IHTC-16 Fourier Lecture
HISTORICAL DEVELOPMENT IN THE THOUGHT
OF THERMAL SCIENCE—HEAT AND ENTROPYPREFACE AND CHAPTERS 18‒20(250 YEARS AFTER JAMES WATT AND
200 YEARS AFTER SADI CARNOT)
Yoshitaka Yamamoto
Surugadai Gakuen, Tokyo, Japan
English Translation & Presentation
Hideo Yoshida
Kyoto University, Kyoto, Japan53

Yoshitaka Yamamoto (1941‒)

52

Historian of Science, Natural Philosopher, Educator
1964:

BS, Department of Physics, The Univ. of Tokyoalso studied at YukawaInst. for Theoretical Physics, Kyoto Univ.

1968‒9: Representative of The Univ. of Tokyo
All-Campus Joint Struggle League
https://www.youtube.com/watch?v=WBVXMXx_p_Y
https://www.youtube.com/watch?v=WBVXMXx_p_Y

1976‒ : Sundai (Surugadai) Preparatory SchoolYamamoto’s Literary Works of “Science”

51

Original Works- Gravity and Mechanical World (1981)- Formation of Classical Mechanics: From Newton to Lagrange (1997)- Analytical Mechanics (1998)- The Pull of History—Human Understanding of Magnetism and Gravity through the Ages, 1-3(2003 in Japanese, 2018 in English)- Cultural Revolution in the 16th Century, 1-2 (2007)- On Fukushima Nuclear Accident: What I learned and considered (2011)- Conversion of World View, 1-3 (2014)- Canonical Theory of Geometrical Optics (2014)- Atom, Nucleus and Nuclear power: What I wanted to tell young students (2015)- 150 Years of Modern Japan: Bankruptcy of All-out War Based on Science and Technology (2018）- Discovery of Decimal and Logarithm (2018)

Ernst Cassirer (Japanese translation)- Zur Einsteinschen Relativitätstheorie. Erkenntnistheoretische Betrachtungen (1976)- Substanzbegriff und Funktionsbegriff. Untersuchungen über die Grundfragen der Erkenntniskritik(1979)- Das Erkenntnisproblem in der Philosophie und Wissenschaft der neueren Zeit 4 (1996)

Niels Bohr (Japanese translation)- Niels Bohr Collected Works, 1-2 (1999, 2000)

The Pull of History—Human Understanding of Magnetism
and Gravity through the Ages

50

Original in Japanese (2003)
English translation (2018)
 Part 1. Antiquity and the Middle Ages
Chapters 1‒8
 Part 2. Renaissance
Chapters 9‒16
 Part 3. The Dawn of the Modern Age
Chapters 17‒22https://www.worldscientific.com/worldscibooks/10.1142/10540

Historical Development in the Thought of Thermal Science—Heat and Entropy (1986, 2009)
Preface
Part 1 Materialism and Dynamical Reductionism
Chapter 1‒6
Part 2 Formation of Caloric Theory
Chapter 7‒12
Part 3 Calorimetry and Conservation of Heat
Chapter 13‒17
Part 4 Motive Power of Heat—Carnot and Joule
Chapter 18‒20‒24
Part 5 Proposal of the Principles of Thermodynamics
Chapter 25‒29
Part 6 Energy and Entropy
Chapter 30‒34
Postscript49

3 volumes,
more than
1000 pages

1600

1700

1800

1900

from Galilei
to Nernst

Separate condenser
1769: 249 years ago

James Watt
1736‒1819“250 Years after James Watt and
200 Years after Sadi Carnot”
Carnot’s theorem
1824: 194 years ago

Sadi Carnot
1796‒1832

1600

1700

1800

48

47

Contents of Original Book
Chap. 18 Setting of a New Problem—“Motive Power” of Heat
—Carnot and Watt

I.
Neglect of Carnotʼs paper during his lifetime
II.
Carnotʼs premise and problem setting
III. Caloric theory and cosmology based on heat
IV. Carnotʼs view of nature and of society based on heat
V.
Develop. of steam engine and its defects before Watt
VI. Wattʼs improvement—separate condenser
VII. Wattʼs improvement—expansive principle
VIII. P‒V diagram and expression for work (Watt)
IX. High-pressure engine and its development in France

46

Chap. 19 Theory of Ideal Heat Engine—Carnotʼs Theorem
I.
II.
III.
IV.

