prac-
tices to their students and passed on to them their particular sense of the important
problems in mathematics and physics, as well as how to attack them. Distinct
traditions in mathematics were crystallized in the same decades. 4 Equally distinct
traditions of research arose in physics in the universities of Konigsberg, Berlin and
Bonn. The older historiography of physics in the German university system placed
the mathematics-physics seminar at Konigsberg, begun in the 1830s, as the unique
source for the development of research in German physics and mathematics. It
was the first, but its influence on the subsequent development of the disciplines is
more difficult to trace. The actual impact of the physics half of this seminar and
the laboratory, also conducted by Franz Neumann, is harder to judge than even the
impact of Jacobi's training in the mathematics seminar.5 Neumann trained future
gymnasium teachers as he was creating the methods to do so. He inspired many
students but student publications emerging from that seminar were few compared
to those from the mathematics seminar. Recent research indicates that modern
forms of physics developed more gradually and in several centers. 6
The model from which both physicists and mathematicians began was that of
4 The main rival traditions in mathematics developed at the universities of Berlin and
Gi:ittingen. Joseph Dauben, "Mathematics in Germany and France in the Early Nineteenth
Century: Transmission and Transformation," in, Epistemological and Social Problems,
Jahnke and Otte, eds. 371-399. Dauben restricts his comments on German mathematics
to Berlin. For later decades see Thomas Hawkins, "The Berlin School of Mathematics,"
in Social History of Nineteenth-Century Mathematics, Mehrtens, Bas, and Schneider,
eds. 233-245. On Gi:ittingen see David E. Rowe, "Klein, Hilbert and the Gi:ittingen
Mathematical Tradition," Osiris, 5 (1989): 186-213.
5 The number of Jacobi's students were small but some lived the same ideal of research
within the German university system, even though the next generation of mathematics
seminars was not established until the 1860s. His research approach was repudiated by
later generations of mathematicians centered at the University of Berlin who dominated
the profession from the 1860s to the 1890s. Assessment is also clouded by the symbolic
value of Jacobi for liberal historians and for mathematicians. During the revolution of
1848 Jacobi ran afoul of the King of Prussia and lost the support of other academics in
the Prussian university system. His jewish background added to his problems, although
that is softpedalled in all the accounts. See Alexander von Humboldt and Carl Gustav
Jacob Jacobi, Briefwechsel, (Berlin: Akademie Verlag, 1987).
6 The range of the training in physics offered by different universities is given in Jungnickel
and McCormmach, Intellectual Mastery of Nature. See Kathryn Mary Olesko, The
German States, 1830-1870 263
the French. The French gave German experimental physicists a model of practice,
from measurement techniques to interpretations of experimental results. French
mathematics and mathematical physics offered both German physicists and math-
ematicians a series of research problems and forms for their solutions. 7
Carl Gustav Jacob Jacobi epitomized the opportunities of the reform university
and redefined the mathematics of the university as "pure," then successfully trained
students in this new brand of mathematics. Jacobi captured the ideology of the new
system and then demonstrated his commitment to this ideal in his early research
into transcendental functions, his methods for the solution of partial differential
equations, and his teaching in his mathematical seminar through research. Students
in the seminar produced publishable research that usually appeared in Crelle's jour-
nal. The origins of Jacobi's research were in the mathematics of Euler, Lagrange,
and Legendre, among others. Physical problems and their direct expression in
mathematical form was not the starting point for his mathematics. 8 Physics was
reduced to the role of supplying illustrative examples for his methods.
Jacobi was not initially interested in mechanics and was only drawn to it by the
work of William Rowan Hamilton. Jacobi had labored for some years to develop
a general method of solving partial differential equations of the first order in n
unknowns. Building his methods as he went he generalized his findings for three,
five, then more unknowns. In 1834, just as his research was beginning to bear
fruit, it looked as though he had been anticipated by Hamilton. Hamilton began
with Lagrange's analytical formulation of mechanics, based upon the principle
of Least Action, restricting himself to time-independent central forces. He then
examined the variation of the time-Action integral V written as a function of the
initial and final coordinates of the path of a particle. He then varied the end-points
of the path and found linear, partial differential equations in V, the initial and final
coordinates of the system, and H, a constant. Hamilton reduced the problem of
9 Hamilton's final version was also reached through successive attacks on a specific prob-
lem, the mathematical problem of the perturbed orbit of a planet. His goal was to develop
a conceptually simpler mathematical method than those currently available. See Robert
Percival Graves, Hamilton.
10 Jacobi, "Uber die Reduction der Integration der Partielle Differentialgleichungen erster
Ordnung zwischen irgend einer Zahln Variabeln auf die Integration eines einzigen Sys-
tems gewohnlicher Differentialgleichungen," J. ReineAngew. Math. 17 (1837): 97-162,
and, "Note sur l'integration des equations differentielles de la dynamique," Comptes
Rendus 5 (1837): 61-67. The paper was translated in full as, Jacobi, "Sur la reduction
de l'integration des equations differentielles partielles du premier ordre," J. Math. Pures
Appliques 3 (1838): 60-96, 161-202. It was still important for German mathematicians
to have their worked noticed by the French. Jacobi was careful to send his earlier work
to Legendre, whose informed appraisal of it eased Jacobi's appointment at Konigsberg.
11 These lectures were edited and then published posthumously as, "Lectures on Dynamics."
This title might well be that of the editor rather than Jacobi. See Carl Gustav Jacob
Jacobi, Vorlesungen iiber Dynamik, Alfred Clebsch, ed. In Jacobi, Gesammelte Werke,
Karl Theodor Wilhelm Weierstrass, ed. (New York: Chelsea Pub. Co., reprint of second
edition, 1969), vol. 8.
German States, 1830-1870 265
12 Such claims were made initially in histories of mathematics written largely by mathemati-
cians themselves. Many of those mathematicians were strongly allied to Felix Klein's
views of mathematics and its relationship with physics. See also Paul L. Butzer, "Dirich-
let and his Role in the Founding of Mathematical Physics," Arch. Int. Hist. Sci. 37 (1987):
49-82. Butzer's argument cannot be sustained on two grounds. In considering only the
German context the French origins of mathematical physics is ignored. In addition, given
the definitions of mathematics of the era, the work is mathematical, not physical.
13 Not even Fourier as secretary of the Academie was able to obtain a post for Dirichlet in
Paris and he returned to Berlin.
14 For the interaction of German mathematicians with Fourier's work, see Garber, "Reading
Mathematics, Constructing Physics," in No Truth Except in the Details, Kox and Siegel,
eds.31-54.
15 Dirichlet, "Bedingungen der Stabilitat des Gleichgewichts," J. Reine Angew. Math. 32
(1846): 85-88.
266 Physics and Mathematics
16 Dirichlet, "Uber ein neuen Ausdruck zur Bestimmung der Dichtigkeit einer unendlich
dunnen Kugelschale, wenn der Werth des Potentials derselben in jedem Puncte ihrer
oberflache gegeben ist," Ber. Berlin (1850): 99-116, published in fuJI in J. Math. Pures
Appliques 2 (1851): 57-80. Dirichlet also improved the methods of the new calculus.
In 1829 he critiqued Cauchy on his work on the convergence of trigonometric series
and then made convergence tests more rigorous. Dirichlet, "Sur la convergence des
series trigonometriques qui servent a representer une fonction arbitraire entre des limites
donnes," J. Reine Angew. Math. 5 (1829): 157-169, and, "Solution d'une question
relative a la theorie mathematique de la chaleur," J. Reine Angew. Math. 5 (1829):
287-295.
17 Dirichlet, Vorlesungen fiber die im umgekehrten Verhiiltnis des Quadrats der Entfernung
wirkenden Kriifte, F. Grube, ed. (Leipzig: B. G. Teubner, 1876).
German States, 1830-1870 267
ematical work was recognized later as important for physics. This subsequent
importance cannot be used in retrospect to color either his intentions or our own
reading of his papers.18
Bernhard Riemann's goal of producing a mathematics that united the whole of
physics was a mathematical exercise and seems remote from the mathematical
needs of physicists in the middle decades of the nineteenth century. Riemann
studied physical questions to develop a "self-contained mathematical theory" that
encompassed all of mechanics, thermodynamics, electricity and magnetism and he
did not distinguish between them. 19 Physicists were left to themselves to decipher
the physics locked within this mathematical physics and reorient that language for
their own purposes.
Much the same can be said of Riemann's text on partial differential equations and
"their application to physical problems." These were lectures on definite integral
solutions to the partial differential equations of Fourier's theory of heat. Riemann
initially established all the required mathematical techniques, then launched into a
section on ordinary differential equations before turning to the issue of linear partial
differential equations of the second order. He turned his attention to Fourier by
setting up the most general form of the partial differential equation for the flow of
heat in three dimensions. He took a series of mathematically special cases that lead
to definite integral solutions. All were mathematically defined through systematic
restrictions-mathematical boundary conditions on the generality of the equation
he first started with. None of these mathematical developments were accompanied
by a hint of a physical explanation. The physicist would have to figure out what the
various sets of boundary conditions meant in physical terms, then chase down the
physical meanings of the processes represented by the accompanying mathematical
details. What use this might be to a physicist in this era is far from obvious and rather
is a counterexample to the claims of historians that many German mathematicians
worked on physical problems or in areas that physicists would find useful. 2o
A more accessible example of Riemann's approach to the solution of physi-
cal problems was in the essay of 1860 submitted in a prize competition of the
18 As a antidote to this see Oystein Ore, "Gustav Peter Lejeune Dirichlet, 1805-1859," Diet.
Sci. Bio., vol. 4, 127, where the mathematical focus of Dirichlet's work is emphasized,
even as the writer noted where the solutions were important for physics, without stating
where or how.
19 See Thomas Archibald, "Riemann and the Theory of Electrical Phenomena: Nobili's
Rings," Centaurus, 34 (1991): 247-271,259.
20 B. Riemann, Partielle Differentialgleichungen und deren Anwendung auf physikalisehe
Fragen, K. Hattendorff, ed. (Brunswick: Friederich Viewig und Sohn, 1896) The text
consisted mainly of Riemann's lectures for the winter semester of 1860-1861. Because
they were edited after Riemann's death, the title mayor may not have been his choice.
The lectures are another example of the impact of Fourier and physical problems on the
development of mathematics.
268 Physics and Mathematics
Academie des Sciences in Paris on the theory of heat. The essay demonstrates
the cryptic nature of Riemann's mathematics, criticized by the Academie, and the
reason he was not awarded the prize. Riemann stated the general equation for the
conduction of heat in three dimensions but did not obtain particular solutions to
the general partial differential equation as the starting point of the mathematical
exercise. Instead he investigated, on the most general level possible, methods
to reach particular solutions. He was interested in the characteristics of classes
of particular solutions rather than the solutions themselves. These classes were
then described in geometrical terms. In some cases Riemann demonstrated how a
certain type of solution might be obtained in principle. Riemann apologized for
the incompleteness of his solutions and blamed his already deteriorating health.
This approach shows Riemann's concerns. His interests lay in function theory and
mathematics, not the physical meaning of the solutions he obtained. 21
These arguments can be extended to a discussion of Rudolph Friedrich Alfred
Clebsch's work in physical subjects, particularly elasticity, one subject of interest
to contemporary physicists working on light and the properties of the aether.22 No
contemporary physicist working in this area refers to, or appears to use, Clebsch's
work.
Finally we must consider the reputation of Gauss for publishing significant
results in "theoretical" physics. If we look at Gauss's mathematical work on
physical problems, we find that they are mathematical exercises. His paper on
inverse square forces was an exercise in the mathematics of the potential function.
After working on a mathematically general level, Gauss turned to the particular
cases of gravitation electrostatics and magnetism and lumped them all together. The
mathematical results covered all physical cases as a "special case of a particular
solution," a phrase only a mathematician could use. 23 In the body of the paper
21 Ruth Farwell and Christopher Knee, "The missing Link: Riemann's 'Commentatio',
Differential Geometry and Tensor Analysis," Hist. Math. 17 (1990): 223-255. The
authors' main contention is that this essay is not an early intimation of his work in
differential geometry and tensor analysis. They present an English translation of the
essay in an appendix.
22 Alfred Clebsch, Theorie der Elasticitiit Jester Korper (Leipzig: Teubner, 1862).
23 Gauss, "Allgemeine Lehrsatze in Beziehung auf die im verkehrten Verhaltnisse des
Quadrats der Entfernung wirkenden Anziehungs- und Abstossungs-Krafte," Resultate
(1840): 1-51, reprinted in Gauss, Werke vol. 5, 196-242, 241. Translated in Taylor's
Scientific Memoirs 3 (1843): 153-196. Ch.-J. de la Vallee Poussin, "Gauss et la tMorie
du potentiel," Rev. Quest. Sci. 23 (1962): 315-330, also sees Gauss' work in potential
theory as a branch of "pure mathematics." Kenneth O. May, "Gauss, Carl Friederich,
1777-1855," Diet. Sci. Bio., vol. 5, 298-314, argues that this was "the first systematic
treatment of potential theory as a mathematical topic." May also considers that it was im-
portant for the rigor Gauss introduced into the subject. More recently Thomas Archibald,
in "Physics as a Constraint on Mathematical Research," in The History ojModern Math-
ematics, David Rowe and John McCleary, eds. 2 vols, vol. 2, 29-75, argues that this
German States, 1830-1870 269
were a number of theorems proving the existence and continuity of the potential
function. There were also theorems for transforming volume integrals into integrals
of functions over corresponding surfaces. The paper was the formal demonstration
of the mathematics he had published on the characteristics of the magnetic potential
over the surface of the earth.
Gauss' work in geomagnetism fell into two types. The first, done in conjunction
with Wilhelm Weber, was in the development and use of more accurate methods for
observing the components of the earth's magnetic field. In reporting these methods,
Gauss included long tables of the results of these observations. Along with these,
copious tables of scrupulously observed data were exhaustive discussions of the
errors of observation of very small quantities and methods to minimize them. These
demonstrate the general German concern with accuracy and Wilhelm Weber's
passion for "measurement physics." The papers also convey a sense of a developing
confidence in data that grew from such discussions of errors and their careful
minimization. Gauss was one of the first to initiate such extensive discussions of
observational error in his work on geodesy for the Hanover government. 24 At the
same time Gauss was supervising the establishment of the observatory at G6ttingen.
Until the year of his death Gauss observed regularly, reduced his own data, and
reported his results.
Gauss was unusual in that he undertook imaginative work in physics, considered
as the complex of observational and experimental sciences, and mathematics. 25
This eighteenth-century definition of disciplinary boundaries fitted Gauss' own
sense of his relationship to the State of Hanover and his attitude towards the publica-
tion of his mathematical work. 26 None of these activities suggest that Gauss sought,
let alone established, any changes in the disciplinary boundary between the obser-
vational and the mathematical sciences. Studies within the observational sciences
gave him ample opportunity to explore mathematical problems. To change the
boundaries required the injection of specific physical notions that Gauss rejected.
This division of labor was reinforced in his mathematical paper on the earth's
magnetic field. Given that there were two magnetic poles, Gauss discussed the
mathematical characteristics of the magnetic potential. His discussion included
the general, closed form of the equipotential lines on the surface of the globe. Al-
gebraically he represented the potential by an infinite series of spherical functions.
The lengthy tables of data included from the world-wide network of magnetic
observations enabled Gauss to evaluate the first twenty-four coefficients in this
series. Thus, the mathematical mapping of the magnetic potential was possible.
Its physical meaning remained a mystery. Gauss quite specifically separated his
theory from all current models of magnetism that might have led to the specific
form he explored mathematically.27
We can perhaps now understand the mixed reception given his work in terrestrial
magnetism. The carefully collected data and his observational methodology fitted
the criteria for quantitative work within "Humboltian" science. He did not offer
the physical qualitative explanatory interpretation that usually accompanied such
observational results. The "theory" was not a physical theory at all but an exercise
in mathematics and a piece of research at that. No wonder the group interested in
the geosciences were mystified. 28
The one area in which Gauss was actually a physicist in the modern sense was
in his work on the measurement of the earth's magnetic force in absolute units.
In this he acknowledged the help of Weber who extended his own interests in
the measurement of non-mechanical phenomena in his work on electricity. This
was, however, a paper on the problems of experimental physics and included no
theory on the nature of the earth's magnetic field. Terrestrial magnetism remained
a mystery.29 Therefore, the arguments made by Jungnickel and McCormmach
cannot be sustained with respect to Gauss, Dirichlet, or Riemann. They give only
general statements as to the necessity of higher mathematics being recognized by
physicists, but no statements from specific physicists on this subject. Nor do they
give specific examples of physicists using the mathematical methods they claim as
important for them in this era. 30
Most historians of mathematics and physics follow the statements of the earliest
historical accounts and accept all work in mathematical physics as related to and
pertinent for the development of physics. This is not the case for physicists working
in the middle decades of the nineteenth century and the research of their mathe-
matical colleagues using the results of their experiments to generate mathematical
puzzles. Even if that mathematical research began with a problem of interest to
contemporary physicists, the solution was not the source for the development of
physical understanding of the problem. Only in retrospect was such abstract math-
31 Later in the century, Felix Klein wrote of the work of mathematical physics done by
the above mathematicians as a part of mathematics. He regarded the work of Clausius,
Kirchhoff, and Helmholtz as research that required mathematics but was of a different
order and did not call it mathematics. Felix Klein, Vorlesungen iiber die Entwicklung der
Mathematik im neunzen Jahrhundert (Berlin: Springer, 1927.) Klein did not investigate
those differences.
32 By foundational issues I mean the examination of the mathematical meanings of the
foundational ideas of mathematics, such as number, space, function etc.
272 Physics and Mathematics
physics within, teach themselves those methods and how to use them in their own
areas of research. They also needed to create the institutional forms to train the
next generation of German physicists in those methods and interpretations. While
experiment remained the disciplinary core of physics, the relationship between
experiment, speculation, and the mathematics that grew out of those experiments
and speculations was about to change dramatically.
Some of the first indications of change came in Gustav Theodor Fechner's Reper-
torium der Experimental Physik. 33 This was a report on the state of research in
physics. Fechner focussed upon those references that were difficult to acquire, and
he had clearly as he claimed read the originals. The table of contents promised
a survey that seems extremely modern. The layout of the text betrayed his own
preferences for French methodology and Kantian metaphysics. Yet the foundation
of physics remained as experiment coupled with vernacular interpretations. He
recounted Poisson's theory of matter as atomic, with bodies made up of particles
of "imperceptible size" which even in the aggregate were still "imperceptible" and
between which were the material stuff of electricity, magnetism, and heat. 34 These
particles attracted each other and the particles of heat, but this aspect of the theory
was not emphasized. Cauchy's ideas on matter were dealt with similarly. Fech-
ner's account of the mathematics of heat, electricity, magnetism, and mechanics
were given separately. However, Fechner claimed that Poisson and Cauchy treated
the equations of motion of bodies on the basis of their physical assumptions.
Fechner's description of the physical ideas and their mathematical development
were necessarily short and schematic, and he did not address how the two were
connected. What Fechner did give was a comparison of the content and the results
of various mathematical theories. He also related the mathematical benefits of
the various approaches. Cauchy treated elasticity very generally and never got
down to "particular" problems. Fechner reported that Navier obtained differential
equations without bothering to integrate them to find further cases for application.
In comparison Poisson put these differential equations into a more general form,
then integrated them anew and found various applications for them although the
results were "less than practical." Fechner also included a bibliography of the
literature on the integration of partial differential equations.
Fechner began to compare the results of the mathematics directly with those of
experiment. He confronted the results of French mathematics and tried to extract
physical significance from that mathematics. In this he had to argue which phys-
ical cases were encapsulated in the mathematics of Cauchy, Navier, and Poisson
without giving the analytical details. He claimed to get mathematically deduced
frequencies of the various oscillation of bodies from Poisson and Cauchy. He
used one or the other as it suited his needs without telling the reader how he got
the frequencies he reported. From Poisson he took the expression for the ratio
of the longitudinal to translational elastic moduli and frequencies of oscillation.
Fechner took data from various experiments and directly compared the mathe-
matical and experimental numbers, without giving the criteria he used to decide
whether the mathematical cases actually matched the idealized experimental con-
ditions. The numbers deduced from the two sources were listed in tables without
comment.
It is clear that Fechner worked through the mathematics to extract the physics.
However, he used results from any and all mathematical sources without referring
back to theory and passing on a sense of a theoretical image carried through the
mathematical deductions. He had taken mathematics and transformed it where
possible into a discussion about physical entities. Mathematical results were made
over into physical characteristics and compared directly with what physicists actu-
ally measured in their experiments. There is no sense that mathematical physics is
actually a part of physics not mathematics. Lest we expect too much, many pages
of the Repertorium is unremitting reportage of experiments, especially in the areas
of his own research, galvanic electricity and its connection with chemistry.35
This was an important change but was not echoed in the work of his immediate
contemporaries. Both Henri-Gustav Magnus and Heinrich Dove, two of the more
prominent examples, were experimentalists. Helmholtz's biographer noted that
Magnus,
regarded experimental and mathematical physics as separate depart-
ments and warned him [Helmholtz] repeatedly against undue partiality
for mathematics, and the attempt to bring the remote provinces of physics
together by its means. 36
35 In this respect it is interesting that Fresnel's work on light appeared only in the experi-
mental section, not in his account of the mathematics of light where Fechner discussed
only the work of Poisson and Cauchy.
36 Leo Koenigsberger, Hermann von Helmholtz (New York: Dover Pubs., reprint of 1906
edition, 1965). Translated by Francis A. Welby, 38. See Jungnickel and McCormmach,
274 Physics and Mathematics
Franz Neumann
It is not with Fechner that we see a sustained confrontation between the needs
of German experimental physicists and the works of the French mathematicians.
Franz Neumann succeeded in connecting mathematical to experimental results
in the context of a new form of mathematical physics. 38 Yet it took Neumann
decades to develop a method that made the characteristics of the physical phenom-
ena limit the mathematics and guide its development. He developed one approach
to bending the mathematics of the French to the needs of the physicist, based on
his assumption that physical meaning lay in the mathematics without the need for
speculations about the operations of nature. He rejected the hypothetical ram-
blings of Naturphilosophie and eschewed, as far as he could, all speculations about
physical processes. If he used hypotheses, he kept them to a minimum and they
were of the most general kind. While Fourier was clearly a major influence on his
mathematics, he used the results and methods of Poisson and others as it suited his
immediate purpose. Physics was more important than mathematical consistency.
Despite his development of a new form of mathematical physics, for Neumann
the core of physics was measurement. Mathematical physics was a complement to
this core. His first accomplishment in mathematical physics was the extension of
Fourier's methods into the related physical domain in crystals. Fourier generalized
his mathematics through idealized physical examples. Neumann took the physical
specifics from the laboratory. He first generated a mathematical expression of an
experimentally defined relationship or process, then moved on to a more general
physical and geometrical arrangement. The physical elements changed from linear
to circular to a more general shape and their relationships changed from planar
to three-dimensional. Neumann put these experimental, geometric possibilities
directly into mathematical form. In general he did not explore the mathematics
any further than necessary to delve into those particular physical cases.
Neumann was practicing a form of mathematical physics grounded in physics.
Yet there was, for example, no discussion of flux as a physical concept. All such
physical ideas were treated as mathematical functions. In crystals these were de-
pendent on the geometric symmetries of the crystal. Unlike Fourier, Neumann
usually could not develop mathematical expressions from which all physical cases
would then follow. He hedged the mathematics about with the limitations of his
chosen physical cases. Neumann stuck closely to possible experimental configu-
rations. His examples were physically connected yet sat in mathematical isolation
from one another. This necessarily limited what he allowed the mathematics to
accomplish. The mathematics was also dense, less than elegant, and always tied
to particulars.
