Substantia. An International Journal of the History of Chemistry 3(2): 73-96, 2019

Firenze University Press 
www.fupress.com/substantia

ISSN 1827-9643 (online) | DOI: 10.13128/Substantia-638

Citation: P.R. Salvi, V. Schettino 
(2019) Sadi Carnot’s Réflexions and 
the foundation of thermodynamics. 
Substantia 3(2): 73-96. doi: 10.13128/
Substantia-638

Copyright: © 2019 P.R. Salvi, V. 
Schettino. This is an open access, 
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Historical Article

Sadi Carnot’s Réflexions and the foundation of 
thermodynamics

Pier Remigio Salvi, Vincenzo Schettino

Dipartimento di Chimica “Ugo Schiff ”, Università di Firenze via della Lastruccia 3, 
50019 Sesto Fiorentino (FI), Italy

Abstract. The purpose of this article is to present a short review of Sadi Carnot work 
on heat engines and on the role his adherence to the caloric theory may have had. The 
essential points developed in the Réflexions are reviewed as forerunners of the science 
of thermodynamics. The antecedents that may have inspired the brilliant scientific 
insights of Carnot are reviewed together with the reception of the Carnot principles in 
the engineering and in the scientific community until the formulation of the two prin-
ciples of modern thermodynamics.

1. INTRODUCTION

In several cases new important scientific theories have been outlined 
starting from models or interpretative schemes that later developments have 
shown groundless or partially incorrect. The limits of the starting bases were 
overcome by the intuition or the imagination of the scientists. An example of 
this twisted way in the advancement of science is the discovery of the peri-
odic system of the elements by Dmitrij Ivanovich Mendeleev (1834-1907). In 
a meeting of the newly founded Russian Chemical Society (held on March 6, 
1869) Mendeleev presented his periodic table of the elements, later published 
in the journal of the Society [1] and in a German edition [2] and included in 
Mendeleev’s treatise Principles of Chemistry (1868-1870). Mendeleev arranged 
the 63 known elements in order of increasing atomic weight and the table 
showed the periodic recurrence of their physical and chemical properties, 
identifying group of elements with similar properties. The really innovative 
aspect of the table was in its heuristic power. In fact, in his ordering Men-
deleev was forced to leave empty places corresponding to unknown chemi-
cal elements whose physical and chemical properties were predicted by Men-
deleev. These unknown elements were actually discovered a few years later 
[3] and their properties were found to be in good agreement with Mendeleev 
predictions. Almost simultaneously a similar periodic table, including only 
28 elements, was published by Lothar Meyer [4].

Today we know that the ordering of the elements in the periodic table is 
based on the atomic number and that the chemical and physical properties 
of the elements depend on the electronic structure of the atom. Neverthe-
less, the general idea of Mendeleev’s periodic table has remained unchanged 



74 Pier Remigio Salvi, Vincenzo Schettino

surpassing, almost unscathed, the revolution of quan-
tum mechanics apart from the adaptations required to 
accommodate the numerous new elements discovered. 
Sadi Carnot’s contribution to the foundation of thermo-
dynamics can be analyzed along the same lines. In a his-
torical period in which the first and second principle of 
thermodynamics and the equivalence between heat and 
work had not yet been established Nicolas Léonard Sadi 
Carnot (1796-1832), in his famous booklet Réflexions sur 
la puissance motrice du feu et sur les machines propres à 
développer cette puissance [5], published in 1824, (Fig. 1), 
was able to arrive at a substantial definition of the sec-
ond principle, starting from the assumption of the calor-
ic theory which attributed a character of materiality to 
heat. Even though, in addition to the erroneous nature 
of the current heat theory, various aspects of gas prop-
erties, such as the pressure-volume relationship along 

adiabatic transformations or the specific heats of gases, 
were not completely defined at the time on the basis of 
available experiments and theories, Carnot, starting 
from the study of the general characteristics of the ther-
mal engines and the conditions for optimizing their per-
formances, succeeded in defining general principles that 
would open the way to the establishment of thermody-
namics as an autonomous science .

The work of Carnot and his Réflexions have been 
the subject of extensive and detailed studies (as it will 
be reported and discussed in the following) regarding, 
on the one hand, the original type of scientific reasoning 
underlying his conclusions and on the other the previ-
ous scientific knowledge and the later developments in 
thermodynamics. The aim of this work is to present a 
review of Carnot’s contribution to thermodynamics and 
the various possible interpretations of his work. After a 
brief biographical profile of Sadi Carnot and an over-
view of the theories of heat, the essential points of the 
Réflexions will be revisited and subsequently examined 
with reference to possible scientific backgrounds and to 
the subsequent reception of the Réflexions in the scien-
tific and engineering community.

2. BRIEF BIOGRAPHY OF SADI CARNOT

To better frame the work of Carnot in its histori-
cal context a brief biographical profile may be appropri-
ate. The first information we have on Sadi Carnot can 
be found in the note of one of his fellow students of the 
École Polytechnique, and probably his friend, Michel 
Chasles [6] and in the obituaries of Claude-Pierre Robe-
lin [7] and of Adolphe Gondinet [8], this latter reported 
by Pietro Redondi [9]. More substantial biographical 
information has been reported later by Paolo Ballada 
count of Saint-Robert, a Piedmontese engineer inter-
ested in thermal machines and industrialization pro-
cesses [10], based on a letter from the grandson Adolphe 
of Sadi Carnot [11]. A similar but more informative let-
ter dated 1878 from the brother Hippolyte Carnot has 
been reported by R.H. Thurston [12]. On the biography 
of Carnot, Birembaut has returned with new documen-
tation [13] noticing various inaccuracies in his brother’s 
story.

Sadi Carnot (Fig. 2) was born in Paris, June 1st, 
1796. His father, Lazare Carnot (1753-1823), was a lead-
ing political figure during and after the French Revolu-
tion, deserving the name of Organizer of Victory due to 
the military successes during the revolutionary period. 
He was also a great mathematician and physicist and a 
cultivated poet and in honor of the persian poet Sadi of 

Figure 1. Front page of the original work of Carnot.



75Sadi Carnot’s Réflexions and the foundation of thermodynamics

Shiraz he gave the name to his first son. The mother was 
a gifted pianist. Carnot had a very reserved character 
but, whenever necessary, he was able to show great ener-
gy and decision. This results already from an anecdote of 
his childhood as reported by the brother Hippolyte [12]. 
His father often brought Sadi with him. On one occa-
sion, when Napoleon Bonaparte enjoyed throwing rocks 
in the water to splash a group of ladies, including Mad-
ame Bonaparte, who were on a boat, the little Sadi did 
not hesitate to turn to the First Consul decisively: Beast 
of a First Consul, will you stop tormenting those ladies? 
The great tension of the moment vanished when Napo-
leon, followed by everyone present, burst into laughter. 
Sadi Carnot, after being initially instructed directly by 
his father Lazare [14], at age 16 was enrolled at the Poly-
technic School having as professors, among others, Pois-
son, Thenard, Arago, Petit and Dulong [15]. After gradu-
ation he was admitted in 1814 in the Artillery and Engi-
neering Application School of Metz as a cadet sub-lieu-
tenant. He entered the military career in April 1817 with 
typical duties like inspecting fortifications, proposing 
and reporting on engineering plans. In 1818 he applied 
successfully for a position in a newly formed engineering 
corps at the Army Headquarters in Paris. This allowed 

him to attend courses of mathematical sciences, natu-
ral history, industrial art and political economy held in 
the College of France and Sorbonne. He had also the 
opportunity, as it will be discussed in the following, to 
make acquaintance with Clément at the Conservatory 
of Arts and Crafts. In 1821 he visited his father in exile 
at Magdebourg. Back in Paris and after completing vari-
ous tasks as a military engineer he resigned as captain 
of the Military Engineering and developed a more direct 
interest in heat engines which led, in the following years, 
to publication of the Réflexions. In 1831 he resumed the 
study on the properties of gaseous substances encour-
aged by the appearance of two memoirs on the subject 
by Dulong. Unfortunately, in the same year he took 
scarlet-fever and fell seriously ill. Sadi Carnot died at age 
36 in Paris by a violent attack of cholera on August 24, 
1832.

3. CALORIC VERSUS MECHANISTIC THEORY OF 
HEAT

The type of reasoning used by Carnot in the Réflex-
ions was based on the acceptance, albeit with significant 
distinctions, of the caloric theory. The theory of heat, 
with its evolution and oscillation between a materialistic 
and mechanistic view, has been discussed in great detail 
in many texts [16-19] and in many articles in scientific 
journals [20-23]. In this section only some points of this 
long history will be recalled to highlight how, even if 
the caloric theory was dominant in France at the time 
of Carnot, the two aspects of the heat theory tended 
to overlap in general and sometimes even in the same 
author, as it was indeed the case for Carnot.

Intuitively, the concept of heat is linked to that of 
fire. Fire was assumed as the prime element in the phi-
losophy of Heraclitus, to explain the continuous becom-
ing of natural phenomena, and had then become one of 
the four constitutive elements of Empedocles’ philoso-
phy. The fire instinctively arouses the idea of the motion 
of elementary particles emitted by the bodies and capa-
ble of producing the physical sensation of heat. From 
this point of view, it is remarkable that in the title of his 
Réflexions Carnot refers to fire: la puissance motrice du 
feu, a diction that in English translations will become 
the motive power of heat. Redondi [10] attached a par-
ticular significance to the use by Carnot of the word feu 
instead of chaleur as an attempt to give to the term a 
wider generality.

The concept of heat has remained scientifically 
undefined for a very long time because of a lack of 
experimental tools to measure and quantitatively define 

Figure 2. Sadi Carnot at the age of 17.



76 Pier Remigio Salvi, Vincenzo Schettino

heat and make a clear distinction between heat and tem-
perature. These shortcomings were overcome by ther-
mometry and calorimetry. Confining the attention to 
chemistry, Hermann Boerhaave (1668-1738), a physician 
and chemist, introduced in the chemical laboratory a 
thermometer, built by Daniel Gabriel Fahrenheit (1686-
1736), allowing to go beyond the sensory abilities in the 
control and understanding of heat [24]. For Boerhaave 
heat, or fire as he called it, was a subtle and imponder-
able fluid that interacted with matter to give rise to all 
that concerned heat [25]. However, the materialistic 
vision of Boerhaave had a dynamic character: the parti-
cles of heat were constantly moving and the increase in 
heat produced an increase in the movement of the par-
ticles.

Further progress in the study of heat was made with 
Joseph Black (1728-1799) who highlighted the conceptual 
difference between temperature and heat and invent-
ed the calorimeter to measure the amount of heat that 
develops in a chemical reaction [26]. An important dis-
covery of Black was the observation that in the process 
of melting or boiling a substance absorbed heat with-
out changing temperature arriving at the distinction 
between latent heat and free (or sensible) heat. Black also 
established that the specific heat differs for various sub-
stances. It is remarkable to note that James Watt (1736-
1819), the instrument maker who perfected Newcomen’s 
steam engine, was a student of Black.

