A Dim Portrait

This has been a portrait of Helmholtz the scientist and famous intellect. What
was he like as a human being? In spite of his extraordinary prominence, that
question is difficult to answer. The authorized biography, by Leo Ko¨nigsberger,
is faithful to the facts of Helmholtz’s life and work, but too admiring to be reliably
whole in its account of his personal traits. Helmholtz’s writings are not
much help either, even though many of his essays were intended for lay audiences.
His style is too severely objective to give more than an occasional
glimpse of the feeling and inspiration he brought to his work. We are left with
fragments of the human Helmholtz, and, like archaeologists, we must try to
piece them together.
We know that Helmholtz had a marvelous scientific talent, and an immense
capacity for hard work. Sessions of intense mental effort were likely to leave
him exhausted and sometimes disabled with a migraine attack, but he always
recovered, and throughout his life had the working habits of a workaholic.
He was blessed with two happy marriages. The death of his first wife, Olga,
after she spent many years as a semiinvalid, left him incapacitated for months
with headaches, fever, and fainting fits. As always, though, workwas his tonic,
and in less than two years he had married again. His second wife, Anna, was
young and charming, “one of the beauties of Heidelberg,” Helmholtz wrote to
Thomson. She was a wife, wrote Ko¨nigsberger, “who responded to all [of Helmholtz’s]
needs . . . a person of great force of character, talented, with wide views
and high aspirations, clever in society, and brought up in a circle in which intelligence
and character were equally well developed.” Anna’s handling of the
household and her husband’s rapidly expanding social commitments contributed
substantially to the Helmholtz success story in Heidelberg and Berlin.
To achieve what he did, Helmholtz must have been intensely ambitious. Yet
he seems to have traveled the road to success without pretension and with no
question about his integrity, scientific or otherwise. Max Planck, a man whose
opinion can be trusted on the subjects of integrity and intellectual leadership
without pretension, wrote about his friendship with Helmholtz in the 1890s in
Berlin:
I learned to know Helmholtz . . . as a human being, and to respect him as a
scientist. For with his entire personality, integrity of convictions and modesty
of character, he was the very incarnation of the dignity and probity of science.
These traits of character were supplemented by a true human kindness, which
touched my heart deeply. When during a conversation he would lookat me
with those calm, searching, penetrating, and yet so benign eyes, I would be overwhelmed by a feeling of boundless filial trust and devotion, and I would
feel that I could confide in him, without reservation, everything I had on my
mind.
Others, who saw Helmholtz from more of a distance, had different impressions.
Englebert Broda comments that Boltzmann “had the greatest respect for
Helmholtz the universal scientist, [but] Helmholtz the man . . . left him cold.”
Among his students and lesser colleagues, Helmholtz was called the “Reich
Chancellor of German Physics.”
There can hardly be any doubt that Helmholtz had a passionate interest in
scientific investigation and an encyclopedic grasp of the facts and principles of
science. Yet something contrary in his character made it difficult for him to communicate
his feelings and knowledge to a class of students.We are again indebted
to Planck’s frankness for this picture of Helmholtz in the lecture hall (in Berlin):
“It was obvious that Helmholtz never prepared his lectures properly. He spoke
haltingly, and would interrupt his discourse to lookfor the necessary data in his
small notebook; moreover, he repeatedly made mistakes in his calculations at the
blackboard, and we had the unmistakable impression that the class bored him at
least as much as it did us. Eventually, his classes became more and more deserted,
and finally they were attended by only three students; I was one of the
three.”
Helmholtz viewed scientific study in a special, personal way. The conventional
generalities required by students in a course of lectures may not have been
for him the substance of science. At any rate, Helmholtz was not the first famous
scientist to fail to articulate in the classroom the fascination of science, and (as
those who have served university scientific apprenticeships can attest) not the
last.
The intellectual driving force of Helmholtz’s life was his never-ending search
for fundamental unifying principles. He was one of the first to appreciate that
most impressive of all the unifying principles of physics, the conservation of
energy. In 1882, he initiated one of the first studies in the interdisciplinary field
that was soon to be called physical chemistry. His workon perception revealed
the unity of physics and physiology. Beyond that, his theories of vision and
hearing probed the aesthetic meaning of color and music, and built a bridge
between art and science. He expressed, as few had before or have since, a unity
of the subjective and the objective, of the aesthetic and the intellectual.
He had hoped to find a great principle from which all of physics could be
derived, a unity of unities. He devoted many years to this effort; he thought that
the “least-action principle,” discovered by the Irish mathematician and physicist
William Rowan Hamilton, would serve his grand purpose, but Helmholtz died
before the workcould be completed. At about the same time, Thomson was failing
in an attempt to make his dynamical theory all-encompassing. In the twentieth
century, Albert Einstein was unsuccessful in a lengthy attempt to formulate a unified
theory of electromagnetism and gravity. In the 1960s, the particle physicists
Sheldon Glashow, Abdus Salam, and StevenWeinberg developed a unified theory
of electromagnetism and the nuclear weakforce. The search goes on for stillbroader
theories, uniting atomic, nuclear, and particle physics with the physics
of gravity. We can hope that these quests for a “theory of everything” will eventually
succeed. But we may have to recognize that there are limits. Scientists may
never see the day when the unifiers are satisfied and the diversifiers are not busy.

Physics

By 1871, the year he reached the age of fifty, Helmholtz had accomplished more
than any other physiologist in the world, and he had become one of the most
famous scientists in Germany. He had worked extremely hard, often to the detriment
of his mental and physical health. He might have decided to relax his
furious pace and become an academic ornament, as others with his accomplishments
and honors would have done. Instead, he embarked on a new career, and
an intellectual migration that was, and is, unique in the annals of science. In
1871, he went to Berlin as professor of physics at the University of Berlin.
The conversion of the physiologist to the physicist was not a miraculous rebirth,
however. Physics had been Helmholtz’s first scientific love, but circumstances
had dictated a career in medicine and physiology. Always a pragmatist,
he had explored the frontier between physics and physiology, earned a fine reputation,
and more than anyone else, established the new science of biophysics.
But his fascination with mathematical physics, and his ambition, had not faded.
With the death of Gustav Magnus, the Berlin professorship was open. Helmholtz
and Gustav Kirchhoff, professor of physics at Heidelberg, were the only candidates;
Kirchhoff preferred to remain in Heidelberg. “And thus,” wrote du Bois-
Reymond, “occurred the unparalleled event that a doctor and professor of physiology
was appointed to the most important physical post in Germany, and
Helmholtz, who called himself a born physicist, at length obtained a position
suited to his specific talents and inclinations, since he had, as he wrote to me,
become indifferent to physiology, and was really only interested in mathematical
physics.”
So in Berlin Helmholtz was a physicist. He focused his attention largely on
the topic of electrodynamics, a field he felt had become a “pathless wilderness”
of contending theories. He attacked the work of Wilhelm Weber, whose influence
then dominated the theory of electrodynamics in Germany. Before most of his
colleagues on the Continent, Helmholtz appreciated the studies of Faraday and
Maxwell in Britain on electromagnetic theory. Heinrich Hertz, a student of Helmholtz’s
and later his assistant, performed experiments that proved the existenceof electromagnetc waves and confirmed Maxwell’s theory. Also included among
Helmholtz’s remarkable group of students and assistants were Ludwig Boltzmann,
Wilhelm Wien, and Albert Michelson. Boltzmann was later to lay the
foundations for the statistical interpretation of thermodynamics (see chapter 13).
Wien’s later workon heat radiation gave Max Planck, professor of theoretical
physics at Berlin and a Helmholtz prote´ge´, one of the clues he needed to write
a revolutionary paper on quantum theory. Michelson’s later experiments on the
velocity of light provided a basis for Einstein’s theory of relativity. Helmholtz,
the “last great classical physicist,” had gathered in Berlin some of the theorists
and experimentalists who would discover a new physics.

Physiology

After 1847, Helmholtz was only intermittently concerned with matters relating
to thermodynamics. His worknow centered on medical science, specifically the
physical foundations of physiology. He wanted to build an edifice of biophysics
on the groundworklaid by Mu¨ ller, his Berlin professor, and by his colleagues du
Bois-Reymond, Ludwig, and Bru¨ cke, of the 1847 school. Helmholtz’s rise in the
scientific and academic worlds was spectacular. For six years, he was professorof physiology at Ko¨nigsberg, and then for three years professor of physiology and
anatomy at Bonn. From Bonn he went to Heidelberg, one of the leading scientific
centers in Europe. During his thirteen years as professor of physiology at Heidelberg,
he did his most finished workin biophysics. His principal concerns were
theories of vision and hearing, and the general problem of perception. Between
1856 and 1867, he published a comprehensive workon vision, the three-volume
Treatise on Physiological Optics, and in 1863, his famous Sensations of Tone, an
equally vast memoir on hearing and music.
Helmholtz’s workon perception was greatly admired during his lifetime, but
more remarkable, for the efforts of a scientist working in a research field hardly
out of its infancy, is the respect for Helmholtz still found among those who try
to understand perception. Edward Boring, author of a modern text on sensation
and perception, dedicated his bookto Helmholtz and then explained: “If it be
objected that books should not be dedicated to the dead, the answer is that Helmholtz
is not dead. The organism can predecease its intellect, and conversely. My
dedication asserts Helmholtz’s immortality—the kind of immortality that remains
the unachievable aspiration of so many of us.”