Neumann repeated this pattern in his work on crystal optics in which he relied
upon the experimental results of Fresnel and others. He could no longer avoid using
some hypothesis about the nature of light, but he avoided any assumptions about
the interaction of light and matter in the interior of solids. Neumann could only
replicate the experimental results established by Fresnel. He also concluded, as had
his predecessors, that Fresnel's wave front was necessary for the production of such
phenomena. Neumann's mathematical attempts at understanding the polarization
phenomena of solids shared the problem of physical implausibility with those of
the mathematicians that preceded him whose work became the starting point for
his own research. 39
Neumann chose Navier's hypothesis that the displacement force acting on a
particle of a solid was proportional to its displacement. For crystals Neumann
generalized this force to a function of the angle between the direction of the dis-
placement and the crystal axis. He then treated this function with all mathematical
generality, apart from the mathematical simplifications introduced by assuming the
displacement was very small. 4o He arrived at a generalized equation of motion for
39 For a statement of the generic problems of Neumann's approach to the theory of light see
Whittaker, A History of the Theories ofAether and Electricity vol. 1, 136-139. Whittaker
points to the arbitrary physical nature of theories that tried to reconstruct experimental
results mathematically, beginning with Cauchy and finishing with McCullagh and Neu-
mann. To see the physical content of Neumann's argument, see his rejection of Fresnel's
assumption that refraction is due to changes in the density of the aether in different me-
dia. Neumann, "Theoretische Untersuchung der Gesetze, nach weJchen das Licht, an der
Grenze zweier vollkommen durchsichtigen Medien refiectirt und gebrochen wird," Abh.
Akad. Wissen. Berlin (1835): 1-160, 7-8.
40 Neumann, "Theorie der dopplten Strahlenbrechung, abgeleitet aus den GJeichungen der
Mechanik," Ann. Phy. 25 (1832): 418-454.
276 Physics and Mathematics
a disturbance propagated through the medium which was characterized by six con-
stants. He deduced the relationships between the constants, but not their numerical
values. The latter Neumann noted was done through experiment. When Neumann
examined the case of a homogeneous solid, the equation of motion reduced to
Navier's. So far this was mathematics.
Neumann explored the problem further only for physically plausible cases. He
investigated homogeneous, non-crystalline and crystalline solids where functions
reflected the symmetry characteristics of particular materials. He then faced inte-
grating experiment with mathematics more directly. He began with empirical laws
and the same assumptions with which he structured his mathematics and tried to
establish Fresnel's expression for the polarization of light reflected off metals. 41
Neumann hoped to replicate Fresnel's trigonometric expressions for the ratio of the
intensities and amplitudes of the incident and reflected rays along with their known
directions of relative polarization. As with previous mathematicians Neumann had
to replicate Fresnel's expression for the shape of the wave front within the solid.
Neumann then extended this to crystalline substances by generalizing the behavior
of the elasticity of the aether. Elasticity became a function of direction.
What separated Neumann's work from earlier mathematical theories of light
was the way he constructed his mathematics. He considered a series of specific
physical cases, each more complicated than the last. He moved from homogeneous
solids, where he could directly replicate Fresnel's results, to crystals. In both he
dealt with incident, reflected, refracted rays with concomitant intensities, ampli-
tudes and polarization directions. While more complicated to visualize and keep
track of on the page, Neumann constructed those quantities for some of the solids
whose characteristics Fresnel had already measured. Neumann could compare
his mathematical results directly with experiment. Mathematically, he did not go
beyond reconstructing these quantities. Thus his mathematics was limited.
Neumann's method becomes clearer in his 1841 paper on light. His handling of
the problems of optics was surer when he connected them, first analogically, then
directly with the conduction of heat. Here there were mathematically more general
cases. But because Neumann would not investigate hypothetical mechanisms that
might account for the change in the character of the light in its passage through
the solid, many critical constants in his equations were taken from observations
already at hand. Only the consistency of these equations could be tested. It was
by no means a complete theory or mathematical description. 42
The same limitations surfaced in his work on electromagnetic induction. Neu-
41 Neumann, "Theorie der elliptischen Polarisation des Lichtes, weIche durch Reflexion
von Metallfliichen erzeugt wirden," Ann. Phy. 26 (1832): 89-122.
42 Neumann, "Die Gesetze der Doppelbrechung des Lichtes in comprimirten oder ungleich-
formig erwiirmten uncrystallinischen K6rpern," Abh. Akad. Wissen. Berlin (1841) Part
II: 1-254.
German States, 1830-1870 277
mann developed mathematical expressions for the interaction of primary and sec-
ondary circuits and of circuits and moving magnets of evermore geometrical com-
plexity. He did this through consideration of the specific geometries of the circuits,
or magnets, and the geometries of their motions. His cases were limited to closed
circuits, or their equivalent magnetic fields. The induced emfs were from the
movement of either circuit or magnet. Neumann assumed that the induced current
was proportional to the velocity of the motion. The proportionality factor, L, was a
function of the Amperean force on the secondary circuit because it changed its sign
with any change in direction of the motion of the moving element. The simplest
form of dependence of L on this force was linear. Therefore, "the intensity of the
induced current is proportional to the component of the electrodynamic force in
the direction of motion." His justification for this assumption was that it worked.
Neumann could reproduce the known laws of induction. 43
Lenz's and Ampere's experimental laws allowed Neumann to set up a general
equation for a series of relationships between the elements of two circuits moving
with respect to one another in a completely general geometrical sense. The result
of the movement of one or both of the elements was an induced emf. Neumann
took the simplest case of linear circuit elements. Having got an expression for the
induced emf for that case, he considered what happened when each expression for
the induced emf changed in time, that is, if the velocity of the moving element was
not constant, or the current in the primary was a function of time. He then escalated
the cases to non-linear circuits. Here his expression for the induced emfs were all
local. To obtain the effect of the whole circuit line integrals were necessary. This
took his mathematical case from infinitesimal elements to the physically probable
and observable. His analysis began with carefully chosen directions to the motions,
and coordinate system etc., to give him known currents in known directions. Any
physical constants in these initial equations came from experiment. However, any
coefficients introduced as the mathematics developed or constants introduced into
the integration were defined mathematically, their physical significance remaining
unexpressed. Their values are left in general integral form. Neumann deduced
known, physical cases and could replicate known experimental results. His work
here paralieled his earlier accomplishments in mineralogy using geometry. What
Neumann did accomplish was the transformation of the physical problem directly
into differential form without the intervention of any physical hypothesis.
In this first paper he did not rise above the consideration of particular cases
until he reworked the whole by following the mathematical implications of one
of his examples. Here Neumann's use of Fourier was explicit and successful.
He constructed an expression for the induced current, then reconstructed it by
considering the flow of electricity into an element, ds, of a closed circuit, in this
case a ring, in a short period of time. The primary induced a difference in "electric
tension," U, between two elements of the circuit that produced an emf, E. The
fluid thus had a tendency to move from a region of high to one of lesser tension,
and
dU
E=--.
ds
The quantity of electricity traversing a cross section q of the element ds in unit
time was, -q k (dU Ids), where k was the velocity of flow of the electricity for a
unit emf in unit time.
Neumann then used Fourier's argument for constructing the flow of heat in a ring,
substituting electricity for heat, and tension for temperature. 44 He then constructed
an expression for the accumulation of electricity in this section as
d 2-
qk ( - U - -dE) ds,
ds 2 ds
where, U = U(s, t), was the potential effective at ds at time t. Here U was anal-
ogous to temperature in Fourier's work. At this point Neumann parted company
with Fourier. He did not examine the flow out of this element. Neumann simply
equated the increase in electricity to q ds(dU Idt) and equated the two expressions,
obtaining
dU = k (d 2 U _ dE).
dt ds 2 ds
He then constructed mechanical expressions that related the induced current to
the electrodynamic force that produced the induced current. This was one of the
simpler examples where the circuit and its motion were geometrically restricted.
More general motions and circuit geometries led to more intricate mathematical
expressions.
When he examined the expression for the current induced in a circuit in motion
in the field of a magnet, Neumann found that it depended only on the changes in
44 Compare, Fourier, Heat, 86-87, for the flow of heat in a closed ring, and Neumann,
"Allgemeine Gesetze," (1845),18-19.
German States, 1830-1870 279
a function caused by the motion itself. 45 This function represented the potential
between the circuit and the magnet, when a unit of current traversed the circuit.
Neumann therefore reformulated his ideas on the basis of this potential function. 46
By the end of the 1840s, Neumann rose above the mathematical particulars of
each physical case when he recognized a mathematical function the potential which
allowed him to reorganize his description of induction. He used this function as
the organizing principle to recreate the series of physical appearances which he
had previously used to construct his mathematical cases. This is an example where
Neumann was able to shake off the limitation of the particular physical cases
that produced only isolated mathematical results. It remained the only example
where a consistent mathematical point of view generated a connected series of
physically significant and known examples. Elsewhere, his insistence upon a
ready reference to the physically particular and the needs of the empirical referent
stifled investigations into the physical questions of his research. His mathematical
descriptions of physical processes was dogged by the particular and evidential.
Neumann did not produce traditional mathematical physics. It was not possi-
ble through particular physical examples to arrive at a mathematical theory that
would cover all possible cases without jettisoning the particularities of the physics.
Neumann achieved this just once. He did refashion mathematical physics to the
needs of physicists by considering only those cases of known physical significance.
Despite these limitations, his legacy to his colleagues and students was important.
Neumann deciphered the mathematically particular cases of his French predeces-
sors in physical terms. However, his approach did not allow him to develop the
physical implications of some of the mathematics developed. He limited his ideas
on theory to the improvement of experiment and enmeshed the results of the latter
in a context of mathematical description. The focus of his work in physics was
experiment, with the mathematics too tightly controlled to enable Neumann to
explore regions of physics not already visualized experimentally.47
Neumann was exploring new ways of doing physics. Much of it was in the realm
of the mathematical expression of specific, complicated physical cases in which
both physical cases and mathematical descriptions of them were obscure. This is
one reason why he had a marginal role in developing theoretical physics as a distinct
domain of research. The other reason was the development of electrodynamics that
was more in keeping with metaphysical principles accepted by German physicists,
45 Neumann, "Allgemeine Gesetz," (1845), sect. 9,57. The equation to which he refers is
on p. 56.
46 Neumann, "Uber ein allgemeines Princip der mathematischen Theorie indukirter elec-
trischer Strome," Abh. Akad. WissenBerlin (1847), Part 11: 1-72.
47 See Carl Neumann in his commentary on Franz Neumann's paper on induction in Franz
Neumann, Allgemeines Princip.
280 Physics and Mathematics
even while they denied that they used those principles at all. 48
In his published lectures on mathematical physics, his closeness to the French
mathematical tradition reasserted itself. In the section on hypotheses from his lec-
tures on elasticity, Neumann assumed bodies were made up of moving mass points
between which forces act. Yet after setting up a molecular level expression for
pressure and tension, he dropped this approach and only considered macroscopic
pressures and forces. This was the starting point for his mathematical theory.
When he got down to physical specifics, he discussed cases that emerged from
mathematical particulars, not physically significant cases that imposed limits on
the mathematics. 49 In the case of Elasticity, Neumann followed the French model
of constructing a highly abstract mathematical theory with few connections to ex-
periment and no systematic pattern of connecting the two. It is immaterial here
that, in the eyes of his students who were better trained in mathematics than he,
his mathematics was not "the most elegant or the most general from the point of
view of the mathematician."so
In his research he gave his students and contemporaries an example of how to
confront the problem of relating the constructs of mathematicians with the empir-
ical evidence of physicists. For the next generation, better trained in mathematics
and in the measurement physics he helped to create, Neumann's was a legacy that
was discarded in detail but one that was richly suggestive.
Tracing Neumann's influence within the work of his younger colleagues and
students is more than usually problematic. There is no question that he inspired
many students, and in general established a model of how to be a physicist. The
particular kinds of mathematical methods and solutions he developed in his research
left few traces in the work of others.
The one exception was in the research approach of Gustav Kirchhoff and partic-
ularly in his use of the potential. Kirchhoff's work on current networks was done
while he was a student in Neumann's Physics Seminar. Even as a student Kirchhoff
was better prepared mathematically than Neumann. His first paper was published
in the form of a general mathematical theory of the distribution of electricity in a
48 Olesko, Physics as a Calling notes that we must largely ignore accounts of Neumann's
work published by his students. They overstated his importance in an effort to gain
what they thought was his rightful place in an already crowded profession in the late
nineteenth century. While they, better trained in mathematical physics, could see physical
implications in his work, we cannot attribute those physical insights to Neumann himself.
As an example of such a student assessment see Woldemann Voigt, "Gedachtnissrede,"
in Neumann, Werke, vol. 1,3-19. Voigt claimed that many of Neumann's papers were
groundbreaking but was unable to state exactly what ground they actually broke.
49 Franz Neumann, Vorlesungen iiber mathematische Physik gehalten an den Universitiit
Konigsberg, 7 vols (Leipzig: 1883).
50 Voigt, "Gedachtnissrede," 10, Wangerin, "Neumann als Mathematiker," in Franz Neu-
mann, Werke, vol. 1.
German States, 1830-1870 281
thin infinite plane. This paper was followed by an account of his own experiments
to establish the form of the curves that represented equipotential surfaces across
a disc. Kirchhoff tried to use Neumann's method of literally using experiment
to establish the forms of the mathematical functions that represented the physical
entity he would then examine mathematically. In ways strongly reminiscent of
the French and Neumann, his physical concepts were measurables, and relation-
ships with which he began his analysis were those that emerged from his own
experiments.
His mathematical analysis was not limited to just those experimental cases.
His work also covered the distribution of current electricity in three-dimensional
networks. 51 Kirchhoff assumed only the validity of Ohm's law rewritten in terms
of a function u that determined the flow of the current. This function had the
properties of the potential function and allowed Kirchhoff to use some of the
mathematical results of Gauss' paper on the subject. He then looked for closed
curves of equal "tension." To get results back down to physically plausible cases
and experiment Kirchhoff took the example of a circular plate. Details of his
measurement techniques and results followed together with their comparison with
his mathematics.
Kirchhoff extended this work to the physical conditions under which Ohm's law
was valid and mathematically generalized these conditions. He started by asserting
that Ohm's demonstration of his law was only true when the density of electricity
within the conductor was the same in all directions. He considered an electrical
conductor in equilibrium and the force of "free" electricity on a point within the
conductor. Kirchhoff mathematically identified Ohm's electroscopic force and
electrostatic potential. He also demonstrated mathematically that Ohm's condition
was the only one possible. What Kirchhoff achieved was to show the mathematical
compatibility between electrostatics, Ohm's work on current electricity, and his
own work in the same domain. 52
Separated from this mathematical exercise were some of Kirchhoff's rumina-
tions on the various conjectures about the nature of electricity.53 These conjectures
51 Gustav Robert Kirchhoff, "Uber der Durchgang eines elektrischen Stromes durch eine
Ebene, insbesondere durch eine kreis formige," Ann. Phy. 64 (1845): 497-514. Kirch-
hoff, "Nachtrag zu dem Aufsatze: Uber den Durchgang eines elektrischen Stromes durch
eine Ebene, insbesondere durch eine kreisfOrmige," Ann. Phy. 65 (1846): 344-349,
Kirchhoff, "Uber die Anwendbarkeit der Formeln fUr die Intensitaten der galvanischen
Strome in einem Systeme linearer Leiter auf Systeme, die zum Theil aus nieht !inearen
Leitern bestehen," Ann. Phy. 75 (1818): 188-205.
52 Kirchhoff, "Uber eine Ableitung der Ohm'schen Gesetze welche sieh an die Theorie
der Electrostatik anschleisst," Ann. Phy. 76 (1849): 506-513, Phil. Mag. 37 (1850):
463-468.
53 Jungnickel and McCormmach, Intellectual Mastery, vol. 1, 154-155, see these papers as
physics and hence consider the conjectures of Kirchhoff at the end of his paper as part
282 Physics and Mathematics
played no role in the mathematical parts of the paper. Kirchhoff's later work in
electrodynamics and elasticity show the same reluctance to use specific hypothe-
ses about molecular processes in nature. He preferred to base his mathematical
physics on general principles and statements of laws deduced from experiments.
His was mathematical physics directed towards the needs of physicists. 54
Kirchhoff combined the two paths along which Neumann's students appear
to direct their careers. Some became "measurement" physicists who replicated
his concern with error analysis and carefully constructed experiments to measure
small quantities. Others used the emerging methods of mathematics to construct
a mathematical physics that soared far above the concerns of the empirical base
that gave them the starting point for their mathematical developments. What they
did share with Neumann was an aversion to hypotheses. While his students might
succumb to the blandishments of highly abstract theory, Neumann's insistence
that the physically plausible lay at the root of mathematical physics meant that this
discipline became the mathematics of physicists and was no longer the exclusive
province of mathematicians.
Wilhelm Weber
Wilhelm Weber began his career in the same decade as Neumann with the same
French models and resources before him for both experimental and mathematical
physics. However, because of his chosen research problems, Weber followed other
models of performing experiments and mathematical physics. The interests of
Neumann and Weber coincided in the 1840s when both converged on the problem
of induction and the work of Ampere. The end results of that particular convergence
of interests were very different. They presented to their colleagues contrasting
ways of approaching an experimental and theoretical problem and visions of what
constituted their solution.
Weber's interest in physics began in exact experiment withoutseriously branch-
ing out into mathematical physics. He analyzed the results of Poisson, Cauchy,
and Navier in their theories of elasticity to extract physically meaningful results.
He explicitly compared the results of the mathematicians with the experiments
of physicists. However, Weber's passion was measurement. Beginning with the
papers published with his brother, he examined every aspect of the act of obser-
vation, measurement, and the reduction of data. They extended this research into
consideration of the psychological aspects of the relationship between observer
and observed. Both Weber and Neumann were struggling with the same problems
of interpreting the French and creating a methodology for a discipline.
Weber's interests did not remain with instrumentation, measurement and data
reduction. His role in the development of Gauss' ideas on absoiute measurements
is unclear but seems substantial. 58 The issue of absolute measurement became
more important as he investigated how to use the magnetometer as a galvanometer
using the phenomenon of induction. 59 This led him to a study of induction and the
question of the measurement of electrodynamic quantities that converged in his
first massive paper on the subject. 6o
Within "measurement physics" Weber continually confronted the problem of
making plausible connections between what he observed with his instruments,
motions due to mechanical forces, and what he was supposedly measuring, mag-
netic then electrical quantities. 61 Weber confronted the need for a theory connecting
the three domains of physics-the mechanics of measuring instruments, magnetism
and then galvanic electricity.62 The phenomena that drew these three domains of
experimental physics together were those of electrodynamics and induction. The
systematic expression of their relationships were in Ampere's empirical laws on
electrodynamical phenomena and Faraday's on induction. Weber began his search
for a theoretical net that connected these different domains of physics together in
a theory of the measurement in absolute, i.e., mechanical terms, of the measur-
abIes of Ampere's and Faraday's experiments. This was a theory that originated
in the new measurement physics. Measurement did not merely serve to present
mathematicians with expressions that formed the initial and/or final points of their
Bifilar-Magnetometers," Resultate (1837), "Uber den Einfluss der Temperatur auf den
Stabmagnetismus," Resultate (1838): 38-57.
58 See May's remarks in May,"Gauss," Diet. Sci. Bio. and Archibald, "Tension and Poten-
tial," Centaurus, 31 (1988): 141-163.
59 Weber, "Der Induktor zum Magnetometer," Resultate, (1839): 86-101, "Der Rotations-
inductor," Resultate, (1839): 102-117, and "Unipolare Induktion," Resultate, (1840):
63-90, and "Messung starker galvanischer Strome bei geringem Widerstande nach ab-
solutem Maasse," Resultate, (1841): 83-90.
60 Weber, "Elektrodynamische Maassbestimmungen," Abh. Leipzig (1846): 209-378, and
in Weber, Werke, 6 vols., (Berlin: Springer, 1892-1894) vol. 3, 27-214. Abstracted
in Ann. Phy. 73 (1848): 193-240. The abstract was translated as Weber, "On the
Measurement of Electrodynamic Forces," in Taylor's Scientific Memoirs 5 (1852): 489-
529. The painstaking experiments to examine Ampere's law were cut short in the abstracts
as were Weber's descriptions of his instrumentation. Only the skeleton of his theory
remained.
61 This is different from Neumann's measuring problems in his work on heat and light
because thermal and optical properties were measured directly by the instrumentation.
62 For an account of the political complications of Weber's academic career and his rela-
tionship with Gauss and the role of Gauss in the development of his ideas, see Jungnickel
and McCormmach, Intellectual Mastery, vol. 1, 130-140. Their account pps. 140-144,
of the contents of Weber's first electrodynamics paper discussed here is different from
my own and assumes a great deal that I am exploring as problematic.
German States, 1830-1870 285
analysis. The analysis of the problems of measurement themselves were the step-
ping stones into a discussion of a theory that joined distinct types of phenomena
together with imagery and mathematics. Experiment was enmeshed finally in spe-
cific physical imagery, expressed mathematically, whose physical meaning was
investigated using mathematics as the language of exploration. Physical imagery,
as Weber knew, did not come with measurement; it was hypothetical and necessary.
In the abstract to his paper Weber made this explicit by comparing the devel-
opment of electrodynamics with astronomy. In the former, no principle linked
the laws of magnetism and electromagnetism together. This was unlike astron-
omy where Kepler's laws were joined together through Newton's principle of
gravitation. Newton's theoretical leap led to much research in astronomy. In elec-
trodynamics Ampere's work had not led to such research. The known induction
phenomena were discovered independently of Ampere's research. To press the
point home Weber cited Faraday's work. If one could develop the "true" laws of
Electrodynamics, they would serve as a "guide to different classes of phenomena"
as had happened in astronomy. Unlike astronomy, there was no serviceable com-
bination of theory and observation available in electrodynamics. While Ampere
provided the mathematics, there was no idea equivalent to gravitation that joined
disparate aspects of electrodynamics together. A vital link was missing,
in the development of electrodynamics no such combination of observa-
tion with theory has occurred as in that of the general theory of gravi-
tation. Ampere who was rather a mathematician than an experimenter,
very ingeniously applied the most trivial experimental results to his sys-
tem and refined this to such an extent, that the crude observations im-
mediately in question no longer appeared to have any direct relation to
it. 63
Ampere's mathematics was remote from its experimental foundations. Its origins
could no longer be traced through the refinements of the mathematics.
Part of Weber's objection was that Ampere did not actually measure the quan-
tities he examined in his experiments because his were nUll-point experiments.
Ampere had not demonstrated what he claimed. The solution lay in improving
methods of observation and in the careful comparison of specific points of theory
with experiment. This would provide for the introduction of the "spirit of theory in
observation, without the development of which no unfolding of its powers is possi-
ble." This meant that Weber was committed to the development of a theory that did
not lose sight of its physical, empirical foundations. In addition Weber had further
ambitions, to join into one theoretical whole electrostatics and electrodynamics as
well as induction.
His massive paper had many layers to it. To characterize it as the presentation
of a theory of electrical action misses many of Weber's intentions, and most of the
text. Weber devoted much space to reestablishing Ampere's law experimentally.
His problem was to construct the fundamental laws of electrodynamics through the
actual measurement of electrodynamic forces, not through the precarious balancing
of such forces against one another. By investigating the rotation of one coil within
another, while both carried currents, Weber reconstructed Ampere's law. 64
Weber immediately inverted the role of Ampere's law. It became the foundation
for the measurement of all electrodynamic quantities. To do this Weber had to
demonstrate that using Ampere's law he could construct the moment of rotation
for a coil in motion induced by a current. By comparing experimental results to an
extended form of Ampere's law, Weber established that the moment of inertia for
this rotation gave a direct measure of the current. 65 Mechanical quantities, move-
ments, rotations, moments of inertia, were a direct measure of electrical quantities.
Weber then launched himself into an empirical examination of induction. What
he hoped to establish, although he could only state it not prove it, was that the
induction phenomena Faraday had uncovered could not be used as a foundation
for understanding Ampere's law. Only the inverse was possible. 66 Weber thus se-
cured the absolute status of mechanical measurement and quantities over electrical
or magnetic ones. Weber contended that if we begin with Faraday's results and
induction, we could only reach Ampere's law on a case by case basis by replacing
the electrodynamic activity of the current by an equivalent magnet. Constant cur-
rent phenomena were not encompassed in this analysis. In contrast, Ampere's law
assumed constant currents but offered the possibility of striking out beyond this
case by assuming the general mathematical case of currents as functions of time,
and hence contained the possibility of explaining induction.