Purpose of Carnotʼs paper
Carnotʼs preliminary theorem and its background(the first and second premises)
Carnot cycle
Carnotʼs theorem (the third premise)

Chap. 20 Structure of Carnotʼs Theory and its Extension—The Start of Thermodynamics
I.
II.
III.
IV.
V.

Premises of Carnotʼs theory (premises A, B, and C)
Analytical expression for Carnotʼs theorem (expressions I and II)
Experimental determination of Carnot function
Extension of Carnotʼs theory and gas theorem (gas theorems 1, 2, and 3)
Further discussion and significance of thermodynamics

Contents of This Lecture45
46

1. Development of Steam Engine in Englandand its Impact on France ..........................................44
2. Sadi Carnot and Neglect of his Paper
during his Lifetime .................................................... 33
3. Carnot’s Theorem ...................................................... 22
4. Structure of Carnotʼs Theory and its Extension...... 8

1. Development of Steam Engine in Englandand its Impact on France

Matthew Boulton

James Watt

44

Effective means of removing and lifting spring waterfrom mines 16–17th century
Increased demand for metals
owing to the modernization of weapons
Development of a money economy
Many lifting pumps in“De re Metallica”
by Agricola (German)“The Various and Ingenious Machines”
by Ramelli (Italy)
Also, in England“the fifty-five patents for inventions granted during
the reign of Elizabeth, 1561-99, one in seven is forthe raising of water” [Dickinson, 1938]

43

Denis Papin(1647–1713)(France)
Piston steam
engine (1690)

A cylinder filled withwater was heated, anda piston lifted by steam
was then cooled by water.

Thomas Savery (ca.1650–1715)

42

 In England, water drainage was particular criticalproblem Thomas Savery (later, & Thomas Newcomen)in the southwest of England developed an actual
water-pumping machine by using steam

Outside of P1/P2 is cooled by water

Defects
 not work in principle more than
10m column of water, practical limitations owing to
the immature high-pressuretechnology
 the low efficiency associatedwith cooling from the outside of
the container could not practically contributeThomas Newcomen (1664–1729)
 On the basis of Papinʼs idea, Thomas Newcomen separated
the cylinder and boiler.
 Newcomenʼs practically useful engine (around 1712) have revitalized
the coal mines in England“There is no known image of Thomas Newcomen”
http://www.newcomen.com/about-thesociety/thomas-newcomen/

MISCONCEPTION

41

Newcomen’s steam engine & Smeaton’s improvement

40

John Smeaton(1724‒1792)“father of civil engineering”

 When the cylinder is full of steam, cold water issprayed into it so that a vacuum is formed.
 At that time, it was understood that the vacuum inside
the cylinder jacks up the piston, rather than that
atmospheric pressure pushes down the piston.

 also, tried to improve
its efficiency
 but, no essential
improvements until
those by Watt a halfcentury later.

James Watt—Philosophical instrumentmakerto the University of Glasgow since 1757

39

 Analyticalbalance Latent heat Specific heat
Joseph Black(1728‒1799)

John Robison(1739–1805)

James Watt
(1736‒1819)

 The fact that in the university scientists and a craftsman could have discussions
and cooperate was due to not only the intellectual atmosphere in Glasgow but
also Wattʼs own temperament.
 Watt (1809) “Every thing I learnt from him (=Black) was in conversation and by
doing small mechanical jobs for him. These Conversations and those I had withyou (=Robison) served to give me true notions in Science.” Robison (1758) “I saw a workman and expected no more:
I found a philosopher.”Watt encountered Newcomenʼs engine by chance38“in the Winter of 1763, having occasion to repair a Model of Newcomenʼs Engine
which belonged to the Natural Philosophy class of the University of Glasgow;”“I found that the Boiler, though large in proportion to the Cylinder could notsupply it with Steam to work at a proper rate unless the fire was violently urged
with bellows.”
A the net amount of steam required for the operation
V the volume of the cylinder∆A an amount of steam
By observing the
S the inner surface area of the cylinder
miniature steam engine
model, the essential
ΔA aS a∝∝defect of NewcomenʼslAV
engine was brought tol the characteristic length of the model
the fore.
a is a constant with the dimension of length→ the ratio increases with decreasing size of the modelSeparate condenser (1769: 249 years ago) To reduce fuel consumption: the cylinder at a high temperature
 for high mechanical performance: effective cooling
Watt found a method to simultaneously satisfy both requirements by spatiallyand substantially separating the expansion process and contraction process.