Black was a follower of the phlogiston theory that 
had been developed by Johann Joachim Becher (1635-
1682) and his pupil Georg Ernst Stahl (1660-1734). 
According to Becher [27] there were three elements, the 
terra fluida (or mercurial), the terra pinguis (or fat or 
combustible) and the terra lapidea (or vitrifiable). The 
combustible earth produced oils and fuels. Stahl [28] 
developed the master’s ideas and called the combustible 
earth phlogiston. The phlogiston was volatile and tended 
to rise upwards. According to the theory, the metals 
were rich in phlogiston which was liberated during the 
calcination and their transformation into calxes (oxides). 
The process was reversible and by burning the oxides 
with coal the metal was regenerated with the reabsorp-
tion of the phlogiston. The phlogiston theory spread 
among chemists because of its ability to explain the 
phenomena of combustion, despite considerable incon-
sistencies. For example, since metals during calcination 
increase in weight, it was necessary to hypothesize that 
the phlogiston had a negative weight. Despite this, the 
phlogiston theory held up until Antoine Laurent Lavois-
ier (1743-1794) correctly interpreted the phenomena of 
combustion as reactions of substances with oxygen, the 
dephlogisticated air discovered by Joseph Priestley (1733-

1804) and Carl Wilhelm Scheele (1742-1786) [29]. In dis-
cussing his new theory of chemistry and the critique of 
the phlogiston theory [30] Lavoisier was unable to aban-
don the theory of caloric although he still considered the 
caloric as one of the chemical elements. The caloric the-
ory still survived for its extraordinary ability to explain 
many physical or chemical phenomena in a simple way. 
For example, Pierre Simon Laplace (1749-1817), a staunch 
supporter of this theory [31], on the basis on the theo-
ry of caloric was able to calculate the velocity of sound 
in gases. It is, however, remarkable that Lavoisier and 
Laplace in a joint article [32,33] adopt the caloric theory 
but, preliminarily, express severe doubts about the same 
theory with respect to a theory based on atomic move-
ments.

In the Renaissance, with resumption of atomism, a 
more convinced connection of heat with the movement 
of the microscopic particles constituting matter gradu-
ally makes its way. Francis Bacon (1561-1626) adopts the 
atomistic philosophy of Democritus and in the Novum 
Organum [34] explicitly expresses himself on the nature 
of heat:

from the instances taken collectively, as well as singly, the 
nature whose limit is heat appears to be motion. This is 
chiefly exhibited in flame, which is in constant motion, and 
in warm or boiling liquids, which are likewise in constant 
motion… the very essence of heat, or the substantial self of 
heat, is motion and nothing else.1

Also Galileo Galilei (1564-1642) did not disdain the 
atomistic theory of the constitution of matter and in the 
Saggiatore [35] he writes about heat:

…I incline very much to believe that [...] those materials 
that produce and make us feel the heat, which we call with 
general name of fire, they are a multitude of little bodies, in 
such a way figured out, moved with so much speed, which, 
meeting our body, penetrate it with their subtlety, and that 
their touch, made in their passage through our substance 
and felt by us, generates the effect that we call hot.2

These conceptions, and similar ones we can find, 
for example, in Robert Boyle and Isaac Newton, must 
be considered intuitions rather than scientific theories. 
A progress in this direction will take place with Dan-

1 [30], book 2, aphorism XX.
2 original sentence in [35], section 48, which in Italian reads: … incli-
no assai a credere che […] quelle materie che in noi producono e fanno 
sentire il caldo, le quali noi chiamiamo con nome generale fuoco, siano 
una moltitudine di corpicelli minimi, in tal modo figurati, mossi con tanta 
e tanta velocità; li quali, incontrando il nostro corpo, lo penetrino con la 
loro somma sottilità, e che il lor toccamento, fatto nel lor passaggio per la 
nostra sostanza e sentito da noi, sia l’affezzione che noi chiamiamo caldo.



77Sadi Carnot’s Réflexions and the foundation of thermodynamics

iel Bernoulli (1700-1782) and the publication in 1738 of 
Hydrodynamica [36] a treatise on the dynamics of fluids 
which, in chapter X, proposes a kinetic model of a gas, 
consisting of spherical particles in rectilinear motion. 
For the interest of the present paper, the model assumes 
that heat increases the velocity υ of the particles and that 
both the pressure of the gas and its temperature are pro-
portional to υ2, that is to the kinetic energy. Even if the 
work of Bernoulli did not immediately undergo the reso-
nance that deserved, it constituted an anticipation of the 
kinetic theory of gases that would take place only a cen-
tury after its publication.

From an experimental point of view, doubts about the 
theory of the caloric had been advanced in 1798 by Benja-
min Thomson (1753-1814), Count of Rumford, who, wit-
nessing the reaming of the cannon barrels in the Munich 
arsenal, observed that large (apparently inexhaustible) 
quantities of heat developed in the process both in the 
cannon and in the boring shavings [37] without changes 
in the properties (and in particular of the specific heat) 
of the cannon or shavings. Similarly, in 1799 Humphry 
Davy (1778-1829) reported that the fusion of ice occurred 
by simply making friction between two blocks of ice at a 
temperature lower than the melting point [38]. Later, in 
1842, Julius Robert Mayer (1814-1878) showed that the 
water temperature could be increased by one degree by 
simple mechanical stirring [39]. Although these experi-
ments were not able to undermine the caloric theory, a 
preliminary form of kinetic theory continued to affirm 
its uncertain presence thank to work by John Herapath 
(1790-1868) and John James Waterston (1811-1883), with 
considerable hostility in the scientific community.

The seminal work of Bernoulli saw a definitive flow-
ering with the work of James Prescott Joule (1818-1889) 
[40] and Rudolf Clausius (1822-1888) [41] and with the 
complete elaboration in statistical terms by James Clerk 
Maxwell (1831-1879) [42,43] and Ludwig Boltzmann 
(1844-1906) [44,45].

4. THE THERMODYNAMICS OF CARNOT

The Réflexions sur la puissance motrice du feu et sur 
le machines propres à développer cette puissance were ini-
tially printed in 1824 by Bachelier in Paris [5]; this edi-
tion can be easily accessed online. A second French edi-
tion was published in 1872 [46] and can be accessed in 
the Annales Scientiques de l’École Normale Supérieure 
at the site www.numdam.org/item/ASENS_1872. Among 
the English versions we already mentioned the transla-
tion edited by R.H. Thurston in 1897 [12]. An English 
critical edition by R. Fox [47], containing also the sur-
viving manuscripts including the Recherche d’une for-

mule propre à représenter la puissance motrice de la 
Vapeur d’Eau, first published by Gabbey and Herivel 
[48], has been published in 1986. Other notable English 
translations have been edited by Mendoza [49] and Mag-
ie [50].

The Réflexions begin by extolling the contribution 
of steam engines to the progress and wealth of England 
and the further advantages that could be foreseen for the 
development of civilization if technical improvements 
were able to increase their efficiency. However, Carnot 
realizes that

their theory [of steam engines] is very little understood, 
and the attempts to improve them are still directed almost 
by chance.3

and remarks that

the phenomenon of the production of motion by heat has 
not been considered from a sufficiently general point of 
view.4

Hence, the declared purpose of the work is of a the-
oretical nature, i.e., the identification of the principles 
and laws that regulate the phenomenon. The extraordi-
nary nature of the Réflexions lies in the critical discus-
sion of general principles without being anchored to a 
corresponding mathematical formulation so that the 
conclusions lend themselves to be framed in the scheme 
of the subsequent theory of thermodynamics.

4.1 The steam engine

After establishing the issues to be analyzed, namely:
a) if the motive power of heat is limited;
b) if the improvement of the steam engine can go 

beyond a certain limit;
c) if there is an agent more efficient than water vapor,
Carnot initially focuses the attention on the steam 
engine, schematically represented in Fig. 3. The water 
vaporizes in the boiler and the steam is admitted in the 
cylinder, thus causing the piston to move, and then, by 
further expansion cools back to water at the condenser 
temperature.

The first general statement rules out the possibil-
ity of a thermal engine, like the one depicted in Fig. 4a, 
in which heat from a single source is transformed in 
motive power. For the production of motive power two 
heat reservoirs at different temperatures are necessary:

3 [12], p. 42.
4 [12], p. 43.



78 Pier Remigio Salvi, Vincenzo Schettino

the production of heat alone is not sufficient to give birth 
to the impelling power: it is necessary that there should 
also be cold; without it, the heat would be useless. And in 
fact, if we should find about us only bodies as hot as our 
furnaces, how can we condense steam? What should we do 
with it if once produced? We should not presume that we 
might discharge it into the atmosphere, as is done in some 
engines; the atmosphere would not receive it. It does receive 
it under the actual condition of things, only because it ful-
fils the office of a vast condenser, because it is at a lower 
temperature; otherwise it would soon become fully charged, 
or rather would be already saturated.5

The heat flow from the furnace at high temperature 
TH to the condenser at lower temperature TC would not 
be by itself effective in producing motive power unless 
the heat transfer occurs through the mediation of an 
agent, the steam in the present case, able to expand 
under the action of heat. The temperature difference 

5 [12], p. 46-47.

between the two reservoirs plays the role of a potential 
energy difference like the height in the waterfall:

according to established principles at the present time, we can 
compare with sufficient accuracy the motive power of heat 
to that of a waterfall. Each has a maximum that we can-
not exceed, whatever may be, on the one hand, the machine 
which is acted upon by the water, and whatever, on the other 
hand, the substance acted upon by the heat. The motive pow-
er of a waterfall depends on its height and on the quantity of 
the liquid; the motive power of heat of heat depends also on 
the quantity of caloric used, and on what may be termed, on 
what in fact we will call, the height of its fall, * that is to say, 
the difference of temperature of the bodies between which the 
exchange of caloric is made. In the waterfall the motive pow-
er is exactly proportional to the difference of level between the 
higher and lower reservoirs. In the fall of caloric the motive 
power undoubtedly increases with the difference of tempera-
ture between the warm and the cold bodies; but we do not 
know whether it is proportional to this difference.6

In transmitting the caloric from the hot to the cold 
reservoir the volume of the steam changes and this gen-
erates the motion of the piston in the cylinder. Besides 
steam, any substance that expands due to heat could be 
employed as an agent in the cyclic operation. In the con-
text of the caloric theory as an indestructible fluid, the 
heat engine would work according to the scheme of Fig. 
4c [51], while the correct thermodynamic functioning is 
that of Fig.4b.

After establishing these general criteria, Carnot 
moves on to the examination of the steam engine, iden-

6 [12], p. 60-61.

Figure 3. Schematic representation of the steam engine.

Figure 4. (a) a heat engine with a single heat source; (b) a heat 
engine operating between two reservoirs at TH and TC tempera-
tures; (c) the steam engine according to the caloric theory of heat. 
In the three schemes q is the heat expended, -q the heat absorbed 
by the reservoir and -w the work done by the engine.