Weber approached the problem of induction in several steps, while also intro-
ducing his fundamental assumptions about the nature of electricity and electric
currents in as low key and unobjectionable a way as possible. At the same time
he developed more and more general mathematical cases as his argument became
less tied to laboratory visualization. In so doing Weber drew together all three
aspects of electrical science. At work in this section of the paper are three levels
of concern, mathematical abstraction, physical hypotheses, and the development
of the imagery made possible by expressing the hypotheses in the language of
mathematics.
It was no accident that Weber began in electrostatics and from there deduced
64 After his series of experiments Weber concluded that the agreement between observed
and calculated values could not have been better. Ampere's fundamental law was con-
firmed in its most general and important consequences. Weber, "Elektrodynamische
Maassbestimmungen," 249, Werke, vol. 3, 69.
65 Weber, "Elektrodynamische Maassbestimmungen," 249-268.
66 Weber, "Elektrodynamische Maassbestimmungen," sect. 18, 305-307, Werke, 132.
German States, 1830-1870 287
Ampere's law. The introduction of electric charge as a center of force was un-
objectionable to his German audience. Weber was not the first to assume that
electric current was the motion of oppositely charged particles within a conductor.
Fechner had introduced the idea a year earlier. Fechner's unlike charges attracted
when they moved in opposite directions. Fechner had explained induction for the
case of parallel wires starting in Ampere's electrodynamic law. 67 Weber set his
charges in motion parallel to one another and then took only their relative motions
into account. The resultant force that emerged from all possible combinations of
moving charge interactions was
ee' 2
+8-
I
2 a uu,
r
where a 2 was a constant, e and e' were charges with velocities u and u ' at a distance
r apart. He then generalized this particular case to one with relative accelerations
of his point charges as well as velocities. A term of the form,
2
I-a 2(dr)2
- d r
+b-,
dt dt 2
had to be multiplied into the expression for the force.
To get from electrostatics to induction, Weber introduced mathematical results
deduced from Faraday's work to show the plausibility ofthe approach, if not a cast-
iron mathematical case. Weber was looking for a general differential mathematical
expression for induction. Therefore, he started with the most general expression of
Ampere's law, reworked into a form deduced from his starting point, electrostatics.
He had to construct expressions for each possible moving charge in one conducting
element acting on the two possible moving charges in the other element separating
from one another to produce the current. By translating both the velocities and the
cosines of the angles involved into differentials of distance and manipulating them
algebraically, Weber obtained an expression for the force between two charges that
contained accelerations along the conducting elements
motions and accelerations of the charges as well as their masses and distances
apart.
Weber required one additional assumption to deduce the laws of induction from
his basic imagery of moving electric charges; the induced emf was measured as the
difference between the forces acting on the moving charges of the secondary by the
moving charges of the primary. He took the simplest case, the imbalance of force
from the unequal motions, however produced, of the charges moving in the primary
circuit induced the emf in the secondary circuit. Since the emf was measured
along the circuit element of the secondary, he needed only that component of the
resultant force. 7o Assuming constant currents and one circuit moving with respect
to another and through some ingenious algebraic manipulations of the velocities as
derivatives with respect to time of their distances from one another, Weber obtained
an expression for the induced emf for parallel closed circuits where currents were
flowing parallel and antiparallel. In the last section of his paper Weber confronted
the results of Fechner's paper and Neumann's first paper on induction.
Both Weber and Neumann built their mathematical case through the consider-
ation of physical particulars. Yet, Weber's is truly a physical theory of electrical
and electrodynamical phenomena, developed in the language of the calculus. The
physical imagery was powerful, developed mathematically as far as was necessary
to demonstrate a particular known physical case, and in certain circumstances the
general case. Like Neumann, Weber started from empirical laws and constructed
other, known laws directly. However he used the specifics of a physical model
and the actions of moving electric charges. Neumann's path was mathematically
convoluted and offered no sense of the underlying physical process. In contrast,
Weber constructed a specific physical model and traced physical processes through
that model. Beginning with his ideas on electric currents as the separation of
charges, Weber constructed known empirical laws. The consistency, simplicity
and Kantian familiarity of the imagery were powerful. Finally, Weber kept math-
ematics subservient to the requirements of the physical cases and the physicality
of the mathematics was obvious. He introduced into physics that which Neumann
avoided, specific imagery. In his commentary on his mathematics, Weber wrote
in terms of the physical content of the mathematics, velocities, distances, forces,
emfs, not in terms of functions, coefficients, etc. The latter were part of Neumann's
commentary on his developing mathematics even as the physical case that set up
the mathematics is visually explicit.
One of Weber's criticisms of Neumann's induction paper was physical. While
Neumann could replicate the known laws of induction, there was essentially no
physical sense of what was actually going on. To Weber, induction arose from a
"mutual exchange of electric currents" and any explanation of these phenomena
"must be based on the consideration of this mutual exchange." This pointed to the
foundations of what physical theory should consist of and its relationship to process.
There was, for Weber, no "inner coherence" in Neumann's work on induction.
The inner coherence of mathematics was no longer sufficient. For Weber it was
"remarkable" that Neumann's law agreed with known empirical rules.1 1 Whether
we take this as irony or not, Weber had not spotted the coherence and conjunction
between the mathematics of their approaches that lay in the potential.
More importantly, Weber was calling for a new foundation upon which to build
physical theory. What Weber constructed here, through simple powerful imagery
based on philosophically familiar assumptions, was a theory of the processes of
nature that drew together a wide range of phenomena that had, until then, remained
isolated on an explanatory level. Neumann's mathematical description also drew
some of these same phenomena together, but as Neumann acknowledged, there
was a mystery at the bottom of his mathematics which eluded him.
There was only one aspect of theoretical physics not encompassed in Weber's
massive paper, prediction. There was also a price to pay for using specific im-
agery, alternatives that appeared quickly and often after the publication of Weber's
work. The basis of Weber's work remained an hypothesis that could, at best, be
compatible with phenomena, but could never be established as necessary beyond
its usefulness. 72
Whatever the ontological difficulties of using specific physical imagery, or the
direction of Weber's career after this 1846 paper, the terms of this achievement
were the basis for theory within German physics. 73
The terms of the debate shifted. Arguments were about specific physical models,
their adequacy as images of the operation of nature, their implications as images
of nature, and whether they were understood by their protagonists or not. Mathe-
matical prowess was no longer an issue within the community of physicists. It was
assumed as a prerequisite for entry into the discipline. The grounds upon which its
use were judged had also changed. Was the mathematics suitable and employed
well enough for the physical cases at hand? Physicists had established criteria
of judging theories within physics expressed in the language of mathematics that
were no longer dominated by the criteria of mathematicians. German physicists
used mathematics in a variety of ways to explore the structures and processes of
nature, from the formal and highly abstract to the exploration of the implications
As Weber and Neumann reached the midpoints of their careers, a new genera-
tion of physicists, some of whom they trained, began to publish their first pieces of
research. This later generation had before them a series of research problems, to-
gether with examples of solutions that incorporated the new standards and methods
of physics. This younger generation also could take advantage of the systematic
training now offered in both the experimental and mathematical aspects of this
new discipline, together with training in higher mathematics offered at German
universities. However, the profession and discipline of physics was not so well
defined that the unorthodox might not enter. Hermann von Helmholtz, one of the
most important members of the generation that came to maturity in the late 1840s,
received no systematic training in higher mathematics or mathematical physics.
At the other end of this educational spectrum, Rudolf Julius Emmanuel Clausius
received all the training now offered for an aspiring physicist. At the University
of Berlin he worked with Dove and Magnus in experimental physics and heard
Dirichlet's courses in mathematics. 74
Clausius' research was never the familiar nineteenth-century mix of experiment
and mathematics. He was a theoretical physicist and never published any exper-
imental research although he was always well aware of it. Experimental results
were the starting point of all his research, and incorporated into his explanatory
compass. 75 His career and the courses he taught reflect the deviant and difficult pro-
fessional path he chose.76 He was also one ofthe first German physicists to be fully
74 The courses from Dirichlet included potential theory and differential equations. He also
heard Dirksen's lectures on analytical functions and Steiner on function theory.
75 He included in his research the problems of engineers, publishing a text on the steam
engine and on the design of electric motors. See Clausius, "Uber die Anwendung der
mechanischen Warmetheorie auf die Dampfmaschine," Ann. Phy. 97 (1856): 441-476,
533-558. Clausius, "Zur Theorie der dynamoelectrischen Maschinen," Ann. Phys. 20
(1883): 353-391.
76 From 1844-1850 Clausius taught at a Gymnasium in Berlin and from 1850 was professor
of physics at the Royal Artillery and Engineering School. He received a call as Ordinarius
to the Polytechnic in Zurich in 1855 and then Wiirzburg in 1867, returning to Prussia
and the University of Bonn in 1869. In Berlin and Zurich he taught physics courses to
engineers.
German States, 1830-1870 291
77 See Clausius, "Uber die mechanische Wiirmetheorie," Poly. J. 150 (1858). The mathe-
matical parts of this paper were expanded and published as Chapter 1 of Clausius, Ab-
handlungen iiber die mechanische Wiirmetheorie (Braunschweig: Vieweg, 1864--1867)
2 vols, translated into English by T. Archer Hirst as The Mechanical Theory of Heat
(1867).
78 Clausius, "Uber die Lichtzerstreuung in der Atmosphiire," J. Reine Angew. Math. 34
(1847): 122-147: "Uber die Intensitiit des durch die Atmosphiire reflectirten Sonnen-
Iichts," same journal 36 (1848): 185-215.
79 See Frederick Gregory, Scientific Materialism in Nineteenth Century Germany (Hingham
MA.: Reidel, 1977).
292 Physics and Mathematics
was cut short by his posting from Berlin to Potsdam, far enough to cut him off from
necessary laboratory resources of Muller. Helmhoitz then turned to the study of
animal heat and the heat generated by muscular action. Here he demonstrated that
chemical changes occurred in working muscles and a year later, 1848, that heat
was generated by muscle contractions.
In 1847, in the middle of this research Helmholtz published "On the Conservation
of Force." The extent of his physicalist image of life became clear, as well as the
Kantian foundation of his physics. His physical explanations were based upon the
idea that matter was made up of point masses between which were attractive or
repulsive forces. Helmholtz's work in the 1880s on the Principle of Least Action
served to reemphasize his commitments. 8o
In 1848, the considerable pressure friends at the center of Prussian academic
life in Berlin brought to bear on the army and the Kultus-Ministerium, together
with Helmholtz's research, led to his release from his army obligations. He never
practiced medicine again, although his research over the next twenty years was
mainly in physiology beginning in his experimental work on the velocity of nerve
impUlses. His research again was guaranteed to undermine confidence in vitalism
and build his own physicalist ideas. It also led him into the instrumentation of
electrical experiments and their improvement to measure small time intervals and
electric pulses. Helmholtz treated physiological problems with the quantitative
methods of experimental physics. 81
These same physiological problems became the starting point for many of
Helmholtz's forays both experimental and theoretical physics. The experiments
on nerve impulses led him to consider electric pulses and this expanded into an
80 Helmholtz, Uber die Erhaltung der Kraft, eine physikalische Abhandlung vorgetragen
in der Sitzung der physikalischen Gesellschaft zu Berlin am 1847 (Berlin: G. Reimer,
1847), translated as "On the Conservation of Force," Taylor's Scientific Memoirs 2 (1853):
114-162, trans., John Tyndall. This essay was republished throughout Helmholtz's
life. In the edition of 1881 Helmholtz reaffirmed his commitment to Kantianism. See
Helmholtz Uber die Erhaltung der Kraft, eine physikalische Abhandlung in Helmholtz,
Wissenschaftliche Abhandlungen (Leipzig: Barth, 1882-1895) 3 vols., vol. 1, 12-68,
Appendix. On Helmholtz's Kantianism see, Peter Heiman, "Helmholtz and Kant: The
Metaphysical Foundations of Uber die Erhaltung der Kraft," Studies Hist. Phil. Sci. 5
(1974): 205-238. On Helmholtz's monocycIes see Gunther Bierhalter, "Zu Hermann
von Helmholtzens mechanischer Grundlegung der Warmelehre aus dem Jahre 1884,"
Arch. Hist. Exact Sci. 25 (1981): 71-84, and "Die von Helmholtzschen Monozykel-
Analogien zur Thermodynamik und das Clausiussche Disgregationskonzept," same jour-
nal 29 (1983): 95-100.
81 Helmholtz, "Uber die Dauer und den Verlauf der durch Stromesschwankungen inducirten
elektrischen Strome,"Ann. Phy. 83 (1851): 505-540. Helmholtz needed a theory of his
measuring instruments to convince his colleagues, even Muller, of the physiological va-
lidity of his work. His work using this instrumentation led later to a specific image of color
reception. See Timothy Lenoir, "Helmholtz and the Materialities of Communication,"
Osiris, 9 (1994): 185-207.
German States, 1830-1870 293
interest in the problems of induction and his critical overview of the whole domain
of electrodynamics of the 1870s. He followed the same general path in his inves-
tigation of the sense of hearing. Hearing led him to the consideration of the actual
motion of air in open-ended pipes to the motion of air at the end of those pipes
and problems of gases with internal friction and hydrodynamics. For Helmholtz,
physiology led to more general experimental physical issues and more abstract
questions of "theoretical" physics.
It was not until 1870 that Helmholtz received the call to Berlin and a chair in
Physics. His research work in physiology ceased. This is a reminder of just how
long it took him to formally enter the profession he had published in with great
distinction for twenty years. Helmholtz's career is also a reminder of how fluid the
term "Physics" still was in mid century. All of this is to set Helmholtz's research
in physics in the context of the constraints of his education, then opportunities that
his research gave him to transcend those limitations and encroach on the turf of the
field that he worked to make his own. Throughout the 1840s Helmholtz educated
himself in higher mathematics, initially to understand how to use mathematics
in physiology. Helmholtz realized that his vocation was physics and took every
opportunity to drive his research into physics.
The closeness of his work in physiology and physics is illustrated in his work
on conservation of force. The principle itself was actually stated in a review
paper on animal heat of 1845.82 After surveying work done by Davy and Lavoisier
on the issue, he examined Leibig's paper on the origins of animal heat. Stating
that it was of interest to physics in general as much as physiology, Helmholtz
asserted that the principle of the constancy of force-equivalence was already used
as the foundation of mathematical theories. As examples he cited Carnot's and
Clapyeron's determination of the work contained in a given quantity of heat and
Neumann's theory of currents induced by moving magnets. Helmholtz took his
principle of "conservation of force" as empirically grounded and "theoretically
stated and well known." The material theory of heat was doomed. Helmholtz
then conjectured that, "if we substitute the motion theory of heat for the material
theory of heat, we see heat as originating from mechanical force." From this
it followed that chemical, electrical, and mechanical force were equivalent, and
Helmholtz cited some of the experimental evidence he would use in his 1847 paper
to demonstrate this.
Ifwe accept heat as motion we can firstly assume mechanical, electrical,
and chemical forces as equivalent to one another, as complement to a kind
of transformation of one force into another. For mechanical force exists
yet no experiment demonstrates this; the work of Carnot and Clapyeron
and Holtzmann seem not to point to the production but to the diffusion
of heat. In the case of chemical forces the heat equivalent (latent heat)
has been determined for a series of chemical processes and the law of
the constancy of heat production which bind two substances together is
known. For constant electric current it follows from the law of Ohm and
Lenz and established empirically by Becquerel from the heat developed
during electromechanical change. 83
This was the outline of his argument in the paper on the conservation of force.
What was missing, and added to the 1847 paper, was a vision of matter that
allowed him to illustrate mathematically what he believed was the universality of
his principle. Helmholtz's paper on the conservation of force was ambitious. In
describing this paper as one in physics, Helmholtz was using the term in its older
sense, not in the sense that had been built up through the research, discipline and
profession called physics over the previous two decades. "Physics" included all the
experimental sciences and he drew on evidence from chemistry etc., in arguing for
the conservation of force. Helmholtz read his paper on the conservation of force
to the Berlin Physical Society. Similar papers were read after he left the city. Even
though the physicists from the University of Berlin also attended, few understood
the implications of the paper until it was explained to them by Dubois-Reymond.
Reactions to the paper seem to reflect the idea that this was a paper in a discipline
not defined by the contents of the Annalen der Physik to which it was submitted.
While Poggendorffwas busy narrowing the definition of the discipline, Helmholtz
was trying to urge a broadening of it in directions that Poggendorff had repudiated
as a young editor in the 1820s.
Mathematical physicists might begin with the generalized mathematical solution
to a problem from which particular solutions were extracted, Helmholtz began
from the most general metaphysical principles, from which he extracted specific
physical results in mathematical form. This led him from mechanics to other
important research areas within physics, namely electricity and the nature of heat.
Forces brought about change yet they were themselves conserved. For Helmholtz
the physical question was the measurement of this conservation. In his model force
was reduced to the mechanical forces of attraction and repulsion. In a closed system
of mass particles, change was measured by alterations in vis viva. Helmholtz
reexpressed this change in terms of the changes in the "intensity of the force," that
is in the potential of the forces that acted between such particles,
1
-mv
2
2
I
1
- -mv
2 2
2
= lR
r
cpdr
'
where, m is the mass of the particle, whose velocity changes from VI to V2. cp
was the intensity of the force constructed by considering changes in v 2 . Both
the velocity and force X, Y, Z were functions of the coordinates x, y, z only and
Helmholtz expressed this as
1
"2md(v2) = Xdx + Ydy + Zdz.
Defining the x-component of the intensity of the force as X = (x jr)</J Helmholtz
showed that for central forces, if the vis viva was conserved, then so was the
intensity of the force. He then generalized this to a system of an arbitrary number
of such centers of force.
In his discussion of heat Helmholtz used examples from physiology, chemistry,
and electricity to argue that heat was not a substance but a measure of the vis
viva of thermal motions. (Latent heat, a measure of the forces between atoms that
changed with the changes in position of those atoms.) The nature of atomic mo-
tions were unknown and unknowable. It was sufficient simply to understand that
heat was motion. To make this plausible, Helmholtz turned to systems in phys-
iology in which vis viva was not conserved, and Joule's experiments on friction.
In both cases heat was generated but until now neither the increase in "tensional"
force within the body, nor the extraction of mechanical effect, had been taken into
account. Helmholtz asked whether in these cases the "force" developed equaled
the mechanical force lost and when mechanical force disappeared was a definite
amount of heat always developed? If so, then there was a quantity of heat equiv-
alent to mechanical force. The evidence Helmholtz introduced for this argument
included experiments on exothermic chemical reactions and Joulean heating. To
counter the objection that in induction no heat was generated, Helmholtz stated that
there was no source of heat because there was no transfer of material substance.
To reach an estimate for the mechanical equivalent of heat, he cited Joule's and
Holtzmann's experiments on the compression of gases, those on the velocity of
sound, latent heat, and the expansion of water vapor with temperature.
The unifying concept throughout this paper was "tensional" force and its inten-
sity. Helmholtz used this concept to extend his argument into electricity to obtain
the "force equivalent" of electrical processes. Thus, using his principle, the change
in the vis viva of two charges moving from distance r to R apart was
12~md(v2) = _JR
1 2 r
</Jdr,
He identified </J with Gauss' potential function. With his paper Helmholtz gave the
potential function physical meaning. He then drew into this new conceptual net of
force equivalence the heat generated by galvanic currents and a physical analysis
of the results of Neumann's first paper on induction.
The physiological intention of the paper surfaced in Helmholtz's afterword. He
wanted to address issues for live matter, but could only show such principles for
inanimate processes. Hypotheses, and the condensation of the meaning of disparate
296 Physics and Mathematics
phenomena into one general principle, had once again entered German physics. In
this paper Helmholtz displayed a pattern of argument that he repeated throughout
his life. He surveyed an existing set of known phenomena and their explanations
in a domain replete with conceptual ambiguity and explanatory confusion and
contestation. He then cut through the confusion to bring out the physical essentials
of the cases and pinpointed a method, experimental, mathematical, or in this case
conceptual, to sweep away the ambiguities and confusions and open up new ways
of dealing with whole domains of physics. He then explored these new domains
himself, mathematically and experimentally. 84
Sweeping metaphysical principles, and undemonstrable models of matter even
if they led to useful results were the antithesis of the systematic, experimental
quantitative approach and the mathematical description of these results that consti-
tuted many pages of the Annalen der Physik. Weber, in the theoretical sections of
his electrodynamics paper, had kept close to his own experimental results. He had
introduced his moving, charged particles as a principle only after discussing the
problems of the approaches of physicists that denied the necessity for hypotheses.
He also used hypotheses only to unify phenomena he had investigated himself.
Helmholtz had done none of the experiments he cited. The mathematical content
of his paper was minimal, just enough to demonstrate the physical point but no
more. There were no sophisticated developments of its implications, and certainly
not of the mathematical caliber already displayed by Clausius. 85 When Clausius
addressed the same mechanical problem of deducing a conservation law for a
system of mass points his mathematical understanding of the problem was much
deeper and more carefully stated. 86
The place of Helmholtz's paper within the body of the discipline of physics in
the 1840s was peculiar. It simply set aside all the standards of the discipline and
was a measure of Helmholtz's distance from it. Reaction to the paper was less
than enthusiastic. It was not surprising that this brilliant but rambling paper was
rejected for theAnnalen der Physik. Many physicists could not follow his argument.
84 See his work in hydrodynamics, of the 1850s, that had implications for both mathemat-
ics, as well as acoustics, and the physics of gases. He performed experiments on the
behavior of gases and examined fluid behavior mathematically. This was repeated in his
reexamination of electrodynamics. He was less successful in his attempt to bring unity
to the physical sciences using the Principle of Least Action.
85 This is not to say that Helmholtz was incapable even at this early stage in his career of
understanding or producing such mathematics. See his analysis of Challis' theory of
sound published the following year. Helmholtz, "Bericht tiber die theoretische Akustik
betreffenden Arbeiten von Jahre 1819-1848," Fort. Phys. 4 (1849): 101-118, 124-125,
and 5 (1850): 93-98. He recognized Challis' work as an exercise in mathematics rather
than physics. Challis was chasing a particular integral in the equations of hydrodynamics.
86 Clausius, "Uber das mechanischen Aequivalent einer elektrischen Entladung und' die
dabei stattfindende Erwarmung des Leitungsdrahles," Ann. Phy. 86 (1852): 337-375.
Translated in Tay/or's Scientific Memoirs (1853): 1-32.
German States, IS30-1S70 297
Helmholtz's use of the term force was ambiguous. It was a mUltipurpose word
to cover much the same ground as Ohm's usage of it thirty years beforehand. By
force Helmholtz meant vis viva, potential, a term he took from Gauss but only saw
in relation to galvanic electricity, and mechanical potential. Kirchhoff had only
recently explicated Ohm's law in terms of potential. Helmholtz had to personally
win over Neumann to his ideas .
... after a severe struggle, I have converted a bold mathematician, who
gets confused over non-mathematical logic, and is himself a lecturer in
mechanics, to the doctrine of conservation of force, so that it is now
official doctrine in this University. Neumann is rather difficult to get at;
he is hypochondrical and shy, but a thinker of the first order. 87
Older physicists such as Magnus were dubious, even hostile to his work. 88
In the period immediately following its publication, Helmholtz's memoir was
not mentioned in the debates over the nature of heat and the mathematics in which
to express it. Of the younger physicists Clausius was the one who understood
Helmholtz's work, its limitations, and the challenge that it offered his own work in
the domain of heat theory. Clausius' approach to the theory of heat was similarly
dependent on the experiments of others. His analysis of the problems with current
ideas on heat depended on the ambiguities inherent in the experimental record
itself. In addition, he did not leap from these contradictions to a grand principle
of nature. Instead he built up his case for his assumptions about the nature of heat
through a series of well considered, special physical cases presented in a succession
of mathematical papers.
In these mathematical papers Clausius reinterpreted Clapyeron's mathematical
version of Camot's work. 89 His reworking meant that Clausius kept much of the
mathematical analysis developed by Clapyeron. Its physical foundation required
reexamination and Clausius gave many of Clapyeron's results physical meaning.