Although this is very simple from the present perspective, it was“the greatest single improvement ever made in the engine.” [Dickinson, 1938]

37

36

Boulton & Watt
Royalty: 1/3 of fuel saved
Business partner
Matthew Boulton(1728–1809)

 In 1782, Watt invented a double acting engine,
operated by steam in both the upward anddownward directions. A 100% steam engine,
whose operation was independent of
atmospheric pressure and gravity, had beendeveloped.
 “Watt & Boulton decided that their engine should be paid for by a royalty, oras they termed it a ʻpremiumʼ, based very appropriately on the saving in fueleffected by the engine as compared with the consumption of a common
engine (Newcomen) doing the same work. Boulton & Watt stipulated that theyshould receive one-third of the value of the fuel saved.” [Dickinson, 1938]

1800 Expiration of Watt’s patent
1802 Trevithick: High-press. engine
1807 Fulton: Steamboat

Advance in steam engine

35

1814 Carnot: graduated from École Poly.
1815 Napoleonic Wars ended

1814 Woolf: Compound engine“From the very start the Woolf compound engine ... which made use of highpressure to operate two cylinders
alternately and offered a fuel economy
over the Watt engine of about 50 per centfound its greatest market in France.”[Landes, 1965]

Railway & its impact
on France (1820‒30s)
GeorgeStephenson(1781–1848)34

1825 Stockton-Darlington“By them, he turns each dropof water into a reservoir ofsteam, .....Is there any thingwhich gives a higher idea ofthe power of man, than thesteam-engine under the form
Saintin which it is applied toSimonianism produce motion on railroads?”
Michel Chevalier(1836)(1806‒79)

Marc Seguin(1786–1875)“ just there a drop of water turned into steam
acts to supplement his weakness, to create forhim a power of which we cannot now, nor yet fora long time to come, measure the extent. From
now on with the help of this agent ..... ” (1839)

In western Europe, which rushed headlong toward a highly industrializedsociety, the predominance of human beings over nature was first recognized;
this presumptuous philosophy was based on the development of steam engines.

2. Sadi Carnot and Neglect of
his Paper during his Lifetime33

In 1832, Galois & Carnot—The light of science from Paris
France experienced successive early deaths of
two great geniuses in mathematics and physics

https://www.youtube.com/watch?v=WBVXMXx_p_Y
https://www.youtube.com/watch?v=WBVXMXx_p_Y

Évariste Galois(October 25, 1811‒May 31, 1832)
20 years old, killed by duel

Nicolas Léonard Sadi Carnot(June 1, 1796‒August 24, 1832)
36 years old, died of cholera

The driving force of their work was the French intellectual atmosphere of thatage, as if“the whole of Europe was illuminated by the light of science which emanatedfrom Paris during the first third of this century.” [Merz, 1904]

32

Carnot Family

31
https://fr.wikipedia.org/wiki/Sadi_Carnot_(physicien)“the Organizer of Victory”
in the French Rev. Wars

“He had such a repugnance to bringing himself
forward that, in his intimate conversations with a few
friends, he kept them ignorant of the treasures ofscience which he had accumulated.” [Hippolyte]
LazareHippolyte

Despite his reticence, however, he was a quarter ofa century ahead of the physicists at that time.

Sadi Carnot’s three manuscripts

30

Graduated from the École Polytechnique in Paris (1814).
 “Réflexions sur la Puissance Motrice du Feu(Reflections on the Motive Power of Fire)” (1824)
 “Notes sur les Mathématiques, la Physique et autres
sujets” (belatedly 1878)
 “a draft of the Réflexions” (discovered in 1964)
In physics, if a problem has been properly defined, its solution is generally
near at hand.
In this sense, it is no exaggeration to say thatthe birth and consolidation of thermodynamics and its foundations were almost
entirely due to a single paper: the Réflexions

Carnot’s motivation and problem setting

29

 “If you were now to deprive England of her steam engines, you would
deprive her of both coal and iron; you would cut off the sources of all her
wealth, totally destroy her means of prosperity, and reduce this nation ofhuge power to insignificance.” First, “The question whether the motive power of heat is limited or whether
it is boundless has been frequently discussed. Can we set a limit to theimprovement of the heat engine, a limit which, by the very nature of things,
cannot in any way be surpassed? Or, conversely, is it possible for theprocess of improvement to go on indefinitely?” Second, “is the motive power of heat fixed in quantity, or does it vary withthe working substance that is used? Does it vary with the intermediarymaterial that we subject to the action of heat?”Carnot’s radicalism at the age of just 28—théorie complète28