79Sadi Carnot’s Réflexions and the foundation of thermodynamics

tifying three successive phases described in detail in the 
Recherche [48]:
a) steam generation in the boiler absorbing heat from 

the high temperature source at TH and expansion 
into the cylinder equipped with a movable piston by 
the opening of the upper valve;

b) further steam expansion and piston motion with 
upper and lower valves closed;

c) steam condensation at the refrigerant temperature 
TC after the opening of the lower valve and return of 
the piston to the initial position.
It can be seen that, from the beginning, Carnot 

includes the adiabatic expansion (process b) according 
to the expansive principle of Watt introduced explicitly 
by Clément [52,53]. Supposing that the steam engine 
works without dispersion of heat and the conditions for 
maximum power output are satisfied, the mode of oper-
ation of the steam engine can be reversed. Calling the 
hot and cold reservoirs A and B, respectively, the direct 
and inverse operation of the steam engine can be com-
pared: in the first the caloric is transferred from A to B 
and motive power is produced, in the second the caloric 
flows from B to A and motive power is expended. It is 
evident that acting on the same quantity of vapor and 
with no loss of caloric or motive power, the A → B and 
B → A amounts of caloric are equal as well as the direct 
and inverse motive powers, apart from the sign, so that 
the overall balance is zero. Alternating the two processes 
in opposite directions in an indefinite number of opera-
tions neither motive power is produced nor caloric is 
transferred. If a different process were available produc-
ing more motive power than that produced by the steam 
engine, all other conditions being equal, it would be pos-
sible to couple this process with the steam engine, to 
return at the initial conditions and to divert a portion of 
the motive power at the end of the reversed process. The 
net result would be creation of motive power from noth-
ing. This is perpetual motion, contrary to the laws of 
mechanics and as such inadmissible. The conclusion is:

the maximum of the motive power resulting from the 
employment of steam is also the maximum of motive power 
realizable by any means whatever.7

Carnot realizes that the proposition should be consid-
ered only as an approximation8 and that a more rigorous 
demonstration is necessary. An important point is that 
the described process of the steam engine is not revers-
ible since the agent at the end of the process has not 
recovered the initial state, which is a basic requirement 

7 [12], p. 55.
8 [12], p. 56.

for the comparison of the performances of engines with 
different agents. The closure of the cycle cannot be sim-
ply obtained by the direct contact of the cold liquid with 
the high temperature reservoir since this direct contact 
between bodies at different temperatures will cause a 
loss of motive power and the reverse process would be 
impossible. This problem is circumvented when the tem-
perature difference between A and B is indefinitely small 
since in such a case the heat necessary to raise the cold 
liquid to the initial temperature is also negligibly small 
compared to the caloric producing power. In the more 
general case of a finite temperature difference one may 
imagine that a series of other reservoirs, C, D, E, …… 
could be inserted between A and B with infinitely small 
spacing between two adjacent reservoirs such that the 
caloric transfer from A to B occurs through intermediate 
steps each developing maximum motive power.

4.2 The Carnot cycle

The analysis continues to arrive at a more exhaustive 
demonstration of the general principle derived from the 
study of the steam engine which Carnot himself defined 
as approximate. To this end Carnot proposes an ideal 
thermal engine, the famous Carnot engine, which works 
in a perfectly cyclical manner and which uses a perma-
nent gas, air, as an agent. This choice corresponded to a 
need felt in the environment of thermal engines to use 
an agent other than water that could be used at higher 
pressures and, hopefully, with fuel savings [16].

The starting experimental observation is that expan-
sion causes a temperature fall, and compression a tem-
perature rise, which can be compensated by absorption 
and release of caloric, respectively. The series of opera-
tions can be described with reference to the reproduc-
tion of the original Carnot drawing shown in Fig. 5a:
I) The gas, initially enclosed in the abcd volume (with 

cd the actual position of the piston), is in contact 
with the wall of the cylinder which freely transmits 
the caloric from furnace A. The gas is thus taken at 
the temperature TH of the furnace.

II) The piston gradually moves isothermally up to the 
position ef.

III) The furnace is removed and the gas is fully isolated 
from external bodies. The piston moves from posi-
tion ef to gh. During this adiabatic expansion the 
gas temperature decreases until it reaches the tem-
perature of the condenser B.

IV) The gas is now placed in contact with the condens-
er B and isothermally compressed until the piston 
moves back from the position gh to cd recovering 
the initial volume but at the condenser temperature.



80 Pier Remigio Salvi, Vincenzo Schettino

V) The condenser B is removed. An adiabatic compres-
sion of the gas is carried out until the temperature 
rises to reach again the temperature of the furnace 
A. The piston moves from cd to ik.

VI) The gas is now placed in contact with the furnace A 
and the piston goes from ik to ef.
The cycle is successively repeated along the steps 

III, IV, V, VI. For the sake of clarity in Fig. 5b the usual 
representation of the Carnot cycle on a p-V diagram is 
shown. Problems connected with the correct closure of 
the cycle in the p-V diagram have been discussed by 
Klein [54], Kuhn [55], La Mer [56,57] and Tansjo [58]. 
Useful motive power is obtained since the elastic force 
(i.e., pressure) of the gas in the isothermal expansion is 
greater than in the isothermal compression so that the 
power produced in the first operation exceeds that con-
sumed for compression:

the quantity of motive power produced by the movements 
of dilatation is more considerable than that consumed to 
produce the movements of compression.9

By a line of similar reasoning as employed to show 
the impossibility to produce motive power greater than 
that by a reversible steam engine, the general conclusion, 
known as Carnot’s principle, is reached

the motive power of heat is independent of the agents 
employed to realize it; its quantity is fixed solely by the 
temperatures of the bodies between which is effected, final-
ly, the transfer of the caloric.10

9 [12], p. 65.
10 [12], p. 68.

The next basic question concerns the dependence of 
the motive power on the temperature of the two reser-
voirs and, in particular, whether a difference of motive 
power should be expected for a fall of caloric from 
100oC to 50oC and from 50oC to 0oC. To this purpose, 
Carnot considers two air engines working between 
100oC and (100 – h)oC and 1oC and (1 – h)oC, respec-
tively, with h extremely small. The motive power results 
from that supplied by the air in the V1→V2 expansion 
minus that expended in the opposite compression and 
is the same for the two engines. Carnot easily comes to 
this conclusion, valid also within the later thermody-
namic theory, in a long note ([12], p. 98). For the com-
parison between q100, the heat necessary to keep air 
at 100oC during the expansion, and q1, the equivalent 
quantity at 1oC, Carnot considers two different paths 
from the starting point, 1oC and V1, to the final point, 
100oC and V2. One is performed by heating at V1 up to 
100oC and then expanding isothermally to V2, the other 
by the reverse combination, i.e., expanding isothermally 
at 1oC to V2 and then heating at V2 up to 100oC. Accord-
ing to the caloric axiom the two amounts of heat are 
independent on the path and therefore

qV1 + q100 = q1 +qV2

where qV1 and qV2 are heats to increase the air tempera-
ture from 1oC to 100oC at the two different volumes V1 
and V2, respectively. It was incorrectly established from 
measurements reported in previous years by Delaro-
che and Bérard on several gases that their specific heats 
depend on density, decreasing with increasing density 
[59]. Carnot acknowledges the result by saying that “the 
capacity of gases for heat changes with their volume” 
([12], p. 78), increasing as the volume increases. Since qV2 
> qV1,it follows that

the quantity of heat due to a change of volume of a gas is 
greater as the temperature is higher11

and as a consequence

the fall of caloric produces more motive power at inferior 
than at superior temperatures12

because the amounts of heat are different (q100 > q1), 
while the motive power is the same for the two engines. 
The conclusion happens to be correct, as everybody of 
us knows looking at the efficiency of thermal engines 
reported in all thermodynamic textbooks, but, as 

11 [12], p. 96.
12 [12], p. 97.

Figure 5. (a) the Carnot cycle as shown in the original drawing; (b) 
the usual representation of the Carnot cycle for a gas on the p-V 
diagram. Points A and 4 correspond to the initial volume and to 
completion of the isothermal compression.



81Sadi Carnot’s Réflexions and the foundation of thermodynamics

already noted [16], the justification rests on the false 
assumption of the caloric conservation and on the mis-
leading volume dependence of the specific heat of gas-
es [59]. As to the latter point, it is correct to say that a 
word of caution about the experimental observations by 
Delaroche and Bérard in favor of this dependence was 
advanced also by Carnot and an invitation to further 
investigate about the law relating the motive power and 
temperature was clearly expressed in the Réflexions. In 
retrospect, the q100 > q1inequality holds not because qV2 
> qV1 but because the heat absorbed in an isothermal 
expansion is equal to the work performed (for an elas-
tic fluid behaving as an ideal gas) and the latter depends 
linearly on temperature.

The second problem analyzed is concerned with the 
evaluation of the motive power developed by the same 
amount of heat absorbed by agents such as air, steam or 
alcohol vapor at the same or at different temperatures. 
Here we mention only the case of steam. To this purpose 
the cycle of Fig. 5a is simplified to include only the two 
isothermal steps, expansion abcd → abef and compres-
sion abef → abcd, joined by two cooling/warming steps 
at constant volume. Starting with 1 kg of water, with 
specific volume ≈ 10-3m3Kg-1, and expanding, abcd → 
abef, under atmospheric pressure at 100oC, it was well 
known that the vaporization leads to a volume increase 
≈ 1700 times the initial value, resulting in an increment 
∆V ≈ Vsteam= 1.7 m3kg-1. The reverse process, the com-
pression abef → abcd, is assumed to occur in the cycle at 
99oC at a slightly smaller pressure inducing steam con-
densation to water and volume decrement 1.7 m3kg-1. 
The motive power is ∆V times the difference ∆p of the 
water vapor pressure at 100oC and 99oC, which accord-
ing to data available to Carnot amounts to 26 mmHg or 
0.36 m.w. (meter of water, 760 mmHg = 10.4 m.w.)13. The 
product ∆V·∆p is

1.7 m3kg-1· 0.36 m.w. = 0.611 units

The hot source delivers heat to the cycle since (a) 
at constant volume the temperature of the water must 
increase from 99oC to 100oC and (b) at 100oC the expan-
sion step must absorb heat in order to be isothermal. The 
first contribution is much smaller than the second and is 

13 Data on vapor pressure of water at discrete values of temperature 
from 0°C to 100°C were already known [60]. Assuming that steam 
obeys the ideal gas law in the form p = c(267 + t)/v [p(mmHg), t(°C), 
v(liters) and c = 3.52 solving for c with v(steam) = 1700 liters at 100°C 
and 760 mmHg ] Carnot found v(steam) at these temperatures and then 
fitted the calculated values to a known function of t(°C) in the range 
0-100°C . The vapor pressure at the desired t(°C) was obtained apply-
ing the gas law equation and solving for p. For instance, p results 734 
mmHg at 99°C [61].

neglected by Carnot in the calculation of the total heat. 
Being experimentally known that 550 units of heat, i.e., 
550 kcal, are necessary to vaporize 1 kg of water under 
atmospheric pressure, the conclusion is, through the 
simple proportion 550/0.611 =1000/x,

thus 1000 units of heat transported from one body kept at 
100 degrees to another kept at 99 degrees will produce, act-
ing upon vapor of water, 1.112 units of motive power14

Next, the steam engine working between 1oC and 
0oC is considered. Carnot was able to estimate15 ∆p = 
0.358 mmHg and ∆V = 174 m3kg-1 following the com-
putational procedure described in footnote 13. The heat 
of vaporization of 1 kg water at 1oC is determined under 
the vapor tension at that temperature, p(1oC)=5.418 
mmHg. According to Carnot, this is the same heat nec-
essary to raise under atmospheric pressure the water 
temperature from 1oC to 100oC and then to vaporize 
completely water. The total heat delivered by the hot 
source to the engine at 1oC (and transmitted to the cold 
source at 0oC) is therefore (100 + 550) kcal·kg-1= 650 
kcal·kg-1. It is easily found after convenient unit conver-
sion of the ∆p∆V product from m3kg-1mmHg to m3kg-
1m.w. that 1000 units of heat will produce 1.290 units of 
motive power. The soundness of this last number is obvi-
ously related to the estimate of the vaporization heat at 
1oC but the strength of Carnot physical insight is shown 
by the comparison with the actual value: the enthalpies 
of vaporization at 100oC and 1oC are [61] 549.5 kcal·kg-
1and 597 kcal·kg-1, respectively, so that the 650 kcal· kg-1 
value differs from the last one only by ≈1/11.

