Published online by Cambridge University Press: 05 January 2009
In the history of thermodynamics, two dates stand out as especially important: 1824, when Sadi Carnot's brilliant memoir Réflexions sur la puissance motrice du feu appeared in print; and 1850, when Rudolf Clausius published his similarly titled paper ‘Ueber die bewegende Kraft der Wärme’. In this paper Clausius narrowly beat the Scottish physicist William Thomson to the solution of a puzzle which had been highlighted in the latter's recent publications: how could Carnot's theory, with all its intellectual attractions, be reconciled with the newly discovered principle of the inter-convertibility of heat and work? Clausius's solution (as is well known) was to replace Carnot's axiom of heat conservation, with the axiom now known as the second law of thermodynamics.
This paper is based on my DPhil thesis of the same name, University of Oxford, 1976.
I would like to thank the Cowrie Trust Fund of Sydney, and the Stiftung FVS of Hamburg, for postgraduate scholarships which supported part of the research reported here. I am also grateful to Professor H. B. Sutherland of Glasgow for a loan of his photocopy of the Synopsis of Rankine's thermodynamics lectures; to the University Library, Edinburgh for tracing Rankine's 1840 paper on geothermal heat; and to the libraries of the University of Cambridge and the Royal Society for access to their collections of Rankine manuscripts.
1 For the general historical background to this paper, see Cardwell, Donald S. L., From Watt to Clausius, London, 1971Google Scholar, especially pp 186–294.
2 But see: Parkinson, E. M., ‘Rankine’, in Gillispie, C. C. (ed.), Dictionary of scientific biography, 16 vols, New York, 1970–1980, xi, 291–5Google Scholar (hereafter cited as DSB); Channell, D., ‘Rankine, Aristotle and potential energy’. The philosophical journal, 1978, 14, 111–4Google Scholar; Daub, E. E., ‘Atomism and thermodynamics’, Isis, 1967, 58, 293–303CrossRefGoogle Scholar; idem, ‘Waterston, Rankine, and Clausius on the kinetic theory of gases’, Isis, 1970, 61, 105–6Google Scholar; idem, ‘The regenerator principle in the Stirling and Ericsson hot-air engines’, British journal for the history of science, 1974, 7, 259–77Google Scholar; Hutchison, K. R., ‘Der Ursprung der Entropiefunktion bei Rankine und Clausius’, Annals of science, 1973, 30, 341–64CrossRefGoogle Scholar; idem, ‘W.J.M. Rankine and the entropy function’, Proceedings of the XIV international congress of the history of science, Tokyo-Kyoto, 1974, Tokyo, 1975, ii, 281–4Google Scholar; Moyer, D. F., ‘Energy, thermodynamics, hidden machinery: Rankine, Thomson and Tait, Maxwell,’ Studies in the history and philosophy of science, 1977, 8, 251–68CrossRefGoogle Scholar; Oison, Richard, Scottish philosophy and British physics, Princeton, 1975, pp. 271–86Google Scholar; Smith, C. W. ‘William Thomson and the creation of thermodynamics’, Archive for the history of the exact sciences, 1977, 16, 231–88CrossRefGoogle Scholar (253–61); Steffens, H. J.James Prescott Joule and the concept of energy, London, 1979, pp. 112–19.Google Scholar The purpose of the present paper, based on my DPhil thesis of the same name, University of Oxford, 1976, is to present a non-technical account of the central results of that thesis: detailed argument is excluded from the paper. I use the notes to provide something like an annotated bibliography of the literature on Rankine's thermodynamics. I am publishing separately a detailed analysis of Rankine's thermodynamical ideas and calculations: ‘Rankine, atomic vortices, and the entropy function’, forthcoming in Archives internationales d'histoire des sciences.
3 Biographical information on Rankine is rather meagre, but see: Parkinson, , DSB, op. cit. (2)Google Scholar; the two principal contemporary sources are Tait, P. G., ‘Memoir’Google Scholar [of Rankinel, W. J. M., on pp. xix–xxxviGoogle Scholar of Rankine, , Miscellaneous scientific papers (ed. by Millar, W. J.), London, 1881Google Scholar, (hereafter cited as MSP), and Gordon, L., ‘Obituary notice of Professor Rankine’, Proceedings of the Royal Society of Edinburgh, 1875, 8, 296–306CrossRefGoogle Scholar; there are two recent biographical sketches in Channell, David, A unitary technology: the engineering science of W.J. M. Rankine Case Western Reserve University PhD dissertation, 1975Google Scholar, and Sutherland, Hugh, Rankine: his life and limes, London, 1973.Google Scholar Further biographical sources are listed in these last two works.