Pros and Cons

Helmholtz’s youthful effort in his paper (he was twenty-six in 1847), read to the
youthful members of the Berlin Physical Society, was received with enthusiasm.
Elsewhere in the scientific world the reception was less favorable. Helmholtz
submitted the paper for publication to Poggendorff’s Annalen, and, like Mayer
five years earlier, received a rejection. Once again an author with importantthings to say about the energy concept had to resort to private publication. With
du Bois-Reymond vouching for the paper’s significance, the publisher G. A. Reimer
agreed to bring it out later in 1847.
Helmholtz commented several times in later years on the peculiar way his
memoir was received by the authorities. “When I began the memoir,” he wrote
in 1881, “I thought of it only as a piece of critical work, certainly not as an
original discovery. . . . I was afterwards somewhat surprised over the opposition
which I met with among the experts . . . among the members of the Berlin academy
only C. G. J. Jacobi, the mathematician, accepted it. Fame and material reward
were not to be gained at that time with the new principle; quite the opposite.”
What surprised him most, he wrote in 1891 in an autobiographical
sketch, was the reaction of the physicists. He had expected indifference (“We all
know that. What is the young doctor thinking about who considers himself called
upon to explain it all so fully?”). What he got was a sharp attackon his conclusions:
“They [the physicists] were inclined to deny the correctness of the law . . .
to treat my essay as a fantastic piece of speculation.”
Later, after the critical fog had lifted, priority questions intruded. Mayer’s papers
were recalled, and obvious similarities between Helmholtz and Mayer were
pointed out. Possibly because resources in Potsdam were limited, Helmholtz had
not read Mayer’s papers in 1847. Later, on a number of occasions, he made it
clear that he recognized Mayer’s, and also Joule’s, priority.
The modern assessment of Helmholtz’s 1847 paper seems to be that it was, in
some ways, limited. It certainly did cover familiar ground (as Helmholtz had
intended), but it did not succeed in building mathematical and physical foundations
for the energy conservation principle. Nevertheless, there is no doubt
that the paper had an extraordinary influence. James ClerkMaxwell, prominent
among British physicists in the 1860s and 1870s, viewed Helmholtz’s general
program as a conscience for future developments in physical science. In an appreciation
of Helmholtz, written in 1877, Maxwell wrote: “To appreciate the full
scientific value of Helmholtz’s little essay . . . we should have to askthose to
whom we owe the greatest discoveries in thermodynamics and other branches
of modern physics, how many times they have read it over, and how often during
their researches they felt the weighty statements of Helmholtz acting on their
minds like an irresistible driving-power.”
What Maxwell and other physicists were paying attention to was passages
such as this: “The task[of theoretical science] will be completed when the reduction
of phenomena to simple forces has been completed and when, at the
same time, it can be proved that the reduction is the only one which the phenomena
will allow. This will then be established as the conceptual form necessary
for understanding nature, and we shall be able to ascribe objective truth to
it.” To a large extent, this is still the program of theoretical physics.

Die Erhaltung der Kraft

If medicine was not Helmholtz’s first choice, it nevertheless served him (and he
served medicine) well, even when circumstances were trying. His medical scholarship
stipulated eight years of service as an army surgeon. He tookup this
service without much enthusiasm. Life as surgeon to the regiment at Potsdam
offered little of the intellectual excitement he had found in Berlin. But to an
extraordinary degree, Helmholtz had the ability to supply his own intellectual
stimulation. Although severely limited in resources, and unable to sleep after
five o’clockin the morning when the bugler sounded reveille at his door, he
quickly started a full research program concerned with such topics as the role of
metabolism in muscle activity, the conduction of heat in muscle, and the rate of
transmission of the nervous impulse.
During this time, while he was mostly in scientific isolation, Helmholtz wrote
the paper on energy conservation that brings him to our attention as one of the
major thermodynamicists. (Once again, as in the stories of Carnot, Mayer, and
Joule, history was being made by a scientific outsider.) Helmholtz’s paper had
the title U¨ ber die Erhaltung der Kraft (On the Conservation of Force), and it was
presented to the Berlin Physical Society, recently organized by du Bois-Reymond,
and other students of Mu¨ ller’s, and Gustav Magnus, in July 1847.
As the title indicates, Helmholtz’s 1847 paper was concerned with the concept
of “force”—in German, “Kraft”—which he defined as “the capacity [of matter] to
produce effects.” He was concerned, as Mayer before him had been, with a composite
of the modern energy concept (not clearly defined in the thermodynamic
context until the 1850s) and the Newtonian force concept. Some of Helmholtz’s
uses of the word “Kraft” can be translated as “energy” with no confusion. Others
cannot be interpreted this way, especially when directional properties are assumed,
and in those instances “Kraft” means “force,” with the Newtonian
connotation.
Helmholtz later wrote that the original inspiration for his 1847 paper was his
reaction as a student to the concept of “vital force,” current at the time among
physiologists, including Mu¨ ller. The central idea, which Helmholtz found he
could not accept, was that life processes were controlled not only by physical
and chemical events, but also by an “indwelling life source, or vital force, which
controls the activities of [chemical and physical] forces. After death the free action
of [the] chemical and physical forces produces decomposition, but during
life their action is continually being regulated by the life soul.” To Helmholtz
this was metaphysics. It seemed to him that the vital force was a kind of biological perpetual motion. He knew that physical and chemical processes did not
permit perpetual motion, and he felt that the same prohibition must be extended
to all life processes.
Helmholtz also discussed in his paper what he had learned about mechanics
from seventeenth- and eighteenth-century authors, particularly Daniel Bernoulli
and Jean d’Alembert. It is evident from this part of the paper that a priori beliefs
are involved, but the most fundamental of these assumptions are not explicitly
stated. The science historian Yehuda Elkana fills in for us what was omitted:
“Helmholtz was very much committed—a priori—to two fundamental beliefs: (a)
that all phenomena in physics are reducible to mechanical processes (no one
who reads Helmholtz can doubt this), and (b) that there be some basic entity in
Nature which is being conserved ([although] this does not appear in so many
words in Helmholtz’s work).” To bring physiology into his view, a third belief
was needed, that “all organic processes are reducible to physics.” These general
ideas were remarkably like those Mayer had put forward, but in 1847 Helmholtz
had not read Mayer’s papers.
Helmholtz’s central problem, as he saw it, was to identify the conserved entity.
Like Mayer, but independently of him, Helmholtz selected the quantity “Kraft”
for the central role in his conservation principle. Mayer had not been able to
avoid the confused dual meaning of “Kraft” adopted by most of his contemporaries.
Helmholtz, on the other hand, was one of the first to recognize the ambiguity.
With his knowledge of mechanics, he could see that when “Kraft” was
cast in the role of a conserved quantity, the term could no longer be used in the
sense of Newtonian force. The theory of mechanics made it clear that Newtonian
forces were not in any general way conserved quantities.
This reasoning brought Helmholtz closer to a workable identification of the
elusive conserved quantity, but he (and two other eminent thermodynamicists,
Clausius and Thomson) still had some difficult conceptual ground to cover. He
could follow the lead of mechanics, note that mechanical energy had the conservation
property, and assume that the conserved quantity he needed for his
principle had some of the attributes (at least the units) of mechanical energy.
Helmholtz seems to have reasoned this way, but there is no evidence that he got
any closer than this to a full understanding of the energy concept. In any case,
his message, as far as it went, was important and eventually accepted. “After [the
1847 paper],” writes Elkana, “the concept of energy underwent the fixing stage;
the German ‘Kraft’ came to mean simply ‘energy’ (in the conservation context)
and later gave place slowly to the expression ‘Energie.’ The Newtonian ‘Kraft’
with its dimensions of mass times acceleration became simply our ‘force.’ ”
I have focused on the central issue taken up by Helmholtz in his 1847 paper.
The paper was actually a long one, with many illustrations of the conservation
principle in the physics of heat, mechanics, electricity, magnetism, and (briefly,
in a single paragraph) physiology.

Medicine and Physics

Helmholtz, like Mayer, was educated for a medical career. He would have preferred
to study physics and mathematics, but the only hope for scientific training,
given his father’s meager salary as a gymnasium teacher, was a government scholarship in medicine. With the scholarship, Helmholtz studied at the Friedrich-
Wilhelm Institute in Berlin and wrote his doctoral dissertation under Johannes
Mu¨ ller. At that time, Mu¨ ller and his circle of gifted students were laying the
groundworkfor a physical and chemical approach to the study of physiology,
which was the beginning of the disciplines known today as biophysics and biochemistry.
Mu¨ ller’s goal was to rid medical science of all the metaphysical excesses
it had accumulated, and retain only those principles with sound empirical
foundations. Helmholtz joined forces with three of Mu¨ ller’s students, Emil du
Bois-Reymond, Ernst Bru¨ cke, and Carl Ludwig; the four, known later as the “1847
group,” pledged their talents and careers to the taskof reshaping physiology into
a physicochemical science.

Medicine and Physics

Helmholtz, like Mayer, was educated for a medical career. He would have preferred
to study physics and mathematics, but the only hope for scientific training,
given his father’s meager salary as a gymnasium teacher, was a government scholarship in medicine. With the scholarship, Helmholtz studied at the Friedrich-
Wilhelm Institute in Berlin and wrote his doctoral dissertation under Johannes
Mu¨ ller. At that time, Mu¨ ller and his circle of gifted students were laying the
groundworkfor a physical and chemical approach to the study of physiology,
which was the beginning of the disciplines known today as biophysics and biochemistry.
Mu¨ ller’s goal was to rid medical science of all the metaphysical excesses
it had accumulated, and retain only those principles with sound empirical
foundations. Helmholtz joined forces with three of Mu¨ ller’s students, Emil du
Bois-Reymond, Ernst Bru¨ cke, and Carl Ludwig; the four, known later as the “1847
group,” pledged their talents and careers to the taskof reshaping physiology into
a physicochemical science.