4.3 Carnot and the caloric theory

In a note of the Réflexions Carnot explicitly states 
that a basic principle of his scientific reasoning is the 
assumption of the theory of caloric as an imponderable 
and indestructible fluid that, in a modern diction, is a 
function of state:

we tacitly assume in our demonstration that when a body 
has experienced any changes, and when after a certain 
number of transformations it returns to precisely its origi-
nal state, that is, to that state considered in respect to den-
sity, to temperature, to mode of aggregation – let us sup-
pose, I say, that this body is found to contain the same 
quantity of heat that it contained at first, or else that the 
quantities of heat absorbed or set free in these different 

14 [12], p.104.
15 Full details about the Carnot calculations on this as well as on all the 
others heat engines considered may be found in ref. 61.



82 Pier Remigio Salvi, Vincenzo Schettino

transformations are exactly compensated. This fact has 
never been called into question.16

However, in the conclusion of the same note, Car-
not explicitly expresses profound doubts about the same 
theory:

For the rest, we may say in passing, the main principles 
on which the theory of heat rests require the most careful 
examination. Many experimental facts appear almost inex-
plicable in the present state of the theory.17

In a subsequent step of the Réflexions, after calcu-
lating the motive power generated when 1000 units of 
caloric experience a thermal fall of 1°C in air, steam and 
alcohol engines, to demonstrate its independence from 
the agent, Carnot is again openly critical of the caloric 
theory:

The fundamental law that we propose to confirm seems to 
us to require, however, in order to be placed beyond doubt, 
new verifications. It is based upon the theory of heat as it is 
understood today, and it should be said that this founda-
tion does not appear to be of unquestionable solidity.18

Therefore, while adopting the caloric theory, it is 
apparent that the theory does not appear to be well 
founded to Carnot [62]. For instance, with reference to 
the principles of the caloric theory he explicitly states 
that:

these theories furnish no means of comparing the quanti-
ties of heat liberated or absorbed by elastic fluids which 
change in volume at different temperatures.19

and in another passage he writes:

we do not know what laws it [the caloric] follows relative 
to the variations of volume: it is possible that its quantity 
changes ….. with its temperature.20

The extraordinary character of the Réflexions lies in 
the fact that, while officially adopting the caloric theo-
ry, Carnot is able to reach general conclusions that go 
beyond the starting hypothesis. This circumstance may 
well be highlighted by the closing mechanism of the 
thermodynamic cycle adopted by Carnot [54-58]. The 
set of six transformations that activate the ideal engine 
starts from point A in Figure 5b and the achievement of 

16 [12], p. 67.
17 [12], p. 67.
18 [12], p. 107.
19 12], p. 84
20 [12], p. 62.

point 4, at the end of the isothermal compression and 
from which the adiabatic compression starts, is defined 
exclusively in terms of volume, without any reference to 
the heat exchanged. On the contrary, in the Clapeyron 
discussion [54,63] the cycle starts at point 1 of the p-V 
diagram and the end of the isothermal compression 3-4 
is defined when the heat transferred to the condenser 
equals that absorbed during the expansion at high tem-
perature, with explicit reference to the theory of caloric.

A certain ambiguity in the adhesion of Carnot to 
the caloric theory was already noted by Clausius [64]. In 
fact, after reporting the experimental tests showing that 
heat could not be considered as an indestructible fluid, 
he writes:

these circumstances, of which Carnot was also well aware, 
and the importance of which he expressly admitted, press-
ingly demand a comparison between heat and work, to 
be undertaken with reference to the divergent assumption 
that the production of work is not only due to an alteration 
in the distribution of heat, but to an actual consumption 
thereof; and inversely, by the consumption of work heat 
may be produced.21

After further discussing experiments in favor of 
the dynamic theory of heat, Clausius defines the Car-
not’s principle that “no heat is lost” only as an additional 
statement in his logical reasoning not affecting the con-
clusions drawn:

on a nearer view of the case, we find that the new theory is 
opposed, not to the real fundamental principle of Carnot, but 
to the addition “no heat is lost;” for it is quite possible that in 
the production of work both may take place at the same time; 
a certain portion of heat may be consumed, and a further 
portion transmitted from a warm body to a cold one; and 
both portions may stand in a certain definite relation to the 
quantity of work produced. This will be made plainer as we 
proceed; and it will be moreover shown, that the inferences to 
be drawn from both assumptions may not only exist together, 
but that they mutually support each other.22

However, Callendar [20] notes that Carnot’s state-
ment concerning a perfectly reversible cyclical process 
was actually misquoted by Clausius when he reports that 
Carnot

expressly states that no heat is lost in the process, that the 
quantity (transmitted from the fireplace to the condenser) 
remains unchanged23

21 [64], p. 2.
22 [64], p. 4.
23 The original Carnot statement: Les quantités de chaleur absorbées ou 
degagées dans les diverses transformations sont exactement compensées, is 



83Sadi Carnot’s Réflexions and the foundation of thermodynamics

According to Callendar [20] the bad interpretation 
of Clausius to identify “compensated” with “equal” may 
have been induced by the work of Clapeyron [63]. Cal-
lendar’s conclusion is that the principles that Carnot 
reaches with regard to reversible processes are independ-
ent of the caloric theory. Considerations of this type can 
be applied to the description of Carnot’s work by Max-
well [65]. Maxwell starts the cycle from the point 2 in 
the p-V diagram with an adiabatic compression up to 
the temperature of the cold source, again without refer-
ence to the heat exchanged.

Apart from the unpublished manuscript reported in 
ref. 48, Carnot did not publish anything else on the the-
ory of heat and thermal engines after the Réflexions and 
some considerations or speculations have been advanced 
as an explanation [16,47]. It is possible that the increas-
ing dissatisfaction with the caloric theory caused in Car-
not some embarrassment when discussing with influen-
tial figures of the scientific milieu, which did not sup-
port the idea of destroying the fundamental axioms of 
the theory inherited from the founding fathers, Lavois-
ier and Laplace [30-33]. A second concern was perhaps 
bound to the assessment of the validity of the Réflexions 
and to some uncertainty about the parts of the work 
which could be saved after the collapse of the old theory. 
Although most of Carnot’s ideas overcame untouched 
tens of years, it was necessary to wait the experimen-
tal results of Joule and the theoretical considerations of 
Kelvin and Clausius to reach the final objective, a giant 
effort for a single man. It has been observed [47] that in 
the remaining years of his life Carnot was probably dis-
appointed and frustrated not being able to reconcile the 
published work with the ideas, freshly growing in his 
mind, tightly relating heat and work. A sad conclusion 
has been drawn that these years had elements of tragedy 
more than of triumph for Carnot, contrary to what we 
are inclined to think on the basis of his anticipation of 
the future laws of thermodynamics.

The doubts of Carnot on the theory of caloric 
are expressed more explicitly in his scientific notes 
which are more or less contemporary to the Réflexions 
[12,47,49]. With reference to the radiant heat, which is 
clearly associated to motion, Carnot poses the problem:

could a motion (that of radiant heat) produce matter 
(caloric)? Undoubtedly no; it can only produce motion. 
Heat is then the result of motion.24

translated as: the quantities of heat lost and gained in the various pro-
cesses cancel one another out, by Fox [47] and as: the quantities of heat 
absorbed or set free in these different transformations are exactly compen-
sated , by Thurston [12].
24 [49], p. 63, Selection from the posthumous manuscripts of Carnot.

From a more general point of view Carnot position 
is as follows:

is heat the result of a vibratory motion? If this is so, quan-
tity of heat is simply quantity of motive power. As long as 
motive power is used to produce vibratory movements, the 
quantity of heat must be unchangeable; which seems to fol-
low from experiments in calorimeters; but when it passes in 
movements of sensible extent, the quantity of heat can no 
longer remain constant.25

4.4 The physics of gases

The physics of gases presented in the Réflexions has 
been critically reviewed and discussed [47] (see notes 42, 
46, 53, 61 and 63 of the Commentary). The sharp insight 
into the matter, despite the Carnot’s adherence to the 
conservation of caloric as a fundamental axiom of the 
theory, is shown by the following examples. Consider-
ing a cycle where the two isothermal operations occur 
at temperatures differing only slightly, the adiabatic con-
tributions to the total motive power may be legitimately 
ignored with respect to those from the isothermal opera-
tions. If different gases are used in the cycle, ensuring 
that they go exactly through the same states of pressure 
and volume, the same motive power will be obtained 
since the gases obey the same law. By the Carnot prin-
ciple this means that the caloric absorbed at higher and 
released at the slightly lower temperature is the same 
whichever the gas used. The proposition follows:

when a gas passes without change of temperature from one 
definite volume and pressure to another volume and anoth-
er pressure equally definite, the quantity of caloric absorbed 
or relinquished is always the same, whatever may be the 
nature of the gas chosen as the subject of the experiment.26

In modern terms, the same follows from the first 
principle, ∆U = q + w, and the fact that the internal 
energy U of an ideal gas depends only on temperature. 
In an isothermal process ∆U = 0 and q = -w. The state-
ment follows since all ideal gases perform exactly the 
same amount of work in the same reversible isothermal 
process. Proceeding further, for one mole of an ideal gas 
expanding isothermally from VA to VB the heat absorbed 
from the surroundings is given by RT ln(VB/VA). Carnot 
expresses the same result with the proposition

when a gas varies in volume without change of tempera-
ture, the quantities of heat absorbed or liberated by the 

25 [49], p. 67, Selection from the posthumous manuscripts of Carnot.
26 [12], p. 72.