4 Cf. Rankine, , ‘On the laws of the conduction of heat and on their application to some geothermal problems’Google Scholar, The Edinburgh academic annual for MDCCCXL, consisting of contributions in literature and science by alumni of the University of Edinburgh. This, Rankine's earliest known publication, is in Edinburgh University Library MS. JA. 2385.
5 Shairp, John, Tait, P. G., and Adams-Reilly, A. (eds.), Life and letters of James David Forbes FRS, London, 1873, pp. 127, 129.Google Scholar
7 For MacCullagh's work see Jellet, J. H. and Haughton, S. (eds.), The collected works of James MacCullagh, Dublin, 1880.Google Scholar For the claim that MacCullagh introduced a model of the aether involving microscopic rotations, see Whittaker, E. T., A history of theories of the aether and electricity, London, 1910, i, 154–7Google Scholar; Silliman, R., ‘William Thomson: smoke rings and nineteenth-century atomism’, Isis, 1963, 54, 461–74Google Scholar (468); Moyer, D. F., ‘MacCullach’, DSB, viii, 591–3Google Scholar (592). It is not clear from the primary sources that such claims are justified. See also note 16 below.
10 See for example, Fox, Robert, The caloric theory of gases from Lavoisier to Regnault, Oxford, 1971Google Scholar, passim, especially pp. 9–19, 110–13; Cardwell, , op. cit. (1), pp. 57–8Google Scholar, and Donovan, Arthur, Philosophical chemistry in the Scottish enlightenment, Edinburgh, 1975, pp. 20–5, 141–2.Google Scholar
11 MSP, p. 152Google Scholar, and ‘On the hypothesis of molecular vortices’, Philosophical magazine, 1864, 27, 313.Google Scholar These sources seem to be conclusive evidence against a view proposed by Olson, , op. cit. (2), p. 279Google Scholar, that Rankine was actually rejecting the aether in adopting this model.
12 The microscopic analogues of these peripheral pressure increments are calculated on pp. 22–3 of MSP. Rankine's final result, equation 5a, is correct, though a pair of compensating errors were made in the calculation leading to it.
16 For Joule and Davy, see: ibid., pp. 142–4, 146–8; Mendoza, E., ‘The surprising history of the kinetic theory of gases’, Memoirs and proceedings of the Literary and Philosophical society of Manchester, 1962–1963, 105, 15–28Google Scholar; The scientific papers of J. P. Joule, London, 1884, i, 186–8, 265–76, 294.Google Scholar For Rankine's attribution, see MSP, pp. 17, 235Google Scholar, and Rankine, ‘On the use ot mechanical hypotheses in science, especially in the theory of heat’, Proceedings of the Philosophical Society of Glasgow, 1864, 5, 126–32Google Scholar (129). Rankine did not mention MacCullagh in a thermal context, but this may only be because MacCullagh did not apply his theory to thermal phenomena. MacCullagh was cited by Rankine for work on optics and electricity: see, for example, MSP, pp. 102, 104, 157.Google Scholar
19 For Rankine's views on the kinetic theory, see MSP, pp. 321–3, 428–30Google Scholar; ‘On the use of mechanical hypotheses …’, op. cit. (16), pp. 128–9Google Scholar; ‘On the thermal energy of molecular vortices’, Philosophical magazine, 1870, 39, 211–21Google Scholar (213–14); ‘On thermodynamics’, ibid., 1870, 40, 103–4, 291–3; and Daub, , ‘Waterston, Rankine, and Clausius…’, op. cit. (2).Google Scholar
21 Cf. MSP, pp. 22ffGoogle Scholar, and nn. 29, 39, below, and associated text, and the caption to my illustration. Again I disagree with Olson, , op. cit. (2), p. 271Google Scholar, who may have been misled by the Cartesian vortex picture, or the fact that some of Rankine's numerical results (as those on MSP, pp. 237–8)Google Scholar agree with those which would have been obtained from nucleus-centred rotation.