For Clausius the mathematical characteristics of the functions that entered his
equations determined the physical characteristics of his system. The only functions
and equations he pursued were those he saw as having physical utility. In his
analysis of the Camot cycle, Clausius chose the ideal gas as his physical system.
He reduced the general problem of the ideal heat engine to analyzing the changing
state of a gas as it traversed an infinitely small cycle. Clausius constructed the
expression for the heat added or expelled for each leg of this cycle. 9o He then
constructed the expression for the inverse of the mechanical equivalent of heat, A,
as the ratio of the heat expended over the work produced, "the equivalent of heat
for the unit of work." He then added all his expressions for the heat added going
around the cycle. Using the accepted mathematical expression for the work done
by an ideal gas, the area within the cycle in the P-V diagram, A became
where d Q was the heat added, a + t the absolute temperature of the gas, and V
its volume. R was the gas constant. Clausius continued that the above expression
showed that,
Q cannot be a function of V and t as long as the two latter are indepen-
dent of one another. For otherwise, according to the known principles of
the differential calculus, that when a function of two variables is differ-
entiated according to both, the order in which this takes place is a matter
of indifference, the right side of the equation must be equal to zero. 91
The equation could be brought under the form of a complete differential,
a +t
dQ = dU +A.R--VdV,
where U was an arbitrary function of volume and temperature. The above expres-
sion was not integrable until the relationship between V and t was established.
Clausius gave both U and the other terms in this equation physical significance.
U was the heat necessary for internal work, and depended only on the initial and
final condition of the gas. The second term, the external work, depended on the
initial and final states of the gas and the path taken between those two states. In
his succeeding papers on the second law, Clausius introduced liT, where Twas
the temperature, as a multiplier of d Q to make a complete differential of the form
Xdx + Ydy.92
This was the antithesis of Helmholtz's approach. Clausius avoided philosophical
explanations of any kind and hid his particular theory of matter. He drew his
theoretical conclusions on the nature of heat directly from the results of experiment
analysis is less general than Thomson's two years later. Mathematically this is true.
Physically it is irrelevant because Clausius reasoned, as had Carnot, that there was a
unique maximum to the mechanical work equivalent of a unit of heat. The working
substance in the ideal engine was irrelevant.
91 For an ideal gas V and t were related through the ideal gas laws. Clausius, "lIber die
bewegende Kraft der Warme und die Gesetze, welche sich daraus fUr die Warmelehre
selbst abl~iten lassen," Ann. Phy. 79 (1850: 368. Translated in Phil. Mag. 2 (1851):
1-21, 102-120,12.
92 Clausius, "lIber eine veranderte Form des zweiten Haupsatzes der mechanischen Warme-
theorie," Ann. Phy. 93 (1854): 481-506.
German States, 1830-1870 299
and a physical interpretation of the properties of the terms in his equations. The
mathematical path of a function became the physical path of the physical entity
represented by the mathematical symbol. Like Helmholtz, Clausius had to make
this reevaluation of Carnot plausible. He therefore turned to experiments on the
latent heat of vapors and the velocity of sound to draw them into a single explanatory
net.
On a more abstract level Clausius showed that the mechanical theory of heat
was conceptually better than Carnot's. Carnot had to assume that perpetual motion
could not exist to argue for the conservation of heat. The mechanical theory
of heat ruled out the possibility of perpetual motion from the beginning. If the
mechanical theory was not accepted, heat could move from a colder to a hotter
body, which went against all observational evidence. Arguing mathematically
from these assumptions Clausius established that Carnot's function C was simply
the absolute temperature. To make this conclusion plausible Clausius compared
values of C deduced from his theory with William Thomson's experiments. To
drive the point home he examined the behavior of vapors as they deviated from
the gas laws, comparing temperatures of maximum density from his theory and
Regnault's experiments.
This long excursion into Clausius' methods demonstrates that his papers on
the mechanical theory of heat were as speCUlative as Helmholtz's. However, his
presentation and methodology lay well within the standards of German physics
while extending those standards into new domains of explanation. In the next
three years Clausius published a series of papers exploring various phenomena to
demonstrate the range and significance of the principle underlying his work. This
included both electrical and thermoelectrical phenomena. Helmholtz, meanwhile,
pursued physiological research and only began publishing review articles on the
mechanical theory of heat after 1855.
In Clausius' pursuit of the mechanical theory of heat there was a consistency
of perception and a systematic methodology tied closely to experimental results.
This makes Clausius' argument more compelling than Helmholtz's of 1847 that
he did not follow up until much later. 93
Clausius criticized the work of Helmholtz and Holtzmann in the theory of heat
over the consistency of their physical interpretations and their usage of mathemat-
ics. In 1853 Clausius argued that Helmholtz's demonstration of his conservation
93 Helmholtz later claimed that he saw his 1847 paper as simply a review of the literature;
his principle not being so remarkable a thing to come by. However, while ceding the
credit for the idea to Mayer he accepted all the credit for its development. See, Helmholtz,
"Erhaltung der Kraft," (1881), Appendix. See also his letter to Tait used by the latter as
an "impartial" account of the early history of thermodynamics to counter Clausius' com-
plaints of mistreatment in Tait Sketch of Thermodynamics (Edinburgh: David Douglas,
1877), chap. 1. For more details see Garber, Brush and Everitt, Maxwell on Heat and
Statistical Mechanics 34-44.
300 Physics and Mathematics
law was only valid for his particular model of matter. Helmholtz had not, math-
ematically, established it in general. In addition Heimholtz had not understood
the notion of the potential or used it consistently in the electrical examples he
chose to illustrate his principle. Clausius added an illustration, inherent in a paper
Helmholtz had missed. 94
In his criticism of Helmholtz on the potential, Clausius separated the physical
potential from its mathematical expression, Gauss' potential for which Clausius
preferred Green's expression the potential function. Helmholtz was stung by the
criticism. It undermined his position within physiology as well as the profession
he was fast discovering he ought to be in, physics. His reply then acknowledged
the validity of Clausius' criticism, while demonstrating he could do his sums as
well as anyone. On the question of his model being particular, Helmholtz could
only reply that he was concerned with "real" forces, not those abstract, generalized
concerns of "mechanicians [Mechaniker ]," i.e., mathematicians. A nice putdown
but hardly an answer to the nub of Clausius' point. 95
Clausius clearly knew both the mathematical and physical aspects of this ques-
tion. He also had a particular vision of the structure of matter that surfaced at the
end of the decade. He had suppressed it earlier for the sake of an analysis based on
more acceptable foundations and experimentally demonstrable assumptions. Later
Clausius was to explore the actual molecular motions that constituted heat through
a theory of gases. He developed the concepts of disgregation and the virial while
deepening his own understanding of the physical significance of the second law of
thermodynamics. 96
In the early 1850s both men were equally able to manipulate modern mathematics
and express the results of their mathematics in the language of a chosen physical
imagery. While Helmholtz had begun this in a manner guaranteed to disturb the
very audience he wanted to reach, by 1860 he demonstrated his control of their
methods in such a way that he could no longer be ignored. In 1858 Helmholtz
published a paper in Crelle's journal that put together a mathematical argument
through physical illustration. This was not new, but the implications he drew from
the solutions to the partial differential equations were put in purely physical terms.
94 Clausius, "Uber einige Stellen der Schrift von Helmholtz 'Uber die Erhaltung der Kraft' ,"
Ann. Phy. 89 (1853): 568-579, and 91 (1854): 601-604. Helmholtz's reply is sand-
wiched between in Helmholtz, "Erwiederung auf die Bemerkungen von Her. Clausius,"
Ann. Phy. 91 (1854): 241-261. Clausius' criticism of Helmholtz's inconsistent under-
standing of the idea of the potential was stated in his earlier papers on galvanic electricity.
Clausius' criticism of Holtzmann appeared as, "Erwiederung auf die im Marz-hefte der
Annalen enthaltenen Bemerkungen des Hrn Holtzmann," Ann. Phy. 83 (1851): 118-125.
On both counts Clausius was correct and later Helmholtz acknowledged this.
95 Helmholtz, "Erwiederung ," and "Erhaltung," (1881), Appendix.
96 See Garber, Brush and Everitt, Maxwell on Heat and Statistical Mechanics, 45-46, and
the literature cited there.
German States, IS30-1S70 301
The problem was in hydrodynamics. Helmholtz argued that the solutions to the
general hydrodynamical equations offered from Euler to Stokes ignored friction,
both internal to the fluid, and between the fluid and fixed bodies. Helmholtz
demonstrated mathematically that if a "velocity potential" existed, that is a perfect
fluid, there could be no rotations within the fluid. While not being able to investigate
the question in general, Helmholtz was able to demonstrate the mathematical and
physical characteristics of such rotations, should they exist. 97 Helmholtz had no
notion of the mathematical forms of two types of friction he identified and it was
unlikely, even if he could do so, that the resulting differential equation would be
integrable. For the particular cases that he could investigate, Helmholtz used both
Green's theorem and the analogies he could draw from the forms of his equations
when they resembled those of electrodynamics. At each stage in the development
of his mathematical cases, Helmholtz referred to a physical description of what
was going on in the fluid, sometimes in analogy to an electrodynamical case. And
he was only interested in physical cases. Purely mathematical ones were not his
concern, although he knew full well that he was solving a previously unsolved
mathematical problem.
A more limited domain for mathematics was evident in Clausius' later work,
none more so than in his 1859 treatise on the potential. This was a text in mechanics
in which conservation of energy was seen as a less general way of understanding
physical processes than the potential. For both Helmholtz and Clausius, mechanics
and electrodynamics were expressed in terms of potential and force, and eventu-
ally least action, in preference to energy. In this text we see Clausius' training in
mathematics, especially his understanding of Dirichlet's work on the potential. 98
Just as Helmholtz had done the previous year, Clausius developed the mathematics
only so far as it was useful for making his physical point. In fact the mathemat-
ics is somewhat sloppy, although Clausius argued in such a way that mechanics,
electricity, and magnetism were separated simply by the value of a constant in his
fundamental equation for the force law. The potential as a physical concept unified
97 Helmholtz, "Uber die Integrale der hydrodynamischen Gleichungen we1che den Wirbel-
bewegung entsprechen," J. Reine Angew. Math. 55 (1858): 25-55. Helmholtz then
investigated fluids and their internal friction experimentally. See, Helmholtz and G.
von Piotrowski, "Uber Reibung tropfbarer Fliissigkeiten," Ber. Berlin 40 (1860): 607-
658. This was followed by Helmholtz, "Uber discontinuirliche Fliissigkeitsbewegungen,"
same journal (1868): 215-228, translated into Phil. Mag. 36 (1868): 337-346. This list
does not include the work that followed on gases, acoustics and sound.
98 Thomas Archibald, "Physics as a Constraint on Mathematical Research," has argued that
the development of understanding of the potential was hampered by its investigation by
physicists rather than mathematicians. My point is that physicists had their own purposes
for investigating the potential. Those of mathematicians they now left to mathematicians.
Perhaps it would be more useful to ask why mathematicians during the 1850s and 1860s
did not see the potential function as offering them interesting mathematical problems to
solve.
302 Physics and Mathematics
99 Clausius, Die Potentialfunction und das Potential: ein Beitrag zur mathematischen Physik
(Leipzig: Barth, 1859), 14. The text went through four editions in Clausius' lifetime.
100 Clausius, Die Potential, 8, 10.
101 Clausius Die Potential, 149-158.
German States, 1830-1870 303
not physics. In Clausius' case we have a prolific author who published in every
field of importance in research during the three decades from 1850 to 1880. With
Helmholtz there was a clarity of physical vision that allowed him to control the
mathematical language lacking in previous generations of German physicists. Af-
ter having made the mathematical point the physical one had to follow. If it did not,
the mathematics was of no interest to them. The adopted languages of mathematics
now described physical processes.
In the German States we also have a series of approaches to the issue of the inter-
pretation of natural processes and structure that parallel those of British physicists,
yet were set upon different foundations. Force and potential were the conceptual
basis for German physics, energy conservation and engineering mechanical models
for the British. In both groups there were individuals who appreciated the need to
rise above the particulars of models to establish more defensible grounds for this
new enterprise of theoretical physics. The British and the Germans produced com-
plementary, yet quite different visions of the same processes when solving shared
research problems. In their work they collectively created a range of possible ap-
proaches to the interpretation of physical phenomena for which mathematics, and
especially the calculus, was the crucial, common language, reshaped to the needs
of the discipline they were at the same time creating.
While we might disagree with some of their premises, much of their methodology
is familiar. Too familiar, for we forget that it had first to be recognized, practiced,
then molded to purposes that were defined by the very research problems they
chose to pursue. Twentieth-century physicists extended and developed the power
of these practices, and their success obstructs our view of the very processes through
which this physics came into existence. This is the process that this exploration
has attempted to render visible once again.
Part IV
Theoretical physics did not come into existence as a subfield of physics until
the 1860s. By 1870 physicists had accepted mathematics as the natural language
of physics and put into place their own ways of training and using the diverse lan-
guages of mathematics. Physicists such as John Tyndall were anachronisms within
the profession. While he performed quantitative experiments, he was not obsessed
with accuracy, even though trained within the German academic system. He also
did not deduce algebraic relationships from his results that were by that time ex-
pected of physicists. 1 Tyndall's statements about the structure and functioning of
nature were qualitative and in the vernacular. And his audiences consisted of the
general public, as well as his colleagues within the profession. His career harkened
back to the era before the formalized, academic and professional structure of the
discipline which he entered in the 1860s. Physicists had withdrawn into a profes-
sion of peers that largely addressed each other. The general public was not privy
to the research process as they had been in the first half of the nineteenth century.
The mathematics now necessary to penetrate the theories of physicists meant that
only the most general of ideas and sketchiest of plans of their understanding of
nature were available to the vast majority of the general public.
However, theoretical physics was not just mathematics. Mathematics encapsu-
lated a physical situation, or process in symbolic form. It represented a relation-
ship between physical concepts, or an operation, interpreted in a particular case
expressed in algebraic or geometrical form. The direction and depth in which the
symbolic forms of this language were developed through mathematical operations
and transformations were now firmly controlled through the physical meanings
embedded in the symbolic forms and operations of the mathematics. The possible
directly from the physical extrapolations of a theory, were now possible. Analogies
developed from the mathematical forms appearing within distinct physical situa-
tions were no longer satisfactory as a guide to predictions. 2 Maxwell was surprised
with the results of his kinetic theory of gases. In general, the internal friction of
gases appeared to be independent of pressure and varied as the square root of the
temperature. This prediction did not include any details of the specific geometry
of an experiment to mimic this result. Maxwell had to develop the details of his
experiment through the mathematical example of circular plates oscillating in a
horizontal plane about a vertical axis.
The elegance of the experiment and the precision with which he captured the
results of his deductions did much to validate his ideas on gases. 3 They also were
instrumental in changing the function of experiments. Because theories now con-
tained precise physical imagery and led to results that were directly reproducible
in the laboratory, experimentalists could literally test the predictions of theories.
Their work became necessary for the validation of theories. 4 Mathematicians'
dismissals of experimental results were no longer sufficient to stifle criticism of
mathematical derivations. If an expected phenomenon was not detected, the theory
was in more trouble than if it was found to be of the wrong order of magnitude.
Yet theories might survive, given ongoing disputes between experimentalists over
methods and accuracy.5 However, experiment could refute theory.6
Experiment was the other half of a new enterprise of physics. While problematic
experiments reflected upon theories, not mathematics: Mathematics had become a
given. The power that the mathematics of the calculus brought to physics changed
the very nature of the theoretical structures physicists could use to interpret the
operations of nature. The range of phenomena that could be encompassed within
the net of an hypothesis broadened. The most dramatic example was a theoretical
that made the mathematics physically visible with a specificity that generalized
principles did not offer.
The use of specific mechanical models to constrain mathematical language led
to compromises in mathematicians' definitions and understanding of the terms and
operations that were the elements of that language. The extent of the compromises
the physicist allowed himself also was a matter of choice. Yet even in the case
where, after 1870, in Boltzmann's work in statistical mechanics, physics again be-
came mathematics, the context of that choice changed how the work was received
and what was then done with it. Boltzmann's work on the second law was contro-
versial within physics because there was no mechanical imagery to visualize the
physical processes that the H-theorem was meant to express.? Boltzmann himself
felt the need to reexpress his theorem in mechanical terms, to give it a real physical
meaning. He finally abandoned these efforts. 8
Mathematics was both structural and expressive of ideas. It was structural in that
it was used to express relationships. At the same time mathematics also allowed
for the exploration of what could happen, given the limitations of mathematical
structures and operations, and the processes allowable through the consideration
of physical hypotheses. Mathematics also limited and tamed speculations with the
necessity for mathematical consistency. The characteristics of mathematical func-
tions or coefficients could suffer immediate physical interpretation with subsequent
consequences for the visualization of how the physical system could behave. Clau-
sius' understanding of internal energy, mechanical work and heat are obvious cases
in point. Mathematics was also itself used as the source for analogies for under-
standing the mathematics of one domain of physics from another. Mathematically
identical structures were also richly suggestive of physical behavior in physically
isolated cases. Both could and did evolve together and were thus processes of
interpretation in mathematical language and physical imagery simultaneously.9
Experiment, hypotheses, and mathematics were the foundations for new ways of
investigating and interpreting the processes of nature. The fusion of these aspects
of physics can be seen in the changes in textbooks during the last third of the nine-
teenth century. German texts of lectures in mathematical physics barely mention
experiment. Mathematical methods, consistency and their manipulation to obtain
theoretically interesting results were the focus of attention. Those in theoretical
physics, starting with Thomson and Tait's text, joined concepts, experiment and
mathematics together. They offered students an introduction to both concepts and
7 For the H-theorem see, Brush, The Kind of Motion we Call Heat, vol. 1, chap. 6.
8 See Martin J. Klein, "Boltzmann, MonocycJes, and Mechanical Explanation," in Boston
Stud. Phil. Sci. 11 (1974): 155-175.
9 In this sense theories are about themselves as well as the external world. See Enrico
Bellone, A World on Paper: Studies in the Second Scientific Revolution (Cambridge
MA.: MIT Press, 1980).
312 Physics
mathematical methods. This was coupled to numerous examples of how the lan-
guage of mathematics couid and should be manipulated and interpreted to yield
physically meaningful results. The practice, used within French mathematical
physics, of beginning in specific examples was taken over into theoretical physics.
However, the purpose of the mathematical exercises was now the investigation and
interpretation of nature, not the generation of mathematics.
French physics, and most of the other sciences, fit awkwardly into any account
of nineteenth-century science under the assumptions that historians make about
the markers of excellence or intellectual development. What French scientists did
in the nineteenth century does not easily fall into line with the work of scientists
in the same fields in either Germany or Britain. The easy way out is to omit them
altogether, or, simply mention those men and their work who are necessary in
marking the intellectual development of a field. Either way historians avoid the
issue of French Science altogether. However, French names and research crop up
too often to ignore the question of what makes French science different from that
of Germany or Britain in the nineteenth century?
Accepting the sociologist's solution of labeling French science as in "decline"
hardly solves the problem. Sociological categories define the sciences using twenti-
eth century criteria. These categories are not the best instruments for understanding
the intellectual differences that existed in the sciences across national and cultural
boundaries over a century ago. Sociological factors and political circumstance
dominate theories on nineteenth-century French science, although the adequacy
of this approach has recently been questioned. lO These factors and circumstances
are used to measure the intellectual place of French scientists and mathematicians
amongst the other European nations. 11
One recent examination of physics and mathematics of France in the nineteenth
century focussed upon research productivity in an effort to draw together an inte-
grated picture of French science during this era. 12 The political economy of science
can point to restrictions in opportunities, and hence decline in the numbers of sci-
entists and their productivity. However, quality is not necessarily equal to quantity.
Putting the issue of how to measure "quality" aside, a decline in quantity of pub-
lished research does not address the continuing importance of the research results
of French scientists to scientists in Germany and Britain in the nineteenth century.
British and German physicists had to take the work of French experimentalists into
account especially in the study of heat and light. To learn the practices of exper-
imental physics, William Thomson spent months toiling in Regnault's laboratory
in the 1840s. German experimentalists and mathematicians still made pilgrimages
to Paris to meet their peers. French assessments of German experimentalist's work
were germane in their files for promotion. This was even more true in mathematics
and mathematical physics. A measure of the importance of French physics to their
German and British colleagues can be made by looking at the reports carried in
German and British journals of French scientific work in French scientific journals
and the publications of scientific societies across France. It is also vividly reflected
in the footnotes to, remarks and reports on, and uses made by British and German
physicists of those works in their own journals. They also pepper their private cor-
respondence. French journals were still required reading. French experimentalists
and mathematicians were important colleagues.
The work of French physicists still mattered to their British and German col-
leagues. Simply to point out that before 1830 there was one major center for
scientific research in Europe, namely Paris, and after that date there were several,
namely London, Cambridge and the Scottish Universities as well as the Universi-
ties of Berlin, Heidelberg, Bonn and Konigsberg, names the phenomenon without
explaining it. Perhaps we need to consider how French scientists practiced their
crafts during the nineteenth century before we declare "decline."
During the first three decades of the nineteenth-century French physicists and
mathematicians developed a highly successful set of practices that defined the dis-
ciplines of experimental physics and mathematics. The intellectual boundaries
of these disciplines were fixed even as the institutional setting for their pursuit
changed from a vocation and the Academie to the Universite and the Ecole Poly-
technique and a profession. These intellectual boundaries were stable throughout
the nineteenth and into the twentieth century. French physics and mathematics
did not decline. The practices of the members of the disciplines conformed to the
highly successful practices developed during that first third of the century.13
French experimental physicists were skeptical of "speculation" and hypotheses,
other than those that had become so accepted within the community of French
physics as to not seem hypothetical at all. The subject matter of their experiments
also related to areas that had been successfully explored during those decades of the
early nineteenth century. These areas included the phenomena of light, within the
13 The social institutions also formed a continuum with those of the early nineteenth century.
See Maurice Crosland, Science under Control: The French Academy o/Sciences, 1795-
1914 (Cambridge: Cambridge University Press 1992).
314 Physics
context ofthe wave theory, and heat, within the context ofthe caloric theory of heat.
Many experiments also related to other areas of strength developed by the French
in those same early decades of the century, including astronomy.14 Astronomy and
precision experiments in optics were closely related. Nineteenth-century French
emphasis on particular aspects of optics begins to make sense.
Within France, the ideological and social barriers erected in the German States
between research that was esoteric and "pure" and that which was practical and
by implication of lesser intellectual value, did not exist. While universities were
teaching institutions, research both esoteric and useful was pursued there even
before the 1870s. After the Franco-Prussian War stringencies of the budgets from
Paris required that physicists and chemists seek local sources of support. The utility
of science was pursued and made manifest in the work of physicists in industry
and for industry. There was a constant flow of scientists from industry to the
university and back again. I5 Henri Victor Regnault's work on the physical constants
of gases was commissioned by the French government in an effort to improve the
design of steam engines. Regnault completed this research at the College de
France as professor of physics. Regnault was by training and previous research a
chemist. Regnault's experiments included redetermining the composition of air and
respiration, a remarkably broad range of experiments that crossed the disciplinary
lines being drawn in both Germany and Britain, yet remaining inside physics within
the borders of France.
In France, the foundations for the practices and standards of experimental physics
established in the early decades of the nineteenth century deliberately excluded hy-
potheses. Knowledge was based on observation and measurement. Experimental
physicists in France regarded their work as purely empirical and devoid of all hy-
potheses. These physicists were reduced to narrow domains of endeavor because
they could not embrace the speculations being investigated as fast as possible by
British and German experimentalists. This trait was particularly marked in exper-
iments on, and speculations about, the nature of heat. Clapyeron's mathematical
explorations ofCarnot's ideas on heat of the 1830s had not contained any deductions
that drew them into contact with experiment, or observation, or the development
of the caloric theory of heat. Regnault's work was empirical and his conclusions
based on phenomenological reasoning. While French experimentalists reported
work that indicated the equivalence of mechanical work and heat they were largely
16 For a discussion of Reech's work see Clifford Truesdell, The Tragicomical History of
Thermodynamics, chap. 10, and, Appendix.