84 Pier Remigio Salvi, Vincenzo Schettino

gas are in arithmetical progression, if the increments or the 
decrements of volume are found to be in geometrical pro-
gression.27

and represents the volume dependence in analytical 
form by the equation

s = A + B log V28

As a second result, it was known at his time that 
by adiabatic compression the temperature of the atmos-
pheric air rises by 1oC when the volume V reduces to V 
– (1/116) V while, on the other hand, the isobaric heat-
ing of air by 1oC increases the volume to [V + (1/267) 
V]. The amount of heat absorbed in the last process 
is cp, the specific heat of air at constant pressure, since 
∆t = 1oC. The final state of the isobaric process may be 
reached alternatively through a second trajectory which 
involves first the adiabatic compression by 1oC and then 
the isothermal expansion to the final volume. Due to 
the conservation axiom the amount of heat remains cp 
but now is entirely expended in the isothermal process, 
being the compression adiabatic. A second point along 
the isotherm curve may be certainly reached at constant 
volume heating air by 1oC and increasing pressure from 
p to [p + (1/267) p] and in this case the heat absorption 
is equal to cV, the specific heat of air at constant volume. 
Going again through the second trajectory but stop-
ping now along the isotherm at a volume equal to the 
initial volume, the heat absorbed in this portion of iso-
therm expansion is cV. As the variations are small with 
respect to the original volumes the amount of heat may 
be reasonably taken as proportional to these variations 
and therefore cp/cV= (1/116 + 1/267)/(1/116) = 1.43, not 
far from the value measured by Gay-Lussac and Wel-
ter, 1.3748, reported elsewhere [31]. It should be noted 
that the argument is valid also in later thermodynamics 
and made explicit by the expression cp/cV= 1 – (∂V/∂T)
p/(∂V/∂T)ad [47]. In addition, taken cp as unity, cV ≈ 0.7. 
The difference, 0.3, represents the amount of heat due to 
the increase of volume when air is heated by 1oC at con-
stant pressure. Since this increase of volume is the same 
for all gases, also the heat absorbed, cp– cV, is the same 
whichever the gas. Provided that the gases are at the 
same pressure and temperature it follows that

the difference between specific heat under constant pressure 
and specific heat under constant volume is the same for all 
gases.29

27 [12], p. 81.
28 [12], p. 90.
29 [12], p. 76.

4.5 The mechanical equivalent of heat

As already noted, in the unpublished notes Carnot 
clearly refuses the caloric theory to such an extent to 
identify heat as a form of work (or energy):

heat is simply motive power, or rather motion which has 
changed its form. It is a movement among the particles of 
bodies. Wherever there is destruction of motive power there 
is at the same time production of heat in quantity exactly 
proportional to the quantity of motive power destroyed. 
Reciprocally, wherever there is destruction of heat, there is 
production of motive power.30

and goes as far as to propose a numerical estimate of the 
mechanical equivalent of heat:

according to some ideas which I have formed on the theory 
of heat, the production of a unit of motive power necessi-
tates the destruction of 2.70 units of heat.31

The reported value (which rigorously is the thermal 
equivalent of work), once the appropriate conversion fac-
tor is inserted, is equivalent to a mechanical factor of 3.7 
joule/cal, quite close to the actual value, 4.184 joule/cal. 
The theoretical justification of this number was however 
not advanced and successively various reconstructions 
have been attempted [47,49]. One possible procedure, 
suggested by Décombe [66] and cited in ref. [49], is par-
ticularly simple and makes use of the only data present 
in the Réflexions. It has been seen in the previous Sec-
tion that (cp– cV) is the difference between the quanti-
ties of heat expended for 1oC increase under constant 
pressure and volume, respectively, and that this differ-
ence fully accounts for the increase of volume in the 
first case. This difference results to be 0.3 if cp is taken 
as unit heat. Since cp of air is 0.267 that of water ([12], 
p. 100), the heat (cp– cV) absorbed by the air for a 1oC 
increase under constant pressure is 0.267·0.3 = 0.081 cal. 
On the other hand, work is performed by the air due to 
heat absorption. Starting with 1 kg of air, the volume at 
0oC and 1 Atm, 0.77 m3([12], p. 99), increases by 1/267 
for a temperature increase of 1oC at the constant pres-
sure of 1 Atm. The work is 1·0.77·103/267 ℓ Atm = 2.88 
ℓ Atm. With the conversion factor from ℓ Atm to tonne- 
meter (the unit of work to which Carnot refers [66]) the 
result is 0.03 tonne-meter. Since the heat and work esti-
mates are relative to 1 kg of air, it follows that a work of 
1 tonne-meter is performed when air absorbs 0.081/0.03 
= 2.7 kcal of heat.

30 [49], p. 67, Selection from the posthumous manuscripts of Carnot.
31 [49], p. 68, Selection from the posthumous manuscripts of Carnot.



85Sadi Carnot’s Réflexions and the foundation of thermodynamics

5. SCIENTIFIC ANTECEDENTS OF SADI CARNOT

For a historical overview of the work of scientific 
innovators, it is important to identify the background 
that may have inspired or facilitated their discover-
ies. The problem, from a general point of view, can be 
framed by paraphrasing the famous line of John Donne 
that no man is an island, entire of itself. Indeed, Isaac 
Newton, the most famous of the innovators of science, 
said of himself that he had seen farther because he was 
travelling on the shoulders of giants.

In this perspective it seems unlikely that Sadi Car-
not was a solitary innovator as claimed, for example, by 
Cimbleris [67]. Considering the topic of the work of Car-
not, the thermal engines, any antecedent must be sought 
primarily in the world of technical and engineering lit-
erature [52,68-72]. Moreover, the formation of Carnot 
at the École Polytechnique and, above all, at the École 
de Metz was mainly of a technical nature, although, as 
described by Taton [15], considerable attention was also 
paid to a formation of a scientific character. The impor-
tance of the formation process of Sadi Carnot for his 
subsequent scientific work has been discussed by Payen 
[73] and by Taton [15]. It is easy to assume that Sadi had 
a more exquisitely scientific preparation also in the ini-
tial phase of his formation under the guidance of his 
father Lazare.

A possible scientific influence of his father on Sadi 
was already taken into consideration in the memory of 
Saint-Robert [11] in 1868 drawing attention to the anal-
ogy between the fall of water in hydraulic machines 
[74,75] and the transfer of heat between the heat source 
and a refrigerant. This connection has since been dis-
cussed in detail by various authors [14,23,76-79]. 
According to Gillispie [76] and Gillispie and Pisano 
[77] the Réflexions by Sadi would have been inspired or 
would even be a continuation of the work of the father 
Lazare on mechanics and on hydraulic engines [74,75]. 
The authors reach this conclusion through the discus-
sion of available documents as well as with a complex 
treatment that involves an epistemological and semantic 
analysis of the writing of Sadi Carnot [78]. The elements 
deriving from the father Lazare [75] would be, in par-
ticular, the idea of a cyclic character in the functioning 
of ideal engines and of the reversibility of the involved 
processes, the need to avoid improper dispersion of 
the work by friction and, correspondingly, of heat by 
direct contact, the denial of the possibility of a per-
petual motion, the extension of the physical principles 
of operation of particular engines to general cases and 
the discursive nature of the arguments. In fact, in the 
Réflexions an analysis or mathematical deduction of the 

principles enunciated by Carnot is found only in a long 
note [79]. It is now clear that, since mechanics was one 
of Sadi Carnot’s scientific interests, he certainly had to 
know his father’s work. In fact, the clearest correspond-
ence between the Réflexions and the work of Lazare Car-
not is found in the explicit analogy between the fall of 
water from a certain height in hydraulic engines and the 
transfer of heat between a high temperature source and 
a low temperature sink in thermal engines. The analogy 
has been discussed in some detail by Muller [14] as the 
real scientific inheritance of Sadi Carnot from the father. 
Apart from this, the conclusions of Gillispie and Pisano 
[77] appear absolutely plausible but in many cases they 
seem to be based mostly on circumstantial evidence. For 
example, when it is recalled that Sadi subjected some 
points of the Réf lexions to his brother Hippolyte to 
check their readability for non-experts [12] the authors 
conclude in a dubitative or presumptive way: The broth-
ers could scarcely have failed to talk then of their father’s 
science32. In this regard, it should be noted that Lazare 
Carnot is never quoted or mentioned in the Réflexions, 
a strange circumstance in the normal scientific practice. 
Gillispie and Pisano [77] take this circumstance as a 
possible evidence that perhaps the Réflexions were actu-
ally the work of Lazare Carnot, which would then have 
been simply completed by the son who would have con-
sidered it useless to quote his father, the true author of 
the work.

The possible influence on the thought of Carnot by 
Nicolas Clément and Charles Bernard Desormes, but 
above all the first, is based on more certain documentary 
elements. In the first instance, we find three quotations 
of Clément and Desormes in the Réflexions. The first is 
related to an experiment, confirming previous Poisson 
data, on the gas temperature during compression33. The 
second concerns the experimentally established law (in 
English known as the Watt law) which states that the 
saturated water vapor, with the same weight, always con-
tains the same amount of caloric whatever the tempera-
ture at which it is formed34. In modern terms this law is 
equivalent to say that the enthalpy of saturated steam is 
conserved at all temperatures [49] and implies that the 
vapor, adiabatically expanded or compressed, main-
tains the initial saturation state. The third quotation, the 
most important from our point of view, is in reference 
to adiabatic expansion and occurs when Carnot states 
that for better performance of a steam engine not only 
is an initial high pressure important but also, subse-

32 [77], p. 78.
33 [12], p. 73.
34 [12], p. 92.



86 Pier Remigio Salvi, Vincenzo Schettino

quently, progressively decreasing pressures35. In a note, 
Carnot acknowledges that the related Clément’s law is 
indeed fundamental in the steam machine theory and 
he has come to the knowledge of the unpublished arti-
cle of Clément by the author’s kindness. With the help 
of this law, and of the one mentioned above, it is possible 
to calculate the work in an adiabatic expansion or com-
pression of the saturated vapor. In absence of a caloric 
flow between vapor and surroundings the heat content is 
constant and the pressure and temperature of the vapor 
change in such a way to maintain the saturation condi-
tions. The two parameters may thus be related following 
well known saturation tables such as those of Dalton. 
Then, the volume is found assuming the vapor obeys 
the Boyle and Gay- Lussac laws. Once the correspond-
ence between pressure and volume is established at each 
temperature the strategy to calculate the adiabatic work 
is straightforward. The note in question clearly indicates 
a relationship of frequentation and, perhaps, of friend-
ship between Carnot and Clément. In fact, even in the 
biographical note of the brother Hippolyte [12] we find 
that Carnot was familiar with Clément.

The possible debt of Carnot to Clément has been 
discussed in detail by Fox [52] and associated with the 
idea of the expansive principle, first conceived by Watt 
and then developed by Clément. The principle concerns 
the advantages that can be obtained in the efficiency of 
the steam engine allowing to continue the expansion 
after the initial supply of steam. While Watt considered 
this further expansive phase to be substantially isother-
mic, Clément, departing from the commonly accepted 
view, clearly defines it as adiabatic. It comes to this con-
clusion on the basis of a thought experiment in which a 
vapor bubble is introduced to the bottom of a cylinder, 
generating mechanical work measured by the water that 
flows from the top of the cylinder. The bubble continues 
to rise in the cylinder, expanding and letting other water 
to flow out of the cylinder corresponding to additional 
motive power.

