Abstract
The mathematical footnote in Réflexions sur la puissance motrice du feu can be considered the highest mathematical level presented by Sadi Carnot in his book. It also represents his way of mathematically interpreting physical phenomena and the relationship between physics and mathematics in thermodynamics . Without proposing new concepts or sophisticated functions in trying to explain the modernity of the result obtained by Sadi Carnot , here we will present an historical analysis of the footnote. Moreover, strictly based on primary sources and documents, we will provide an epistemological interpretation of the arguments that Sadi Carnot used to calculate the formula of the efficiency of a heat machine ; and centring on the role of shared knowledge, of challenging objects, and of knowledge reorganization. The idea is presenting an integrated history and epistemology of scientific methods which combine epistemological and historical approaches to identify significant historical hypotheses. Such epistemological interpretations are subjected to historical facts of scientific activity and original documents in order to trace their historical development. In this case the discussion regards Carnot’s mathematical foundations, considering both the relationship between physics and mathematics and the mathematical (infinitesimal) analysis of the nineteenth century.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
- 2.
Thomson read it on 30 April, 1849. The content is discussed in an Appendix by the author (Thomson [1890] 1943, p 179, line 3; see also: Id., 1852, XXXV, 1882–1911, I, pp 113–155; et al. editions).
- 3.
A meaningful study on the experimental data used by Sadi Carnot is available in Fox (1988), pp 288–299.
- 4.
In 1827 César–Mansuète Despretz (1798?–1863) showed the uncertainness of the Clément and Desormes law (Fox 1988, ft 26, p 301).
- 5.
Laplace resumed Gay–Lussac and Welter ’s experiments in a part of his work. In particular, he properly focused on the constancy of γ (Laplace 1822, ft 18, p 268).
- 6.
We let note that the manuscripts edited by Gauthier –Villars (Carnot 1878b) is not always integrally reproduced. For any complete English consulting of the Sadi Carnot’s manuscripts, please see Fox (1986).
- 7.
Tait (1877).
- 8.
Carnot (1943), p 6, line 27. (Author’s wording and quotation marks).
- 9.
The estimate value proposed by Sadi Carnot in his Notes sur les mathématiques, la physique et autres sujets (folio 7v; see also Carnot 1878b, pp 94–95; Fox 1986, pp 191–192) for the mechanical equivalent of heat of 370 Kg/cal is not so much farther from the correct accepted value (427) than Mayer’s later determination (365). The difference is only 11%. (See below Chapter 10; Hoyer 1975; Edmunds 1902, p 127).
- 10.
Robert Fox in Carnot (1986), p 2, line 9. (Author’s italic).
- 11.
- 12.
Hoyer in Taton (1976), pp 221–228. He showed, by using the cycles , that S. Carnot ’s caloric theory and the modern theory, coincide for an infinitesimal process: that is \( \Delta t \to 0 \). That is correct in the first and second approximation in thermodynamic theory.
- 13.
We will use a modern notation in comparison to that used by Lervig .
- 14.
We will also discuss in the following chapters the details proposed by Reech on Carnot’s theory.
- 15.
Carnot (1978), p 73, ft 1, line 1.
- 16.
Carnot (1978), p 38, line 4. (Author’s italic).
- 17.
Carnot (1978), p 77, ft 1, line 1.
- 18.
Ivi, p 77, ft 1, line 15.
- 19.
Carnot (1978), p 78, ft 1, line 13.
- 20.
Carnot (1978), p 79, line 1, op. cit.
- 21.
Carnot (1943), “Foreword” [by Thurston] line 1.
- 22.
Carnot (1978), p 39, line 7.
- 23.
Poisson proposed another formula.
- 24.
Carnot (1978), p 66, line 7.
- 25.
The first law is usually formulated by saying that the change in the internal energy of a closed thermodynamic system is equal to the difference between the heat supplied to the system and the amount of work performed by the system on its surroundings.
- 26.
Briefly, if a physical system received a certain amount of heat, the same amount of work has been done by the system. Thus the total variation of the internal energy after a cycle is null.
- 27.
The first law of thermodynamics makes no distinction between processes that occur spontaneously and those that do not. The second law of thermodynamics establishes which processes do and which do not occur.
- 28.
- 29.
- 30.
Cfr.: Gillispie and Youschkevitch (1979), pp 251–298, § 13, p 256, ft 1, p 182.
- 31.