23 See, for example, The scientific papers of James Clerk Maxwell, 2 vols, in 1, New York, 1965, ii, 662–4.Google Scholar
26 Silliman, op cit. (7). Silliman does not, however, appear to distinguish between the Rankine atom and the later ‘smoke-ring’ atom, and accordingly tends to see the vortex atom as diametrically opposed to the later orbital atom.
27 See Helmholtz's reports in Die Fortschritte der Physik, 1855, (for 1850–1), 6–7, 561–98Google Scholar; 1855 (for 1852), 8, 374, 380–1; 1856 (for 1853), 9, 406–9; 1957 (for 1854), 70, 366, 374–5, 382; 1858 (for 1855), II, 365, 369–74.
29 Thomson, J. J., ‘The relation between the atom and the charge of electricity carried by it’, Philosophical magazine, 1895, 40, 511–44 (512–15).Google Scholar
32 Rankine, , ‘On the centrifugal theory of elasticity as applied to gases and vapours’, Philosophical magazine, 1851, 2, 509–42Google Scholar; and ‘On the mechanical action of heat, especially in gases and vapours’ (introduction, sections I–IV, and appendix to section IV), Transactions of the Royal Society of Edinburgh, 1853, 20, 147–90.Google Scholar These two papers are reprinted in MSP, pp. 16–48, 234–84. The second was later extended by the addition of supplementary sections.
35 MSP, pp. 321, 326–32Google Scholar; ‘Heat, theory of the mechanical action of, or Thermo-dynamics’ (which Tait, MSP, p. xxiiGoogle Scholar, says was written in 1855), in Nichol, J. P. (ed.), A cyclopaedia of the physical sciences, 1st edn., London, 1857, pp. 338–54Google Scholar (341–2, 347, 353); A manual of the steam engine and other prime movers, 1st edn., London, 1859, pp. 306–7, 323.Google Scholar I have found no differences between editions in the thermodynamical section of this work, and the reader unable to check the 1st edn., may almost certainly use any edition available.
37 For Clausius's acceptance of the result, see his attack on Rankine's partial abandonment of it, in Clausius, Rudolf J. E., ‘On the determination of the disgregation of a body, and on the true capacity for heat’, Philosophical magazine, 1866, 31, 28–33.Google Scholar See also his The mechanical theory of heat, 1st edn., London, 1867, pp. 235–7, 274–7Google Scholar; and ‘The nature of the motion which we call heat’, in Brush, S. G. (ed.), The kinetic theory, Oxford, 1965, i, 111–34Google Scholar (113–14, 130, 132).
39 Many historians of science have tended to confuse the idea of an absolute zero of temperature with that of an absolute scale. The two notions are vitally distinct, and the first was quite common before Thomson introduced his first absolute scale in 1848; Thomson, William, Mathematical and physical papers, Cambridge, 1882, i, 100–6.Google Scholar Rankine used the term ‘absolute temperature’ as early as 1850 (MSP, pp. 30, 239)Google Scholar but it then meant (primarily) gas-temperature measured from the absolute zero of gaseous pressure (which was not, in Rankine's theory, identical with the temperature of utter thermal deprivation). For Thomson's introduction of the modem absolute scale, see op. cit., p. 235; and for Rankine's first absolute scale, see A synopsis of lectures on heat engines delivered at Glasgow in March and April 1855 in connection with Professor Lewis D.B. Gordon's course on civil engineering and mechanics, an undated, privately published lithograph, in Rankine's handwriting, where absolute temperature is described (p. 15) as representing ‘a tendency to perform work by the Energy of Heat’: see also ‘Heat, theory of…’, op. cit. (35), p. 341Google Scholar; cf. MSP, pp. 318–19Google Scholar, for an 1853 discussion which came very close to the later idea of absolute temperature. A photocopy of the Synopsis is held in the library of the Department of the History of Science and Technology, University of Manchester Institute of Science and Technology, but I have been unable to trace the original.