 “The phenomenon of the production of motion by heat has not been treatedfrom a sufficiently general point of view.”
 “In order to grasp in a completely general way the principle governing the
production of motion by heat, it is necessary to consider the problem
independently of any mechanism or any particular working substance.
Arguments have to be established that apply not only to steam engines but also
to any conceivable heat engine, whatever working substance is used andwhatever operations this working substance is made to perform.” “but such a theory is plainly lacking in the case of the heat engine. It will only beachieved when the laws of physics are sufficiently extensive and general for us
to be able to predict all the effects that are produced when heat acts in aparticular way on any given substance.”

27
Possibly the most astonishing fact about Sadi Carnotʼs work is that it wasreceived in complete silence by the world of French science. [Mendoza, 1959]“A few days after its publication, Carnotʼs book was formally presented to the Académie desSciences; It was presented by Girard, a prominent engineer. Among the Academicians present
were:

Arago

Fourier

LaplaceAmpère Gay-Lussac Poinsot

FresnelPoissonCauchyProny

X

mistakenly
usedfor nearly
200 years

Legendre

DulongNavierIn fact, Carnotʼs book made hardly a ripple on the surface of the main stream of science.”[Cardwell, 1971]

Not principal but subsidiary reason of neglect

26

 Main stream
traditionally interested in pure mathematical science—
DʼAlembert, Laplace, and Lagrange, Relatively minor stream
based on practical engineering—Lazare and Sadi Carnot
Exception of non-neglect of the Réflexions
a manuscript book of notes taken by a student who attend the course of lectures
given by Clément at the Conservatoire des Arts et Métiers in 1824‒25 and lateryearsPrincipal reasons of neglect and its significance25

 A major issue not only for Carnot but also for all the researchers andengineers of steam engines—Why high-pressure engines were superior to
low-pressure engines.
 Only Carnot could thoroughly investigate this issue from the principles, anda single paper of Carnot provided the basis for thermodynamics.
 Carnotʼs work was originally stimulated by a technical issue that was not ofinterest to physicists.
 Simultaneously, its approach was so fundamental and theoretical that itwas not understood by engineers, and was therefore neglected.
“The most original study in physical science” [Larmor, 1916]
 Note that the year of 1824 when Carnot published the Réflexions was one
year after the completion of analytical calorimetry by Laplace and Poisson:
the heyday of caloric theory.

Excavation by Émile Clapeyron (1799‒1864)24

 Graduated from the École Polytechnique in Paris (1818)(3 years younger than Carnot)
 In 1834, he took up the Réflexions and gave it graphical andmathematical expressions in Mémoire sur la Puissance Motricede la Chaleur (Memoir on the Motive Power of Heat) Clapeyronʼs paper first expressed the Carnot cycle as
an indicator diagram in the P‒V plane, and also pointed
out that the area enclosed by the cycle lines is equal to
the work.
 “S. Carnot, avoiding the use of mathematical analysis, arrives by a chain ofdifficult and elusive arguments at results which can be deduced easily from amore general law which I shall attempt to prove.” [Clapeyron]
The Réflexions, however, still continued to be neglected.

23

William Thomson (born in 1824, later, Lord Kelvin)
The Réflexions finally reached a reader with a staturecomparable to its author in 1848, 24 years after its publication.
“The perusal of this mémoire incited Thomson to refer to the original tract ofCarnot. In vain did he inquire for it in the Library of the Collège de France. No one
could tell him even where a copy might be seen. But a copy he must have, even if
he searched all Paris to find it. ......
I (= Thomson) went to every book-shop I could think of, asking for the Puissancemotrice du feu, by Carnot. “Caino? Je ne connais pas cet auteur.” With muchdifficulty I managed to explain that it was “r” not “i” I meant. “Ah! Ca-rrr-not! Oui,voici son ouvrage,” producing a volume on some social question by Hippolyte
Carnot; but the Puissance motrice du feu was quite unknown.
....... Not until the end of the year 1848 did he (= Thomson) see the book, a copy ofit being sent by Professor Lewis Gordon.”22