Carnot (1832), p 1. [“Je cherche à savoir en quoi consiste le véritable esprit de l'Analyse infinitesimal” (Id., 1813, p 1, line 1)]. Let us note that the English translation of the 1832 version is differently organized in comparison to the original 1813 version. E.g., the number of paragraphs in the two versions does not correspond.
- 32.
See also its English translations “[…] creatures of reason […]” (Carnot 1832, p 105, line 13).
- 33.
- 34.
- 35.
- 36.
- 37.
- 38.
Berkley ([1734] 2002), sect. XXXV, p 18, line 9.
- 39.
Berkeley ’s lemma claims: “If with a View to demonstrate any Proposition, a certain Point is supposed, by virtue of which certain other Points are attained; and such supposed Point be itself afterwards destroyed or rejected by a contrary Supposition; in that case, all the other points, attained thereby and consequent thereupon, must also be destroyed and rejected, so as from thence forward to be no more supposed or applied in the Demonstration”. (Berkeley 1734, sect. XII–XIV; See also: Berkeley [1734] 2002). Let us consider the function f(x) = x 2; let us pass to the calculation of its derivative: the increment of the variable is dx (which at that time was not distinguished by \( \Delta x \)) and the increment of the function is df (or \( \Delta f \)), that is equal to: df = f(x + dx) – f(x) = (x + dx) 2 – x 2 = x 2 + 2xdx + dx 2 – x 2 = 2xdx + dx 2. The last term is then suppressed, since it is much smaller compared to the others. Dividing df by dx we obtain 2x, which is exactly the derivative the function of the function. Berkeley applied his lemma, pointing out that at the beginning of the calculation \( dx \ne 0 \) has been set; however, subsequently the calculation eliminates the infinitesimal of higher order at dx, considering it equal to zero. Therefore, either dx is zero, as stated at the end of the reasoning – but then the entire calculation is devoid of content – ; or it is not zero, as stated in the middle of the reasoning – but this goes against the aforementioned lemma. This criticism allows Berkeley to suggest an original interpretation of the efficacy of this type of calculation. At the beginning, \( dx \ne 0 \) is set and this, Berkeley claims, constitutes the first error because df as dx does not represent a specified number: nor does zero, nor a number different from zero. So there is no reason to include it in the only calculation that we know very well: elementary algebra. Continuing with the calculation, the last term is suppressed – which we know to be different from zero (otherwise we would have already eliminated it along with all of the dx) – and this clearly represents an algebraic error. Since the first result is correct, this second error cancels out the first one. (Cfr.: Drago 2007a, b; see also Parigi (2010)).
- 40.
- 41.
Carnot (1813), “Note”, p 242, line 7.
- 42.
Ivi, p 243, line 9.
- 43.
- 44.
- 45.
Please refer to Lazare Carnot ’s quotations. The wording seems identical.
- 46.
- 47.
- 48.
Carnot (1813), p 252, line 18.
- 49.
Carnot (1978), p 10, line 2.
- 50.
Ivi, p 25, line 7.
- 51.
Ivi, 18, ft 1, line 1.
- 52.
Carnot (1978), pp 39–40.
- 53.
Ivi, pp 29–38.
- 54.
Martin J. Klein in Taton (1976), p 214, line 17.
- 55.
Very briefly we mention the ancient Moscow Mathematical Papyrus (also called the Golenischev Mathematical Papyrus at Pushkin State Museum of Fine Arts in Moscow), where it remains today demonstrating knowledge of a formula for the volume of a pyramidal frustum. Greek Eudoxus (ca. 370 BC), for areas and volumes by breaking them up into an infinite number of shapes for which the area or volume was known. Further advancements were made in the early seventeenth century by Isaac Barrow (1630–1677) and by Torricelli (Pisano and Capecchi 2010; Capecchi and Pisano 2010a) who provided hints of a connection between integration and differentiation. One of the first Barrow proofs of the fundamental theorem of calculus was provided by Barrow and it is known that Wallis generalized Cavalieri ’s method (Cavalieri 1635).
- 56.
Nevertheless, we agree with the current secondary literature of the history of mathematics that a major advance in integration appeared in ca. the seventeenth century: an independent discovery of the fundamental theorem of calculus by Newton and Leibniz . Here, a strictly connection between integration and differentiation was proposed. Of course, for our aim, and equal in historical importance, is the comprehensive mathematical framework that both Newton and Leibniz developed, since we mainly take in account mathematical analysis calculus in the history until XIX century.
- 57.