42 I give some indication of the role of the equation dL = T.dF in thermal calculations in my ‘Der Ursprung…’, op. cit. (2), pp. 349–51.Google Scholar
44 For such a deduction, see ‘Heat, theory of…’, op. cit. (35), p. 341Google Scholar (where phi is the entropy).
46 For a discussion of the position of this result in early thermodynamics, and for the complications involved in its confirmation, see Rankine, , Steam engine, op. cit. (35), pp. 367–8Google Scholar, and Hutchison, K. R., ‘Mayer's hypothesis: a study of the early years of thermodynamics’, Centaurus, 1976, 20, 279–304.CrossRefGoogle Scholar While Steffens, (op. cit. (2), pp. 118–19)Google Scholar correctly perceives the importance of this result in the acceptance of the new theory of heat, he drastically underestimates the complexity of the process of its confirmation, even suggesting (ibid., 129, contra my ‘Mayer's hypothesis …’,) that Thomson saw Joule's experiments as having confirmed Mayer's hypothesis.
48 Clausius, , ‘A contribution to the history of the mechanical theory of heat’, Philosophical magazine, 1872, 43, 106–15Google Scholar (107–9). Smith, C. (op. cit. (2), p. 277)Google Scholar disagrees with my interpretation here, and accepts Clausius's claim as ‘substantially accurate’. Steffens, on the other hand (op. cit. (2), pp. 115–17)Google Scholar, takes the opposite view, and suggests that Rankine incorporated the Carnot cycle into his 1850 work: Steffens appears to have confused Rankine's use of a cycle, with his patent failure to accommodate the Carnot cycle.
49 Cambridge University Library MS. Add. 7342, PA. 119. At MSP, p. 16Google Scholar, and in his correspondence with Thomson (Cambridge University Library MS. Add. 7342, R16, R17), Rankine indicated that some changes in the 1850 papers were made before printing, as a result of Thomson's perusal. Doran, B. G. suggests, in ‘Field theory in 19th century Britain’, Historical studies in the physical sciences, 1975, 6, 134–260Google Scholar (188), that one of Thomson's criticisms of Rankine's vortex theory may have been that ‘it left unanswered the fundamental questions as to the nature of both aether and matter’. I see no grounds to entertain such a speculation, as there is no hint of this criticism in Thomson's notes; as Rankine's theory made no pretence to attack these questions; and as the nature of the aether did not arise in the sections of the paper where Thomson's criticisms are acknowledged. These criticisms seem rather to have been directed against some of the details of Rankine's calculations.
50 Let us consider Clausius, Helmholtz, Maxwell, and Tait. Clausius has already been discussed in the text. For Helmholtz, see the unequivocal remarks in his Fortschritte, op. cit. (27), 1856, 9, 409.Google Scholar Maxwell's remarks in op. cit. (23) may give the impression that Maxwell had read Rankine's work thoroughly, but I reject this impression because of Maxwell's suggestion that Rankine's mechanisms are perfect. For Tait, see Rankine's remarks in ‘On the hypothesis …’, op. cit. (11); the work discussed by Rankine was written by Tait: see Daub, E. E., ‘Entropy and dissipation’, Historical studies in the physical sciences, 1970, 2, 321–54CrossRefGoogle Scholar (324–30). As an expansion of this work, Tait prepared his two editions of his A sketch of thermodynamics, Edinburgh, 1868Google Scholar, 1877, which also reveal that Taii had not studied Rankine's work closely. And Daub even suggests (ibid., p. 337) that parts of the discussion of Rankine were actually written by Rankine himself.
51 The original 1850 deduction of the ‘general equation of thermodynamics’ for example, appears to depend on a crucial shift in the meaning of one of the symbols (δQ') used by Rankine: compare the description of δQ' given immediately before and immediately after equation (2), MSP, p. 248.Google Scholar
58 For Rankine's argument, see MSP, pp. 301–3.Google Scholar There is a different argument on pp. 342–4 of Steam engine, op. cit. (35), but again the argument is inapplicable to non-fluid engines, and the onus of establishing maximality is still avoided.