17 See MaryJo Nye Molecular Reality: A Perspective on the Life ofJean Perrin (New York:
American Elsevier, 1972). These were also the standards of French chemists who only
reluctantly accepted atomism by 1900. See Terry Shinn, "Orthodoxy and Innovation on
Science: The Atomist Controversy in French Chemistry," Minerva, 18 (1980): 539-555.
18 See Perrin, "Mouvement brownien et grandeurs moleculaires," Ann. Chim. Phys. 18
(1909): 1-114, translated by Frederick Soddy as Brownian Movement and Molecular
Reality (London: Taylor and Francis, 1910). See also Perrin, "Rapport sur les preuves
de la realite moleculaire," in La theorie du rayonnement et les quanta, Paul Langevin and
Louis de Broglie, eds. (Paris: Gauthier-Villars, 1912), 153-250.
19 Pierre Duhem The Aim and Structure of Physical Theory, P. Wiener, trans. (Princeton:
Princeton University Press, 1954.) We have omitted any reference to the religious goals
of Duhem's philosophy of science. See also Bruce Eastwood, "A Second Look: On
the Continuity of Western Science from the Middle Ages, A. C. Crombie's Augustine to
Galileo," Isis, 83 (1992): 84-99, 88.
316 Physics
20 See Jesper Liitzen Joseph Liouville. On French mathematicians' use of Fourier, see
Garber, "Reading Mathematics, Constructing Physics."
21 Poincare's work in celestial mechanics focussed on the three-body problem and examined
the mathematical properties of recurrent orbits. See, Henri Poincare, "Sur Ie probJ(:me
des trois corps et les equations de la dynamique," Acta Math. 13 (1890): 1-270.
About 1870 317
Some Conclusions
We can no longer assume that physics, with its modern standards and practices,
has existed since Newton, Galileo or anyone person. Nor can we claim, as did
Cannon some twenty years ago, that physics "was invented by the French between
1810 and 1830." Historians of physics now agree that social institutions shape
the lives of their practitioners and the functioning of disciplines. The institutions
and standards of modern physics were not in place in Europe until the 1860s. In
addition, the research practices that shaped the discipline into its modern form were
created in the nineteenth century. Those practices, together with the institutional
forms in which they functioned, were the keys to making modern physics.
In this study we have focussed upon the ways in which theories about the structure
and functioning of nature shaped the practices of what we call theoretical physics.
Foundational ideas, the general principles upon which speculations about nature
rested, are insufficient to define what theories are and what physicists did in creating
theoretical physics. Meaning is conveyed only through the exploitation of those
principles in the context of specific problems. The implications of mechanical
principles were interpreted and reinterpreted through the results of explorations of
the behavior of bodies under well defined circumstances. General principles needed
often to be coupled with sets of subsidiary hypotheses to bring those principles to
bear upon the solutions of particular problems. Specific analyses of these particular
problems are the hallmarks of theory.
The language that eased the development of such detailed working out of the
implications of general ideas was the calculus. Without the investigation of how
mathematics became the language of physics, any account of the development of
theory is hollow. Mathematics was necessary to the development of theory. Before
its widespread use within physics, natural philosophy was speculative and closer to
metaphysics than the experiments that formed the core ofthe discipline in the eigh-
teenth century. Mathematics has shaped and reshaped physicists' interpretations
of nature. Different forms of mathematics have allowed physicists to reinterpret,
to literally, envision phenomena and their interpretation in new ways.22
Theory also encompassed experiment in ways that the older speculative natural
philosophy and mathematical physics did not. Experiments were integrated into
the very body, into the detailed implications, of the physical ideas making up that
theory. Mathematics gave physicists the flexibility to develop ideas on high levels
of abstraction, while also allowing them to descend into the detailed functioning
22 The most dramatic nineteenth century example of this lies in the introduction of vector
analysis into the theory of electromagnetism. In another context Ana Millan Gasca has
discussed how different mathematical approaches affected the biological sciences and the
images of biological systems the mathematics brought with them in Gasca, "Mathematical
Theories versus Biological Facts: A Debate in mathematical population Dynamics in the
1930s," Hist. Stud. Phys. Sci. 26 (1996): 347-403.
318 Physics
of specific cases where experiment might be mirrored, in ideal terms, within the
compass of the formalisms of mathematics. Mathematics could be reduced to
particular, numerate cases. As a language mathematics could be used to extrapolate
beyond the confines of known experimental results to predict the results of specific,
theoretically visualizable, yet still unrealized experimental conditions. This was no
longer a deduction from mathematical analogy but specific, physical juxtapositions
that might be put into experimental form. Both the use of mathematics as the
language of speculation and the ability to integrate experimental findings into the
body of theory changed the nature of speculations about nature and what was
acceptable as speculation about nature.
At the same time that physicists were creating theoretical physics, experiment
was reconfirmed as the center of the discipline. Theoreticians did not take over the
discipline. They remained a minority in numbers and their output was subject to the
searching probes of experimentalists who were apt to mold theoreticians results
to their own purposes. They were also apt to modify, if not deny, the validity
of theories. And theoretical physicists felt compelled to follow the dictates of
experiment. 23 The dominance of experiment was also reaffirmed in the 1890s with
the detection and exploration of x-rays and radioactivity. It was not yet plausible
to declare the independence of theory from experiment. Even in the 1930s it was
still possible to state that physics was experiment, the rest was only mathematics. 24
The complex of methods that made up theoretical physics by 1870 and the tan-
gled relationships that developed between mathematicians and physicists can only
be clearly understood if we distance ourselves from the concerns of philosophers
and the histories of physics written by physicists. In the late nineteenth-century
philosophers took as their model of physics descriptions of physics and its de-
velopment written by contemporary physicists. Philosophers have defined the
essentials of physics for historians of physics ever since. And, as these essentials
have changed, so have the narratives of historians. During the late nineteenth
century, physicists in Germany and Britain remade their history to conform to the
new disciplinary boundaries and practices they had created. This makeover was
done both in formal histories of physics and in the reinterpretation of the content
of technical papers written in the eighteenth and early nineteenth centuries. In
this they were aided and abetted by mathematicians also busily rewriting their own
history which discounted the standards of eighteenth-century mathematical prac-
tices. Physicists were able to claim as physics many of the papers written within
mathematics in the eighteenth century, by interpreting the mathematical results of
those paper in physical terms where none existed in the original. Tait gave the
notion of the conservation of energy a pedigree that reinterpreted the meaning of
Newton's work that was historically and technically dubious. In general, New-
ton's significance was redrawn to conform to late nineteenth-century standards of
physics as a discipline. 25 Euler, Lagrange, Poisson, Fourier and a host of others
were soon accepted as working within physics as well as mathematics and became
prodigious heros with deep physical insight as well as exulted mathematicians.
Physics was redefined by their inclusion in its pantheon. 26 Within these narratives
physics, since Galileo and Newton, was a discipline driven by theory and expressed
mathematically.
Historians of physics have taken these histories far too seriously. They have as-
sumed that throughout the eighteenth and nineteenth centuries papers, treatises, and
textbooks bearing titles that place them within the boundaries of late nineteenth-
century physics were written as physics papers, treatises and textbooks. And
because their mathematical methods became part of the practice of theoretical
physicists, historians accepted their designation as "physics" at their time of pub-
lication.
We have to discard the idea that once a method was introduced into physics
It remained part of the practice of the discipline. We also have to rethink The
notion that what we regard as theoretical physics was always read as such in the
past three centuries. Specifically, we need to consider what mathematics meant
in the eighteenth and early nineteenth centuries to judge whether, mechanics for
example, was indeed an aspect of physics or a branch of mathematics in those eras.
We must consider the practices of mathematicians and physicists simultaneously
before such assumptions become historically reasonable. Perhaps mathematics
has only been seen as the "natural" language of physics for the last century and a
half.
By 1900 mathematics had become so essential and integrated into the practices
of theoreticians that the idea of a preestablished harmony between the the two
25 Perhaps the last history of physics published that took experiment as its core was that
of Poggendorff's in the 1870s. See J. C. Poggendorff, Geschichte der Physik (Leipzig:
Zentral-Antiquariat of the DDR reprint of 1879 edition, 1964). The history of Max-
imilien Marie, Histoire des sciences mathematiques et physiques (Paris: Kraus reprint
of Gauthier-Villars edition of 1883-1888, 1979) is a history familiar from eighteenth-
century France. The narrative is biographical with some technical discussion of what
the list of characters did. All biographies are treated strictly chronologically. The only
explicit value judgments that enter are those directed against astrologers and alchemists.
This is another indication of the uniqueness of French mathematics and physics in this
era and their connections to their eighteenth-century roots.
26 Rachel Laudan, "Definitions of a Discipline: Histories of Geology and Geological His-
tory," in Functions of Disciplinary Histories, Loren Graham, Wolf Lepenies and Peter
Weingart, eds. (New York: Reidel, 1983) has followed the same pattern of the ap-
propriation of history by geologists. See also Paula Findlen, Possessing Nature on the
rewriting of the history of natural history in the eighteenth century that rendered the work
of Renaissance naturalists invisible.
320 Physics
27 See Lewis Pyenson, "Relativity in Late Wilhelmian Germany: The Appeal to a Preestab-
lished Harmony between Mathematics and Physics," Arch. Hist. Exact Sci. 27 (1982):
137-155. The author thanks David Cassidy for pointing out this argument and its impor-
tance for twentieth century physicists.
Chapter X
Epilogue:
Forging New Relationships, 1870-1914
By 1870 both physics and mathematics had become distinct academic specialties
within universities across Europe and, in these forms, spread to the United States,
Japan and elsewhere. The research center of physics was in the laboratory and
in the pursuit of quantitative experiments of increasing accuracy tied consciously
to the development of theory. Theory was an accepted research activity and its
language was the calculus, that is, ordinary and partial differential and integral
calculus as it stood within mathematics in the 1830s. This might be the end of the
beginning except that theoretical physics as an accepted subfield within physics,
with theoreticians forming a distinct subcommunity within physics, did not coa-
lesce until the twentieth century. Theoretical physicists formed a loosely connected
set of individuals within the discipline and profession of physics itself. To form a
subfield within academic physics, theoreticians needed to develop a set of practices
that were distinct from those of their experimental peers. They also had to replicate
themselves by training students as theoreticians, rather than students merely taking
courses in theory. Even if they did not develop their own specialist societies and
journals, their work needed the recognition of experimentalists as valuable and
complementary to their own research. These processes as well as the development
of a sense of collective identity as theoreticians unfolded within physics during the
forty odd years between 1870 and World War I. This growing awareness was fos-
tered also through a series of intense, competitive interactions with mathematicians
in the 1890s and early 1900s. While in 1870 both disciplines reached a maturity
marked by autonomy, within forty years members of both disciplines had forged
new relationships across their respective disciplinary boundaries. These were not
so much alliances as sometimes fierce competitive interactions that have marked
the development of theoretical physics throughout the twentieth century.
In the first two decades after 1870, the center of research activity in physics
shifted decisively to universities of the newly established German Empire. How-
ever, within German universities the disciplines of mathematics and physics drew
steadily apart.! Although this separation was only temporary, the terms under which
mathematicians and physicists interacted with one another in the early twentieth
century were different from those of the mid-nineteenth century. By 1900 mathe-
matics was taken for granted as a part of physics and included along with laboratory
courses in the training of physicists. In the same decades physicists developed their
own versions of the calculus for their students and, more significantly, had begun
to develop mathematics beyond the calculus, directed to their own needs without
the mediation of mathematicians.
In the same decades mathematicians in Germany focussed on research prob-
lems that emerged from mathematics, not the problems of physics. Not all math-
ematicians engaged in this form of research, yet, those involved in "foundational
problems" and pure mathematics dominated the departments of the prominent uni-
versities, influenced professional appointments, and sat on the editorial boards
of the major mathematical journals. This state of affairs changed only in the first
decade of this century through the efforts of Felix Klein and others with their belief
that their interests in certain types of mathematical problems coincided with those
of physicists. Thus began a series of interactions of mathematicians and physicists
that were mutually beneficial, yet shot through with mutual misunderstandings.
The patterns discernible in the institutions and discipline within Germany cannot
be superimposed upon the profession or the research produced within France,
Britain, or the United States. 2 These three communities followed their unique paths
of development where theoretical physics held an even less prominent position than
in the German universities.
One common characteristic of all academic disciplines in this era, whatever the
national differences in their internal organization, was their international charac-
ter. Both mathematicians and physicists addressed their respective international
research communities. 3 A second common characteristics was that articles in jour-
nals addressed a small international audience of mathematicians or physicists that
excluded all but the authors' immediate colleagues engaged in the same cluster of
research problems. Addresses to colleagues across physics was becoming more
difficult except in general terms and those to colleagues across the academic cam-
pus were reduced to philosophical issues with minimal technical content.
Significant aspects of the emergence of theoretical physics to a central position
within the discipline of physics lay in the solutions to research problems that
1 The reasons for this isolation were both institutional and intellectual. The relationships
between the disciplines also depended heavily on the particular institution under discus-
sion. See Jungnickel and McCormmach, Intellectual Mastery of Nature vol. 2 chaps.
21-23.
2 This account is skewed towards German universities because the institutions and intel-
lectual development of physics and mathematics elsewhere have been less studied.
3 We should remember the advent of International Conferences in mathematics and physics
began in the early twentieth century.
Forging New Relationships 323
required physicists to develop mathematical languages that led them beyond the
calculus, the mathematical language that seemed to define their subfield at its
inception.
The subject matter of this chapter falls naturally into three overlapping themes;
the range of ways in which physicists used mathematics as the languages of theory
and how these languages related to both general laws of nature and specific models;
the kinds of mathematics they used and/or developed in their research and the
kinds of mathematics that they taught to the next generation of physicists; the
development of a new set of relationships between mathematicians and physicists
within the context of the modern professions in the first two decades of this century.
We confine ourselves to how physicists appropriated the languages of, as well as
developed new ones useful for their own purposes. While it is usual to trace
changes in theory through ideas, imagery and experiments, these enter into this
account only on its margins. We trace mathematics, the language of theory that
had become so thoroughly integrated into practice that it was almost invisible, a
"tool." A tool it has remained in historical accounts. Supposedly the real work
of theorists was in the development of ideas about nature, not the development of
languages in which to express those ideas. 4
After 1870, research produced by the group within physics that we recognize as
theorists integrated mathematics, physical imagery and experimental results into
a form familiar in the twentieth century. The autonomy enjoyed in the German
university systems by mathematicians and physicists changed the ways in which
research within their disciplines was structured, to whom it was addressed, and the
training available for the next generation of professionals. These activities were
geared to maintaining the autonomy of the discipline and to training students as
future professionals. These processes had the cumulative effect of isolating math-
ematicians and physicists from one another. Another component aiding isolation
was limited resources, the budgets of even Prussia could not expand indefinitely
as the university system expanded. The competition for limited resources and the
standards used in distributing those resources led inexorably to several character-
istics shared by mathematics and physics.
In general, career patterns depended on both teaching and research. For advance-
ment to chairs at major universities it was necessary to develop an international rep-
utation within a disciplinary research field. For a successful career the physicist or
4 Throughout this chapter a distinction is understood between mathematical physics and
theoretical physics. This lies essentially in the level of abstraction to which the math-
ematics is carried and its relationship, drawn up by the author, to the physical problem
under consideration. Added to this is the institutional place of the author and the disci-
plinary character assigned him by his colleagues. While being somewhat arbitrary this
distinction is however useful.
324 Epilogue
mathematician had to be well aware of the type of research that would actually lead
to recognition, and hence promotions, and the teaching and research support that
went with them. In physics this meant minutely analyzed precision experiments,
costly in themselves in terms of equipment and human resources. Excursions into
theory needed to be coupled to experience in the laboratory. 5 Although courses
in theoretical physics developed in all universities, systematic training specifically
for theorists with courses of increasing sophistication and difficulty still lay in the
future.
Since theory was an accepted aspect of the research enterprise, it was integrated
into the training of physicists. However, the research interests of the teachers of
such courses often lay in experiment not theory. As significant, the occupants
of chairs in theoretical physics were, extraordinarius not ordinarius, professors.
Even as the sense of identity of "theoretical physics" within physics grew, "theo-
retical physicists" were still a minority within the discipline and profession. For
many physicists their primary field of research was experiment and the theory they
explored was particular to their experimental concerns. Experiment was still con-
sidered the core of the discipline. 6 In the German Empire at the end of the century
some chairs in theoretical physics were offered to experimentalists if they agreed
to teach the smaller, specialized classes in theoretical physics. The larger, more
lucrative classes were for the full professors, and those classes were in general
and experimental physics. 7 Even though theoretical physics flourished during the
last decades of the nineteenth century, we should not overestimate the numbers or
importance of the field in physics during the closing decades of that century.
Experimental precision remained the key that separated the professional from
the amateur. To gain the necessary skills, systematic training was required. Ed-
ucation in the manual skills of precision offered moral training for the specialists
5 Max Planck was the first physicist to develop a career with no experimental research.
Helmholtz, and Boltzmann did experimental research as well as theoretical even though
only Helmholtz contributed significantly to experimental physics. Clausius taught ex-
perimental physics and controlled the resources of an experimental, research laboratory.
Hemholtz also was the only one of this group to head a major research laboratory. Dur-
ing his career in Berlin Helmholtz's experimental skills were well appreciated while his
research throughout the 1870's was theoretical. Boltzmann and Clausius spent time try-
ing to restrict their duties to teaching theory rather than overseeing the time consuming
laboratory courses. Jungnickel and McCormmach, Intellectual Mastery vol. 2., chaps.,
14, 16, vol. 1, chap., 8, 12.
6 David Cahan, "The Institutional Revolution in German Physics, 1865-1914," Hist. Stud.
Phys. Sci. (1985): 1-65 demonstrates that the resources poured into the physics institutes
built at the end of the century was for experimental, not theoretical, research and teaching.
7 See Paul Forman, John Heilbron and Spencer Weart, "Physics circa 1900. Personnel,
Funding and Productivity of the Academic Establishment," Hist. Stud. Phys. Sci. 5
(1975).
Forging New Relationships 325
needed in the developing industrial economies, or so the arguments for support for
these training laboratories and institutes went. 8 Most of the research reports in the
pages of the major physics journals of Europe were experimental, or theoretical
investigations limited to points that emerged from experimental work.
However, by the 1890s young physicists such as Ernest Rutherford abandoned
the precise methods of their immediate predecessors and were ready to risk their
careers in the exploration of the newly discovered phenomena of radioactivity or
x-rays.9 These phenomena emerged from experimental research and were remote
from the concerns of contemporary theoretical physicists.
Physics was still institutionally one field, and theory a decidedly secondary
aspect of it. 1o Support, within the university and at the ministerial level was luke-
warm. Theory had to prove itself to the experimentalists and they had to convince
the various departments of education of its intrinsic worth. This did not come
together until after 1900.
Mathematics as an integral part of theory was also an aspect of the introduc-
tion of students to theoretical physics. Students of physics were even encouraged
to attend lectures in mathematics departments. Yet, after 1870, with few excep-
tions, less and less that was taught in mathematics departments seemed relevant
or even vaguely connected to the needs of physicists, mature or neophyte. During
the same decades that physicists reconstructed physics, mathematicians in both
Germany and Britain changed their own discipline. Mathematicians turned away
from the eighteenth-century practice of using the solution of problems to generate
mathematics. Direct examination of the foundational concepts of mathematics
and development of their linguistic possibilities from within became the hallmark
of a first class mathematician. What constituted the foundations of mathematics
that now required such extensive examinations differed in Britain and Germany.
This new vision of mathematics was shared only by a minority of mathematicians
in Britain and also split the German mathematical community into two hostile
camps.11
8 See Cahan, "Institutional Revolution." This was also true in Britain, see Graeme Gooday,
"Precision measurement and the Genesis of physics teaching laboratories in Victorian
Britain," Brit. J. Hist. Sci. 23 (1990): 25-5l.
9 Isobel Falconer, "J. J. Thomson and 'Cavendish Physics'," in The Development of the
Laboratory, James, ed. 104-117, argues that Thomson abandoned precision experiments
as head of the Cavendish Laboratory in the 1880s. However, he made no attempts to
recruit students or convert them to his approach to experimental physics.
10 See Jungnickel and McCormmach, Intellectual Mastery vol. 2, chap. 15.
11 See David E. Rowe, "Klein, Hilbert, and the Gottingen mathematical Tradition," Osiris,
5 (1989): 186-213 and Lewis Pyenson, Neohumanism and the Persistence ofPure Math-
ematics in Wilhelmian Germany (Philadelphia PA.: American Philosophical Society,
1983). For the debates and tensions the various avenues of developing mathematics
created amongst mathematicians in the late nineteenth and early twentieth centuries see
326 Epilogue
Mathematics had likewise become a profession driven by research and the needs
of younger mathematicians to make research reputations in fields recognized as
significant by the leading faculty at major universities across Germany. After
about 1860 the kinds of mathematical problems deemed important, and the solu-
tions regarded as significant, were defined by the mathematicians associated with
the Berlin "school." This included Karl Weierstrass in analysis, Ernst Kummer in
algebra, and Leopold Kronecker in number theory. Absent from their approach to
mathematics was geometry or any concern with a mathematics that might be gener-
ated through the consideration of problems that originated outside the boundaries
of mathematics itself. Collectively these mathematicians narrowed the impor-
tant research problems for mathematicians. Kronecker ultimately narrowed the
foundations of mathematics down to arithmetic to which all other branches of
mathematics were subordinate. The development of this approach was coupled
with, then justified by, a neo-Kantian philosophy of mathematics. Arithmetic was
a product of the mind and the only purely intellectual foundation for mathematics.
Space and time had a reality that lay outside of their intellectual contemplation,
and were contaminated sources. Mathematical proofs had to stand on rigid evi-
dential grounds that only arithmetic met. This philosophical grounding was not
new but was well suited to the development of an autonomous profession within
the Prussian university system. 12
The dominance of the Berlin approach to mathematics had more than an intellec-
tual impact on the field. Their standards began to affect the assessment of research
and the effectiveness of the teaching of individual mathematicians throughout the
German Empire. Professorships at the more prominent universities went to those
mathematicians whose research met the expectations of the Berlin mathemati-
cal faculty.13 This did not mean that all mathematicians abandoned other lines
of research, or that the Berlin mathematicians agreed completely on how or what
research issues to pursue. However, some research possibilities were judged as sec-
ondary and it was increasingly difficult for their practitioners to gain appointments
to other than provincial universities. 14
Herbert Mehrtens, Moderne Sprache Mathematik: Eine Geschicte des Streits um die
Grundlagen der Disziplin und des Subjekts formaler Systeme (Frankfurt: Suhrkamp
Verlag, 1990).
12 For the development of these changes see Umberto Bottazzini, The Higher Calculus: A
History of Real and Complex Analysis from Euler to Weierstrass (New York: Springer,
1986), Ivor Grattan-Guinness Foundations, and Harold M. Edwards "Kronecker's Views
on the Foundations of Mathematics," in The History of Modern Mathematics, Rowe and
McCleary, eds. 2 vols. (New York: Academic Press, 1989), vol. 1,67-78.
13 See Gert Schubring, "Pure and Applied Mathematics in divergent institutional Settings in
Germany: The Role and Impact of Felix Klein," in Modern Mathematics vol. 2, 171-220.
14 Bernhard Riemann and Hermann Grassmann are cases in point. Their lives and careers
were not blessed by easy appointments to major university departments of mathemat-
Forging New Relationships 327
ics. For Riemann's geometry see J. J. Gray, Ideas of Space: Euclidean, non-Euclidean,
and Relativistic (Oxford: Oxford University Press, 1989). For a recent assessment of
Grassmann and his influence into the twentieth century see Hermann Gunther Grass-
mann (1809-1877), Gert Schubring, ed. (Dordrecht: Kluwer Academic, 1996). For the
philosophical aspects of his mathematics see A. C. Lewis, "Hermann Grassman 1844
Ausdehnungslehre and Schleiermacher's Dialetik," Ann. Sci. 34 (1977): 103-162.