On the first page of the manuscript presented by Hippolyte Carnot to the Académie des sciences on 16 December 1878, one can see (Carnot 1878a, folio 1) Sadi Carnot avoided the ancient use of the word “feu” (fire) in his book (see details Chapters 6 and 7 above). Nevertheless something remained in the book, e.g.: “[…] machine à vapeur de la machine à feu en général […]”. (Carnot 1978, p 8, ft 1, line 1).
- 58.
Carnot (1978), p 8, ft 1, line 1.
- 59.
Robert Fox in Carnot (1986), p 147, En. 78, line 1. (Authors’ quotations).
- 60.
Carnot (1978), p 66, line 10.
- 61.
Martin J. Klein in Taton (1976), p 214, line 19.
- 62.
Einstein (1949 [1970]) vol. VII, p 33; also quoted in: Klein (1967), pp 509–516.
- 63.
Galileo (1890–1909), II, pp 147–191. For a recent review of Le Mecaniche see Romano Gatto ’s work (Galileo 2002) and, of course indispensable Stillmann Drake’ works on the Galileo (e.g., Drake 1973, 1985, 2000; Drake and Drabkin (1969)). See also an interesting French version, of the les mécaniques de Galilée by Egidio Festa and Sophie Roux (forthcoming). In regard to Le Meccaniche, Gatto comments both versione breve (4 manuscripts, Galileo 2002, pp 3–42) that versione lunga (14 manuscripts, Ivi, pp 43–154). The manuscript edited by Antonio Favaro (1847–1922) in Opere nazionali di Galileo Galilei is the versione lunga. It is composed of 10 manuscripts at his time known (Galileo 1890–1909, II, 155–190). In the versione lunga one can see the effort cited in the running text. For recent works concerning Galilei’s method and his mathematical approach one can consult: Pisano (2008, 2009a, b, c; Pisano and Bussotti (2012)).
- 64.
- 65.
[available via http://www.maths.tcd.ie/pub/HistMath/People/Berkeley /Analyst]
- 66.
[also available in .pdf via: http://www.brera.unimi.it/sisfa/atti/index.html]
- 67.
The manuscript Notes sur les mathématiques, la physique et autres sujets is also conserved at Archives of the Académie des Science–Institut de France, Paris. Thus, we used “Carnot S 1878a” to cite both of the two original manuscripts studied. The difference in the running text is presented by the titles of the two manuscripts. We let note that the manuscripts edited by Gauthier –Villars (Carnot 1878b) are not always integrally reproduced. For any complete consulting of the Sadi Carnot’s manuscripts, please see Fox (1986).
- 68.
[also available in .pdf via: http://www.brera.unimi.it/sisfa/atti/index.html]
- 69.
[available in .pdf via: International Galilean Bibliography , Istituto e Museo di Storia delle Scienze. Firenze: http://biblioteca.imss.fi.it/]
References
Berkeley G (1734) The Analyst Dublin, Fuller S, Globe–in–Meath–Street, and J Leathly, Sect: XII–XIV
Berkeley G ([1734] 2002) The analyst, by Wilkins DFootnote
[available via http://www.maths.tcd.ie/pub/HistMath/People/Berkeley /Analyst]
Callen H (1974) Thermodynamics as a science of symmetry. Found Phys 4:423–440
Callendar HL (1910) Caloric theory of heat and Carnot’s principle. Proc Phys Soc Lond 23(1):153–189
Capecchi D, Pisano R (2007) La teoria dei baricentri di Torricelli come fondamento della static. Physis XLIV(1):1–29
Capecchi D, Pisano R (2008) La meccanica in Italia nei primi anni del Cinquecento. Il contributo di Niccolò Tartaglia. In: Tucci (ed) Proceedings of XXV SISFA Congress, Milano, pp C17.1–C17.6 Footnote
[also available in .pdf via: http://www.brera.unimi.it/sisfa/atti/index.html]
Capecchi D, Pisano R (2010a) Reflections on Torricelli’s principle in mechanics. Organon 42:81–98
Capecchi D, Pisano R (2010b) Scienza e Tecnica nell’Architettura del Rinascimento. Cisu, Roma
Carnot L (1780) Mémoire sur la théorie des machines pour concourir au prix que l’Académie Royale des Sciences de Paris doit adjuger en 1781. The manuscript is dated from Béthune 15 July 1780. It is conserved in the Archives de l’Académie des sciences, Institut de France, and consists of 191 sections in 106 folios. Sections 101–160 are reproduced. In: Gillispie 1971, Appendix C, pp 299–343