A similar but more detailed analysis of the relation-
ship between Carnot and Clément is reported by Lervig 
[53]. In particular, Lervig reports on the participation of 
Carnot to at least some lessons of the Clément’s course 
on Industrial Chemistry at the Conservatoire des Arts et 
Métiers. This results from the set of notes to the course 
written in the years 1824-28 by a certain J.M. Baudot 
(partially reported by Lervig) which clearly show that 
Carnot was well acquainted with Clément and his sci-
entific work (and in particular with the aforementioned 
Clément’s law and with the phases of expansion [détente] 

35 [12], p. 115.

and compression in the steam engine). In these notes the 
lecture of January 20, 1825 is reported where Clément 
says that Carnot, one in the audience of the course, has 
dealt with the principles of thermal engines [53]

… mais un des auditeurs de ce cours, M. Carnot, off.erdu 
génie, ancien élève de l’École Polytechnique, a eu le cour-
age et l’ heureuse idée d’aborder cette intéressante question 
dans un ouvrage fort remarquable qu’ il vient de publier 
sous le titre de Réflexions sur la puissance du feu.36

In the note of March 8, 1827 Carnot is further men-
tioned by Clément as distinguished mathematician

la formule algébrique n’est ici que comme sujet d’exercice 
pour ceux qui voudront l’employer … Elle lui a été donnée, 
dit-il, par un mathématicien distingué.37

Lervig, more explicitly than Fox, advances the 
hypothesis that in fact it was Carnot that influenced 
Clément at various points, as in the numerical exam-
ples contained in the notes taken by the mathematician 
L.B. Francoeur attending as a student the 1823-24 course 
(also these reported in part in ref. 53). The remarkable 
conclusion of the Lervig analysis is relative to the con-
densation phase of the steam engine and states that it 
was an idea entirely due to Carnot that in the evalua-
tion of the total work the (negative) contribution of the 
isothermal work of compression must be taken into 
account. This conclusion is supported by an in-depth 
study of the Francoeur notes and by the accurate recon-
struction of calculations present in the long abstract of 
the Clément and Desormes lost memoir describing the 
theory of the steam engines. Also, no hint about the 
condensation term is found in the notes taken by Baudot 
in the successive years [53].

In a simpler and more direct way the search for 
antecedents of Carnot can be conducted on the basis of 
the cites in the Réflexions. Gouzevitch [69] discussed the 
influence Prony and Betancourt (mentioned in Réflex-
ions) may have had on Carnot for the emphasis these 
authors have put both on the need for a theoretical treat-
ment of the thermal engines and on the necessary pres-
ence of a hot source and a low temperature sink [69].

6. THE RECEPTION OF CARNOT’S IDEAS

Carnot’s Réflexions had a very limited initial for-
tune for various reasons. Carnot, like many at the time, 

36 [53], p. 185.
37 [53], p. 188.



87Sadi Carnot’s Réflexions and the foundation of thermodynamics

was an amateur scientist not introduced into the impor-
tant circuits of scientific communication. Moreover, 
already in his presentation in the front page of the book 
he defined himself simply, with an understatement, an 
ancien élève de l’École Polytechnique. The book was pre-
sented by Pierre Simon Girard, a well-known engineer, 
at a meeting of the Académie des Sciences on 14 June 
1824 to the presence of many important scientists but 
only in oral form and therefore the book was not pub-
lished in the Mémoires of the Academy, which would 
have guaranteed the necessary publicity. This was not 
afforded either by a subsequent written presentation by 
Girard himself in the Revue Enciclopédique [80]. In an 
obituary of 1832 Robelin [7] attributed in part the scarce 
diffusion of the work to the difficult style of Carnot:

unfortunately, this writing ….. could be accessed by only 
few readers, and lacked the degree of utility it entailed.38

Actually, the Réf lexions were published at the 
expenses of Carnot in a very limited number of copies 
so that later, in 1845, William Thomson (Lord Kelvin) 
found it impossible to find a copy despite his research at 
all booksellers in Paris [81]

I went to every book-shop I could think of, asking for the 
Puissance Motrice du Feu, by Carnot. ‘Caino? Je ne con-
nais pas cet auteur’ … ‘Ah! Ca-rrr- not! Oui, voici son 
ouvrage’, producing a volume on some social question 
by Hippolyte Carnot [Sadi’s brother]; but the Puissance 
Motrice du Feu was quite unknown.39

and he was initially acquainted with Carnot’s work 
only through Emile Clapeyron [82]

Having never met with the original work, it is only through 
a paper by M. Clapeyron, on the same subject, published 
in the Journal de l’École Polytechnique, Vol. xiv. 1834, and 
translated in the first volume of Taylor’s Scientific Memoirs, 
that the Author has become acquainted with Carnot’s The-
ory.40

In the next ten years after publication the book had 
a footnote citation in a treatise by Jean Victor Poncelet 
[83] where the analogy was made between the properties 
of gases and those of the caloric intended as a gas-like 
material.

Apart the biographical note prepared by his brother 
[12], Sadi was scarcely referenced also in books on the 
Carnot family and in other contexts. In a two-volume 

38 [7], authors’ translation of the French obituary.
39 [81], p. 458.
40 [82], p.100.

biography of the father [84], Hippolyte barely alluded 
to the Sadi’s work. The family history by Maurice Drey-
fous [85] reported primarily on Lazare, Hippolyte and 
Sadi, Hippolyte’s son, which was the fourth President 
of the Third Republic, murdered in 1894 by the Italian 
anarchist Sante Ieronimo Caserio. François Arago, secre-
tary for life of the Academy of Sciences, mathematician, 
physicist and politician wrote a historical note on steam 
engines [86] with the purpose of denying the thesis that 
the steam engine was entirely an English invention and 
emphasized the role of Denis Papin while completely 
ignoring Sadi’s contribution.

As regards the success of the Réflexions, it is of 
course necessary to consider the dual nature of Carnot’s 
work defined by Redondi [9] as “un défi théorique à la 
pratique” (a challenge of theory to practice) and to look 
at its reception both in the engineering environment and 
application and to its impact as a moment of foundation 
of the science of thermodynamics. The first aspect has 
been considered in detail by Redondi [9,87]. On the basis 
of a new documentation, reported as a group of annexes 
accompanying his work, Redondi brought to attention 
numerous explicit references to Carnot, also as explicit 
quotations of the Réflexions, in works by engineers and 
technicians, even though there is no evidence of prac-
tical applications of Carnot’s principles. In particular, 
Redondi mentions an Essai sur le machines à feu (1835) 
by M. Boucherot, an Emploi de l’air comme moteur and 
a Machine à air à effet alternative (1838) both by F. Bres-
son. These are all projects submitted to the Académie des 
Sciences. These, and other technological projects men-
tioned by Redondi, have the common purpose of pro-
posing the air at high pressures as an agent of thermal 
engines and therefore constitute a logical reference to 
the ideal air engine of the Réflexions. Of particular inter-
est may be the air engine proposed by Boucherot, the 
pyraéromoteur, a new variant of the pyreolophore pro-
posed by the Niepce brothers in 1800, an antecedent of 
the internal combustion engine, mentioned by Carnot in 
the Refléxions. Of course, Clapeyron, the first true com-
municator of Carnot’s ideas, was also an engineer but 
his interest in the Réflexions was not really technical. 
But this is another story [88-93] that is discussed in the 
ref. [47], p. 110-111.

6.1 the Clapeyron contribution to the diffusion of the Car-
not theory

It was in 1834 that a detailed exposition of the 
Réflexions appeared in the Journal de l’École Polytech-
nique by Clapeyron [63]. In the Mémoire sur la Puissance 
Motrice du Feu the verbal analysis of Carnot, sometimes 



88 Pier Remigio Salvi, Vincenzo Schettino

cumbersome, was substituted by the symbolism of the 
calculus and use was made of the indicator diagram of 
Watt, since then the familiar p-V diagram, to discuss the 
Carnot cycle. As the law relating pressure and volume 
in an adiabatic process was unknown to Clapeyron, the 
analysis was restricted to cycles with very small temper-
ature difference between isotherms. Making reference to 
Fig. 5(b) Clapeyron assumes that the two isotherms, 1 → 
2 and 3 → 4, are closely approaching each other at tem-
peratures t + dt and t (degrees centigrade), respective-
ly, and that the gas is allowed to expand, 1 → 2, and to 
compress, 3 → 4, by the volume increment/decrement dV. 
Due to the infinitesimal variations the two isotherms, 1 
→ 2 and 3 → 4, as well as the two adiabats joining them, 
2 → 3 and 4 → 1, are essentially parallel segments and the 
area of the minute 1234 parallelogram is the “quantity of 
action” [63], i.e., the work performed due to the absorp-
tion of heat dQ during the 1 → 2 isotherm. As a fervent 
calorist Clapeyron points out that

successive states which the same weight of gas experiences 
are characterized by the volume, the pressure, the tempera-
ture, and the absolute quantity of caloric which it contains: 
two of these four quantities being known, the other two 
become known as a consequence of the former41

Thus, the differential dQ may be defined as a func-
tion of p and V and the ratio between the “quantity of 
action” and dQ, which represents the maximum work 
for a unity of heat falling from t + dt to t, is determined 
by the expression

Rdt/[V(dQ/dV) – p(dQ/dp)]

where (dQ/dV) and (dQ/dp) are partial derivatives, the 
first at constant pressure and the second at constant vol-
ume. The constant R comes from the combination of the 
Boyle-Mariotte and Gay-Lussac laws for a given weight 
of an elastic fluid

pV/(267 +t) = p0V0/(267 + t0) = R

where p, V, t and p0, V0 and t0 are two different sets of 
values of pressure, volume and temperature and 1/267 
is the (then) measured reduction/magnification fac-
tor of volume (1/273.15, actual value) for 1oC lowering/
increasing at constant pressure. Through mathematical 
analysis the Q equation, Q = R(B – C ln p) with B and C 
unknown functions, is determined and, more important, 
the above defined ratio is found to be equal to dt/C. The 
Carnot principle says that C must depend only on tem-

41 [63] middle sect. II.

perature and not on the specific nature of the substance 
working in the cycle. The function B may in addition 
vary from gas to gas [63]. It follows that (1/C), called 
later the Carnot coefficient by Kelvin for its importance 
in the theory of heat and denoted by µ, is the maximum 
work due to a unit heat descending 1oC.

In another passage of the Mémoire, taking in con-
sideration the saturated vapor as working substance, 
Clapeyron was able to derive the now famous “Clapey-
ron equation”, a most remarkable fact in absence of the 
second law and the entropy concept. He observes that 
the maximum work performed with a unit input of heat 
when a liquid is vaporized in a cycle with infinitely close 
isotherms cannot be different from that obtained by 
any other substance between the same temperature lim-
its, which was already shown to be dt/C. The following 
equation is obtained

k = (1 – δ/ρ)·(dp/dt)C

where k is the latent caloric contained in the unit vol-
ume of vapor and δ and ρ are the vapor and liquid den-
sities. The comparison with the Clapeyron equation 
appearing in all textbooks of thermodynamics suggests 
that C coincides with the absolute temperature, a quan-
tity not yet defined at that time.

Clapeyron not only recovered the Carnot Réflexions 
from obscurity but also introduced the point of true 
weakness of the theory, hinting at the possibility of vis 
viva (i.e., kinetic energy) destruction for the special case 
of direct contact of two bodies at different temperatures

caloric passing from one body to another maintained at a 
lower temperature may cause the production of a certain 
quantity of mechanical action; there is a loss of vis viva 
whenever bodies of different temperature come into con-
tact42

The Mémoire was translated into English in 1837 
and into German in 1843, thus making the Carnot 
theory available for further analysis and development. 
As a mining engineer, Clapeyron was engaged in rail-
road engineering construction in France and abroad. 
Later Clapeyron was professor in the École des Ponts et 
Chaussées from 1844 to 1859 but alluded scarcely to the 
Mémoire in his courses and only briefly in 1847 in the 
scientific biography supporting his election to the Acad-
emy of Sciences [70].