Unities and a Unifier Hermann Helmholtz

Unifiers and Diversifiers
Science is largely a bipartisan endeavor. Most scientists have no difficulty identifying
with one of two camps, which can be called, with about as much accuracy
as names attached to political parties, theorists and experimentalists. An astute
observer of scientists and their ways, Freeman Dyson, has offered a roughly
equivalent, but more inspired, division of scientific allegiances and attitudes. In
Dyson’s view, science has been made throughout its history in almost equal measure
by “unifiers” and “diversifiers.” The unifiers, mostly theorists, search for
the principles that reveal the unifying structure of science. Diversifiers, likely to
be experimentalists, workto discover the unsorted facts of science. Efforts of the
scientific unifiers and diversifiers are vitally complementary. From the great bodies
of facts accumulated by the diversifiers come the unifier’s theories; the theories
guide the diversifiers to new observations, sometimes with disastrous results
for the unifiers.
The thermodynamicists celebrated here were among the greatest scientific unifiers
of the nineteenth and early twentieth centuries. Three of their stories have
been told above: of Sadi Carnot and his search for unities in the bewildering
complexities of machinery; of Robert Mayer and his grand speculations about
the energy concept; of James Joule’s precise determination of equivalences among
thermal, electrical, chemical, and mechanical effects. Continuing now with the
chronology, we focus on the further development of the energy concept. The
thermodynamicist who takes the stage is Hermann Helmholtz, the most confirmed
of unifiers.

A Joule Sketch

Osborne Reynolds, who met Joule in 1869, gives us this impression of his manner
and appearance in middle age: “That Joule, who was 51 years of age, was rather
under medium height; that he was somewhat stout and rounded in figure; that
his dress, though neat, was commonplace in the extreme, and that his attitude
and movements were possessed of no natural grace, while his manner was somewhat
nervous, and he possessed no great facility of speech, altogether conveyed
an impression of simplicity, and utter absence of all affectation which had characterized
his life.”
Joule married Amelia Grimes in 1847, when he was twenty-nine and she
thirty-three; they had two children, a son and a daughter. Amelia died in 1854,
and “the shocktook a long time to wear off,” writes Joule’s most recent biographer,
Donald Cardwell. “His friends and contemporaries agreed that this never
very assertive man became more withdrawn.” About fourteen years later, Joule
fell in love again, this time with his cousin Frances Tappenden, known as
“Fanny.” In a letter to Thomson he writes “an affection has sprung up between
me and my cousin you saw when last here. There are hindrances in the way so
that nothing may come of it.” The “hindrances” prevented marriage, and eventually
Fanny married another man.
Joule’s political leanings were conservative. He had a passionate, sometimes
irrational, dislike of reform-minded Liberal politicians such as William Gladstone and John Bright. In a letter to John Tyndall, he wrote, “The fact is that Mr. Gladstone
was fashioning a neat machine of ‘representation’ with the object of keeping
himself in power. . . . Posterity will judge him as the worst ‘statesman’ that England
ever had and the verdict with regard to that Parliament will be ditto, ditto.”
Joule had a personality that was “finely poised,” as another biographer, J. G.
Crowther, puts it. On the one hand he was conducting experiments with unlimited
care and patience, and on the other hand fulminating against Liberal politicians.
He feared that too much mental effort would threaten his health. In 1860,
a new professorship of physics was created at Owens College in Manchester, and
Joule could have had it, but he decided not to apply, as he explained in a letter
to Thomson: “I have not the courage to apply for the Owens professorship. The
fact is that I do not feel it would do for me to overtaskmy brain. A few years
ago, I felt a very small mental effort too much for me, and in consequence spared
myself from thought as much as possible. I have felt a gradual improvement, but
I do not thinkit would be well for me to build too much on it. I shall do a great
deal more in the long run by taking things easily.”
Joule’s life was hectic and burdensome at this time, and he may have felt that
he was near breakdown. Amelia died in 1854, the brewery was sold in the same
year, and the experiments with Thomson were in progress. During the next six
years, he moved his household and laboratory twice. After the second move, he
was upset by an acrimonious dispute with a neighbor who objected to the noise
and smoke made by a three-horsepower steam engine Joule included in his apparatus.
The neighbor was “a Mr Bowker, an Alderman of Manchester and chairman
of the nuisances committee, a very important man in his own estimation
like most people who have risen from the dregs of society.”
During this same period, Joule narrowly escaped serious injury in a train
wreck, and after that he had an almost uncontrollable fear of railway travel. At
the same time, he loved to travel by sea, even when it was dangerous. In a letter
to Fanny, he described a ten-mile trip to Tory Island, in the Atlantic off the coast
of Ireland, where his brother owned property: “Waves of 4 to 600 feet from crest
to crest and 20 feet high. Dr Brady who was with us and had yachted in the
ocean for 25 years said he was never in a more dangerous sea. However the
magnificence of it tookaway the disagreeable sense of danger which might have
prevailed.”
In some measures of scientific ability, Joule was unimpressive. As a theorist,
he was competent but not outstanding. He was not an eloquent speaker, and he
was not particularly important in the scientific establishment of his time. But
Joule had three things in extraordinary measure—experimental skill, independence,
and inspiration.
He was the first to understand that unambiguous equivalence principles could
be obtained only with the most inspired attention to experimental accuracy. He
accomplished his aim by carefully selecting the measurements that would make
his case. Crowther marvels at the directness and simplicity of Joule’s experimental
strategies: “He did not separate a quantity of truth from a large number of
groping unsuccessful experiments. Nearly all of his experiments seem to have
been perfectly conceived and executed, and the first draft of them could be sent
almost without revision to the journals for publication.”
For most of his life, Joule had an ample independent income. That made it
possible for him to pursue a scientific career privately, and to build the kind of intellectual independence he needed. Crowther tells us about this facet of Joule’s
background:
As a rich young man he needed no conventional training to qualify him for a
career, or introduce him to powerful future friends. His early researches were
pursued partly in the spirit of a young gentleman’s entertainment, which happened
to be science instead of fighting or politics or gambling. It is difficult to
believe that any student who had received a lengthy academic training could
have described researches in Joule’s tone of intellectual equality. The gifted
student who has studied under a great teacher would almost certainly adopt a
less independent tone in his first papers, because he would have the attitude
of a pupil to his senior, besides a deference due to appreciation of his senior’s
achievements. A student without deference after distinguished tuition is almost
always mediocre.
Joule was not entirely without distinguished tuition. Beginning in 1834, and
continuing for three years, Joule and his brother Benjamin studied with John
Dalton, then sixty-eight and, as always, earning money teaching children the
rudiments of science and mathematics. The Joules’ studies with Dalton were not
particularly successful pedagogically. Dalton tookthem through arithmetic and
geometry (Euclid) and then proceeded to higher mathematics, with little attention
to physics and chemistry. Dalton’s syllabus did not suit Joule, but he benefited
in more-informal ways. Joule wrote later in his autobiographical note, “Dalton
possessed a rare power of engaging the affection of his pupils for scientific truth;
and it was from his instruction that I first formed a desire to increase my knowledge
by original researches.” In his writings, if not in his tutoring, Dalton emphasized
the ultimate importance of accurate measurements in building the
foundations of physical science, a lesson that Joule learned and used above all
others. The example of Dalton, internationally famous for his theories of chemical
action, yet self-taught, and living and practicing in Manchester, must have convinced
Joule that he, too, had prospects.
Joule’s independence and confidence in his background and talents, natural
or learned from Dalton, were tested many times in later years, but never shaken.
His first determination, in 1843, of the mechanical equivalent of heat was ignored,
and subsequent determinations were given little attention until Thomson
and Stokes took notice at the British Association meeting in 1847.
When Joule submitted a summary of his friction experiments for publication,
he closed the paper with three conclusions that asserted the heat-mechanicalworkequivalence
in the friction experiments, quoted his measured value of J,
and stated that “the friction consisted in the conversion of mechanical power to
heat.” The referee who reported on the paper (believed to have been Faraday)
requested that the third conclusion be suppressed.
Joule’s first electrochemistry paper was rejected for publication by the Royal
Society, except as an abstract. Arthur Schuster reported that, when he asked Joule
what his reaction was when this important paper was rejected, Joule’s reply was
characteristic: “I was not surprised. I could imagine those gentlemen sitting
around a table in London and saying to each other: ‘What good can come out of
a town [Manchester] where they dine in the middle of the day?’ ”
But with all his talents, material advantages, and intellectual independence, Joule could never have accomplished what he did if he had not been guided in
his scientific workby inspiration of an unusual kind. For Joule “the study of
nature and her laws” was “essentially a holy undertaking.” He could summon
the monumental patience required to assess minute errors in a prolonged series
of measurements, and at the same time transcend the details and see his work
as a quest “for acquaintance with natural laws . . . no less than an acquaintance
with the mind of God therein expressed.” Great theorists have sometimes had
thoughts of this kind—one might get the same meaning from Albert Einstein’s
remarkthat “the eternal mystery of the world is its comprehensibility”—but experimentalists,
whose lives are taken up with the apparently mundane tasks of
reading instruments and designing apparatuses, have rarely felt that they were
communicating with the “mind of God.”
It would be difficult to find a scientific legacy as simple as Joule’s, and at the
same time as profoundly important in the history of science. One can summarize
Joule’s major achievement with the single statement
J 778 ft-lb per Btu,
and add that this result was obtained with extraordinary accuracy and precision.
This is Joule’s monument in the scientific literature, now quoted as 4.1840
kilogram-meters per calorie, used routinely and unappreciatively by modern students
to make the quantitative passage from one energy unit to another.
In the 1840s, Joule’s measurements were far more fascinating, or disturbing,
depending on the point of view. The energy concept had not yet been developed
(and would not be for another five or ten years), and Joule’s number had not
found its niche as the hallmarkof energy conversion and conservation. Yet Joule’s
research made it clear that something was converted and conserved, and provided
vital clues about what the something was.