3. Carnot’s Theorem

Carnot starts
from here

Expansive principle—Watt & Carnot

21

In Watt’s patent of 1782, one more improvement, which is very important in termsof thermodynamics, was based on
 by adiabatic expansion of the steam, the steam will be gradually cooled simultaneously carrying out work with the energy contained in the steam

Carnot appreciated this ‘expansive principle’“Watt, who was responsible for nearly all the main
improvements of the steam engine and who so perfected theengine that further advances would now be difficult, was also
the first to use steam at progressively diminishing pressures.”“The mark of a good steam engine, therefore, must be not onlythat it uses steam at a high pressure but also that it uses it atpressures which are not constant but which vary substantiallyfrom one moment to the next and progressively decrease.”Watt’s recognition later theorized by Carnot

20

First, a heat engine consists of three elements, a furnace, cooler, and working
substance; the working substance can freely be separated from and forced tocome in contact with the furnace and cooler.
Second, to perform work, not only a high temperature but also a low temperature is necessary. That is, heat is carried by the working substance from a hightemperature region (furnace) to a low-temperature region (cooler).
Third, work is performed by the volume change of the working substance. Thatis, the existence of a high temperature and a low temperature is a necessary but
nor sufficient condition for performing work; to perform work, the potential
power of the high-temperature working substance should be extracted byexpanding its volume.

Carnotʼs premise—conservation of heat19

 “We think it unnecessary to explain here what is meant by the termsʻamount of caloric (calorique)ʼ or ʻamount of heat (chaleur)ʼ (we use the twoterms interchangeably) or to describe how to measure these quantities
with a calorimeter. Nor shall we explain the meaning of latent heat,temperature, specific heat. etc. The reader should be familiar with theseterms from elementary treatises on physics or chemistry.” “In our proof, we make the implicit assumption that when a body has
undergone its various changes and after passing through a number ofstages, has returned precisely to its original state (its state here being
defined in terms of its density, temperature, and mode of aggregation), it
contains the same quantity of heat as it did at the start.”

dq
=
0
∫

Carnot’s motivation and problem setting (same with 29 )18

 “If you were now to deprive England of her steam engines, you would
deprive her of both coal and iron; you would cut off the sources of all her
wealth, totally destroy her means of prosperity, and reduce this nation ofhuge power to insignificance.” First, “The question whether the motive power of heat is limited or whether
it is boundless has been frequently discussed. Can we set a limit to theimprovement of the heat engine, a limit which, by the very nature of things,
cannot in any way be surpassed? Or, conversely, is it possible for theprocess of improvement to go on indefinitely?” Second, “is the motive power of heat fixed in quantity, or does it vary withthe working substance that is used? Does it vary with the intermediarymaterial that we subject to the action of heat?”17

Motive power (work)
At that time, however, those who recognized
the importance of motive power (work) andhad grasped its concept clearly wereresearchers of applied mechanics and engineers
rather thanthose of mathematical physics.

Lazare Carnot Hachette

NavierPoncelet

1 [hp]≡ 180[lb]×2π×12[ｆｔ]×144/3600[s]= 550 [ft・lb/s] = 0.75 [kW]WattCarnotʼs preliminary theorem (1st)16“Wherever there is a difference in temperature, motive power can be produced.”
This can be summarized to mean that a temperature difference is a necessary
condition to produce motive power. The detailed description is as follows:“So the production of motive power in a steam engine is due not to an actual
consumption of caloric but to its passage from a hot body to a cold one. It isdue, in other words, to a restoration of the equilibrium of caloric after thatequilibrium has somehow been disturbed, for example by a chemical reaction
such as combustion, or by some other means.”
The discovery that not only a high temperature but also a low temperature isnecessary was Carnotʼs breakthrough.