42 [63], end sect. II.



89Sadi Carnot’s Réflexions and the foundation of thermodynamics

6.2 the Joule - Kelvin controversy and the approach to the 
second principle

A full recognition of the ideas contained in the Car-
not Réflexions was granted only by the two founders of 
the second principle, Lord Kelvin and Rudolf Clausius. It 
it is worthwhile to first refer shortly to the point of view 
of Lord Kelvin’s brother, James Thomson, about the Car-
not’s theory since it heavily influenced Kelvin’s ideas in 
the following years. According to James [94], heat and 
work are proportional to one another in the sense that 
a given quantity of heat produces a given quantity of 
work and vice versa but the two entities cannot intercon-
vert. It may help to go back to the waterfall analogy: as 
the fall from upper to lower height produces work with 
no loss of water, so the transfer of heat from high to low 
temperature produces work with heat conservation. This 
view was a source of strong debate, the two main actors 
being Joule, who in a series of experiments [95] in the 
years 1843-1844 had conclusively shown that work is con-
verted into heat at a fixed ratio, and Lord Kelvin. Indeed, 
the defective point in the Clapeyron report on the Carnot 
theory was caught with penetrating criticism by Joule

I conceive that this theory, however ingenious, is opposed 
to the recognized principles of philosophy, because it leads 
to the conclusion that vis viva may be destroyed by an 
improper disposition of the apparatus. Thus Mr. Clapeyron 
draws the inference that “the temperature of the fire being 
from 1000oC to 2000oC higher than that of the boiler, there 
is an enormous loss of vis viva in the passage of the heat 
from the furnace into the boiler” ([63], sect. VIII). Believ-
ing that the power to destroy belongs to the Creator alone, 
I entirely coincide with Roget and Faraday in the opinion 
that any theory which, when carried out, demands the 
annihilation of force, is necessarily erroneous43

The Joule’s idea about the steam engine, substantial-
ly coincident with the modern interpretation, was clearly 
expressed

the steam expanding in the cylinder loses heat in quantity 
exactly proportional to the mechanical force which it com-
municates by means of the piston and on condensation of the 
steam the heat thus converted into power is not given back.44

and led necessarily to the dramatic confutation of 
heat conservation, the basic principle of the caloric the-
ory:

the theory here advanced demands that the heat given 
out in the condenser shall be less than that communicated 

43 [95], p. 188.
44 [95], p. 189.

to the boiler from the furnace, in exact proportion to the 
equivalent of mechanical power developed.45

These considerations represent a turning point in 
the science of thermodynamics: the conversion of heat 
into work is apparently incompatible with the transmis-
sion of heat associated with the production of work. Kel-
vin knows Joule’s results but the first reaction is of oppo-
sition [82]

In the present state of science no operation is known by 
which heat can be absorbed, without either elevating the 
temperature of matter, or becoming latent and producing 
some alteration in the physical condition of the body into 
which it is absorbed; and the conversion of heat (or calor-
ic) into mechanical effect is probably impossible, certainly 
undiscovered.46

adding in the footnote that Joule has reported

… some very remarkable discoveries which he has made 
with reference to the generation of heat by the friction of 
fluids in motion … seeming to indicate an actual conver-
sion of mechanical effect into caloric. No experiment how-
ever is adduced in which the converse operation is exhib-
ited; but it must be confessed that as yet much is involved 
in mistery with reference to these fundamental questions of 
natural philosophy.47

Successively, Kelvin had a more cautious approach 
to Joule’s conclusions trying, in a long note of [96], to 
answer the core question about thermal engines; what 
happens when heat flows by conduction from the hot 
to the cold body or in other words when the thermal 
engine has zero mechanical effect?

When thermal agency is thus spent in conducting heat 
through a solid what becomes of the mechanical ef fect 
which it might produce? Nothing can be lost in the opera-
tions of nature – no energy can be destroyed. What effect 
is then produced in place of the mechanical effect which 
is lost ? A perfect theory of heat imperatively demands an 
answer to this question; yet no answer can be given in the 
present state of science. It might appear that the difficulty 
would be entirely avoided by abandoning Carnot’s funda-
mental axiom; a view which is strongly urged by Mr. Joule. 
If we do so, however, we meet with innumerable other dif-
ficulties, insuperable without further experimental investi-
gation, and an entire reconstruction of the theory of heat 
from its foundation.”48

45 [95], p. 189.
46 [82], p. 102.
47 [82], p. 102.
48 [96], note 7.



90 Pier Remigio Salvi, Vincenzo Schettino

As it is evident from these considerations, Kelvin 
maintained an open mind on the issue. On one hand he 
did not venture in the complete rejection of the Carnot’s 
theory for the above mentioned difficulties to adequately 
replace the caloric theory; on the other he brought to 
completion three major achievements, the calculation 
of the maximum work in the cycle as a function of the 
temperature [96], the successful proposal for the absolute 
scale of temperature [82] and the discovery of the pres-
sure dependence of the water freezing point [97,98], all 
of them representing brilliant results coming from the 
application of Carnot’s theory.

The key work [96]: Account of Carnot Theory of the 
Motive Power of Heat: with Numerical Results derived 
from Regnault Experiments on Steam, already in title 
indicates that the author not only reviews the original 
study but also provides a strong basis for the theory 
using the data on latent heat of vaporization and pres-
sure of saturated vapors collected by the great experi-
mentalist Victor Regnault. Two expressions are obtained 
for the mechanical effect due to “the transference of heat 
from one body to another at a lower temperature” ([96], 
paragraph 11) in an engine operating with steam or air. 
By the Carnot principle the maximum work M is the 
same in the two cases. If H units of heat are allowed to 
fall from the body A at temperature t + τ to B at t, the 
result is, in the Kelvin notation,

M = (1 – σ) (dp/dt)(1/k)Hτ = E[p0V0/(Vdq/dV)]Hτ

where the Hτ coefficient is denoted by µ and has the 
usual meaning of maximum work for a unit heat trans-
mitted from A to B with 1oC gap (measured by an air 
thermometer). Thus, µ is given by

µ = (1 – σ) (dp/dt)(1/k) = E[p0V0/(Vdq/dV)]

where the left expression is appropriate for the saturated 
steam (with σ the ratio of steam and water densities, k 
the latent heat of water vaporization per unit volume) 
and the right expression for air. Using Regnault data for 
(a) the pressure p of saturated steam in the range 0oC - 
230oC and (b) the latent heat of vaporization per unit 
weight in the same temperature range; and assuming 
that the density of the vapor follows the law of ordinary 
gases up to 100oC and beyond may be estimated from 
pressure data, µ was found from 0oC to 230oC. The coef-
ficient steadily diminishes increasing the temperature, 
consistently with the few scattered points obtained by 
Clapeyron using boiling water, sulphuric ether, alcohol 
and turpentine [63]. Kelvin was fully aware of the great 
generalization embodied in this calculation but, at the 

same time, he worried about its physical basis, empha-
sizing the request of experimental confirmation

in paragraph 30 some conclusions drawn by Carnot from 
his general reasoning were noticed; according to which 
it appears, that if the value of µ for any temperature is 
known, certain information may be derived with refer-
ence to the saturated vapor of any liquid whatever, and, 
with reference to any gaseous mass, without the neces-
sity of experimenting upon the specific medium considered. 
Nothing in the whole range of Natural Philosophy is more 
remarkable than the establishment of general laws by such 
a process of reasoning. We have seen, however, that doubt 
may exist with reference to the truth of the axiom on which 
the entire theory is founded, and it therefore becomes 
more than a matter of mere curiosity to put the inferences 
deduced from it to the test of experience.49

The second important point is concerned with a 
fundamental quantity like temperature which is expect-
ed to be defined in a general way rather than looking at 
specific properties of a substance, so as to make its defi-
nition independent of any kind of material [82]. On the 
basis of the Carnot theory the mechanical effect due to 
the transmission of heat from a hot to a cold body does 
not depend on the nature of the working medium but 
only on the temperatures of the two bodies. Further, the 
maximum work done by a unit heat falling 1oC is given 
by µ. From µ data on steam and few others on different 
substances [96], µ is found to decrease as the tempera-
ture, measured by the air thermometer, increases. The 
Kelvin proposal was that µ, rather than other physical 
properties, must be used to fix the temperature scale. 
The central point of the proposal is that a degree is 
defined by the amount of maximum work done by a unit 
heat falling down this degree, irrespective of the temper-
ature value. This is equivalent to say that µ becomes con-
stant through the whole temperature range. In Kelvin’s 
own words:

In M. Clapeyron paper various experimental data, confess-
edly very imperfect, are brought forward, and the amounts 
of mechanical effect due to a unit of heat descending a 
degree of the air-thermometer, in various parts of the scale, 
are calculated from them, according to Carnot’s expres-
sions. The results so obtained indicate very decidedly, 
that what we may with much propriety call the value of a 
degree (estimated by the mechanical effect to be obtained 
from the descent of a unit of heat through it) of the air-
thermometer depends on the part of the scale in which it 
is taken, being less for high than for low temperatures. The 
characteristic property of the scale which I now propose is 
that all degrees have the same value; that is, that a unit of 

49 [96], paragraph 41.