Living Force and Heat

Living Force and Heat
Joule believed that water at the bottom of a waterfall should be slightly warmer
than water at the top, and he made attempts to detect such effects (even on his
honeymoon in Switzerland, according to an apocryphal, or at any rate embellished,
story told by Thomson). For Joule this was an example of the conservation
principle that “heat, living force, and attraction through space . . . are mutually
convertible into one another. In these conversions nothing is ever lost.” This
statement is almost an expression of the conservation of mechanical and thermal
energy, but it requires some translation and elaboration.
Newtonian mechanics implies that mechanical energy has a “potential” and a
“kinetic” aspect, which are linked in a fundamental way. “Potential energy” is
evident in a weight held above the ground. The weight has energy because work
was required to raise it, and the workcan be completely recovered by letting the
weight fall very slowly and drive machinery that has no frictional losses. As one
might expect, the weight’s potential energy is proportional to its mass and to its
height above the ground: if it starts at a height of 100 feet it can do twice as
much workas it can if it starts at 50 feet.
If one lets the weight fall freely, so that it is no longer tied to machinery, it does no work, but it accelerates and acquires “kinetic energy” from its increasing
speed. Kinetic energy, like potential energy, can be converted to work with the
right kind of machinery, and it is also proportional to the mass of the weight. Its
relationship to speed, however, as dictated by Newton’s second law of motion,
is to the square of the speed.
In free fall, the weight has a mechanical energy equal to the sum of the kinetic
and potential energies,
mechanical energy kinetic energy potential energy. (1)
As it approaches the ground the freely falling weight loses potential energy, and
at the same time, as it accelerates, it gains kinetic energy. Newton’s second law
informs us that the two changes are exactly compensating, and that the total
mechanical energy is conserved, if we define
mv2
kinetic energy (2)
2
potential energy mgz. (3)
In equations (2) and (3), m is the mass of the weight, v its speed, z its distance
above the ground, and g the constant identified above as the gravitational acceleration.
If we represent the total mechanical energy as E, equation (1) becomes
mv2
E mgz, (4)
2
and the conservation law justified by Newton’s second law guarantees that E is
always constant. This is a conversion process, of potential energy to kinetic energy,
as illustrated in figure 5.2. In the figure, before the weight starts falling it
has 10 units of potential energy and no kinetic energy. When it has fallen halfway
to the ground, it has 5 units of both potential and kinetic energy, and in the
instant before it hits the ground it has no potential energy and 10 units of kinetic
energy. At all times its total mechanical energy is 10 units.
Joule’s term “living force” (or vis viva in Latin) denotes mv2, almost the samething as the kinetic energy , and his phrase “attraction through space” means
mv2
2
the same thing as potential energy. So Joule’s assertions that living force and
attraction through space are interconvertible and that nothing is lost in the conversion
are comparable to the Newtonian conservation of mechanical energy.
Water at the top of the falls has potential energy only, and just before it lands in
a pool at the bottom of the falls, it has kinetic energy only. An instant later the
water is sitting quietly in the pool, and according to Joule’s principle, with the
third conserved quantity, heat, included, the water is warmer because its mechanical
energy has been converted to heat. Joule never succeeded in confirming
this waterfall effect. The largest waterfall is not expected to produce a temperature
change of more than a tenth of a degree. Not even Joule could detect that
on the side of a mountain.
Joule’s mechanical view of heat led him to believe further that in the conversion
of the motion of an object to heat, the motion is not really lost because heat
is itself the result of motion. He saw heat as the internal, random motion of the
constituent particles of matter. This general idea had a long history, going back
at least to Robert Boyle and Daniel Bernoulli in the seventeenth century.
Joule pictured the particles of matter as atoms surrounded by rapidly rotating
“atmospheres of electricity.” The centrifugal force of the atmospheres caused a
gas to expand when its pressure was decreased or its temperature increased.
Mechanical energy converted to heat became rotational motion of the atomic
atmospheres. These speculations of Joule’s markthe beginning of the development
of what would later be called the “molecular (or kinetic) theory of gases.”
Following Joule, definitive workin this field was done by Clausius, Maxwell,
and Boltzmann.

Mechanical Equivalents

Joule made the crucial addition of mechanical effects to his system of equivalences
by following a time-honored route to scientific discovery: he made a fortunate mistake. In the fourth of his electrochemistry papers he reported electric
potential data (voltages, in modern units) measured on voltaic cells whose
electrode reactions produced oxidation of zinc and other metals. He believed,
mistakenly, that these reaction potentials could be used in much the same way
as reaction heats: that for a given reaction the potential had the same value no
matter how the reaction was carried out. This interpretation is not sanctioned by
modern thermodynamics unless cell potentials are measured carefully (reversibly).
Joule and his contemporaries were unaware of this limitation, however, and
the mistake led Joule to calculate electrical and thermal equivalents for the process
in which dissolved oxygen is given “its elastic condition,” the reaction
O2 (solution) O2 (gas).
Joule’s result was an order of magnitude too large. But mistaken as it was quantitatively,
the calculation advanced Joule’s conceptual understanding immensely,
because he believed he had obtained electrical and thermal equivalents for a
mechanical effect, the evolution of oxygen gas from solution. In Joule’s fertile
imagination, this was suggestive. In the fourth electrochemistry paper, he remarked
that he had already thought of ways to measure mechanical equivalents.
He hoped to confirm the conclusion that “the mechanical and heating powers of
a current are proportional to each other.”
In this serendipitous way, Joule began the determinations of the mechanical
equivalent of heat for which he is best known today. The first experiments in
this grand series were performed in 1843, when Joule was twenty-four. In these
initial experiments, he induced an electrical current in a coil of wire by rotating
it mechanically in a strong magnetic field. The coil was contained in a glass tube
filled with water and surrounded by insulation, so any heating in the coil could
be measured by inserting a thermometer in the tube before and after rotating it
in the magnetic field. The induced current in the coil was measured by connecting
the coil to an external circuit containing a galvanometer. Although its
origin was entirely different, the induced current behaved the same way as the
voltaic current Joule had studied earlier: in both cases the current caused heating
that followed the I 2R-law.
In the final experiments of this design, the wheel of the induction device was
driven by falling weights for which the mechanical effect, measured as a mechanical
workcalculation, could be made directly in foot-pounds (abbreviated
ft-lb): one unit was equivalent to the workrequired to raise one pound one foot.
Heat was measured by a unit that fit the temperature measurements: one unit
raised the temperature of one pound of water 1 Fahrenheit (F). We will use the
term later attached to this unit, “British thermal unit,” or Btu.
In one experiment, Joule dropped weights amounting to 4 lb 12 oz ( 4.75 lb)
517 feet (the weights were raised and dropped many times), causing a temperature
rise of 2.46 F. He converted the weight of the glass tube, wire coil, and water
in which the temperature rise occurred all into a thermally equivalent weight of
water, 1.114 lb. Thus the heating effect was 2.46 F in 1.114 lb of water. If this
same amount of heat had been generated in 1 lb of water, the heating effect would
have been . Joule concluded that in this case (517)(4.75)
(2.46)(1.114)
2.74 F
1
ft-lb was equivalent to 2.74 Btu. He usually determined the mechanical workequivalent to 1 Btu. That number, which Thomson later labeled J to honor Joule,
was
(4.75)(517)
J 896 ft-lb per Btu
(2.74)
for this experiment. This was one determination of the mechanical equivalent of
heat. Joule did thirteen experiments of this kind and obtained results ranging
from J 587 to 1040 ft-lb per Btu, for which he reported an average value of 838
ft-lb per Btu. The modern “correct” value, it should be noted, is J 778 ft-lb per
Btu.
If the 27% precision achieved by Joule in these experiments does not seem
impressive, one can sympathize with Joule’s critics, who could not believe his
claims concerning the mechanical equivalent of heat. But the measurements Joule
was attempting set new standards for experimental difficulty. According to Reynolds,
the 1843 paper reported experiments that were more demanding than any
previously attempted by a physicist.
In any case, Joule was soon able to do much better. In 1845, he reported another,
much different determination of the mechanical equivalent of heat, which
agreed surprisingly well with his earlier measurement. In this second series of
experiments, he measured temperature changes, and calculated the heat produced,
when air was compressed. From the known physical behavior of gases
he could calculate the corresponding mechanical effect as workdone on the air
during the compression.
In one experiment involving compression of air, Joule calculated the workat
11230 ft-lb and a heating effect of 13.628 Btu from a measured temperature rise
of 0.344 F. The corresponding mechanical equivalent of heat was
11230
J 824 ft-lb per Btu.
13.628
Another experiment done the same way, in which Joule measured the temperature
change 0.128 F, gave the result J 796 ft-lb per Btu. Joule’s average for the
two experiments was 810 ft-lb per Btu. This was in impressive, if somewhat
fortuitous, agreement with the result J 838 ft-lb per Btu reported in 1843.
Joule also allowed compressed air to expand and do workagainst atmospheric
pressure. Temperature measurements were again made, this time with a temperature
decrease being measured. In one of these expansion experiments, Joule
measured the temperature change 0.1738 F and reduced this to 4.085 Btu. The
corresponding workcalculation gave 3357 ft-lb, so
3357
J 822 ft-lb per Btu.
4.085
Joule did two more experiments of this kind and measured the temperature
changes 0.081 F and 0.0855 F, giving J 814 and J 760 ft-lb per Btu.
When Joule’s colleagues looked at these results, the first thing they noticed
was the accuracy claimed for measurements of very small temperature changes.
In Joule’s time, accurate measurement of one-degree temperature changes was difficult enough. Joule reported temperature changes of tenths of a degree with
three or four significant digits, and based his conclusions on such tiny changes.
As William Thomson remarked, “Joule had nothing but hundredths of a degree
to prove his case by.” Yet, most of Joule’s claims were justified. He made temperature
measurements with mercury thermometers of unprecedented sensitivity
and accuracy. He told the story of the thermometers in an autobiographical note:
“It was needful in these experiments to use thermometers of greater exactness
and delicacy than any that could be purchased at that time. I therefore determined
to get some calibrated on purpose after the manner they had been by
Regnault. In this I was ably seconded by Mr. Dancer [J. B. Dancer, a well-known
Manchester instrument maker], at whose workshop I attended every morning for
some time until we completed the first accurate thermometers which were ever
made in England.”
Joule demonstrated the heat-mechanical-workequivalence with a third gas
expansion experiment that incorporated one of his most ingenious experimental
designs. In this experiment, two constant-volume copper vessels, one evacuated
and the other pressurized with air, were connected with a valve. The connected
vessels were placed in a calorimeter, the valve opened, and the usual temperature
measurements made. In this case, Joule could detect no net temperature change.
Air expanding from the pressurized vessel was cooled slightly, and air flowing
into the evacuated vessel was slightly heated, but no net temperature change was
observed.
This was what Joule expected. Because the combined system consisting of the
two connected vessels was closed and had a fixed volume, all of the workwas
done internally, in tandem between the two vessels. Workdone by the gas in one
vessel was balanced by workdone on the gas in the other; no net workwas done.
Heat equivalent to zero workwas also zero, so Joule’s concept of heatmechanical-
workequivalence demanded that the experiment produce no net
thermal effect, as he observed.
The next stage in Joule’s relentless pursuit of an accurate value for the mechanical
equivalent of heat, which he had begun in 1847, was several series of
experiments in which he measured heat generated by various frictional processes.
The frictional effects were produced in a water-, mercury-, or oil-filled
calorimeter by stirring with a paddle-wheel device, the latter being driven by
falling weights, as in the 1843 experiments. The workdone by the weights was
converted directly by the paddle-wheel stirrer into heat, which could be measured
on a thermometer in the calorimeter.
Of all Joule’s inventions, this experimental design, which has become the bestknown
monument to his genius, made the simplest and most direct demonstration
of the heat-mechanical-workequivalence. This was the Joule technique reduced
to its essentials. No complicated induction apparatus was needed, no
calculational approximations, just falling weights and one of Joule’s amazingly
accurate thermometers.
With the paddle-wheel device and water as the calorimeter liquid, Joule obtained
J 773.64 ft-lb per Btu from a temperature rise of 0.563 F. Using mercury
in the calorimeter, he obtained J 773.762 and 776.303 ft-lb per Btu. In two
further series of experiments, Joule arranged his apparatus so the falling weights
caused two cast-iron rings to rub against each other in a mercury-filled calorimeter;
the results J 776.997 and 774.880 ft-lb per Btu were obtained.
Joule described his paddle-wheel experiments in 1847 at an Oxford meeting of the British Association for the Advancement of Science. Because his previous
papers had aroused little interest, he was asked to make his presentation as brief
as possible. “This I endeavored to do,” Joule recalled later, “and a discussion not
being invited the communication would have passed without comment if a
young man had not risen in the section, and by his intelligent observations created
a lively interest in the new theory.”
The silence was finally broken. The young man was William Thomson, recently
installed as professor of natural philosophy at Glasgow University. Thomson
had reservations about Joule’s work, but he also recognized that it could not
be ignored. “Joule is, I am sure, wrong in many of his ideas,” Thomson wrote to
his father, “but he seems to have discovered some facts of extreme importance,
as for instance, that heat is developed by the friction of fluids.” Thomson recalled
in 1882 that “Joule’s paper at the Oxford meeting made a great sensation. Faraday
was there, and was much struckby it, but did not enter fully into the new views.
. . . It was not long after when Stokes told me he was inclined to be a Joulite.”
George Stokes was another rising young physicist and mathematician, in 1847 a
fellow at Pembroke College, Cambridge, and in two years to be appointed Lucasian
Professor of Mathematics, the chair once occupied by Newton.
During the three years following the Oxford meeting, Joule rose from obscurity
to a prominent position in the British scientific establishment. Recognition came
first from Europe: a major French journal, Comptes Rendu, published a short
account of the paddle-wheel experiments in 1847, and in 1848 Joule was elected
a corresponding member of the Royal Academy of Sciences at Turin. Only two
other British scientists, Faraday and William Herschel, had been honored by the
Turin Academy. In 1850, when he was thirty-one, Joule received the badge of
British scientific acceptance: election as a fellow of the Royal Society.
After these eventful years, Joule’s main research effort was a lengthy collaboration
with Thomson, focusing on the behavior of expanding gases. This was one
of the first collaborative efforts in history in which the talents of a theorist and
those of an experimentalist were successfully and happily united.