59 Rankine, , ‘On the centrifugal theory of elasticity and its connection with the theory of heat’, Transactions of the Royal Society of Edinburgh, 1853, 20, 425–40CrossRefGoogle Scholar; ‘On the second law of thermodynamics’, Philosophical magazine, 1865, 30, 241–5Google Scholar; ‘On the thermal energy …’, op. cit. (19). The first two of these papers were reprinted in MSP, pp. 49–66, 427–31.Google Scholar
60 See in particular, ‘On the thermal energy …’, op. cit. (19).
61 The fourteen principal members of this series were; ‘On the general law of the transformation of energy’, Philosophical magazine, 1853, 5, 106–17Google Scholar (reprinted in MSP, pp. 203–8)Google Scholar; ‘On the mechanical action of heat; section VI …’, Transactions of the Royal Society of Edinburgh, 1853, 20, 565–89Google Scholar (MSP, pp. 310–38)Google Scholar; ‘On the geometrical representation of the expansive action of heat …’, Philosophical transactions, 1854, 114, 115–76Google Scholar (MSP, pp. 339–409)Google Scholar; ‘On the mechanical action of heat … Of a correction applicable to the results of the previous reduction of the experiments of Messrs. Thomson and Joule’, Proceedings of the Royal Society of Edinburgh, 1857, 3, 223–4Google Scholar; ‘On the expansion of certain substances by cold’, Philosofihical magazine, 1854, 8, 357–8Google Scholar; ‘On the mechanical action of heat; Supplement to the first six sections, and section seventh’, (abstract only), Proceedings of the Royal Siciety of Edinburgh, 1857, 3, 287–92Google Scholar; ‘Outlines of the science of energetics’, MSP, pp. 209–28Google Scholar (read on 2 May 1855); A synopsis of lectures on heat-engines …, op. cit. (30); ‘Heat, theory of …’, op. cit. (35); Steam engine, op. cit. (35), pp. 299–345Google Scholar; ‘On thermodynamic and metamorphic functions, disgregation, and real specific heat’, Philosophical magazine, 1865, 30, 407–10Google Scholar; ‘On the theory of the influence of friction upon the mechanical efficiency of steam’, British Association for the Advancement of Science report (transactions), 1867, p. 147Google Scholar; ‘On the want of popular illustrations of the second law of thermodynamics’, The engineer, 28 06 1867Google Scholar (MSP, pp. 432–8).Google Scholar
62 Cf. the first equation (a) on p. 302 of MSP;
. Following contemporary practice; Rankine did not use the partial derivative notation; I have used T for Rankine's τ-κ. In Rankine's theory, the total heat received in an isothermal expansion is equal to the total work done, so my use of L in place of Rankine's H is reasonable, and W is simply I do not know of any case of Rankine's explicitly introducing the Q.dW/dQ expression before 1853, but it is clear from the tone of Rankine's various discussions (such as those referred to in nn. 64, 66 below) that my use of it here is reasonable.
64 Cf. MSP, pp. 203–5, 220–1, 311–12, 347–8.Google Scholar The use of the term ‘catalysis’ is my own, not Rankine's; it is introduced to explain Rankine's idea.
65 Rankine, , ‘Heat, theory of…’, op. cit. (35), p. 341.Google Scholar For other statements of Rankine's version of the second law beyond those quoted in the text, see Steam engine, op. cit. (35), pp. 307–8Google Scholar; MSP, pp. 418–9, 427, 430, 434–5Google Scholar; ‘On the expansion … by cold’, op. cit. (61), p. 357Google Scholar; and n. 87, below.
66 For the statement of this law see Steam engine, op. cit. (35), p. 306.Google Scholar For evidence for my interpretation of the thinking which led to Rankine's enunciation, see MSP, pp. 203–5, 220–1, 311–13, 347–8.Google Scholar The restriction to isothermal changes is explicitly imposed on p. 347, but not mentioned in the general law on p. 348. On p. 312, Rankine deduces the total work done (in equation 67) from the variation in external work (given in the unnumbered top equation). And dQ.is initially an increment of total heat, then, three lines later, any indefinitely small portion of the total heat. For other commentaries on this law, see Achard, A., ‘Exposé du second principe de la theorie mécanique de la chaleur’, Bibliothèque universelle, 1865, 22, 214–41Google Scholar (227–8); Ewing, James A., The steam-engine and other heat-engines, 3rd edn., Cambridge, 1910, pp. 109–110Google Scholar; and n. 89 below.