91Sadi Carnot’s Réflexions and the foundation of thermodynamics

heat descending from a body A at temperature To of this 
scale, to a body B at the temperature (T−1)o, would give out 
the same mechanical effect, whatever be the number T. This 
may justly be termed an absolute scale, since its character-
istic is quite independent of the physical properties of any 
specific substance.50

Finally, a curious question was raised by Kelvin, 
possibly representing a fatal argument to endanger Car-
not’s theory. It is known that water ices at 0oC under 
atmospheric pressure with volume expansion. In this 
process the latent heat is released while in the oppo-
site process, i.e., melting under the same conditions of 
temperature and pressure, an equal amount of heat is 
absorbed with volume contraction. Therefore, at least in 
principle, it may be thought of an ice engine in which 
heat does not flow from a hot to a cold body but between 
bodies at the same temperature, i.e. 0oC, with the result 
that “mechanical work would be given out without any 
corresponding expenditure” ([97] p.156). It was the 
brother James, who succeeded in showing that under 
pressure the melting point of water is lowered allowing 
Kelvin to escape from this impasse [97]. Thus, for an ice 
engine properly operated, it is necessary to run with the 
cold body at a temperature lower than 0oC otherwise 
the freezing process stops when freezing water starts to 
exert a pressure. The theoretical estimate of the tempera-
ture lowering with pressure was proposed considering a 
cyclic ice engine, analogous to the steam engine, which 
was working with the following steps:
1) isothermal (and isobaric) compression of ice at 0oC 

and 1 Atm until one cubic foot of water is obtained 
from ice, absorbing heat from a reservoir (“an indefi-
nite lake of water at 0oC”) ([97], p. 160);

2) adiabatic compression of the water/ice mixture to 
pressure pa(pounds/squarefoot) above that of the 
atmosphere. At the end of the process the tempera-
ture of the mixture is −t(oC);

3) isothermal (and isobaric) expansion causing the 
complete freezing of water and the heat release to a 
reservoir at − t(oC) (“a second indefinitely large lake 
at −t(oC)”) ([97], p.160). According to the caloric the-
ory, of which James Thomson was a follower, “con-
tinue the motion till all the heat has been given out to 
the second lake at −t(oC), which was taken in during 
Process 1 from the first lake at 0oC” ([97], p. 160);

4) adiabatic expansion to the original values of temper-
ature and pressure to close the cycle.
It should be noted that James Thomson predicted 

the temperature lowering with two independent strate-
gies and not making recourse to the laws of thermody-

50 [82], p. 104.

namics. In the first the work is calculated considering 
the area enclosed by the cycle in the p-V diagram, i.e., 
pa·(Vice– Vwater); in the second, being known to James 
the thermal units Q to melt one cubic foot of ice and the 
value of µ at 0oC, the same work was calculated as the 
product Q·µ·t. The final expression is [97]

t = 0.0075n

where t (degrees centigrade) is the lowering of the water 
freezing point with respect to 0oC and n is the pressure 
(atmospheres) above one atmosphere. The theoretical 
estimate was confirmed by the experimental measure-
ments performed by Kelvin [98]. It may be concluded 
that the validity of the Carnot theory was supported 
also by the discovery of an unsuspected new physical 
effect and the admiration of Kelvin for this result was 
expressed by words which go beyond the brotherhood 
relation

In this very remarkable speculation, an entirely novel 
physical phenomenon was predicted in anticipation of any 
direct experiments on the subject; and the actual observa-
tion of the phenomenon was pointed out as a highly inter-
esting object for experimental research.51

6.3 the second principle of thermodynamics: the final state-
ments

Given the circumstances, it may be conjectured that 
a critical revision of the Carnot theory was not a pri-
mary objective for Kelvin. It was Clausius in a histori-
cal paper [64] that conclusively solved the problem of the 
Joule – Carnot antinomy at the expenses of the principle 
of heat conservation. While Kelvin sees insurmountable 
difficulties if the caloric theory is abandoned, Clausius 
in a quite illuminating passage of the paper states:

I believe, nevertheless, that we ought not to suffer ourselves 
to be daunted by these difficulties; but that, on the contrary, 
we must look steadfastly into this theory which calls heat a 
motion, as in this way alone can we arrive at the means of 
establishing it or refuting it. Besides this, I do not imagine 
that the difficulties are so great as Thomson considers them 
to be; for although a certain alteration in our way of regard-
ing the subject is necessary, still I find that this is in no 
case contradicted by proved facts. It is not even requisite to 
cast the theory of Carnot overboard; a thing difficult to be 
resolved upon, inasmuch as experience to a certain extent 
has shown a surprising coincidence therewith.52

51 [98] p. 165.
52 [64], p. 3.



92 Pier Remigio Salvi, Vincenzo Schettino

The option by means of which Joule’s conversion of 
heat to work and Carnot transmission of heat from a hot 
to a cold body are reconciled is rejection of heat conser-
vation. However, a question remains: how can the Car-
not principle still be valid if the caloric theory is “cast 
overboard”? According to Clausius, the production of 
work in a thermal engine is due to the transmission of 
heat from a warm body A to a cold body B with heat 
consumption. Following Carnot, the maximum work is 
obtained if the two bodies never come in contact each 
with the other (in our terms, if the cycle is reversible). 
Reversing the engine, i.e., by consumption of the max-
imum work, heat is transferred from B to A. Alternat-
ing the direct and reverse cycles the work production 
(direct) and consumption (reverse) are equal. The same 
may be repeated for the heat consumption and produc-
tion. The two bodies go back to the initial conditions 
and no total work is done. Let us now consider two dif-
ferent working substances K and K’ with the former 
producing a larger amount of maximum motive power. 
Equivalently, we may assume that if the two substances 
develop the same amount of work, K’ transfers from A 
to B a larger amount of heat, QB’, than K, QB. Operat-
ing the engine with K and K’ in the direct and reverse 
cycle, respectively, works are cancelled but B will trans-
fer in the reverse operation more heat to A than received 
in the direct operation. In conclusion, an amount of heat 
QB’-QB is passed from a body at low temperature to a 
body at high temperature without any other change

Hence by repeating both these alternating processes, with-
out the expenditure of force or other alteration whatever, 
any quantity of heat might be transmitted from a cold body 
to a warm one; and this contradicts the general deportment 
of heat, which everywhere exhibits the tendency to annul 
differences of temperature, and therefore to pass from a 
warmer body to a colder one.53

This constitutes the first historical statement of the 
second principle of thermodynamics. As a consequence, 
the Carnot principle is justified even if the principle 
of heat conservation does not hold anymore. With its 
elimination, other concepts such as “latent heat” and 
“total heat of a body” must be dismissed or critically 
revised. The “latent heat” of vaporization, for instance, 
had in the old theory the meaning of caloric fluid sur-
rounding the particles of vapor as if a composite parti-
cle was formed. According to Clausius heat actually dis-
appears and is converted into the expansion work from 
liquid to vapor

53 [64], p.103.

… we can form a notion as to the light in which latent heat 
must be regarded. Referring again to the last example [the 
liquid – vapor transition] we distinguish in the quantity 
of heat imparted to the water during the change the sensi-
ble and the latent heat. Only the former of these, however, 
must we regard as present in the produced steam; the sec-
ond is, not only as it name imports, hidden from our per-
ception, but has actually no existence; during the alteration 
it has been converted into work.54

As to the “total heat of a body”, i.e., the sum of the 
sensible and latent heat, this property is dependent, 
according to the caloric theory, on the parameters which 
characterize the state of the body. It follows that, going 
from one state to another and then back to the original, 
the total heat is zero. On the contrary, Clausius argues 
that during the cyclical transformation work may be 
done or absorbed by the body and the total work may 
not be necessarily equal to zero, as it is indicated by the 
occurrence of volume change in the body. This work 
must correspond to a well defined amount of heat, on 
the basis of the Joule principle of equivalence.

Clausius summarized the theory of heat by means of 
the two principles [64, 99]:

1) in all cases where work is produced by heat, a quantity 
of heat proportional to the work done is consumed; and 
inversely, by the expenditure of a like quantity of work, 
the same amount of heat may be produced.55

2) heat cannot by itself pass from a colder to a warmer 
body.56

Kelvin acknowledged the dynamical theory of heat 
one year later [100]. From his point of view the two basic 
propositions are

1) When equal quantities of mechanical effect are produced 
by any means whatever from purely thermal sources, or 
lost in purely thermal effects, equal quantities of heat are 
put out of existence or are generated.57

2) It is impossible, by means of inanimate material agency, 
to derive mechanical effect from any portion of matter 
by cooling it below the temperature of the coldest of the 
surrounding objects.58

The first proposition is essentially the Joule principle 
of equivalence, as it is in the Clausius statement. As to 
the second, it is a fair acknowledgment to declare Clau-
sius’ priority:

54 [64], p. 5.
55 [64], p. 4.
56 [99], p. 45.
57 [100], p. 178.
58 [100], p. 181.



93Sadi Carnot’s Réflexions and the foundation of thermodynamics

It is with no wish to claim priority that I make these state-
ments, as the merit of first establishing the proposition 
upon correct principles is entirely due to Clausius.59

At the same time it is a point of honor to say

I may be allowed to add, that I have given the demonstra-
tion exactly as it occurred to me before I knew that Clausi-
us has either enunciated or demonstrated the proposition.60

and to note that the two formulations of the second prin-
ciple are different only in the form, either of them being 
a consequence of the other. The crucial argument of the 
Kelvin second proposition is that heat absorbed cannot 
be integrally converted to work performed in a cyclic pro-
cess. Suppose that a thermal engine is operating between 
temperatures t1 and t2, being t1 > t2, and both higher than 
the temperature t0 of the coldest of the surrounding bod-
ies, for the sake of clarity the environment. The amount of 
heat delivered to the body at t2 is wasted unless it acts as 
input heat in a second thermal engine operating between 
t2 and t3, being t2 > t3 and both still higher than t0. It is a 
result of the Kelvin enunciation that this step-by-step con-
version of heat to work may proceed until the temperature 
of the environment is reached and that the work produced 
is Q1 + Q0, where Q0 is the (negative) amount of heat deliv-
ered to the environment. As the proposition may be of no 
immediate comprehension, it was exemplified by a note

“If this axiom be denied for all temperatures, it would have 
to be admitted that a self-acting machine might be set to 
work and produce mechanical effect by cooling the sea or 
earth, with no limit but the total loss of heat from the earth 
and sea or, in reality, from the whole material world”.61

Probably, Kelvin was motivated to reformulate the 
principles of Clausius to express his own ideas in his 
own way on the issue. In the second place the enuncia-
tion contains the Kelvin answer to the difficult question 
concerning the sort of the mechanical effect which does 
not appear when the two bodies are put in direct ther-
mal contact. The work is “ irrecoverably lost to man, and 
therefore wasted although not annihilated”.62

7. CONCLUSIONS

A question that has frequently been debated in the 
history of science and technology is whether science has 

59 [100], p. 181.
60 [100], p. 181.
61 [100], p. 181.
62 [100], p. 189.

been the driving force behind technological development 
or whether, on the contrary, the development of technol-
ogies has been the stimulus for new scientific knowledge. 
Even if the question, posed in this way, appears too sche-
matic, it has aroused the interest of many scientists and 
historians of science. A case in point is the statement by 
Lawrence Joseph Henderson (1878- 1942), reported by 
Charles Coulston Gillispie [101] that:

Science owes more to steam engine than steam engine owes 
to science.63

On the other side, Hermann von Helmholtz is more 
cautious on the immediate or direct transfer of scientific 
findings to technology [102]:

Whoever in the pursuit of science seeks after immediate 
practical utility may rest assured that he seeks in vain.64

but Ludwig Boltzmann [103], and others as well [104], 
seem more convinced of the primacy of science:

There is nothing more practical of a good theory.

In such a hypothetical dispute Carnot and his 
Réflexions place themselves in an intermediate and more 
balanced position. In fact, as we have already discussed, 
even if Carnot’s initial inspiration is derived from the 
consideration of technology and the practice of steam 
engines, an object that is so eminently technological, its 
line of reasoning is anchored on a logical and principle 
level. So much so that, even with long induction times, 
the work of Carnot has influenced and oriented the defi-
nition of the principles of thermodynamics rather than 
an immediate improvement of the thermal machines.

The truly extraordinary aspect of the work of Car-
not is that, although starting from a theory of heat that 
already known results and subsequent experiments 
would have proved wrong, has led to the identification 
of extremely fruitful principles for the elaboration of the 
theory of thermodynamics.

ACKNOWLEDGMENT

The authors wish to thank Dr. Cristina Gellini for 
figure drawings and suggestions.

63 [101], p. 357.
64 [102], p. 93.



94 Pier Remigio Salvi, Vincenzo Schettino

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