Equivalences

The theme that dominated Joule’s research from beginning to end, and served as
his guiding theoretical inspiration, was the belief that quantitative equivalences
could be found among thermal, chemical, electrical, and mechanical effects. He
was convinced that the extent of any one of these effects could be assessed with
the units of any one of the other effects. He studied such quantitative connections
in no less than eight different ways: in investigations of chemical effects converted
to thermal, electrical, and mechanical effects; of electrical effects converted
to thermal, chemical, and mechanical effects; and of mechanical effects
converted to thermal and electrical effects.
At first, Joule did not fully appreciate the importance of mechanical effects in
this scheme of equivalences. His earliest workcentered on chemical, electrical,and thermal effects. In 1840, when he was twenty-two, he started a series of five
investigations that was prompted by his interest in electrochemistry. (Joule was
an electrochemist before he was a physicist.) First, he demonstrated accurately
that the heating produced by an electrical current in a wire is proportional to the
square of the current I and to the electrical resistance R—the “I2R-heating law.”
His experimental proof required temperature measurements in a “calorimeter” (a
well-insulated, well-stirred vessel containing water or some other liquid), electrical
current measurements with an instrument of his own design, and the invention
of a system of absolute electrical units.
Joule then invested considerable effort in various studies of the role played
by his heating law in the chemical processes produced in electric cells. He
worked with “voltaic cells,” which supply an electrical output (the modern flashlight
battery is an example), and “electrolysis cells,” which consume an electrical
input (for example, a cell that decomposes water into hydrogen gas and oxygen
gas). In these experiments, Joule operated an electrolysis cell with a battery of
voltaic cells. He eventually arrived at the idea that the electrical currents generated
by the chemical reaction in the voltaic cell carried the reaction’s “calorific
effect” or “chemical heat” away from the primary reaction site either to an external
resistance where it could be converted to “free heat,” according to the I 2Rheating
law, or to an electrolysis cell where it could be invested, all or partly, as
“latent heat” in the electrolysis reaction.
To determine the total chemical heat delivered to the electrolysis cell from the
voltaic cells, call it Qe, Joule found the resistance Re of a wire that could replace
the electrolysis cell without causing other electrical changes, measured the current
I in the wire, and calculated Qe with the heating law as I2Re. He also measured
the temperature rise in the electrolysis cell doubling as a calorimeter, and
from it calculated the free heat Qt generated in the cell. He always found that
Qe substantially exceeded Qt; in extreme cases, there was no heating in the cell
and Qt was equal to zero. The difference Qe Qt represented what Joule wanted
to calculate: chemical heat converted to the latent heat of the electrolysis reaction.
Representing the electrolysis reaction’s latent heat with Qr, Joule’s calculation
was
Qr Qe Qt.
This is the statement Joule used in 1846 to determine several latent heats of
electrolysis reactions with impressive accuracy. It is a complicated and exact
application of the first law of thermodynamics, which Joule seems to have understood
in terms of inputs and outputs to the electrolysis cell. That is evident
in the last equation rearranged to
Qt Qe Qr,
with Qe an input to the cell, Qr an output because it is lost to the reaction, and
Qt the difference between the input and output (see fig. 5.1). This was a balancing
or bookkeeping kind of calculation, and it implied a conservation assumption:
the balanced entity could not be created or destroyed within the cell. Joule did
not have a name for the conserved entity. It would be identified six years later
by Rudolf Clausius and William Thomson, and called “energy” by Thomson.
Although he had not arrived at the energy concept, Joule clearly did have, wellahead of his contemporaries, a working knowledge of the first law of
thermodynamics.
Joule’s electrochemistry papers aroused little interest when they were first
published, neither rejection nor acceptance, just silence. One reason for the indifference
must have been the extraordinary nature of Joule’s approach. The
input-output calculation was difficult enough to comprehend at the time, but in
addition to that, Joule used his measured heats of electrolysis reactions to calculate
heats of combustion reactions (that is, reactions with oxygen gas). For
example, he obtained an accurate heat for the hydrogen combustion reaction,
2 H2 O2 2 H2O,
which is just the reverse of the water electrolysis reaction,
2 H2O 2 H2 O2,
and therefore, Joule assumed, its heat had the same magnitude as that of the
electrolysis reaction.
This was an exotic way to study a combustion reaction. Joule’s first biographer,
Osborne Reynolds, remarks that “the views they [the electrochemistry papers and
others of Joule’s early papers] contained were so much in advance of anything
accepted at the time that no one had sufficient confidence in his own opinion or
was sufficiently sure of apprehending the full significance of the discoveries on
which these views were based, to venture an expression of acceptance or rejection.”
We can imagine a contemporary reader puzzling over the papers and finally
deciding that the author was either a genius or a crank.
But for Joule—apparently unconcerned about the accessibility or inaccessibility
of his papers for readers—the complicated method was natural. His primary
interest at the time was the accurate determination of equivalences among thermal,
electrical, and chemical effects. He could imagine no better way to tackle
this problem than to use electrical and calorimetric measurements to calculate
the thermal effect of a chemical effect.