71 For Rankine's methodological writings, see: ‘On the general law …’, op. cit. (61); ‘Outlines of the science of energetics’, op. cit. (61); On the nature and abjects of the Institution [of Engineers in Scotland], Glasgow, , 1857; A manual of applied mechanics, London, 1858, pp. 1–11Google Scholar; referee's report on a paper by Moon, Royal Society of London Library Ms. RR.5.160: ‘On the use of mechanical hypotheses …’, op. cit. (16); ‘On the phrase ‘Potential Energy’, and on the definition of physical quantities’, Philosophical magazine, 1867, 33, 88–92Google Scholar (MSP, pp. 29–33)Google Scholar; various fragments, especially those on MSP, pp. 317, 321, 375–7, 427Google Scholar; and ‘Heat, theory of …’, op. cit. (35), pp. 338, 353.Google Scholar There are brief discussions of Rankine's methodology on p. 430 of Kargon, R., ‘Models and analogy in Victorian science …’, Journal of the history of ideas, 1969, 30, 423–36CrossRefGoogle Scholar (430), and in Smith, , op. cit. (2)Google Scholar; and extended discussions in Moyer, , op. cit. (2)Google Scholar, and Olson, , op. cit. (2).Google Scholar Olson's sources are very frugal (only ‘Outlines of the science …’, and ‘On the use of mechanical hypotheses’), and I myself adopt a substantially different interpretation. For an important study of the background to the methodology of thermodynamics, see Cantor, G., ‘The reception of the wave theory of light in Britain …’, Historical studies in the physical sciences, 1976, 6, 109–32.CrossRefGoogle Scholar
75 Op. cit. (61).
77 On MSP, pp. 207–8, 223Google Scholar, Rankine indicates some branches of physics other than thermodynamics which his laws of energetics may describe. I am unable to estimate the significance of the brief reference to thermoelectricity, but Rankine's remarks about electromotive engines are amplified in an open letter to Joule, , ‘On the mechanical effect of heat and of chemical forces’, Philsophical magazine, 1853, 5, 6–9Google Scholar, where Rankine draws attention to an analogy between the formulae for the efficiencies of perfect thermodynamic engines. This analogy would provide Rankine with one meta-law for both classes of engines, abstracted from the two individual laws. But it hardly justifies the whole family of alleged meta-laws presented in the ‘Outlines of… energetics’, op. cit. (61).
79 See MSP, pp. 220–1, 229–33, 317, 375, 377Google Scholar, for elaboration of this ‘definitional’ methodology.
85 See MSP, pp. 226–7, 449.Google Scholar In the second of these passages, dated 1867, Rankine acknowledged the truth of a principle which we recognize as Clausius's 1850 version of the second law, but he did not point out that this truth is normally called the second law, even though his paper was discussing the second law. This confirms my claim that Rankine saw his second law as making an assertion slightly different from that of the conventional law. I am, however, in fundamental disagreement here with Daub, ‘Atomism …’, op. cit. (2), and Smith, , op. cit. (2)Google Scholar, who see Rankine, 's theory as equivalentGoogle Scholar to conventional thermodynamics: cf. n. 93, below.
86 See MSP, p. 233Google Scholar, and ‘On the reconcentration of the mechanical energy of the Universe’, Philosophical magazine, 1852, 4, 358–60Google Scholar (MSP, pp. 200–2).Google Scholar There are good reasons for supposing that Rankine would not have insisted on these views, but the fact that he even speculated in their direction indicates how little he felt the force of the conventional second law.
87 The statement ofthe second law in Rankine, 's Synopsis of lectures, op. cit. (39), p. 18Google Scholar, is an explicit assertion of the existence of the entropy function: ‘The whole heat transferred and transformed, during a given variation of pressure and volume of an elastic substance [note the exclusion of devices such as thermo-couple], is equal to the product of the variation of a quantity [which at ibid., p. 19, is said to be the ‘thermodynamic function’ (= entropy)] depending on the pressure and volume, by the absolute temperature at which the variation takes place’.