Carnot L (1785) Dissertation sur la théorie de l’infini mathématique, ouvrage destine it concourir au prix qu’a propose L’Académie Royale des Sciences, arts et belles–lettres de Berlin, pour l’année 1786 [The manuscript is dated by Arras 8 Sept 1785]
Carnot L (1786) Essai sur les machines en général. Defay, Dijon
Carnot L (1813a) Réflexions sur la métaphysique du calcul infinitésimal. Courcier, Paris
Carnot L (1813b) Note. In: Carnot L (ed), pp 213–257
Carnot S (1824) Réflexions sur la puissance motrice du feu sur les machinés propre à développer cette puissance. Bachelier, Paris
Carnot L (1832) Reflexions on the methaphysical princiles of the infinitesimal analysis (trans: Browell WR, Parker et al.), Oxford
Carnot S (1878a) Réflexions sur la puissance motrice du feu sur les machinés propre à développer cette puissance.Footnote
The manuscript Notes sur les mathématiques, la physique et autres sujets is also conserved at Archives of the Académie des Science–Institut de France, Paris. Thus, we used “Carnot S 1878a” to cite both of the two original manuscripts studied. The difference in the running text is presented by the titles of the two manuscripts. We let note that the manuscripts edited by Gauthier –Villars (Carnot 1878b) are not always integrally reproduced. For any complete consulting of the Sadi Carnot’s manuscripts, please see Fox (1986).
Archives of the Académie des Science–Institut de France, ParisCarnot S (1878b) Réflexions sur la puissance motrice du feu sur les machinés propre à développer cette puissance. Gauthier-Villars, Paris
Carnot S (1943) Reflections on the motive power of heat and on machines fitted to develop this power. In: Thurston RH (ed). Waverly Press, Inc., Baltimore
Carnot S (1953) Réflexions sur la puissance motrice du feu et sur les machines propres a développer cette puissance. Blanchard, Paris
Carnot S (1978a) Réflexions sur la puissance motrice du feu sur les machinés propre à développer cette puissance, édition critique par Fox Robert. Vrin J, Paris
Carnot S (1978b) Recherche d’une formule propre à représenter la puissance motrice de la vapeur d’eau. In: Carnot S (ed), pp 223–234
Carnot S (1986) Reflexions on the motive power of fire: a critical edition with the surviving scientific manuscripts (trans and ed: Robert Fox). The Manchester University Press, Manchester
Carnot S Notes sur les mathématiques, la physique et autres sujets. In: Carnot S (ed) Réflexions sur la puissance motrice du feu sur les machinés propre à développer cette puissance. Gauthier-Villars, Paris, pp 89–102, 1878b [The manuscript is conserved In: Carnot S (ed) Réflexions sur la puissance motrice du feu sur les machinés propre à développer cette puissance. Archives of the Académie des Science–Institut de France, Paris, 1878a]
Cavalieri B (1635) Geometria Indivisibilius continuo rum nuova quadam ratione promota. Ferrone, Bologna
Commandino F (1565) Federici Commandini Vrbinatis Liber de centro gravitatis solido rum. Ex Officina Alexandri Benacii, Bononiae
Cropper HW (1987) Carnot’s function: origins of the thermodynamic concept of temperature. Am J Phys 55(2):120–122
Delaroche F, Bérard JE (1813) Mémoire sur la détermination de la chaleur spécifique des différens gaz. Ann Chim Phys 85(72–110):113–182
Descartes R (1897–1913) Œuvres de Descartes. Par Charles Adams et Paul Tannery, Paris, 12 vols; also see Discours de la méthode et Essais, Specimina philosophiae. vol VI; Physico–mathematica vol X, Le Monde ou Traité de la lumière, vol XI (Id, 1964–1974 by Rochot B, Costabel P, Beaude J et Gabbery A, Paris)
Drago A (2007a) The genius of infinitesimal analysis: Berkeley’s criticism and its generalization by Lazare Carnot. Speech In: International conference George Berkeley: religion and science in the age of the enlightenment, international Berkeley society and University of Cassino, Gaeta, 26th–26th, September
Drago A (2007b) There exist two models of organization of a scientific theory. Atti della della Fondazione Ronchi 62(6):839–856