15 Helmholtz oversaw and wrote the introduction to the German translation of Thomson
and Tail's text in the 1870s.
16 William Thomson and Peter Guthrie Tail, Treatise on Natural Philosophy (New York:
Dover reprint, 1962). The mathematics for kinematics begins on p. 3, that for simple
harmonic motion is between, 38-59.
328 Epilogue
on these subjects, current electricity and the theory of the electromagnetic field.
However, Maxweii treated mathematics in a simiiar manner. After a description of
the phenomena of electrostatics came a discussion of its mathematics and proofs
of some fundamental theorems. In other chapters the characteristics of the poten-
tial function were explored and then used in the solution of physical problems.
Green's theorem appeared elsewhere and another section was devoted to spherical
harmonics and surfaces. In the second volume mathematics was inserted again
where and how much Maxwell judged necessary for the physics that surrounded
itY
In the above examples mathematics and physics occur in immediate contact.
Less intimate relations between the two occurred in texts from the continent. Clau-
sius presented the mathematics of the potential function in a chapter separate from
the physics of the potential. Helmholtz and Kirchhoff introduced the mathematics
necessary for the physics in separate lectures. 18
Hendrik Antoon Lorentz took this trend further. He produced a textbook in
mathematics for physics students whose needs he claimed were not met by existing
textbooks. Mathematical definitions and proofs were rigorous enough. However,
Lorentz only took the mathematics as far as was necessary to solve the physical
problems that form the problem sets at the end of each chapter. He built the subject
matter from one chapter to the next with repeated use of mathematical techniques
in examples.
Lorentz used definitions of the derivative and integral of limited use to a mathe-
matician. They were sufficient for him to then deduce expressions for the motion
of falling bodies. In the rest of the text mathematics was presented as a series of
techniques to solve physics problems. The physics presented a unified whole, the
mathematics was fragmented. As important, the mathematics is used then reduced
to a numerical expression to demonstrate the behavior of physical phenomena.
Lorentz began in algebra, and ended with Fourier series and differential
equations. Because he used Taylor series his chapters on calculus from a mathe-
matical point of view were of limited use. He also found it necessary to defend
his extended presentation of Fourier series and complex analysis for physics stu-
dents. The text, clearly for undergraduates, demonstrated the distance between the
calculus of the physicist in the 1870s and that of the mathematician of the same
era. Much of this mathematics harked back to a mathematics that was sixty years
17 James Clerk Maxwell, Treatise on Electricity and Magnetism (New York: Dover reprint
of third edition, 1954), vol. 1 chap. xii on electrostatic equilibrium and conjugate functions
where Maxwell puts various functions to particular physical uses, 284-316.
18 Clausius, Die Potentialfunction und das Potential: ein Beitrag zur mathematischen Physik
(Leipzig: Barth, 1859), Helmholtz, Vorlesungen iiber die theoretische Physik 3 vols.,
(Leipzig: Barth, 1903). In the volume on electrodynamics there is a chapter on the
potential function. Kirchhoff, Vorlesungen iiber mathematische Physik 4 vols. (Leipzig:
Teubner, 1876-1895).
Forging New Relationships 329
19 Hendrik Antoon Lorentz, Lehrbuch der Differential- und Integralrechnung und der An-
fangsgrunde der Analytischen Geometrie (Leipzig: Barth, 1900). The original Dutch
edition appeared in 1882.
20 See Jungnickel and McCormmach, Intellectual Mastery, vol. 2.
21 For the variety of such arrangements, see Jungnickel and McCormmach, Intellectual
Mastery vol. 2.
22 Boltzmann, "On the Development of the Methods of Theoretical Physics in Recent
Times," in Boltzmann, Theoretical Physics and Philosophical Problems, Brian McGuin-
ness, ed. (Dordrecht: Reidel, 1974),77-99.
330 Epilogue
Mathematics in Physics
Mathematics had become almost taken for granted as a skill that was now simply
necessary to become a physicist. That mathematics was in a form that spoke
directly to the needs of physicists and to the solutions to their problems. For
mature physicists and students alike, only aspects of theorems and areas of analysis
or results directly pertinent to the solution of physics problems received detailed
attention. Green's and Stokes' theorem and Fourier analysis were tailored to the
needs of physicists. Mathematics had been domesticated.
No physicist, experimentalists included, could afford to ignore mathematics. It
was the language required to understand the operation of their instrumentation, ap-
paratus, and the meaning of their results. Also physicists now expected theoretical
discussions to be a combination of mathematical language, with aspects of mathe-
matical methods of proof, together with a measure of physical insight coupled with
experimental evidence. Yet the balance of mathematics, imagery and empirical
evidence remained a matter of individual choice. In theory development there re-
mained a spectrum of uses of mathematics. This included a mathematical physics
in which there was more concern with mathematical standards that troubled most
other colleagues within physics. Some physicists would not have been out of place
in a mathematics department. 23 At the other extreme there were experimentalists
who used and required of others only the minimum of mathematics.
In the 1870s for some theoreticians, such as Helmholtz, physical results sprang
directly from mathematics without the intermediary of models. Others used models
to create then guide the development of the mathematics and interpret the results
of their manipulations. Maxwell moved freely from one end of this spectrum to
the other. He was well aware of the limitations of using models and the added
credibility results acquired when grounded in a mathematically expressed physics
based in the general principles of mechanics.
In German physics departments during the last third of the nineteenth century
we can see a trend away from theory based in the specifics of models towards
one based in principles. 24 This trend is illustrated through the physical work of
Helmholtz during this period. By 1870 Helmholtz had acquired the reputation of
an intellect in a class by himself. This was further enhanced by his institutional
position as recently appointed director of the new physics institute at Berlin uni-
versity. During the next decade Helmholtz's research focussed in one of most
intensely researched and competitive areas of physics, electrodynamics. 25 While
23 Examples include Ludwig Boltzmann and Carl Neumann. For a short period of time in
the 1870s Boltzmann was professor of mathematics at Vienna University.
24 One notable exception to this is Rudolph Clausius, but his career began in the 1840s.
The other is Ludwig Boltzmann.
25 This was closely connected with the increasing economic importance of the telegraph, the
Forging New Relationships 331
telephone, electric motors, and finally the development of the grid system. See Thomas
Hughes, Networks of Power.
26 For example Jed Z. Buchwald, The Creation of Scientific Effects: Heinrich Hertz and
Electric Waves (Chicago: University of Chicago Press, 1994), Part I, and Appendices
2 and 3 where Helmholtz's mathematics is transposed into vector form, and Buchwald,
From Maxwell to Microphysics (Chicago: University of Chicago Press, 1985), Part IV.
See also S. D' Agostino, "Hertz and Helmholtz on Electromagnetic Waves," Scientia 106
(1971): 637-648, and "Hertz's Researches on Electromagnetic Waves," Hist. Stud. Phys.
Sci. 6 (1975): 261-323,273-279. These are examples, the practice is commonplace.
27 Helmholtz, "Preface," in Heinrich Hertz, The Principles of Mechanics, D. E. Jones and
J. T. Walley, trans. (New York: Dover reprint of 1896 trans., 1956).
28 Jungnickel and McCormmach, Intellectual Mastery, vol. 2, 22, discuss the links between
Helmholtz's electrodynamics papers and his earlier, experimental work on the timing of
nerve impulses.
332 Epilogue
approach.29 Also it reduced some second order differential equations to first order
making their solution feasible.
His general expression for the potential between two current elements was
1 . . (
-2A21: [1 + k]cos(Ds, Da) + [1 - k]cos(r, Ds)cos(r, Da) ) DsDa,
29 See Helmholtz, "Kritisches zur Electrodynamik," Ann. Phy. 153 (1874): 545-556.
30 Helmholtz, "Uber die Bewegungsgleichungen der Electricitat fur ruhende leitende Kar-
per," J. Reine Angew. Math. 72 (1870): 57-129. This judgment reinforced an earlier one
made in Helmholtz, Erhaltung der Kraft. Other papers in this series include, Helmholtz,
"Uber die Fortflanzungsgeschwindigkeit in elektrodynamischen Wirkungen," Monats.
Akad. Berlin (1871): 292-298. In "Uber die Theorie der Elektrodynamik," J. Reine
Angew. Math. 75 (1873): 35-66, Helmholtz considered induction, in "Die elektrody-
namischen Krafte in bewegten Leitern," same journal 78 (1874): 273-324 where he
introduced ponderomotive forces. Shorter versions of the arguments of these papers
appeared elsewhere.
31 While it is true here and elsewhere, this is only useful after the fact; it was a pattern that
Maxwell followed about the same time. Both were seeking a physics based in general,
physical principles. See Maxwell, A Treatise on Electricity and Magnetism, vol. 2, chap.
vi for his attempt to put his electromagnetic theory in Lagrangian form. The limitations
of Maxwell's attempts are discussed in Tetu Hirosige, "Origins of Lorentz's Theory of
Electrons and the Concept of the Electromagnetic Field," Rist. Stud. Phys. Sci. 1 (1969):
151-209,192.
32 This differs from the analysis of Buchwald in, Scientific Effects, of Helmholtz's work
in electrodynamics. Buchwald interprets Helmholtz's work as resting on a physical
argument that is "implicit" in Helmholtz, that has a "natural energetic interpretation."
Maybe, but only from the perspective of the twentieth century.
Forging New Relationships 333
Helmholtz tied his mathematics in this series of papers directly to principles; in the
first paper to that of the "conservation offorce." Until 1873 the dominant physical
language of the papers in this series is about force and potential, not energy.33
However, energy arguments gave Helmholtz further points upon which to criticize
Weber's electrodynamics.. The terms in any expression for the conservation of
energy must be positive. From the mathematical form of a two-part term in his
interpretation of Weber's work, Helmholtz argued that in Weber's theory energy
could become negative and perpetual motion was possible. The argument hinged
on the mathematical representation of physical quantities that, because they were
physical had to behave according to general physical principles. While appearing
excessively mathematical, and Helmholtz did construct the most general mathe-
matical cases, he followed only the implications of physically significant cases.
There were serious limitations to this approach. Constants appearing in equa-
tions could only be determined by experiment, if such coefficients were amenable
to experiment. There was no direct connection between mathematics and the lab-
oratory because the differential equations and their solutions were so general. To
go further than present experiments required some kind of modelling. Further-
more, mathematics even in known cases was not an infallible guide to physical
consequences. Helmholtz's original argument against Weber's theory was that
the function representing the potential could, in certain circumstances, become
negative. This implied that the velocities of the electrical particles could become
infinite. In the ensuing exchange, others pointed out that any increase in velocity
would lead to an induced force that would decrease the particles' velocities. 34 To
clinch his argument and choose between these theories required experiment. De-
vising these, then carrying them through, or at least getting his students to do so,
was a difficult task that consumed much of the decade of the 1870s.35
Helmholtz passed this potent combination of experiment and mathematics to all
his students and in particular to Heinrich Hertz. Eventually Hertz realized the lim-
itations of Helmholtz's approach. Physical imagery was necessary and in this area
Helmholtz's mathematical approach led to inconsistent physics. With this insight
Hertz began to understand experimental results that had puzzled him and he was
33 While Helmholtz had remarked in 1869 that his conservation of force had been renamed
conservation of energy, potential rather than energy remained the mathematically pre-
ferred form in German physics into the 1870s. See Norton Wise, "German Concepts
of Force, Energy, and the electromagnetic Ether: 1845-1880," in Conceptions of Ether,
Cantor and Hodge, eds., 269-307.
34 See Weber, "Maasbestimmungen," Ann. Phy. 4 (1878): 366-373. This was finally
recognized by Helmholtz in an afterword (1881) added in his collected papers. See
Helmholtz, "Uber die Theorie der Elektrodynamik," 1. Reine Angew. Math. 75 (1873):
35-66, reprinted in Abh. vol. 1,647-687, "Zusatz (1881)," 684-687.
35 For a discussion of this aspect of Helmholtz's work, see Jungnickel and McCormmach,
Intellectual Mastery, Vol. 2, 25-30.
334 Epilogue
free to explore his own theoretical investigations into Maxwell's theories. Under
limiting conditions Helmholtz's equations replicated Maxwell's, but they were not
physically equivalent. There was only mathematics. In this case "the physical ba-
sis of Helmholtz's theory disappears.,,36 Physical imagery was necessary to clothe
the mathematical forms. He set about rethinking the physical underpinnings of
his own and Maxwell's theories. However, Hertz was finally reduced to accept-
ing the situation that he saw as a flaw in Helmholtz's work. Hertz was unable to
reconcile Maxwell's ideas and his mathematics, and he was resigned to accepting
Maxwell's equations as Maxwell's theory. He then went on to claim that the "inner
significance" of his own and Maxwell's equations, although different in form, were
the same. To explain his experimental results in terms of electromagnetic waves
in the ether, Hertz simply assumed Maxwell's equations. His focus then shifted
to explaining their physical implications and the legitimacy of his experimental
results. 37 A fuller version of his theoretical ideas followed, in which he gave gen-
eral mathematical expression to a physically consistent image of the origins of
electric and magnetic forces in the ether. 38
Many German theoretical physicists did not accept Helmholtz's judgments on
Weber's electrodynamics and long a lasting schism opened up between supporters
in both camps that affected careers into the next generation. For some, the issue be-
came Helmholtz's mathematics and whether his differential equations represented
Weber's fundamental laws. Beyond the mathematics was a physical imagery,
which if successfully challenged, would destroy more than Weber's work. 39 De-
spite these tensions Helmholtz renewed a pattern of doing theoretical physics in
Germany that was universal. Mathematics offered a unity for physics through its
abstractions as the analytical language of physics. 4o The principles of mechanics
was the center, the means to draw together the other branches of physics, heat,
36 For the most detailed consideration of Hertz's break with Helmholtz and the develop-
ment of his experimental work and its relations to his rethinking and reformulation of
electrodynamics, see Buchwald, Scientific Effects.
37 Hertz, "The Forces of Electrical Oscillations treated according to Maxwell's Theory,"
(1889) in Hertz, Electric Waves, D. E. Jones trans. (New York: Dover reprint of 1893
edition, 1962), 137-159.
38 Hertz, "On the Fundamental Equations of Electromagnetic Bodies at Rest," (1890) in
Hertz, Electric Waves, 195-240. This demand for the generality of mathematics together
with a defensible, consistent physical imagery also drove Hertz in the production of his
text on mechanics. See Hertz, The Principles ofMechanics, D. E. Jones and J. T. Walley,
trans. (New York: Dover reprint of 1896 trans., 1956).
39 For a discussion of this see Buchwald, Scientific Effects, Appendices 6 and 16. This was
in the same period that Helmholtz was arguing with mathematicians and philosophers
over the foundations of geometry. Helmholtz, "The Origin and Meaning of Geometrical
Axioms," (1870) in Science and Culture: Popular and Philosophical Essays, David
Cahan, ed. (Chicago Ill: University of Chicago Press, 1995), 226-248.
40 For the theme of unity in science and physics in particular and its professional and
Forging New Relationships 335
electrodynamics, light, the study of solids, and gases. The creation of a mathemat-
ical language that transcended the particularities of these separate domains was
sometimes the explicit goal of theoretical physicists. In others it can be seen in the
pattern of their work throughout their lifetimes. 41
This same approach to theory through the grand unifying principles of physics
was reflected in the next generation in the work of Max Planck. While not rederiv-
ing equations familiar to his readers, he discussed only those aspects of the physical
issue at hand that were new. It is a spare style that was unusual in the late nineteenth
century. Theoreticians were apt to rederive in their own way all the equations they
might need. Use of mathematics was an indication of the seriousness and the
professionalization of the theoretical enterprise. Planck simply imported them as
needed. He developed just enough mathematics necessary to make his physical
argument. No superfluity of generalization to demonstrate mathematical prowess.
Planck claimed later that mathematics was a mere instrument, and that his focus
was on a physics based in the most general principles possible. 42 However, all
his physical arguments depend directly on the mathematical forms in which those
principles were expressed. Because his physics was one of principle there was no
intermediary models, or particularist assumptions. The only medium for the de-
velopment of his ideas was in the structure of the language he used-mathematics.
And contrary to later statements, he appears to have understood this. A theory
stood or fell with its equations. The physical imagery was flexible, and might
be omitted altogether; the proof of the equations was another matter. He para-
phrased Heinrich Hertz stating that Maxwell's electromagnetic theory of light was
his equations. Planck went further to claim that equations were essential, all else
besides the mathematics could be discarded, hardly a tool-like judgment. 43 This
was especially the case in his early papers on thermodynamics where he focussed
on entropy. Here sophisticated mathematics was of far less use to Planck than
to his contemporaries enmeshed in the complexities of electrodynamics. 44 Math-
ematical complexity would not lead him into physically fruitful directions. His
initial work was on the significance of entropy and the implications of the first
and second laws of thermodynamics in specific cases. 45 Planck was after physical
connections but gaining those connections directly from the mathematics. And
Planck managed to squeeze as much physical significance as possible out of the
partial differential equations at his disposal. No mathematical expression was left
without physical comment or explanation. He also made no attempt to generalize
them. His mathematics referred only to physical cases and conditions, preferably
linked directly to numerical data. In his papers on critical states Planck specified
those states without any assumptions about the inner structure of matter; thus the
numerical aspects of the papers lent him the specificity necessary for theory in the
1870s.
Even as Planck turned his attention to electromagnetism and black body radia-
tion, these same characteristics mark his papers from those of his colleagues. He
imported equations where he could, using mathematics itself sparingly. His physi-
cal explanations were as spare as his mathematics and stuck to the essential points
of principle and their implications in the example at hand. Planck made as few
assumptions as possible about the nature of his resonators or the electromagnetic
radiation with which they interacted. 46 He introduced the notion of "natural radia-
tion" in his fourth paper on the subject. In the last paper he pulled together the work
he had accomplished thus far on the problem, and here began to use Fourier series
to characterize the radiation impinging on a resonator, as well as investigating the
energy and entropy of the same resonators.
His first two papers in quantum theory exemplify his approach. Initially he
stated only the new mathematical expression for the energy distribution and the
new thermodynamical foundations on which the expression was based. He had
previously established that the energy distribution was determined once the entropy,
S of a resonator was known as a function of its vibrational energy U. As he had also
already determined, the second law was not sufficient to calculate this function. To
arrive at his new law, Planck constructed "completely arbitrary expressions for the
entropy" that would not lead to Wien's law and satisfied both thermodynamical and
electromagnetic considerations. Not specifying what this expression was, Planck
45 See his dissertation, Max Planck,"Uber zweitzen Haupsatz der mechanischen Warme
Theorie," Munich 1879 in Physikalische Abhandlungen und Vortriige 3 vols. (Braun-
schweig: Vieweg und Sohn, 1958), vol. 1, 1-60, and Planck, "Verdampfen, Schmeltzen
und Sublimiren," Ann. Phy. 15 (1882): 446--475,Abh. 1,134-163.
46 The series of papers Planck produced on black body radiation begins with Planck, "Ab-
sorption und Emission elektrischer Wellen durch Resonanz," Ann. Phy. 57 (1896): 1-14,
and continues with "Uber elektrische Schwingungen, welche durch Resonanz erregt und
durch Strahlung gedampf werden," same journal 60 (1897): 577-599, and a five part
series, "Uber irreversible Strahlungsvorgange," Sitz. K. Preuss. Akad., Berlin (1897):
57-68,715-717,1122-1145, (1898): 449-476,440-480.
Forging New Relationships 337
where Ci and {3 are constant and U is the energy of the resonator. From this the
radiation law followed,
CA- 5
E = ---0-:-::--:-
e: 1l:r -1. '
where E is the energy density between the wavelengths A and A + dA. 47
While this might be taken as a short statement establishing Planck's priority
for deriving this expression, his follow up was only a mathematical skeleton of
those aspects of the theory that were new and presented just to give physical
meaning to the above expression. Planck found it necessary to turn to Boltzmann
for a new physical understanding of entropy. Entropy was disorder, and that
meant irregularity in the changes in amplitudes, and phases of the radiation of
the oscillators even in a stationary radiation field. This disorder could only be
understood using probability, introduced into thermodynamics by Boltzmann.
The complete deduction of his final equation for the energy density would re-
quire Planck to recapitulate much of his work in electromagnetism along with
the full thermodynamical deduction of the energy of a resonator. Planck merely
sketched what was new. Viewing resonators as groups as before, he constructed
the distribution of energy not by considering the resonators themselves, but by
looking at the distribution of energy over the frequencies of the oscillations to find
the energy of the whole as a function of the vibrations and the temperature of the
system. Unlike Boltzmann, Planck did not, as was still usual in statistical argu-
ments in physics, treat energy as a continuum, and thus introduced the element of
discontinuity into the mathematics of physics, pushing that mathematics further
beyond the calculus. 48 Planck assumed that his audience was familiar with per-
mutations and probabilities. He gave only one simple numerical example before
presenting the general expression for the number of permutations of P "energy
elements" among N resonators. He then looked for the most probable distribution
of the total energy among all the ways of distributing that energy in P amounts
among N resonators. He further adds the necessary thermodynamic expressions
to reach
47 Planck, "Uber eine Verbesserung der Wien'schen Spektralgleichung," Verh. Dtsch. Phy.
Gesell. 2 (1900): 202-204.
48 The most extensive examination of Planck's use of discontinuity in his expression for
the energy distribution function was that by Thomas Kuhn, Black-Body Theory and the
Quantum Discontinuity, 1894-1912 (New York: Oxford University Press, 1978). Kuhn
revised his argument in Kuhn, "Revisiting Planck," Hist. Stud. Phys. Sci. 14 (1984):
231-252.
338 Epilogue
Two points need to be made about this era in nineteenth century physics. First,
mathematics was taken as a natural aspect of physics. Secondly, the skill of the-
oretical physicists was judged on their creative development and manipulation of
physical imagery. Especially valued was imagery on the highest level of abstrac-
tion that unified ever broadening types of phenomena across different domains of
physics. However, during the 1890s these two characteristics were challenged.
Mathematics came to the foreground. The calculus no longer seemed adequate to
describe some of those domains in physics that were the most promising for uni-
fying physics, that is electrodynamics and thermodynamics. Greater unification in
physics was not reached through abstraction but through the specificity of models,
much of it the work of Lorentz and Boltzmann.
Lorentz worked and then reworked his electrodynamics throughout his career. 51
Like the structure of his papers, his return to the same central issues in theoretical
physics were systematic and driven by the need to incorporate new experimental
work, to clarify, then extend previous work. 52 His basic model was of a continuum
ether with only electromagnetic properties in which were embedded charged par-
ticles. All interaction between matter and the ether were through these particles.
Lorentz began with a physical case that led to equations that were then added
to term by term as he developed the physical situation. He translated physical
characteristics and processes directly into separable mathematical entities that car-
ried with them identifiable physical outcomes. The details of his physical model
were added as the complexity of the interactions of ether and matter developed.
Thus he mathematically and physically separated electrostatic phenomena from
those involving the motion of particles, "ions," within their "holes," i.e., dielectric
displacement, and those from the motion of the molecules themselves, i.e., elec-
trodynamical effects. 53 Each state led to forces that were represented as distinct
terms that were added one to the other. 54
Once the physical situation was represented, Lorentz brought all the devices
of the calculus to solve the equations. He expressed sets of partial differential
equations in terms of potentials, thus reducing them from second order to first order
and possible solution. He used Taylor expansions that might later be simplified to
conform to physical conditions, and Green's theorem, and so on.
He also constructed his papers like mathematical papers. His first conclusions
were a set of straightforward mathematically deduced physical results. These
were called upon later as he built his argument from electrostatics to the motion
of "ions" through the ether and the appearance of electromagnetic waves. While
the mathematics was elegant and might be developed with some generality, it was
also tightly bound to the pursuit of his physical quarry. A function represented
a physical quantity that was named and followed throughout his mathematical
excursions. The only cases he pursued were those that led to physically significant
results. However, mathematics imposed its own limitations on the behavior of the
entities they represented. The functions representing the ether were continuous
and the ether therefore defined rigorously.