A Holy Undertaking James Joule

The Scientist as Amateur
James Joule’s story may seem a little hard to believe. He lived near Manchester,
England—in the scientific hinterland during much of Joule’s career—where his
family operated a brewery, making ale and porter. He did some of his most important
workin the early morning and evening, before and after a day at the
brewery. He had no university education, and hardly any formal training at all
in science. As a scientist he was, in every way, an amateur. Like Mayer, who was
also an amateur as a physicist, Joule was ignored at first by the scientific establishment.
Yet, despite his amateur status, isolation, and neglect, he managed to
probe more deeply than anyone else at the time (the early and middle 1840s) the
tantalizing mysteries of conversion processes. And (unlike Mayer) he did not
suffer prolonged neglect. The story of Joule’s rapid progress, from dilettante to a
position of eminence in British science, can hardly be imagined in today’s world
of research factories and prolonged scientific apprenticeships

Strange Success

The final episode in this life full of ironies will seem like the ultimate irony.
Recognition of Mayer’s achievements finally came, but hardly in a way deserved
by a man who had endured indifference, rejection, breakdown, cruel medical
treatment, and reports of his own death. In the early 1860s Mayer, now peacefully
tending his vineyards in Heilbronn, suddenly became the center of a famous
scientific controversy.
It all started when John Tyndall, a popular lecturer, professor, and colleague
of Michael Faraday at the Royal Institution in London, prepared himself for a
series of lectures on heat. He wrote to Hermann Helmholtz and Rudolf Clausius
in Germany for information. Included in Clausius’s response was the comment
that Mayer’s writings were not important. Clausius promised to send copies of
Mayer’s papers nevertheless, and before mailing the papers he read them, apparently
for the first time with care. Clausius wrote a second letter with an entirely
different assessment: “I must retract the statements in my last letter that
you would not find much of importance in Mayer’s writings; I am astonished at
the multitude of beautiful and correct thoughts which they contain.” Clausius
was now convinced that Mayer had been one of the first to understand the energy
concept and its conservation doctrine. Helmholtz also sent favorable comments
on Mayer, pointing especially to the early evaluation of the mechanical equivalent
of heat.
Tyndall was a man who loved controversy and hated injustice. Because his
ideas concerning the latter were frequently not shared by others who were
equally adept in the practice of public controversy, he was often engaged in
arguments that were lively, but not always friendly. When Tyndall decided to be
Mayer’s champion, he embarked on what may have been the greatest of all his
controversies. As usual, he chose as his forum the popular lectures at the Royal
Institution. He had hastily decided to broaden his topic from heat to the general
subject of energy, which was by then, in the 1860s, mostly understood; the title
of his lecture was “On Force.” (Faraday and his colleagues at the Royal Institution
still preferred to use the term “force” when they meant “energy.”)
Tyndall began by listing many examples of energy conversion and conservation,
and then summarized Mayer’s role with the pronouncement, “All that I have
brought before you has been taken from the labors of a German physician, named
Mayer.” Mayer should, he said, be recognized as one of the first thermodynamicists,
“a man of genius arriving at the most important results some time in advance
of those whose lives were entirely devoted to Natural Philosophy.” Tyndall
left no doubt that he felt Mayer had priority claims over Joule: “Mr. Joule published
his first paper ‘On the Mechanical Value of Heat’ in 1843, but in 1842
Mayer had actually calculated the mechanical equivalent of heat.” In the gentlemanly
world of nineteenth-century scientific discourse, this was an invitation to
verbal combat. It brought quickresponses from Joule and Thomson, and also
from Thomson’s close friend Peter Guthrie Tait, professor of natural philosophy
at the University of Edinburgh, and Tyndall’s match in the art of polemical
debate.
Joule was the first to reply, in a letter published in the Philosophical Magazine.
He could not, he said, accept the view that the “dynamical theory of heat” (that
is, the theory of heat that, among other things, was based on the heat-workconnection)
was established by Mayer, or any of the other authors who speculated on the meaning of the conversion processes. Reliable conclusions “require experiments,”
he wrote, “and I therefore fearlessly assert my right to the position
which has been generally accorded to me by my fellow physicists as having been
the first to give decisive proof of the correctness of this theory.”
Tyndall responded to Joule in another letter to the Philosophical Magazine,
protesting that he did not wish to slight Joule’s achievements: “I trust you will
find nothing [in my remarks] which indicates a desire on my part to question
your claim to the honour of being the experimental demonstrator of the equivalence
of heat and work.” Tyndall was willing to let Mayer speak for himself; at
Tyndall’s suggestion, Mayer’s papers on the energy theme were translated and
published in the Philosophical Magazine.
But this did not settle the matter. An article with both Thomson and Tait listed
as authors (although the style appears to be that of Tait) next appeared in a
popular magazine called Good Words, then edited by Charles Dickens. In it,
Mayer’s 1842 paper was summarized as mainly a recounting of previous work
with a few suggestions for new experiments; “a method for finding the mechanical
equivalent of heat [was] propounded.” This was, the authors declared, a
minor achievement, and they could find no reason to surrender British claims:
On the strength of this publication an attempt has been made to claim for Mayer
the credit of being the first to establish in all its generality the principle of the
Conservation of Energy. It is true that la science n’a pas de patrie and it is
highly creditable to British philosophers that they have so liberally acted according
to this maxim. But it is not to be imagined that on this account there
should be no scientific patriotism, or that, in our desire to do justice to a foreigner,
we should depreciate or suppress the claims of our countrymen.
Tyndall replied, again in the Philosophical Magazine, pointedly directing his
remarks to Thomson alone, and questioning the wisdom of discussing weighty
matters of scientific priority in the pages of a popular magazine. He now relaxed
his original position and saw Joule and Mayer more in a shared role:
Mayer’s labors have in some measure the stamp of profound intuition, which
rose, however, to the energy of undoubting conviction in the author’s mind.
Joule’s labours, on the contrary, are in an experimental demonstration. True to
the speculative instinct of his country, Mayer drew large and weighty conclusions
from slender premises, while the Englishman aimed, above all things, at
the firm establishment of facts. And he did establish them. The future historian
of science will not, I think, place these men in antagonism.
Tait was next heard from. He wrote to one of the editors of the Philosophical
Magazine, first offering the observation that if Good Words was not a suitable
medium for the debate of scientific matters, neither were certain popular lecture
series at the Royal Institution. He went on: “Prof. Tyndall is most unfortunate in
the possession of a mental bias which often prevents him . . . from recognizing
the fact that claims of individuals whom he supposes to have been wronged have,
before his intervention, been fully ventilated, discussed, and settled by the general
award of scientific men. Does Prof. Tyndall know that Mayer’s paper has no
claim to novelty or correctness at all, saving this, that by a lucky analogy he got
an approximation to a true result from an utterly false analogy?” Even if the polemics had been avoided, any attempt to resolve Joule’s and
Mayer’s conflicting claims would have been inconclusive. If the aim of the debate
was to identify once and for all the discoverer of the energy concept, neither
Joule nor Mayer should have won the contest. The story of the energy concept
does not end, nor does it even begin, with Mayer’s speculations and Joule’s experimental
facts. Several of Kuhn’s simultaneous discoverers were earlier, although
more tentative, than Joule and Mayer. In the late 1840s, after both men
had made their most important contributions, the energy concept was still only
about half understood; the modern distinction between the terms force and energy
had not even been made clear. Helmholtz, Clausius, and Thomson still had
fundamentally important contributions to make.
Those who spend their time fighting priority wars should forget their individual
claims and learn to appreciate a more important aspect of the sociology of
science: that the scientific community, with all its diversity cutting across race,
class, and nationality, can, as often as it does, arrive at a consensus acceptable
to all. The final judgment in the Joule-Mayer controversy teaches this lesson. In
1870, almost a decade after the last Tyndall or Tait outburst, the Royal Society
awarded its prestigious Copley medal to Joule—and a year later to Mayer.

Over the Edge and Back

Although by this time Mayer was losing ground in his battle against discouragement,
perseverance still prevailed. In 1846, he wrote another paper (this one, on
celestial mechanics, anticipated workdone much later by William Thomson),
and again had to accept private publication.
Professional problems were now compounded by family and health problems.
During the years 1846 to 1848, three of Mayer’s children died, and his marriage
began to deteriorate. Finally, in 1850, he suffered a nearly fatal breakdown. An
attackof insomnia drove him to a suicide attempt; the attempt was unsuccessful,
but from the depths of his despair Mayer might have seen this as still another
failure.
In an effort to improve his condition, Mayer voluntarily entered a sanatorium.
Treatment there made the situation worse, and finally he was committed to an
asylum, where his handling was at best careless and at times brutal. The diagnosis
of his mental and physical condition became so bleakthat the medical
authorities could offer no hope, and he was released from the institution in 1853.
It may have been Mayer’s greatest achievement that he survived, and even
partially recovered from, this appalling experience. After his release, he returned
to Heilbronn, resumed his medical practice in a limited way, and for about ten
years deliberately avoided all scientific activity. In slow stages, and with occasional
relapses, his health began to return. That Mayer could, by an act of will
it seems, restore himself to comparatively normal health, demonstrated, if nothing
else did, that his mental condition was far from hopelessly unbalanced. To
abandon entirely for ten years an effort that had become an obsession was plainly
an act of sanity.
The period of Mayer’s enforced retirement, the 1850s, was a time of great
activity in the development of thermodynamics. Energy was established as a
concept, and the energy conservation principle was accepted by most theorists.
This workwas done mostly by James Joule in England, by Rudolf Clausius in
Germany, and by William Thomson and Macquorn Rankine in Scotland, with
little appreciation of Mayer’s efforts. Not only was Mayer’s theory ignored during
this time, but in 1858 Mayer himself was reported by Liebig to have died in an
asylum. Protests from Mayer did not prevent the appearance of his official death
notice in Poggendorff’s Handwo¨ rterbuch.