Drago A, Pisano R (2000) Interpretazione e ricostruzione delle Réflexions di Sadi Carnot mediante la logica non–classica. Giornale di Fisica 40:195–217
Drago A, Pisano R (2005) La nota matematica nelle Réflexions sur la puissance motrice du feu di Sadi Carnot: interpretazione del calcolo col metodo sintetico. Quad Stor della Fisica–Giornale di Fisica 13:37–57
Drago A, Pisano R (2007) La novità del rapporto fisica–matematica nelle Réflexions di Sadi Carnot. Atti della Fondazione Giorgio Ronchi 62(4):497–525
Drake S (1973) Galileo’s experimental confirmation of horizonal inertia. Isis 64:291–305
Drake S (1985) Galileo’s accuracy in measuring horizontal projections. Ann Ist Museo di Storia dlle Scienze di Firenze 10(1):3–14
Drake S (2000) Two new sciences. A history of free fall, Aristotle to Galileo. Wall and Emerson, Toronto
Einstein A (1949 [1970]) Autobiographical notes. In: Schilpp PA (ed) Albert Einstein: philosopher–scientist, library of living philosophers, vol VII. The Cambridge University Press, London
Fox R (1988) Les Réflexions sur la puissance motrice du feu de Sadi Carnot et la leçon se leur éditions critique. La vie des sciences – Comptes rendus, série générale 5/4:283–301
Galilei G (1890–1909) Le opere di Galileo Galilei: Edizione nazionale sotto gli auspici di sua maestà il re d’Italia. In: Favaro A (ed) Barbera, Firenze
Galilei G (1954) Dialogues concerning two new sciences (trans: Crew H, de Salvio A and an introduction by Favaro A). Dover Publications INC, Mineola
Galilei G (2002) Galileo Galilei. Le Mecaniche. Gatto R (ed). Olschkil, Firenze
Gillispie CC (1971) Lazare Carnot Savant. A monograph treating Carnot’s scientific work, with facsimile reproduction of his unpublished writings on mechanics and on the calculus, and an essay concerning the latter by Youschkevitch AP. Princeton University Press, Princeton
Gillispie CC (1976) The scientific work of Lazare Carnot, and its influence on that of his son. In: Taton (ed) (1976), pp 23–33
Grattan-Guinness I (1970) Berkeley’s criticism of the calculus as a study in the theory of limits. Janus 56:215–227
Hoyer U (1975) How did carnot calculate the mechanical equivalent of heat? Centaurus 19:207–219
Klein F (1967) Development of mathematics in the nineteenth century (trans: Ackerman M). Math SciPress, Brokline
Klein MJ (1976) Closing the Carnot cycle. In: Taton (ed), pp 213–219
Lagrange JL (1797) Théorie des fonctions analytiques contenant les principes du calcul différentiel, dégagés de toute considération d’infiniment petits, de limites et de fluxions, et réduits à l’analyse algébrique des quantités finies. Journal de l’École Polytechnique, III/9: 91–93
Lagrange JL (1806) Leçons sur le calcul des fonctions, Nouvelleth edn. Courcier, Paris
Laplace PS (1822) Notes sur la vitesse du son. Ann Chim Phys XX:266–268
Lavoisier AL (1789 [1937]) Traité élémentaire de Chimie. Gauthier–Villars, Paris
Lervig P (1972) On the structure of Carnot’s theory of heat. Arch Hist Exact Sci 9(2):222–239
Lervig P (1976) The existence of work function in Carnot’s theory. In: Taton (ed), pp 199–212
Mayer RJ (1842a) Bemerkungen Über die Kräfte der unbelebten Natur. Annalen der Chemie und Pharmazie 42:233–40
Mayer RJ (1842b) Bemerkungen Über die Kräfte der unbelebten Natur. In: Weyrauch JJ (ed) Die Mechanik Warme, Stuttgart
Montbrial AL (1976) Interprétation économiques suggérées par le théorème de Carnot. In: Taton (ed), pp 333–335
Pappus from Alexandria (1588) Mathematicae collectiones, by Federico Commandino. Pesaro. See also: Pappus of Alexandria, Matematical collections
Pisano R (2001) Interpretazione della nota matematica nelle Réflexions sur la puissance motrice du feu di Sadi Carnot. In: Proceedings of XX SISFA Congress, Osservatiorio Astronomico di Capodimonte, Napoli, pp 205–230Footnote
[also available in .pdf via: http://www.brera.unimi.it/sisfa/atti/index.html]
Pisano R (2008) Il ruolo della scienza meccanica nella progettazione degli architetti e degli ingegneri del Rinascimento (The role played by mechanical science in the architects and engineers in the Renaissance), PhD dissertations, Roma, 2 vols. University of Roma la Sapienza, RomaFootnote