With the acceptance of Hertz's experimental findings and the reality of elec-
tromagnetic waves established, Lorentz and other theorists could take that aspect
of Maxwell's theory as an experimental given. In 1892 Lorentz's work on elec-
trodynamics was reoriented and put into a unified mathematical structure where
he continued Maxwell's effort to put field theory into a "dynamical" Lagrangian
formulation. Using the energy equation and the Principle of Least Action (here
reduced to d' Alembert 's principle) Lorentz brought mathematical simplicity to his
theory. In this new mathematical form, he replicated previous results and integrated
dieletric displacement into this view. While claiming its "dynamical" character
Lorentz had actually developed a formal, mathematical theory in which physical
53 See Lorentz, "Concerning the Relation between the Velocity of Propagation of Light
and the Density and Composition of the Media," Verhand. K. Akad. Weten. Amsterdam
18 (1878): 1, in Collected Papers, vol. 2, 1-119. His ions had only electromagnetic
properties from p. 23 until part III of the paper when he introduced dispersion and his
particles were then endowed with inertia.
54 See Lorentz, "Concerning the Relation," p. 35-36 where he reconstructs the forces gen-
erated in the ether from moving "molecules" that he has built up in pieces especially over
pps.21-35.
340 Epilogue
55 Lorentz, "La theorie electromagnetique de Maxwell et son application aux corps mou-
vants," Arch. neerl. 25 (1892): 363, Collected Papers vol. II; 164-343, p. 299. While
this paper is important for the clarification of Lorentz's physical ideas on the nature of the
interaction of matter and the ether, it is its mathematical character that concerns us here.
For the physics see, Hirosige, "Origins," and Buchwald, From Maxwell to Microphysics,
194-196.
56 Lorentz Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten
Korpern (Leiden: Brill, 1895), reprinted in Collected Papers, vol. 5, 1-137.
57 For a discussion of the transition from Maxwell's electromagnetic theory of light to
Lorentz's "microscopic" theory see, Buchwald, From Maxwell to Microphysics.
58 Because of his theoretical commitment to seeing the origins of electromagnetic effect
in electrons and their motions Lorentz had to reinterpret, yet again, crucial aspects of
Maxwell's theory such as the displacement current.
Forging New Relationships 341
He also focussed on one general issue, the motion of ponderable matter through
the ether, not on the front burner for most other theoretical physicists working in
electrodynamics.59
Lorentz demonstrated a mastery of vectors, yet his use of mathematics itself
and its relationship to his physical ideas remained the same. Maxwell's equations
and common relationships in electrodynamics were all translated into vector form.
His focus on the problem of charged bodies moving through the ether meant that
Lorentz only sketched in the aspects of electrodynamics that did not pertain to
this main problem. However, when he reached the main issue, mathematics still
carried much of his argument. Mathematical devices maintained Maxwell's laws
when he transformed his reckoning of them from the "stationary" coordinates of
one set of "ions" to the set of "ions" moving through the ether. 6O The relationship
between time and the coordinates between the two systems were introduced as
"new variables." Only the time-transformation was commented upon physically.
Time measured in the moving system is "local," and he referred the reader back
to the mathematical relationship for the conversion from one to the other. The
relationship is mathematical rather than physical.
Vectors, the algebra that seems to us to make the physics of Lorentz's argu-
ments visible in ways in which his earlier mathematical treatments do not, were
still a marginal mathematical form for physicists in 1895, and of no interest to
mathematicians. Along with statistical arguments, they had been developed by
physicists strictly for their own uses within the context of their own problems. It
was only later that mathematicians were to take up either again as mathematics
and generalize both statistics and vector algebra.
While combinatorial methods were considered a legitimate aspect of mathemat-
ics in the nineteenth century, the calculus of probability and statistics challenged
the prevailing thrust of mathematicians towards rigor. Rigor implied that proofs
were either correct or incorrect, with truth established in some absolute form.
Arguing probabilities did not fit this search for certainty and firm foundations.
Additionally, these methods studied by mathematicians in the eighteenth and early
nineteenth centuries arose in the context of social and experimental concerns in
other disciplines, also recently repudiated as a source for research problems in
mathematics. As recent work has demonstrated, the mathematical development of
statistics shows a hiatus between the work of Laplace and the English biometricians
of the late nineteenth century. 61
Harvard University Press, 1986) is a study of how a field, statistics, came into being from
its origins in several other areas of study, scientific and social. His work overlaps that
of Daston somewhat for mathematicians in the late eighteenth century. Theodore Porter,
The Rise o/Statistical Thinking (Princeton: Princeton University Press, 1986), traces the
use of statistics in the social sciences from Quetelet through Galton and Pearson.
62 For Maxwell's papers on kinetic theory and statistical mechanics with a critical introduc-
tion, see Maxwell on Molecules and Gases, Garber, Brush and Everitt, eds., and Maxwell
on Heat and Statistical Mechanics: On "Avoiding all Personal Enquiries of Molecules,"
Garber, Brush and Everitt, eds. (Bethlehem PA: Lehigh University Press, 1995).
Forging New Relationships 343
Boltzmann's major focus was on the meaning of the second law while his remarks
on transport phenomena, mathematically more thorough, were peripheral to the
main thrust of his research.
There were further differences in the level of mathematical apparatus brought
to bear on their various problems. Maxwell's proofs reflect the mathematics of
Cambridge at mid-century where the curriculum incorporated standards of early
nineteenth-century France. Boltzmann's proofs reflect the mathematical standards
of Germany through Andreas von Ettingshausen at the University of Vienna. His
proofs were more rigorous, and eventually physics became mathematics. 63 Boltz-
mann realized more clearly and quickly than Maxwell that the mathematics actually
removed the necessity of considering the particulars of the molecules' interactions.
The molecules in motion in the gas become just so many "individuals" in differ-
ent states of motion and "if the number which on average have known states of
motion stays constant, the characteristics of the gas remain constant." During a
collision, not specified, the states of the molecules involved changed, represented
by a change in their kinetic energy from x to S but in equilibrium the energy, E,
of the gas remained constant. Boltzmann constructed the quantity E, an integral
of functions of the coordinates, velocities and density of all the molecules in the
gas. Through mathematical manipulations Boltzmann expressed E as 64
He examined the behavior of the right hand side of this equation, and through his
use of particular integration techniques and changes of variables, the above was
reduced to
63 The development of Boltzmann's statistical ideas are discussed in Brush The Kind of
Motion we call Heat, vol. 1,566-616, and Kuhn, Black Body Radiation.
64 Boltzmann, "Weitere Studien tiber das Warmegleichgewicht unter Gasmolekiilen," Ber.
Wein 66 (1872): 275-370, translated in Brush, Kinetic Theory, 3 vols. (New York:
Pergamon, 1966), vol. 2, 88.
344 Epilogue
!(;, t) , !(x+x'-;,t)
a=--- a =
~' ";x +x' - ~
and
r = "J;"Xi1/l(x, x', ~).
The function 1/1 depended on the type of collision and remained unspecified. ! (x , t)
represented the number of molecules in unit volume whose kinetic energy lay
between x and x + dx. The product
log-- aa
55' [
aa'
, - 5S 'J
was intrinsically negative. Whatever mechanical system one chose, a quantity
existed that had the properties of the thermodynamic relation - f d Q/ T < 0, for
irreversible cycles. For such a system in equilibrium,
!(x, t) = Cy'xe- hx •
The properties of the physical system came directly from the behavior of the
mathematics.
From criticisms of his gas theory, Boltzmann was goaded into rethinking the de-
termination of thermal equilibrium in terms of probability alone. He reinterpreted
the expression for entropy in terms of the number of ways the total energy of a
gas could be distributed among the molecules constituting the gas. 65 Boltzmann
assumed that the probability of the energy state for the gas was proportional to
the number of ways it could be constituted on a molecular level. Entropy was
directly related to probability without any considerations of the structure of in-
dividual molecules or their interactions with their fellow molecules in the gas. 66
Even if the gas was initially in an improbable energy state, the system would pass
to a more probable one, and finally, if left undisturbed, to the most probable state,
that of thermal equilibrium. 67
65 Boltzmann, "Uber die Beziehung zwischen dem zweiten Haupsatze der mechanischen
Wiirmetheorie und der Wahrscheinlichkeitsrechnung, respective der Siitzen tiber des
Wiirmegleichgewicht," Ber. Wien 76 (1877): 73. See also Boltzmann, Lectures on Gas
Theory, Stephen G. Brush, trans. (Berkeley CA: University of California Press, 1964)
chap. 1.
66 Boltzmann also published papers on the mathematical aspects of kinetic theory and
statistical mechanics. See Boltzmann, "Uber die Integrallinearer Differentialgleichungen
mit periodischen Koeffizienten," Ber. Wien 18 (1868): 54-59, "Einige allgemeine Siitze
tiber Wiirmegleichgewicht," same journal 63 (1871): 679-711, "Uber die Aufstellung und
Integration der Gleichungen weIche die Molekularbewegungen in Gasen bestimmen,"
same journal 14 (1876): 503-552.
67 Objections to the physical implications of Boltzmann's argument in his 1872 and later
papers are discussed in Brush, The Kind of Motion, vol. 2, 602-608.
Forging New Relationships 345
68 Boltzmann, "On the Methods of Theoretical Physics,"5-12, 8-9. This argument is re-
peated in Boltzmann, "On the Development of the Methods of Theoretical Physics in
Recent Times," 77-100, 87-91 along with arguments about the dangers of relying on
mathematics alone in Boltzmann, Theoretical Physics and Philosophical Problems, Se-
lected Writings, Brian McGuinness, ed. (Boston: Reidel, 1974).
69 Boltzmann, "Uber die Eigenschaften monozyklischer und anderer damit verwandter Sys-
teme," f. ReineAngew. Math. 98 (1884): 68-94: 100 (1887): 201-212. The examination
of these mechanical systems, some of whose internal motions did not affect the sys-
tem's macroscopic thermodynamic properties, was begun by Helmholtz, "Principien der
statik monozyklischer Systeme;' f. ReineAngew. Math. 97 (1884): 111-140,317-336.
See Gunther Bierhalter, "Die von Hermann von Helmholtzschen Monozykel-Analogien
zur Thermodynamik;' Arch. Hist. Exact Sci. 29 (1983): 95-100, "Zu Hermann von
Helmholtzs mechanischer Grundlegung der Wiirmelehre aus dem Jahre 1884," same
journal 25 (1981): 71-84; "Boltzmanns mechanische Grundlagung des zweiten Haup-
satzes der Wiirmelehre aus dem Jahre 1866," same journal 24 (1981): 195-205,207-220.
70 See Boltzmann, "Relationships of Applied Mathematics," in Physics for a New Century:
Papers presented to the 1905 St. Louis Congress, K. R. Sopka, compiler (New York: AlP,
1986), 267-279, Boltzmann, "On the Significance of Theories," and "The Second Law
of Thermodynamics," in Theoretical Physics, 33-36, 12-32 respectively.
71 On Energetics see Boltzmann, "Zur Energetik," Ann. Phy. 58 (1896): 39 and "Uber
die Unentbehrlichkeit der Atomistik in der Naturwissenschaft," same journal 60 (1897):
231.
346 Epilogue
their behavior from that of scalars.?2 For this we again need to return to Maxwell
and his examination of mathematics from the point of view of the physicist. 73
Maxwell pointed to an important distinction for physicists between scalar and
vector quantities, recognized by Hamilton in his work on quarternions. He indi-
cated the mathematical results from quarternions and what they indicated about
the properties of scalars and vectors. However, while quarternions were important
for geometry they were not what was needed by physicists because the distinc-
tions between scalars and vectors needed to be kept in mind at all times. Using
quarternions, kinetic energy for example was always negative. Maxwell focussed
on Hamilton's operators and what they represented physically as well as present-
ing to the mathematicans his own nomenclature for these operators, convergence
(divergence), curl, and the combination of the two.
Maxwell persisted. His correspondence with Peter Guthrie Tait over the next
decade frequently included comments or questions about quarternions or results
he had deduced. They appear most conspicuously in Maxwell's Treatise. He in-
troduced them in the mathematical sections of his introductory chapter, along with
the operators above and Laplace's operator. Throughout both volumes Maxwell
noted the quarternionic equivalent of results. In his second volume vectors were
introduced into his mathematical arguments themselves and he began to stress the
type of quantity being dealt with. However, extensive mathematical manipula-
tions were in Cartesian form, and where appropriate, with references to Hamilton
or Tait's work on quarternions. His most extended discussion was in his chapter on
the electromagnetic field. Maxwell equations appear in vector form. Yet although
appearing throughout the text, vectors were marginal; the main mathematical ar-
guments were in Cartesian form, their results expressed in the equivalent vector
format. 74
For over a decade work proceeded in electrodynamics without the benefit of
vector algebra. Heaviside's introduction of them into the subject was piecemeal.
Judging from reactions from his colleagues their usage was neither obvious nor
easy for physicists accustomed to thinking in Cartesian, or quarternionic terms. 75
Responding to Heaviside's demonstration of the usefulness of vectors in electrody-
72 This section depends heavily on Michael Crowe, A History of Vector Analysis: The
Evolution of the Idea of a Vectorial System (New York: Dover reprint of 1967 edition,
1985), although I disagree on some specific points.
73 Maxwell, "On the Mathematical Classification of Physical Quantities," Proc. London
Math. Soc. 3 (1871): 224-232. He had begun this discussion in Maxwell, "Address to
the Mathematical and Physical Sections of the British Association," Rep. British Assoc.
(1870): 1-9, reprinted in Scientific Papers, vol. 2, 215-229, and Maxwell on Molecules,
Garber, Brush and Everitt, eds. 89-97.
74 Maxwell Treatise, vol. 1, 10-32, vol. 2, 247-259.
75 See also Crowe, Vector Analysis, on George Francis FitzGerald's review of Heaviside's
Electromagnetic Theory, and his use of vectors, 175-176.
Forging New Relationships 347
namics, Hertz replied that it was difficult to follow Heaviside's symbols, especially
as he did not use the vector potential at all. He could not understand Heaviside's
symbols and "mode of expressing yourself. You know mathematical symbols are
like a language and your writing like a remote dialect of it...,,76 By this date in
his papers on electromagnetism, Heaviside had published on vector analysis and
its use in electromagnetism. However, it was not until 1893 with the publication
of Heaviside's Electrical Theory that he treated vector analysis systematically, if
polemically. From this date vectors began to make some impact on physicists. 77
Heaviside's vector analysis was taken up, not in Britain but in Germany in
the textbook of August Foppl on Maxwell's theory. The first part of this text is
a systematic exposition of vector analysis necessary for electrodynamics. The
definition of a vector is followed by that for the unit vector, the vector product
and the transformation of coordinates. Foppl also introduced examples, using as
his vector the velocity of a particle, an indication that he saw extensions of their
use in mechanics. He introduced the key differential operators and, again taking
examples from mechanics, demonstrated their importance and also illustrated their
relationships with one another. In the last section to this chapter he dealt with the
integration of vectors and the potential as a scalar. His final words on the subject
were on the Laplacian operator. The whole text was written in vector form, using
Maxwell's nomenclature throughout. 78
Lorentz's Versuch appeared the following year. Despite the popularity of Foppl 's
textbook, vectors did not replace Cartesian methods easily or quickly even in the
domain of electrodynamics. In 1904 Lorentz still published his equations in both
Cartesian and vector forms. 79 Albert Einstein's initial papers on electrodynamics
and relativity were similarly in Cartesian form. Vector analysis seeped into the
research and probably the teaching of physics in Germany in the last decade of the
nineteenth and the first decade of this century.80 It was not until German mathemati-
76 Quoted in J. G. O.'Hara and Willibrand Pricha, Hertz and the Maxwellians, (London:
Peter Peregrinius, 1987), Hertz to Heaviside, March 21,1889,62-63.
77 See Crowe Vector Analysis 169-174 for Heaviside's system and 174-177 for its reception.
I leave any account of Gibbs' and Grassmann's work on vector analysis until the next
section.
78 AugustF6ppl, Einfiihrung in die Maxwell'sche Theorie der Elektricitiit: mit einem ein-
leitenden Abschnitte iiber das rechnen mit Vectorgrossen in der Physik (Leipzig: B. G.
Teubner, 1894). See also Crowe, Vector Analysis, 176,226-227.
79 Lorentz, "Electromagnetic phenomena in a System moving with any Velocity less than
that of Light," Proc. Amsterdam K. Akad. Sci. 6 (1904).
80 Neither Woldemar Voigt, Kompendium der Theoretischen Physik (Leipzig: Veit, 1895)
2 vols., nor Paul Drude, The Theory of Optics C. Riborg Mann and Robert A. Millikan,
trans. (New York: Dover reprint of 1902 edition) and other theoretical physics texts used
vectors. However, Larmor,Aether and Matter (Cambridge: Cambridge University Press,
1900) devoted a section to vectors and their advantages for electrodynamics. However,
348 Epilogue
The transitions to these new relationships were neither smooth nor easy. There
were early and acrimonious confrontations between mathematicians and physi-
cists. Tait and G. G. Knott, ardent quarternionists, took Gibbs to task because of
his misuse of Hamilton's mathematical invention. Their criticism was that Gibbs
had merely invented a new notation and had actually made mathematical matters
worse by not using Hamilton's quarternions. From the point of view of the math-
ematician's, Gibbs had missed the mathematical point of quarternions. And from
the point of view of the history of mathematics Gibbs work has been judged as
"not highly original."81 Yet from his first publications Gibbs went beyond Tait
and others in his ability to treat physical problems. He demonstrated how vector
analysis could be used in astronomy, even when his main purpose was to teach its
methods to students. 82
Gibbs did not produce a systematic study of vector algebra to really bring out
its power to make physical processes visible. In contrast Oliver Heaviside both
transformed vector algebra and Maxwell's electromagnetism. The latter was sim-
plified into a geometrically vivid form centered on the notions of electric and
magnetic force rather than on the analytical potential functions. Heaviside intro-
duced a much improved notation and developed vector algebra in more detail than
Gibbs. 83 While he derived his vector analysis from Hamilton's quarternions he did
it for the purpose of expressing electromagnetic theory simply and graphically.
During the 1890s quarternionic mathematicians argued against this truncated al-
gebra. In Great Britain Heaviside prevailed while the independent development of
vectors by Hermann Grassmann gained acceptance on the continent. 84
Heaviside was not so fortunate in his development of operational calculus. The
Royal Society refused him publication in its Proceedings even though this was
an accepted perk for society fellows. To explain this refusal Hunt invokes the
shifting boundaries of mathematics, and that mathematicians now claimed the
85 Hunt, "Rigorous Discipline: Oliver Heaviside versus the Mathematicians," in The Liter-
ary Structure of Scientific Argument, Peter Dear, ed. 72-96.
86 See Ivor Grattan-Guinness, "University Mathematics at the Turn of the Century," Ann.
Sci. 28 (1972): 369-384. See also G. H. Hardy A Mathematician 'sApology (Cambridge:
Cambridge University Press, 1920).
87 Paul J. Nahin, Oliver Heaviside: Sage of Solitude (New York: IEEE Press, 1988).
88 For the place of mathematics within the German university system see Gert Schubring,
"Germany to 1933," in Companion to History and Philosophy of Mathematics Grattan-
Guinness ed., vol. 2 Part II Higher Education and Institutions, 1442-1456, 1448-1453.
350 Epilogue
assistant in physics in the 1860s. From these beginnings Klein's research lay in
the then unfashionable and professionally disastrous field of geometry. In this he
joined other mathematicians such as Alfred Clebsch who shared Klein's interest in
the mathematical problems that emerged from technical and physical problems. 89
Klein combined extraordinary mathematical ability with equally forbidding po-
litical and organizational talents that forced long-term changes in the profession.
With the rapid development of an industrial economy, Technische Hochschule be-
gan to replace universities as the mathematical training ground for engineers. In
an effort to halt the hemorrhage of students and maintain a future for mathematics,
Klein set out to recapture the teaching and what he considered the proper place of
mathematics with respect to the exact sciences and engineering. 90
In his efforts Klein successfully led university mathematicians in retaining con-
trol of the teaching of their subject in higher education. He also cultivated and
broadcast the idea that mathematics was the key to all the sciences and engineering.
Physics and engineering became "applied mathematics," although Klein's mean-
ing of this term was crucially vague. 91 However, Klein's notion of mathematics
as being key to the development of these other disciplines did not lead him to see
them as equal in this endeavor. Mathematics was not at the center of some vast
horizontal network binding all the specialist studies together. It was an hierarchi-
cal relationship. Klein saw the connections of mathematics to engineering and
the sciences as the key to revitalizing the discipline and becoming the intellec-
tual center for rapidly fragmenting technical fields that were quickly losing touch
with one another. However, mathematics subsumed all these other fields within
itself. While the applied sciences were a source of new discoveries in mathematics,
they were intrinsically inexact. Only pure mathematics developed from axiomatic
foundations could lead to the structure required by the other technical knowledge
bases.
89 Klein's interest in physical problems as a source for mathematics can be traced back to his
Erlanger Program of 1871. See David Rowe, "Felix Klein's' Erlanger Antrittsrede' ," Hist.
Math. 12 (1985): 123-141, which was his inaugural lecture as extraordinary professor at
Erlangen. The text of his lecture is reproduced here. For more details on Klein's career
and its personal costs see Rowe, "Klein, Hilbert and Gottingen Mathematical Tradition."
90 The details of Klein's twenty year campaign is complicated by fundamental changes in
German higher education. How this entwines with his vision of mathematics and its posi-
tion with respect to the exact sciences and engineering is detailed in Lewis PyensonNeo-
humanism and the Persistence of Pure Mathematics in Wilhelmian Germany (Philadel-
phia PA: American Philosophical Society, 1983). See also Rowe, "Klein, Hilbert, and
Gottingen," Gert Schubring, "Pure and Applied Mathematics in Divergent Institutional
Settings: the Role and Impact of Felix Klein," in History ofModern Mathematics, Rowe
and McCleary, eds. vol. 2,171-220, and Lewis Pyenson, "Mathematics, Education, and
the Gottingen approach to physical Reality, 1890-1914," Europa II (1970): 91-127.
91 Schubring has examined Klein's writings on this term. See Schubring, "Pure and Applied
Mathematics," 192-197.
Forging New Relationships 351
While statements like these in the context of changing educational policies led
to clashes with the very groups Klein hoped to cultivate, he was able to convince
both government and private resources to underwrite these intellectual visions in
more concrete forms. He also gathered a group of mathematicians around him at
G6ttingen that seriously investigated the mathematics of the physical sciences and
developed a program that attracted many bright students to this newly invigorated
study. These efforts, begun in the 1890s, came to fruition during the second decade
of the twentieth century.
Whether physicists or engineers would accept his vision and that of likeminded
mathematicians at G6ttingen, among the most outspoken of which was David
Hilbert, of the role of mathematics in their respective specialities is another matter. 92
Physicists in particular would more than once echo Einstein's complaint that the
mathematical axiomatization of an area of physics could be useful but could only be
done after the fact. So far as the development of physical theory in an unchartered
domain, the axiomatic approach offered by mathematicians was useless. 93
In their renewed encounters between roughly 1900 and the first world war,
a changed relationship between these two disciplines would emerge that would
reshape both disciplines. One of the symptoms of this renewed relationship was
the growing interest of mathematicians in the mathematics physicists now needed
for their research. During the early decades of this century texts by mathematicians
appeared on vectors, the partial differential equations of physics, etc. 94
However, the first example that we find of a mathematician seriously involved
in the scrutiny and then reworking of a contemporary field of physics lay not in
Germany but France. This is not surprising seeing that the structural changes that
the universities underwent in France in the nineteenth century did not include the
separation of the "pure" from the practical. If anything, after the Franco-Prussian
War, political changes in the funding of higher education in France reinforced the
mutual support of theoretical and practical interests of French academics in the
92 See David Hilbert, "Die Grundlagen der Physik," Math. Ann. 92 (1924): 1-32, in
Gesammelte Abhandlungen, 3 vols. (New York: Chelsea reprint, 1965), vol. 3, 258-289
on the general theory of relativity, and, "Naturerkennen und Logik," Naturwissenschaften
(1930): 959-963, Abh. vol. 3, 378-387.