Rejection

Mayer submitted his 1841 paper to Johann Poggendorff’s Annalen der Physik und
Chemie. It was not accepted for publication, or even returned with an acknowledgment.
But, according to one of Mayer’s biographers, R. Bruce Lindsay, the
careless treatment was a blessing in disguise. Mayer’s detailed arguments in the
paper were “based on a profound misunderstanding of mechanics.” Although the
rejection was a blow to Mayer’s pride, “it was a good thing for [his] subsequent
reputation that [the paper] did not see the light of day.”
If Mayer had great pride, he had even more perseverance. With help from his
friend Carl Baur (later a professor of mathematics in Stuttgart), he improved the
paper, expanded it in several ways, and at last saw it published in Justus von
Liebig’s Annalen der Chemie und Pharmacie in 1842. Mayer’s most important
addition to the paper was a calculation of the mechanical effect, workdone in
the expansion of a gas, produced by a thermal effect, the heating of the gas. This
was an evaluation of the “mechanical equivalent of heat,” a concern independently
occupying Joule at about the same time. Whether or not Mayer made the
first such calculation became the subject of a celebrated controversy. One thing
that weakened Mayer’s priority claim was that he omitted all details but the result
in his calculation in the 1842 paper. Not until 1845, in a more extended paper,
did he make his method clear. By 1845, Joule was reporting impressive experimental
measurements of the mechanical equivalent of heat.
In the 1842 paper, Mayer based his ultimately famous calculation on the experimental
fact that it takes more heat to raise the temperature of a gas held at
constant pressure than at constant volume. Mayer could see in the difference
between the constant-pressure and constant-volume results a measure of the heat
converted to an equivalent amount of workdone by the gas when it expands
against constant pressure. He could also calculate that work, and the work-toheat
ratio, was a numerical evaluation of the mechanical equivalent of heat. His
calculation showed that 1 kilocalorie of heat converted to work could lift 1 kilogram
366 meters. In other words, the mechanical equivalent of heat found by
Mayer was 366 kilogram-meters per kilocalorie.
This was the quantity Joule had measured, or was about to measure, in a
monumental series of experiments started in 1843. Joule’s best result (labeled as
it was later with a J ) was
J 425 kilogram-meters per kilocalorie.
Mayer’s calculation was incorrect principally because of errors in heat measurements.
More-accurate measurements by Victor Regnault in the 1850s brought
Mayer’s calculation much closer to Joule’s result,
J 426 kilogram-meters per kilocalorie.
In addition to clarifying his determination of the mechanical equivalent of
heat, Mayer’s 1845 paper also broadened his speculations concerning the conservation
of energy, or force, as Mayer’s terminology had it. Two quotations will
show how committed Mayer had become to the conservation concept: “What
chemistry performs with respect to matter, physics has to perform in the case of force. The only mission of physics is to become acquainted with force in its
various forms and to investigate the conditions governing its change. The creation
or destruction of force, if [either has] any meaning, lies outside the domain
of human thought and action.” And: “In truth there exists only a single force. In
never-ending exchange this circles through all dead as well as living nature. In
the latter as well as the former nothing happens without form variation of force!”
Mayer submitted his 1845 paper to Liebig’s Annalen; it was rejected by an
assistant editor, apparently after a cursory reading. The assistant’s advice was to
try Poggendorff’s Annalen, but Mayer did not care to follow that publication
route again. In the end, he published the paper privately, and hoped to gain
recognition by distributing it widely. But beyond a few brief journal listings, the
paper, Mayer’s magnum opus, went unnoticed.

Conservation of Force (Energy)

In 1841 Mayer, now backin Heilbronn, began a paper that summarized his point
of view in the broadest terms. He wrote that “all bodies are subject to change . . .
[which] cannot happen without a cause . . . [that] we call force,” that “we can
derive all phenomena from a basic force,” and that “forces, like matter, are invariable.”
His intention, he said, was to write physics as a science concerned
with “the nature of the existence of force.” The program of this physics paralleled
that of chemistry. Chemists dealt with the properties of matter, and relied on the
principle that mass is conserved. Physicists should similarly study forces and
adopt a principle of conservation of force. Both chemistry and physics were
based on the principle that the “quantity of [their] entities is invariable and only
the quality of these entities is variable.”
Mayer’s use of the term force requires some explanation. It was common for
nineteenth-century physicists to give the force concept a dual meaning. They
used it at times in the Newtonian sense, to denote a push or pull, but just as
often the usage implied that force was synonymous with the modern term energy.
The modern definition of the word “energy”—the capacity to do work—was not
introduced until the 1850s, by William Thomson. In the above quotations, and
throughout most of Mayer’s writings, it is appropriate to assume the second usage,
and to read “energy” for “force.” With that simple but significant change,
Mayer’s thesis becomes an assertion of the principle of the conservation of
energy.

Voyage of Discovery

One of the first to penetrate this conceptual tangle was Robert Mayer, a German
physician and physicist who spent most of his life in Heilbronn, Germany. Mayer
was a contemporary of James Joule (chapter 5), and like Joule, he was an amateur
in the scientific fields that most absorbed his interest. His university training was
in medicine, and what is known of his student record at the University of Tu¨ -
bingen shows little sign of intellectual genius. He was good at billiards and cards,
devoted to his fraternity, and inclined to be rebellious and unpopular with the
university authorities; eventually he was suspended for a year. With hindsight,
we can see in Mayer’s reaction to the suspension—a six-day hunger strike—
evidence for his stubbornness and sensitivity to criticism, and even some forewarning
of his later mental problems.
Mayer’s youthful behavior was not that of an unmitigated rebel, however;
when the Tu¨ bingen authorities permitted, he returned, finished his dissertation,
and passed the doctoral examination. But he was still too restless to plan his
future according to conventional (and family) expectations. Instead of settling
into a routine medical practice, he decided to travel by taking a position as ship’s
surgeon on a Dutch vessel sailing for the East Indies. He found little inspiration on this trip, either in the company of his fellow officers or in the quality and
quantity of the ship’s food. But to Mayer the voyage was worth any amount of
hunger and boredom.
Mayer tells us, in an exotic tale of scientific imagination, of an event in Java
that set him on the intellectual path he followed for the rest of his life. On several
occasions in 1840, when he let blood from sailors in an East Java port, Mayer
noticed that venous blood had a surprisingly bright red color. He surmised that
this unusual redness of blood in the tropics indicated a slower rate of metabolic
oxidation. He became convinced that oxidation of food materials produced heat
internally and maintained a constant body temperature. In a warm climate, he
reasoned, the oxidation rate was reduced.
For those of us who are inclined toward the romantic view that theoreticians
make their most inspired advances in intuitive leaps, this story and the sequel
are fascinating. Mayer’s assumed connection between blood color and metabolic
oxidation rate was certainly oversimplified and partly wrong, but this germ of a
theory brought an intellectual excitement and stimulation Mayer had never before
experienced. It did not take him long to see his discovery as much more
than a new medical fact: metabolic oxidation was a physiological conversion
process in which heat was produced from food materials, a chemical effect producing
a thermal effect. Mayer was convinced that the chemical effect and the
thermal effect were somehow related; to use the terminology he adopted to express
his theory, the chemical reaction was a “force” that changed its form but
not its magnitude in the metabolic process. And most important in Mayer’s view,
this interpretation of metabolic oxidation was just one instance of a general
principle.

On the Dark Side Robert Mayer

Something Is Conserved
To the modern student, the term energy has a meaning that is almost self-evident.
This meaning was far from clear, however, to scientists of the early nineteenth
century. The many effects that would finally be unified by the concept of energy
were still seen mostly as diverse phenomena. It was suspected that mechanical,
thermal, chemical, electrical, and magnetic effects had something in common,
but the connections were incomplete and confused.
What was most obvious by the 1820s and 1830s was that strikingly diverse
effects were interconvertible. Alessandro Volta’s electric cell, invented in 1800,
produced electrical effects from chemical effects. In 1820, Hans Christian Oersted
observed magnetic effects produced by electrical effects. Magnetism produces
motion (mechanical effects), and for many years it had been known that motion
can produce electrical effects through friction. This sequence is a chain of “conversions”:
Chemical effect electrical effect magnetic effect mechanical effect
electrical effect.
In 1822, Thomas Seebeckdemonstrated that a bimetallic junction produces an
electrical effect when heated, and twelve years later Jean Peltier reported the
reverse conversion: cooling produced by an electrical effect. Heat engines perform
as conversion devices, converting a thermal effect (heat) into a mechanical
effect (work).
Most of the major theories of science have been discovered by one scientist,
or at most by a few. The search for broad theoretical unities tends to be difficult,
solitary work, and important scientific discoveries are usually subtle enough that
special kinds of genius are needed to recognize and develop them. But, as Thomas
Kuhn points out, there is at least one prominent exception to this rule. The
theoretical studies inspired by the discoveries of conversion processes, whichenergy concept, were far from a singular effort. Kuhn lists
twelve scientists who contributed importantly during the early stages of this “simultaneous
discovery.”
The idea that occurred to all twelve—not quite simultaneously, but independently—
was that conversion was somehow linked with conservation. When one
effect was converted to another, some measure of the first effect was quantitatively
replaced by the same kind of measure of the second. This measure, applicable
to all the various interconvertible effects, was conserved: throughout a conversion
process its total amount, whether it assessed one effect, the other effect,
or both, was precisely constant.
The twelve simultaneous discoverers were not the first to make important use
of a conservation principle. In one form or another, conservation principles had
been popular, almost intuitive it seems, with scientists for many years. Theorists
had counted among their most impressive achievements discoveries of quantities
that were both indestructible and uncreatable. Adherents of the caloric theory of
heat had postulated conservation of heat. In the late eighteenth century, Antoine-
Laurent Lavoisier and others had established that mass is conserved in chemical
reactions; when a chemical reaction proceeds in a closed container, there is no
change in total mass.
So it was natural for theorists who studied conversion processes to attempt to
build their theories from a conservation law. But, as always in the formulation
of a conservation principle, a difficult question had to be asked at the outset:
what is the quantity conserved? As it turned out, a workable answer to this
question was practically impossible without some knowledge of the conservation
law itself, because the most obvious property of the conserved quantity, ultimately
identified as energy, was that it was conserved. No direct measurement
like that of mass could be made for verification of the conservation property. This
was a search for something that could not be fully defined until it was actually
found.