[available in .pdf via: International Galilean Bibliography , Istituto e Museo di Storia delle Scienze. Firenze: http://biblioteca.imss.fi.it/]
Pisano R (2009a) On method in Galileo Galilei’ mechanics. In: Hunger H (ed) Proceedings of ESHS 3rd conférence. Austrian Academy of Science, Vienna, pp 147–186
Pisano R (2009b) Continuity and discontinuity. On method in Leonardo da Vinci’ mechanics. Organon 41:165–182
Pisano R (2009c) Il ruolo della scienza archimedea nei lavori di meccanica di Galilei e di Torricelli. In: Giannetto E, Giannini G, Capecchi D, Pisano R (eds) Da Archimede a Majorana: La fisica nel suo divenire. Proceedings of XXVI SISFA congress, Guaraldi Editore, Rimini, pp 65–74
Pisano R (2011a) On Lazare and Sadi Carnot. A synthetic view of a historical epistemological research program. In: Proceedings of XXX SISFA congress, Urbino (in press)
Pisano R (2011b) Historical reflections on the physics mathematics relationship in electromagnetic theory. In: Barbin E, Pisano R (eds) The dialectic relation between physics and mathematics in the XIXth century, Springer, Dordrecht, in press
Pisano R, Capecchi D (2010) On Archimedean roots in Torricelli’s mechanics. In: Ceccarelli M, Paipetis S (eds) Proceedings of the genius of archimedes. Springer, Dordrecht, pp 17–27
Pisano R, Gaudiello I (2009b) On categories and scientific approach in historical discourse. In: Hunger H (ed) Proceedings of ESHS 3rd conference. Austrian Academy of Science, Vienna, pp 187–197
Poisson SD (1823a) Sur la chaleur de gaz et des vapeurs. Ann Chim Phys XXXIII:337–352
Poisson SD (1823b) Sur la vitesse du son. Ann Chim Phys XXIII:5–16
Reech F (1853) Théorie général des effets dynamiques de la chaleur. Journal de Mathématiques pures et appliquées XVIII:357–378
Taton A (ed) (1976) Sadi Carnot et l’essor de la thermodynamique, Table Ronde du Centre National de la Recherche Scientifique. École Polytechnique, 11–13 Juin 1974. Éditions du Centre National de la Recherche Scientifique, Paris
Thomson W (1852) An account of Carnot’s theory of the motive power of heat with numerical results deduced from regnault’s experiments on steam. Annales de Chime et de Physique XXXV:248–255
Thomson W (1882–1911) An account of Carnot’s theory of the motive power of heat with numerical results deduced from Regnault’s experiments on steam. Mathematical and physical papers by Sir William Thomson. The Cambridge University Press, Cambridge, I:113–155
Thomson W ([1890], 1943) Carnot’s theory of the motive power of heat with numerical results deduced from regnault’s experiments on steam. Thurston 1943:127–204
Truesdell CA (1970) The tragicomedy of classical thermodynamics. Springer, Berlin
Truesdell C (1980) The tragicomical history of thermodynamics. 1822–1854. Springer, Berlin
Truesdell CA, Bharatha S (1977) The concepts and logic of classical thermodynamics as a theory of heat engines: rigorously constructed upon the foundation laid by S Carnot and F Reech (Theoretical and mathematical physics). Springer, Berlin/Heidelberg/New York
Parigi S (ed) (2010) George Berkeley. Religion and science in the age of enlightenment. Springer, Dordrecht
Drake S, Drabkin IE (1969) Mechanics in sixteenth century Italy. The University of Wisconsin Press, Madison
Pisano R, Bussotti P (2012). Galileo And Kepler. On Theoremata Circa Centrum Gravitatis Solidorum And Mysterium Cosmographicum. History Research, in press.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Gillispie, C.C., Pisano, R. (2014). A Historical Epistemology of Thermodynamics. The Mathematics in Sadi Carnot’s Theory. In: Lazare and Sadi Carnot. History of Mechanism and Machine Science, vol 19. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-8011-7_9
Download citation
DOI: https://doi.org/10.1007/978-94-017-8011-7_9
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-017-8010-0
Online ISBN: 978-94-017-8011-7
eBook Packages: EngineeringEngineering (R0)