Carnotʼs preliminary theorem (2nd)15“Obviously heat can only be a source of motion in so far as it causes
substances to undergo changes in volume or shape.”
This can be summarized to mean that a change in the volume of substances is
an indispensable condition for producing motive power.
“For there is nothing in nature that does not undergo changes in volume,
contracting and then expanding as it experiences cold or heat; there is nothingwhich, in doing so, cannot act against a resisting force and there by develop
motive power.”This is precisely the theoretical specification of Wattʼs expansive principle.
“Hence the necessary condition for the achievement of maximum effect is thatthe bodies used to produce motive power should undergo no change intemperature that is not due to a change in volume. ”

Analogy with current of water (1)14“From the ideas that have been established so far,
we are sufficiently justified in comparing the motive
power of heat with that of a fall of water. ...... The
motive power of a fall of water depends on its
height and on the amount of liquid. The motive
power of heat likewise depends on the amount ofcaloric that is used and on what might be termed—
in fact on what we shall call—the height of its fall; it
depends, in other words, on the difference intemperature of the bodies between which thepassage of caloric occurs”
Column-of-water engine
by modeling Newcomenʼs engine, a piston is operated by using hydro pressure
instead of steam pressureAnalogy with current of water (2)13

Lazare Carnot: “Essay upon Machines in General” (1782)“We may conclude from this, for example, that the method of producing thegreatest possible effect in a hydraulic machine moved by a current of water, is
not to adapt a wheel to it, the wings of which receive the shock of the fluid. In
fact, two good reasons prevent us from producing in this way the greatest
effects:
the first is, as we have already said, because it is essential to avoid every kindof percussion whatever;the second is, because after the shock of the fluid there is still a velocity which
remains to it as a pure loss, since we should be able to employ this remainder
in still producing a new effect to be added to the first.
We can see the remarkable analogy between an ideal hydraulic machine
which changes its state as slowly as possible and an ideal heat engine
which operates through a quasi-static process.

Not single process but cycle

12

He was wise enough to consider not only a single
process but also an entire cycle to clarify the work
done by heat; at that time, all other scientists andengineers considered only the expansion process
directly producing work.
The most important feature of the cycle
When the working substance has returned to the initial state,
the heat added at the furnace—and the heat exhausted to thecooler in a Carnot cycle—is considered to be completely used
for work to the outside. If the state has not returned to theinitial state, however, part of the heat may be used to induce aphysical internal change of the working substance, orconversely, the physical internal change of the working
substance may be used to produce work.

On the basis of his preliminary theorem, the idealengine must consist of isothermal and adiabatic
processes.

Carnot cycle starts from not A but A0

11

Explanation by Clapeyron:isothermal expansion: A → B
adiabatic expansion: B → C
isothermal compression: C → D
adiabatic compression: D → A
 it might be considered that the existence of theheat function Q(θ ,V ) , i.e., the conservation ofheat, is necessary to close the cycle.
 however, this is not always the case.
 Even if the sum of the heat absorbed
Explanation by Carnot:
during processes A→ A0→B and the heat isothermal expansion: A0 → B
discharged to the cooler in process C→D adiabatic expansion: B → C
are different, this cycle is closed.
 isothermal compression: C → D
 This enabled the Carnot cycle to play a key
 adiabatic compression: D → A
role in the development of thermal science.
 isothermal expansion: A → A0

Note: assumption and limited knowledge at the time (1824)10 Upon reading Carnotʼs original paper after studying thermodynamics, it may
be considered that his description of the above five processes is somewhat
convoluted as well as strange; however, we should recognize that such a
description was necessary.
 If we know the P-V relationship for an adiabatic process—Poissonʼsformula PV γ = const. , we can determine point D as the point which can
adiabatically return to point A without using the relation (the sum of the heatabsorbed during processes A→B and the heat discharged to the cooler inprocess C→D are equal), as employed by Clapeyron. Thus Clausius
succeeded in removing the restriction of the conservation of heat.
 However, if we start from point D or B, the discussion is much simpler.

Carnotʼs theorem

9 “Whether the motive power of heat is limited
or whether it is boundless,”
 “is the motive power of heat fixed in quantity,
or does it vary with the working substancethat is used?”“The whole sequence of operations just described canbe reversed, and carried out in the opposite order.”By assuming that there is a cycle (referred to as C') between two temperatures
and whose efficiency is superior to the Carnot cycle C,“We should then have a case not only of perpetual motion but of motive power
being created in unlimited quantities without the consumption of caloric or of any
other agent. Creation of this kind completely contradicts prevailing ideas, thelaws of mechanics, and sound physics; it is inadmissible.”8

4. Structure of Carnotʼs Theory and its Extension
premises A, B, and C
Carnotʼs theorem(expressions I and II)
gas theorems 1, 2, and 3

Premises A, B, C

7

The basis of Carnotʼs heat theory is the same as that in the work of Laplace
and Poisson except that the taking temperature q and volume V are consideredas the variables; they are summarized by the following three premises:
Premise A: paradigm of specific heat and latent heatdq = CV dθ + ΛD dV

where

ΛD the latnt heat of expansion.