93 For such an axiomatic approach see Hilbert, " Bemerkungen tiber die Begrtindung der
eiementaren Strahlungstheorie," Gottingen Nach. (1912): 773-789, (1913): 409-416,
(1914): 275-298 in Gesammelte Abhandlungen 3 vols. (New York: Chelsea reprint of
1935 edition, 1965), vol. 3, 217-257.
94 See Heinrich Weber, Die partiellen Differential-Gleichungen der mathematischen Physik
nach Riemann's Vorlesungen (Braunschweig: F. Vieweg, 19900-01), 2 vols., Richard
Gans, Einfilhrung in der Vektoranalysis mit Anwendungen auf die mathematische Physik
(Leipzig: Teubner, 1905). Also see the articles in Encyklopiidie der mathematischen
Wissenschaften. Mit Einschluss ihren Anwendungen, Felix Klein ed. (Leipzig: Teubner,
1904-1922).
352 Epilogue
95 Terry Shinn, "The French Science Faculty System, 1808-1914: Institutional Change and
Research Potential in Mathematics and the Physical Sciences," Hist. Stud. Phys. Sci. 16
(1979): 271-332. See also Maurice Crosland, Science Under Control.
96 Poincare, "Analyse de ses travaux scientifiques," Acta Math. 38 (1921): 116-125, in
Oeuvres, 9 vols. (Paris: Gauthier-Villars, 1954), vol. 9., 1-14.
97 Poincare also published papers on the mechanical foundations of thermodynamics, ther-
mal conduction, elasticity, capillarity and the theory of errors as well as on electricity
and optics. Aspects of Poincare's work on electrodynamics and optics initially appeared
as a series of papers, then reappeared as published lectures in the 1890s and early 1900s.
The papers took up particular aspects of the subject matter (electric waves) or the work of
particular physicists including Weber, Larmor, Lorentz and Hertz. The lectures were pub-
lished as Poincare, tlectricite et Optique. La lumiere et les theories electrodynamiques
Lectures at the Sorbonne, 1888, 1889, 1899 (Paris: Carre Naud, 1901).
98 Olivier Darrigol, "Henri Poincare's Criticism of Fin de Siecle Electrodynamics," Stud.
Forging New Relationships 353
523-529. The Abraham paper Poincare refers to is Abraham, "Dynamik des Electrons,"
Gottingen Nach. pt. 1 (1902): 20-41.
104 See Poincare, "The Principles of Mathematical Physics," Physics at St. Louis, 281-299.
The argument he made for the upper limit on velocity was based on the limitations of
measurement techniques. Because measurement techniques were visual and dependent
on the velocity of light we cannot measure any velocity greater than that. See also,
Poincare, "La dynamique de l'electron," Rev. gen. sci. pures et app/iquees 19 (1908):
386-402, in Oeuvres, vol. 9, 551-586, 574. He argued for the elimination of the ether,
575-576.
Forging New Relationships 355
to improve on and them develop his own. From the beginning Einstein was out to
refashion both mechanics and electrodynamics. Both based the idea of the death
of the ether on experimental grounds, the relativity of measurement that leads to
the notion of local time and the velocity of light as an upper limit. In addition,
and not incidentally, there was Einstein's commitment to a particular aesthetic for
physical explanation with which he opened his first relativity paper. Explanations
should reflect the symmetries of the phenomena they are developed to explain.
What then did they accomplish with these ideas? Poincare could not release
himself from the notion that somehow time was a coordinate different from the
other space coordinates. He explored local time and discussed a four-dimensional
space with coordinates x , y, z, t.J=I, but in relation to his discussion ofthe Lorentz
transformation group and the invariance of certain functions that after manipulation
demonstrated the velocity of propagation of gravitation. Einstein merely remarked
on the group character of his transformations, nothing more. He discussed his
transformation equations in a section on "The physical meaning of the equations
concerning moving rigid bodies and clocks.,,105 Einstein made the rest of his first
paper on special relativity an exploration of some of the implications for mechanics,
electrodynamics, and optics of using his postulates, as well as placing time on the
same footing as space coordinates. The mathematical apparatus was minimal. He
imported equations as he needed them. The steps that drove the argument and
the mathematics forward were physical. While the mathematical expression of
electrodynamics might be the same as in earlier theories the physical meanings
of those equations needed rethinking. 106 This was not an exhaustive investigation
but an exploration using particular physical cases such as the Doppler effect, and
radiation pressure.
At this point Whittaker's judgments, and those that followed him, on the respec-
tive merits and claims of the mathematician Poincare versus the physicist Einstein
were not wrong but wrongheaded. 107 His account only stands if the mathemati-
cal form is the solution, no further explanation being necessary. One must also
assume that the statement of physical implications at the end of a string of anal-
ysis is entirely equivalent to its statement as a matter of physical principle. That
equivalence encompasses the theory that follows, worked out in detail, with re-
spect to the changes in the interpretation of physical processes expressed in the
equations that follow from those assumptions. Assuming Poincare's priority here
105 Poincare, "I 'electron," Rendiconti, 541-543. Albert Einstein, "On the Electrodynamics
of Moving Bodies," Ann. Phy. 17 (1905): 891-921, in The Collected Papers of Albert
Einstein, vol. 2, Anna Beck, trans. (Princeton NJ: Princeton University Press, 1989),
140-171,156.
106 Einstein, "On the Electrodynamics," 159.
107 Edmund Whittaker History of the Theories ofAether and Electricity, vol. 2, chap. II The
Relativity Theory of Poincare and Lorentz. In this chapter Whittaker consistently diverts
attention and credit from physicists to mathematicians.
356 Epilogue
108 His early commitment was to the "molecular-kinetic" theory of heat and the fundamental
nature of the laws of thermodynamics, as interpreted statistically. For a discussion of
mechanics at the turn ofthe twentieth century and the role of thermodynamics in Einstein's
work see Martin Klein, "Mechanical Explanation at the End of the nineteenth Century,';
Hist. Stud. Phys. Sci. 1 (1969): 127-149.
109 Lewis Pyenson, "Hermann Minkowski and Einstein's Theory of Relativity," Arch. Hist.
Forging New Relationships 357
work of Poincare and Lorentz. 11o Minkowski located the "theorem of relativity"
in the covariance of the original equations of Lorentz's electrodynamics of 1895
through the transformations explored by Poincare. He took this as "purely mathe-
matical." This theorem depended on the form of the differential equations for the
propagation of waves with the velocity of light.
Minkowski claimed that no one had as yet followed through the implications of
this theorem for matter, hence it was as yet a postulate. Minkowski recognized Ein-
stein's 1905 paper as the clearest on the postulate. However, Einstein's argument
was based on phenomena and new ideas about the concept of time. The principle
of relativity had not yet been formulated for the electrodynamics of moving bodies.
Then, in vector form he stated the Lorentz equations for the electrodynamics of a
moving body. These were rewritten to demonstrate their symmetry with respect
to the indices attached to the four coordinates Xl, X2, X3, X4. Minkowski redefined
the Lorentz transformation as a rotation and demonstrated that the equations of
electrodynamics are invariant under such a rotation. For Minkowski the Lorentz
transformation introduced a modification of the coordinate X4, the "time parame-
ter." He then plunged into a short section on simultaneity. The Lorentz transform
permits us to consider time exactly as we do the other three space coordinates.
This, Minkowski claimed, should be easier for mathematicians as they were used
to dealing with four dimensional and non-Euclidean geometry. Minkowski referred
the reader to Einstein's 1905 paper for an account of the physical explanation.
Minkowski constructed two types of vectors invariant under the Lorentz trans-
formation, then demonstrated that the electromagnetic equations for a body at rest
could be rewritten in terms of these types of vectors and were therefore themselves
invariant. He turned to moving bodies and transposed the coordinate system to the
moving body, with respect to which the electrodynamic equations for a body at
rest must hold. He had already demonstrated that these equations were invariant
under such a transformation, hence so were those for the moving body.
Minkowski generalized his mathematical treatment of the Lorentz transforma-
tion, putting the argument into matrix form.lll Theorems on particular matrices
followed and their invariance under the Lorentz transformations. He pointed out
that many results of electrodynamics simply fall out of the algebraic characteristics
Exact Sci. 17 (1977): 71-96, established how little Minkowski knew or understood of
Einstein's special relativity.
110 In his first paper Minkowski referred to Lorentz, Versuch, and his piece for Klein's
Encyklopiidie der mathematischen Wissenschaften on Maxwell's theory and electrons
and and Poincare's Rendiconti paper. See Hermann Minkowski, "Die Grundgleichungen
fUr die elektromagnetischen Vorgange in bewegten K6rpem," Gottingen Nach. (1908):
53-111, in Gesammelte Abhandlungen, 2 vols (New York: Chelsea, reprint of 1911
edition, 1967), vol. 2, 352-404, 352.
111 The format of the equations in the first sections of the paper give away the direction in
which Minkowski would take the argument.
358 Epilogue
112 For Jacob Laub and Einstein's work in relativity in this era and its mathematical, physical
and institutional context see Pyenson, The Young Einstein: The Advent of Relativity
(Boston: Adam Hilger, 1985).
113 Peter Galison, "Minkowski's Space-Time: From Visual Thinking to the Absolute World,"
Hist. Stud. Phys. Sci. 10 (1979): 85-121, sees Minkowski as a "visual thinker," and like
previous scholars emphasizes the geometrical aspects of his thought. However, there is
only passing mention of the geometrical representation in his first published version of
his work on space-time.
114 Lorentzian rotations.
115 Minkowski, "Raum und Zeit," Phys. Zt. 10 (1909): 104-111, in The Principle of
Relativity, W. Perrett and G. B. Jeffery, trans. (New York: Dover reprint of 1923 edition,
nd), 75-91, 83. This was an address given to the Naturforscher-Versammlung September
1908.
116 Minkowski, "Raum und Zeit," 91.
Forging New Relationships 359
117 Minkowski made this last remark, repeated later by Klein, in the draft notes to a lecture on
relativity in 1907. See Galison, "Minkowski," 95-96, and Pyenson, "Relativity in Late
Wilhelmian Germany: The Appeal to a Preestablished Harmony between Mathematics
and Physics," Arch. Hist. Exact Sci. 27 (1982): 137-155, 147. The idea of preestablished
harmony can be traced back to Leibniz.
118 Einstein, "The Methods of Theoretical Physics," in The World as I see It (London: 1935),
quoted in Pyenson, Einstein, p. 153.
119 It is interesting to note that their paper of 1913 is in two parts, a physical one by Einstein,
and a discussion of the mathematics and proof of crucial theorems by Grossmann. See,
"Entwurf einer verallgemeinerten Relativitatstheorie und einer Theorie der Gravitation,"
Zt. Math. Phys. 62 (1913): 225-261.
360 Epilogue
were limited by physical laws and postulates that set the criteria the mathematics
needed to satisfy.120 The mathematical choices were still iarge. His subsequent
misgivings and then abandonment of his 1913 formulation were based on the ar-
gument that the gravitational field equations were not themselves covariant, but
also that the motion of the perihelion of Mercury deduced from it was too small. 121
His goal, a physics that was independent of all coordinate systems. "The laws of
physics must be of such a nature that they apply to systems of references in any
kind ofmotion."122 His concerns were physical, expressed in mathematical form.
The problem of gravitation and general relativity attracted numerous others
besides Einstein, many of them mathematicians, especially those educated or as-
sociated with Gottingen. One of the most active and intent was David Hilbert. 123
Hilbert recognized that mathematics and physics had drawn together, and both had
changed. Previously mathematics had treated physical problems too mathemati-
cally and physicists had only taken necessary formulae from mathematics. With
the example of his close colleague Minkowski before him, Hilbert concluded that
physics needed pure mathematics. However, since mathematicians could learn
physics easily while physicists found it impossible to follow modern mathematical
papers, mathematicians would complete the union by solving physicists' problems
for them. He would invite prominent physicists, such as Einstein, to give lecture
series, and thus informed mathematicians would solve their problems.
Einstein gave such a series of lectures on the status of general relativity theory
at Gottingen in the summer of 1915. Hilbert's solution to the problem of general
relativity was based on axioms, the first of which was that "the laws of physical
phenomena are determined by a world function H" that had certain mathematical
properties. Hilbert treated H as a generalized Hamiltonian that was invariant under
any transformation of any world coordinate. 124 Hilbert went further in defining H
as the sum of two other functions. Relationships between these functions, K and L,
contained all of electrodynamics and the "equations of gravitation." While Hilbert
explored the mathematical properties of H, the physical conclusions that he drew
120 See Einstein, "Zum gegenwartigen Stande des Gravitationsproblems," Phy. Zt. 14
(1913): 1249-1262, in Collected Papers, vol. 4,198-222,198-200. See also the discus-
sion in Jungnickel and McCormmach, Intellectual Mastery, vol. 2, 325-328. For a full
account of Einstein's path to his general field equations of 1915 see Abraham Pais "Subtle
is the Lord... " The Science and Life ofAlbert Einstein (Oxford: Oxford University Pres,
1982), chaps., 12-14.
121 See John Norton, "How Einstein found his Field Equations: 1912-1915," Hist. Stud.
Phys. Sci. 14 (1984): 253-316,298-299 ..
122 Einstein, "Die Grundlage der allgemeinen Relativitatstheorie," Ann. Phy. 49 (1916):
769-822, in The Principle of Relativity, 109-164, 113. Emphasis is in the original.
123 This account relies heavily on Pyenson The Young Einstein, 183-193.
124 Hilbert's world function clearly derived from Minkowski but its immediate predecessor
was an equally mathematical function in Gustav Mie's electrodynamics.
Forging New Relationships 361
125 See David Hilbert, "Die Grundlagen der Physik," GottingenNach. part I (1915), reprinted
in Gesammelte Abhandlungen vol. 3, 258-289. Hilbert discussed the motion of mass
points, 285-289.
126 For a discussion on Einstein on mathematics, see Jungnickel and McCormmach, Intel-
lectual Mastery, vol. 2, 334-340.
127 See, "Discussion" following the lecture on Einstein "Zum gegenwarten Stande," reported
in Phy. Zt. 14 (1913): 1262-1266, in Collected Papers, vol. 4, 223-230, on Gustav Mie,
Max Abraham, and Gunnar Nordstrom's efforts.
362 Epilogue
The instrument that mediates theory and praxis, thought and experiment
is mathematics: it binds them together and forms their inner essence.
Therefore, it appears that our contemporary culture in as far as it rests on
the contemplation and manipulation of nature, depends on mathematics.
- David Hilbert 128
It still seems to me that you very much overrate the value ofpurely formal
points of view. These are quite precious if there is an already-discovered
truth finally to be formulated, but they almost always fail as a heuristic
aid -Einstein to Felix Klein 129
A full bibliography for a monograph such as this is a monster, even while listing
only those items cited. Since footnotes appear at the bottom of the pages, it seemed
less taxing on the reader to expand the index. Works cited are entered by page
number at their first entry and by author(s) and subject matter separately. This
bibliography is, therefore, a short introduction to available sources that will lead
the reader into the topics covered in the text. The sources are grouped by topic area.
My only regret is that Charles Gillispie, Laplace (Princeton NJ.: Princeton Uni-
versity Press, 1998), Christa Jungnickel and Russell McCorrnmach, Cavendish
(Philadelphia: American Philosophical Society, 1997) and Robert E. Schofield,
The Enlightenment of Joseph Priestley (University Park: University of Pennsyl-
vania Press, 1997) carne to my attention too late to be consulted for this volume.
That these three volumes are biographies indicates the importance of the lives
of individuals play in the history of science where the diversity of approaches to
addressing research issues in both mathematics and physics are highlighted.
The best introduction to how different science was in the eighteenth century
intellectually, socially and culturally and how historians have grappled with those
differences is George S. Rousseau and Roy Porter, eds., Ferment of Knowledge
(New York: Cambridge University Press, 1980). For eighteenth-century physics,
see John Heilbron, Electricity in the Seventeenth and Eighteenth Centuries: A
Study ofEarly Modem Physics (Berkeley CA: University of California Press, 1979)
despite its subject matter being restricted to electrostatics. For physics as natural
philosophy, see Casper Hakfoort, Optics in the Age of Euler: Conceptions of the
Nature of Light (Cambridge: Cambridge University Press, 1995) and Geoffrey
Cantor, Optics after Newton: Theories ofLight in Britain and Ireland, 1704-1840
(Manchester: University of Manchester Press, 1983). On caloric theories of heat,
see Robert Fox, Caloric Theories of Gases from Lavoisier to Regnault (New York:
Cambridge University Press, 1971). For mathematics in the eighteenth century
364 Bibliography
there are the essays in the volumes (still emerging) of Euler's papers that set
his work in their inteiiectuai context. For example, see Clifford Truesdell, "The
Rational Mechanics of Flexible, or Elastic Bodies, 1638-1788," in Euler, Opera
Omnia, series 2, vol. 11, part 2. For the development of the calculus, see Ivor
Grattan-Guinness, The Development of the Foundations of MathematicalAnalysis
from Euler to Riemann (Cambridge MA.: MIT Press, 1970), Judith Grabiner,
The Origins of Cauchy's Rigorous Calculus (Cambridge MA.: MIT Press, 1981)
and Niccolo Guicciardini, The Development of Newtonian Calculus in Britain,
1700-1800 (Cambridge: Cambridge University Press) for the development of the
calculus in Britain. There is no overview of the social history of the sciences in
the eighteenth century. The standard on the most important eighteenth-century
scientific society is still Roger Hahn, The Anatomy of a Scientific Institution: The
Paris Academy of Sciences, 1666-1803 (Berkeley CA.: University of California
Press, 1971). The place of science in the consumer societies of western Europe
has been studied for the case of Britain in Jan Golinski, Science as Public Culture:
Chemistry and Enlightenment in Britain, 1760-1820 (Cambridge: Cambridge
University Press, 1992) and Larry Stewart, The Rise of Public Science: Rhetoric,
Technology andNatural Philosophy in Newtonian Britain, 1660-1750 (Cambridge:
Cambridge University Press, 1992).
The period covering the French revolution and the Napoleonic era is usually
subsumed in accounts of the science under eighteenth century and early nineteenth
century science, or the Romantic era. The collection edited by Andrew Cunning-
ham and Nicholas Jardine, Romanticism and the Sciences (Cambridge: Cambridge
University Press, 1990) addresses the issues raised by the connections between sci-
ences and philosophies in the Romantic era and demonstrates the suggestive nature
and the elusive quality of those ties. Grattan-Guinness, Convolutions in French
Mathematics, 1800-18403 vols. (Basel: Birkhiiuser, 1990) covers the intellectual
changes and the power structures within French mathematics in a crucial period
of its development. The vitality of science within the social and cultural life of
Britain is explored in Ian Inkster and Jack Morrell, eds., Metropolis and Province:
Science in British Culture, 1780-1850 (London: Hutchinson, 1983).
The history of physics in the early years of this century is covered in Jungnickel
and McCormmach, Intellectual Mastery vol. 2. Other volumes are specialist stud-
ies, or biographies, or focus on later decades and the ties of science and scientists to
governments, power and the changing economies across the globe. A very useful
discussion of the personnel, funding, and research of academic physicists about
1900 is in Paul Forman, John Heilbron and Spencer Weart, "Physics circa 1900:
Personnel, Funding and Productivity of the Academic Establishment," Hist. Stud.
Phys. Sci. 5 (1975). For a look into a relationship between a physicist and a math-
ematician that does not seem to be competitive and yet was productive, see Elie
366 Bibliography
Some of the issues in the philosophy of mathematics that are touched on here
are in Thomas Tymoczko ed., New Directions in the Philosophy of Mathematics
(Boston: Birkhiiuser, 1986). The problems of the social constructionists are in
Stephen Cole, Making Science: Between Nature and Society (Cambridge MA.:
Harvard University Press, 1992). Going beyond the purely social constructionist
approach of sociologists of science Bruno Latour examines the intersection be-
tween the natural world, and constructions of it in Latour, "One more Turn after
the Social Turn," in Ernan McMillan ed., The Social Dimensions of Science (Notre
Dame: Notre Dame Press, 1992). The problems of a purely sociological use of the
the term practice is examined in Stephen Turner, The Social Theory of Practices:
Tradition, Tacit Knowledge, and Presuppositions (Chicago: University of Chicago
Press, 1994).
Index
The function of the bibliography has been incorporated into this index. The first
occurrence of a citation is complete and is referenced in the index both by author
and subject. Footnote page references use the usual notation of an appended "n."
See also, Heiden, Albert van and physics at Berlin university, 274n
Hannaway, 0, see Achinstein, Peter as physicist, 296, 330, 333
Harding, M. c., ed., Oersted's Corre- career of, 291-293
spondence, 143n education of, 291-292
Harman, Peter, on Maxwell on imagery Kantianism, 292
and nature, 250n mathematics, 330-333
on Maxwell's physics and Scottish phi- mathematics of, Clausius on, 300
losophy, 249n on animal heat, 292 , 293-294
ed., on physics and mathematics in on caloric theory of heat, 294
nineteenth-century Cambridge, 199n on Challis on sound, 297
See also, Heiman, Peter on Conservation of Energy, 333
Harrison, John, on longitude, 76n on Conservation of Force, 292 , 293-
Harte, Henry, 189n 297, 300n, 331, 333
Hattendorff, K., ed., Riemann's lectures on consonance, 44n
on partial differential equations, 267 on electrodynamics, 331-333, 353n
Hawkins, Thomas, on Berlin school of on electrodynamics and physiology,
mathematics, 262n 331n
Hays, J. N., on London Institution, 171n on gases, 296n
on science lecturing in early on geometry, 8, 334n
nineteenth-century London, 174n on observation and experiment, 8
Heat, conduction and radiation, 218-222 on hydrodynamics, 296n, 301
Heat, mathematics and physics of, 95, on mathematics of physics, 328
111-119,232,267 on Maxwell's electromagnetism, 332
mechanical theory of, 291, 293-297, on monocycles, 345n
296-297, 297 on Neumann's electrodynamics, 332
molecular-kinetic theory of, 356n on physiology and physics, 292-293
theory of Riemann's, 268 on potential, 296, 331
See also, Caloric; Thermodynamics on Principle of Least Action, 292
Heaviside, Oliver, and mathematicians, on transcendence of mathematics, 8,
349n 334
on electromagnetism, 259, 346-347 on velocity of nerve impulses, 293n
on Gibbs' vector analysis, 348n on Weber's electrodynamics, 332, 333,
on operational calculus, 348-349 334
on vectors, 347-362 Helmholtz, Richard, 152n
Heidelberger, Michael, on Ohm, 157n Henry, Thomas, 172n
on Baconian sciences in early Henslow, John Stevens, 199n, 201n
nineteenth-century Germany, 141n Herder, Johann Gottfried von, on
See also, Kriiger, Lorenz physics, 144n
Heilbron, John, on eighteenth-century Herivel, John, on Fourier, 113n
physics, xvii, 16n and Pierre Costabel, on Fourier and his
on early modem physics, 363 critics, 117n
on electricity in eighteenth-century, 65n Hermann, Armin, on physics at Berlin
on Euler on natural philosophy, 61n University, 274n
on physics about 1800, 101 Hermann, Dieter, on early nineteenth-
on quantification in eighteenth-century century German astronomy, 144n
science, 365-366 Hermann, Jakob, on vibrating strings,
See also, Forman, Paul; Frangsmyr, 46n
Tore Hermeticism, in eighteenth century, 71n
Heiman, Peter, on Helmholtz's Kantian- Herschel, John William Frederick, 101,
ism, 265 195n,257
See also, Harman, Peter and Cambridge curriculum, 205n
Heisenberg, Werner Karl, 9n, 52n astronomy of, 201-202
Heiden, Albert van and Thomas L. Han- epistemology of, 201-202
kins eds., on Scientific instruments, mathematics of, 201n
66n on Clairaut, 203
Helmholtz, Hermann, 64, 261n, 271n, on government funding for science,
274, 290, 290-303, 310, 324n, 327n 214
and Clausius, 299-300 on Lagrange's Mecanique Analytique,
and engineering, 152n 203
and mathematics, 301 on light, 201
382 Index from H