Recognition

So, in the end, Sadi Carnot’s theory was resurrected, understood, and used. And
it finally became clear that Carnot, no less than his father Lazare, should be
celebrated as a great revolutionary. Born into a political revolution, Carnot started
a scientific revolution. His theory was radically new and completely original.
None of Carnot’s predecessors had exploited, or even hinted at, the idea that heat
fall was the universal driving force of heat engines.
If Carnot’s contemporaries lacked the vision to appreciate his work, his numerous
successors have, at least for posterity, repaired the damage of neglect.
Science historians now regard Carnot as one of the most inventive of scientists.
In his history of thermodynamics, From Watt to Clausius, Donald Cardwell assesses
for us Sadi Carnot’s astonishing success in achieving Lazare Carnot’s grand
goal, the abstraction of general physical principles from the complexities of machinery:
“Perhaps one of the truest indicators of Carnot’s greatness is the unerring
skill with which he abstracted, from the highly complicated mechanical contrivance
that was the steam engine . . . the essentials, and the essentials alone, of
his argument. Nothing unnecessary is included, and nothing essential is missed
out. It is, in fact, very difficult to thinkof a more efficient piece of abstraction in
the history of science since Galileo taught . . . the basis of the procedure.”Scant records of Carnot’s life and personality remain. In the two published
portraits, we see a sensitive, intelligent face, with large eyes regarding us with a
steady, slightly melancholy gaze. Most of the biographical material on Carnot
comes from a brief article written by Sadi’s brother Hippolyte. (Lazare Carnot
was partial to exotic names for his sons.) Hippolyte’s anecdotes tell of Carnot’s
independence and courage, even in childhood. As a youngster, he sometimes
accompanied his father on visits to Napoleon’s residence; while Lazare and Bonaparte
conducted business, Sadi was put in the care of Madame Bonaparte. On
one occasion, she and other ladies were amusing themselves in a rowboat on a
pond when Bonaparte appeared and splashed water on the rowers by throwing
stones near the boat. Sadi, about four years old at the time, watched for a while,
then indignantly confronted Bonaparte, called him “beast of a First Consul,” and
demanded that he desist. Bonaparte stared in astonishment at his tiny attacker,
and then roared with laughter.
The child who challenged Napoleon later entered the E´ cole Polytechnique at
about the same time the French military fortunes began to collapse. Two years
later Napoleon was in full retreat, and France was invaded. Hippolyte relates that
Sadi could not remain idle. He petitioned Napoleon for permission to form a
brigade to fight in defense of Paris. The students fought bravely at Vincennes,
but Paris fell to the Allied armies, and Napoleon was forced to abdicate.
Hippolyte records one more instance of his brother’s courage. Sadi was walking
in Paris one day when a mounted drunken soldier galloped down the street,
“brandishing his saber and striking down passers-by.” Sadi ran forward, dodged
the sword and the horse, grabbed the soldier, and “laid him in the gutter.” Sadi
then “continued on his way to escape from the cheers of the crowd, amazed at
this daring deed.”
Sadi Carnot lived in a time of unsurpassed scientific activity, most of it centered
in Paris. The list of renowned physicists, mathematicians, chemists, and
engineers who worked in Paris during Carnot’s lifetime includes Pierre-Simon
Laplace, Andre´-Marie Ampe`re, Augustin Fresnel, Sime´on-Denis Poisson, Adrien-
Marie Legendre, Pierre Dulong, Alexis Petit, Evariste Galois, and Gaspard de
Coriolis. Many of these names appeared on the roll of the faculty and students
at the E´ cole Polytechnique, where Carnot received his scientific training. Except
as a student, Carnot was never part of this distinguished company. Like some
other incomparable geniuses in the history of science (notably, Gibbs, Joule, and
Mayer in our story), Carnot did his important workas a scientific outsider. But
there is no doubt that Carnot’s name belongs on anyone’s list of great French
physicists. He may have been the greatest of them all.

After Carnot

The man who rescued Carnot’s workfrom what certainly would otherwise have
been oblivion was E´ mile Clapeyron, a former classmate of Carnot’s at the E´ cole
Polytechnique. It was Clapeyron who, in a paper published in the Journal de
l’E´ cole Polytechnique in 1834, put Carnot’s message in the acceptable language
of mathematical analysis. Most important, Clapeyron translated into differential
equations Carnot’s several verbal accounts of how to calculate his efficiency function
F(t).
Clapeyron’s paper was translated into German and English, and for ten years
or so it was the only linkbetween Carnot and his followers. Carnot’s theory, in
the mathematical translation provided by Clapeyron, was to become the point of
departure in the 1840s and early 1850s for two second-generation thermodynamicists,
a young German student at the University of Halle, Rudolf Clausius,
and a recent graduate of Cambridge University, William Thomson (who became
Lord Kelvin). Thomson spent several months in 1845 in the Paris laboratory of
Victor Regnault. He scoured the Paris bookshops for a copy of Carnot’s memoir
with no success. No one remembered either the bookor its author.
In different ways, Clausius and Thomson were to extend Carnot’s workinto
the science of heat that Thomson eventually called thermodynamics. One of Clapeyron’s
differential equations became a fixture in Thomson’s approach to thermodynamics;
Thomson found a way to use the equation to define an absolute
temperature scale. Later, he introduced the concept of energy, and with it resolved
a basic flaw in Carnot’s theory: its apparent reliance on the caloric theory.
Among Clausius’s contributions was an elaboration of Carnot’s heat engine analysis,
which recognized that heat is not only dropped in the heat engine from a
high temperature to a low temperature but is also partially converted to work.
This was a departure from Carnot’s water engine analogy, and in later research
it led to the concept of entropy.

Publication and Neglect

Sadi Carnot’s workwas presented as a privately published memoir in 1824, one
year after Lazare Carnot’s death, and it met a strange fate. The memoir was published
by a leading scientific publisher, favorably reviewed, mentioned in an
important journal—and then for more than twenty years all but forgotten. With
one fortunate exception, none of France’s esteemed company of engineers and
physicists paid any further attention to Carnot’s memoir.
One can only speculate concerning the reasons for this neglect. Perhaps Carnot’s
immediate audience did not appreciate his scientific writing style. Like his
father, whose scientific workwas also ignored at first, Carnot wrote in a semipopular
style. He rarely used mathematical equations, and these were usually
relegated to footnotes; most of his arguments were stated verbally. Evidently Carnot,
like his father, was writing for engineers, but his book was still too theoretical
for the steam-engine engineers who should have read it. Others of the scientific
establishment, looking for the analytical mathematical language commonly used
at the time in treatises on mechanics, probably could not take seriously this
unknown youth who insisted on using verbal science to formulate his arguments.
It didn’t help either that Carnot was personally reserved and wary of publicity
of any kind. One of his rules of conduct was, “Say little about what you know
and nothing at all about what you don’t know.” In the end, like Newton with the
Principia, Carnot missed his audience.
In time, Carnot probably would have seen his workrecognized, if not in
France, perhaps elsewhere where theoretical research on heat and heat engines
was more active. But Carnot never had the opportunity to wait for the scientific
world to catch up. In 1831, he contracted scarlet fever, which developed into
“brain fever.” He partially recovered and went to the country for convalescence.
But later, in 1832, while studying the effects of a cholera epidemic, he became a
cholera victim himself. The disease killed him in hours; he was thirty-six years
old. Most of his papers and other effects were destroyed at the time of his death,
the customary precaution following a cholera casualty.

Carnot’s Function

To continue with his analysis, Carnot had to deduce what he could concerning
the physical and mathematical nature of ideal engine operation. Here he seems
to have exploited further his idea that heat engines do workby dropping heat
from a higher to a lower temperature. It seemed that the ability of heat to do
workin a heat engine depended on its thermal level expressed by the temperature
t, just as the ability of water to do workin a water engine depends on its
gravitational level.
Carnot emphasized a function F(t) that expressed the ideal heat engine’s operating
efficiency at the temperature t. He made three remarkable calculations of
numerical values for his function F(t). These calculations were based on three different heat engine designs that used air, boiling water, and boiling alcohol as
the working materials. Carnot’s theory required that ideal heat engine behavior
be entirely independent of the nature of the working material and other special
design features: values obtained for F(t) in the three cases had to be dependent
only on the temperature t. Although the primitive data available to Carnot for
the calculation limited the accuracy, his results for F(t) seemed to satisfy this
requirement. No doubt this success helped convince Carnot that his heat engine
theory was fundamentally correct.
To complete his theory, Carnot had to find not just numbers but a mathematical
expression for his function F(t). In this effort, he was unsuccessful; he could see
only that F(t) decreased with increasing temperature. Many of Carnot’s successors
also became fascinated with this problem. Although in the end Carnot’s
function was found to be nothing more complicated than the reciprocal of the
temperature expressed on an absolute scale, it tookno fewer than eight thermodynamicists,
spanning two generations, to establish this conclusion unequivocally;
five of them (Carnot, Clausius, Joule, Helmholtz, and Thomson) were major
figures in nineteenth-century physics.