89 See, for example, Helmholtz, , Fortschritte, op. cit. (27), 1858, 11, 370Google Scholar; Turazza, D., ‘Teoria dinamica del calorico’, Il nuovo cimenta, 1860, 11, 370–96Google Scholar; 1860, 12, 85–145 (109), 262–77 (originally in Memorie dell' I. R. Istituto di scienze, littere ed arti, 1859, 8, 1–86Google Scholar; di San Roberto, Paolo, Principes de thermodynamique, 1st edn., Turin, 1865, pp. 69ff.Google Scholar
91 For Rankine's solution, see Steam engine, op. cit. (35), pp. 383–7.Google Scholar For a discussion of some of the difficulties earlier encountered in calculating the work done during expansion, see Fox, R., ‘Watt's expansive principle in the work of Sadi Carnot and Nicolas Clément’, Notes and records of the Royal Society, 1969, 24, 233–53.CrossRefGoogle Scholar
93 Cf., for example, Rankine's method of Computing the entropy as described on p. 342 of ‘Heat, theory of …’, op. cit. (35), p. 342Google Scholar, or in Steam engine, op. cit. (35), pp. 311–12Google Scholar, with that of Pippard, A. B., The elements of classical thermodynamics, Cambridge, 1966, p. 59.Google Scholar
94 Rankine, ‘On the thermal energy …’, op. cit. (19), p. 220Google Scholar, and ‘On thermodynamic … functions …’, op. cit. (61).
95 Rankine, ibid. For Rankine's opponents, see Clausius, , Mechanical theory, op. cit. (37), pp. 231–5, 239Google Scholar: idem., ‘On the determination …’, op. cit., (37); Roberto, San [Paul de Saint-Robert]Google Scholar, op. cit. (89); idem., ‘Remarques … Sur la détermination de la disgrégation …’, Archives des sciences physiques et naturalles, 1866, 25, 34–43.Google Scholar (Clausius's paper was originally published in French in Archives, 1865, 24, 117–24Google Scholar, and Saint-Robert's paper was a reply to Clausius.)
96 Clausius, , ‘on the application of the mechanical theory of heat to the steam engine; (English translation), in Mechanical theory, op. cit. (37), pp. 137–207.Google Scholar
98 For examples, see Rankine, , Steam engine, op. cit. (35), pp. 318, 320, 327, 330–2, 385.Google Scholar
100 Ewing, , op. cit. (66), pp. 109–10Google Scholar, does mention Rankine's approach, as an aside; and some engineers chose to follow it here and there in works other than textbooks; see, for example, the remarks of Trowbridge, in Babcock, G. H., ‘Substitutes for steam’, Transactions of the American Society of Mechanical Engineers, 1886, 7, 680–741Google Scholar (726–7); or Thurston, Robert H., Heat as a form of energy, London, 1890, pp. 91, 239.Google Scholar It is interesting that Thurston chose to follow Rankine; there is good evidence that Thurston's understanding of the second law was also rather deficient.
101 See the opening remarks of Swinburne, James, Entropy: or, thermodynamics from an engineer's standpoint, and the reversbility of thermodynamics, London, 1904.Google Scholar
102 For a discussion of the introduction of these engines, see: Rankine, , A memoir of John Elder, Edinburgh and London, 1872Google Scholar, passim, especially pp. 37ff: Elder, J., ‘On double cylinder expansion marine engines’ (abstract only), British Association for the Advancement of Science report (transactions), 1858, p. 211Google Scholar: Bramwell, F. J., ‘On the progress effected in the economy of fuel in steam navigation …’, Proceedings of the Institution of Mechanical Engineers, 1872, pp. 125–85Google Scholar; Cardwell, D. S. L. and Hills, R. L., ‘Thermodynamics and practical engineering in the nineteenth century’, History of technology, 1976, 1, 1–20 (6–7, 10–11).Google Scholar
104 Rankine, , Memoir of Elder, op. cit. (102), pp. 16–28Google Scholar; Clark, D. K., ‘On the expansive working of steam in locomotives’, Proceedings of the Institution of Mechanical Engineers, 1852, pp. 60–88.Google Scholar Clark's experiments were performed on the Caledonian Railway, at a time when both Rankine and his father were involved in railway engineering in Glasgow, and they could well have attracted Rankine's especial attention.