Premise B: existence of the equation of state −1
for most ordinary gases,α+θ()
P = P θ ,V
the Boyle‒Gay-Lussac law P = R
VPremise C: conservation of heat

Q(θ ,V ) = ∫(θ ,V )

(θ 0 ,V0 )dq

 In most of the books about the history of thermal science, Premise A andPremise C are inseparably linked to each other.
 Although this is historically true, it is not logically true.
 Premise A is not always associated with Premise C.

Analytical expression for Carnotʼs theorem

6

In general, the work for Carnot cycle is expressed as

W = W (q+ ,θ + ,θ − ,VA ,VB )Carnotʼs theorem: expression IW= Ψ (θ + ,θ − )q+
Ψ a universal function independent of the working substance
Carnotʼs theorem: expression II.
1Ψ (θ ,θ ′) 
lim 
≡

where Θ(θ ) : Carnot function
θ ′ →θ
 θ − θ ′  Θ(θ )ΔWΔθ= Ψ (θ ,θ − Δθ ) =
q+
Θ(θ )

The existence of this universal function is the key point of Carnotʼs theorem, and
it is the ultimate basis of thermodynamics.

Gas theorems

5

To confirm the correctness of a basic theory such as Carnotʼs theorem,
however, a direct proof is not necessarily required. On the contrary, its
extension, i.e., what is theoretically deduced from it, is more important.
Gas theorem 1“When a gas passes from a particular volume and pressure to anotherspecified volume and pressure, without undergoing a change in temperature,
the amount of caloric absorbed or given out is always the same, whatever thegas on which the observation is made.”
Gas theorem 2“When the volume of a gas changes at a constant temperature, the amounts ofheat absorbed or released by the gas will follow an arithmetical progression
when the increases or decreases in volume follow a geometrical progression.”
Gas theorem 3“The difference between the specific heat at constant pressure and the specificheat at constant volume is the same for all gases.”Most of Carnotʼs theory is correct4

 Clapeyron proved the gas theorems firmly on the basis of
the existence of the heat function. Consequently, after theconservation of heat was later rejected, these theorems
were often regarded as having been obtained as a result ofmultiple errors cancelling each other out.
 However, Premise A and Premise C are different; most of Carnotʼs theory
was obtained from Premise A, and thus Premise C is not necessary.
 Even if the conservation of heat does not hold, provided we consider an
infinitesimally small cycle,∫ dq = infinitesimal quantity of the second order ≅ 0

Thermodynamically correct results are often derived in practice.
 Most of Carnotʼs mistakes are based on erroneously measured data relating
to the pressure dependence of specific heat.

Experimental determinationof Carnot function

3

Carnot “In order to test our basic
proposition, to test whether the nature of
the working substance that is used fordeveloping motive power really has no
bearing on the amount of power produced,
we shall consider a number of substancesin turn. These substances are air, steam,and alcohol vapour.”Thomson “The complete theoreticalinvestigation of the motive, power of heatis thus reduced to the experimental
Clapeyron “The function C (Carnot function) determination of the coefficient µ ( = 1 Θ(θ ) );is, as we have seen, of great impor-tance: it and may be considered as perfect, when,is the common link between the phenomena by any series of experimental researchescaused by heat in solid bodies, liquids and. whatever, we can find a value of µ for
every temperature within practical limits.”
gases.”

2

5. Concluding remarks(from Preface of the original book)
The creation of thermodynamics was a giant step toward
understanding terrestrial phenomena. Looking back over the historyof thermal science, from the very beginning, it developed withthe aim of understanding the phenomena on earth, which is a uniqueenvironment for humanity and also a huge thermal engine.
Thermal science has retained its profound significance to this day,even though mainstream physics has shifted from macroscopic tomicroscopic subjects. The existence of humanity depends on theentropy balance of the globe.

1 Thank you very much for your kind attention.
 If you have any comments, please contact me by e-mail(sakura@hideoyoshida.com).
 I will be able to forward them to Mr. Yamamoto by post,
because he does not use any internet.
 Two PDF files of this paper (54 pages) and this presentation will be shortly
uploaded on http://www.aihtc.org/history.html