History of ‘temperature’: maturation of a measurement concept

ABSTRACT

Accounts of how the concept of temperature has evolved typically cast the story as ancillary to the history of the thermometer or the history of the concept of heat. But then, because the history of temperature is not treated as a subject in its own right, modern associations inadvertently get read back into the historical record. This essay attempts to lay down an authoritative record not of what people in the past thought about what we call ‘temperature’ but of what they thought about what they called ‘temperature’ (or one of its cognates), from medieval times to today. It is found that invention of the thermometer had little impact on the concept of temperature. Much more significant were Fahrenheit’s invention of a reliable instrument and William Thomson’s effort to make a degree of temperature a unit of measure. Overlapping definitions of temperature then emerged in the late nineteenth century, and twentieth-century scientific developments forced physicists to reconsider temperature’s conceptual boundaries. It turns out that the concept of temperature has evolved through stages that correspond to four increasingly sophisticated types of measurement. Its maturity sheds light on the philosophy of conceptual change.

KEYWORDS:

• Temperature
• heat
• thermometer
• Daniel Gabriel Fahrenheit
• William Thomson
• Hasok Chang

Acknowledgements

I thank Laura J. Snyder for invaluable feedback and guidance, Travis Norsen for crucial discussions over the years on topics covered in this essay, Pamela Rappaport for help with some French, and an anonymous referee for very helpful suggestions on ways to enrich the essay. It should not be presumed that any of them agrees with what I say here.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 For some examples, see Ernst Mach, Die Prinzipien der Wärmelehre (Leipzig: Barth, 1896) and the English translations of excerpts on temperature published in The Open Court in 1902 and 1903, translated by Thomas J. McCormack; Paul Feyerabend, ‘Explanation, Reduction and Empiricism’, in Scientific Explanation, Space, and Time, ed. H. Feigl and G. Maxwell (Minneapolis: University of Minneapolis Press, 1962), pp. 28–97, and the subsequent literature on incommensurability, much of it influenced by Thomas Kuhn, The Structure of Scientific Revolutions (Chicago: University of Chicago Press, 1962 and 1970); Hilary Putnam, ‘The Nature of Mental States’, Mind, Language, and Reality (Cambridge: Cambridge University Press, 1975); Hasok Chang, Inventing Temperature: Measurement and Scientific Progress (New York: Oxford University Press, 2004); and Travis Norsen, ‘Scientific Cumulativity and Conceptual Change: The Case of “Temperature,”’ preprint, submitted Oct. 14, 2010, http://philsci-archive.pitt.edu/id/eprint/8332, and the commentaries thereon in a forthcoming volume, Concepts, Induction, and the Growth of Scientific Knowledge, ed. Corinne Bloch-Mullins and Theodore Arabatzis, with relevant contributions by Hasok Chang, James G. Lennox, John P. McCaskey, John D. Norton, and Gregory Salmieri.

2 The taxonomy proposed in S. S. Stevens, ‘On the Theory of Scales of Measurement’, Science, 103 (June 7, 1946), 677–80, has been influential and widely adopted though not without challenges. For an up-to-date survey, see Eran Tal, ‘Measurement in Science’, The Stanford Encyclopedia of Philosophy (Fall 2017 Edition), ed. Edward N. Zalta, https://plato.stanford.edu/archives/fall2017/entries/measurement-science/. One challenge has been whether nominal and ordinal gradings deserve to be called measurements at all. Whether they should or not, they certainly played a role in the history of temperature.

3 Prior studies on temperature, often with important information and crucial insights, but generally either limited in scope or focused less on the history of temperature than on the history of thermometers, of thermometry, of heat, or of thermodynamics include the following. Henry Carrington Bolton, Evolution of the Thermometer, 1592–1743 (1900); Kirstine Meyer, Die Entwickelung des Temperaturbegriffs im Laufe der Zeiten (Braunschweig: S. Vieweg und Sohn, 1913); F. Sherwood Taylor, ‘The Origin of the Thermometer’, Annals of Science, 5 (1942), 129–56; C. B. Boyer, ‘History of the Measurement of Heat,’ Scientific Monthly, 57 (1943), 442–52, 546–54; Martin Barnett, ‘The Development of Thermometry and the Temperature Concept,’ Osiris 12 (1956), 269–341; Duane Roller, ‘The Early Development of the Concepts of Temperature and Heat: The Rise and Decline of the Caloric Theory,’ Harvard Case Histories in Experimental Science, 1 (Harvard University Press, 1957); W. E. Knowles Middleton, A History of the Thermometer and Its Use in Meteorology (Baltimore, MD: The Johns Hopkins Press, 1966); T. J. Quinn, ‘The Meaning of Temperature and the Development of Thermometry’, Temperature, 2nd ed. (Elsevier, 1990), pp. 1–23; Hasok Chang, Inventing Temperature; Arianna Borelli, ‘The Weatherglass and its Observers in the Early Seventeenth Century’, Philosophies of Technology: Francis Bacon and His Contemporaries, ed. Claus Zittel et al. (Brill, 2008), pp. 67–130; Matteo Valleriani, ‘Pneumatics, the Thermoscope and the New Atomistic Conception of Heat’, Galileo Engineer (Springer, 2010); Arianna Borrelli, ‘Die Reproduktion des Temperaturbegriffs’, Epistemologie und Differenz: Zur Reproduktion in den Wissenschaften, ed. Ute Frietsch and Bettina Bock von Wulfingen (Bielefeld: Transcript, 2010), pp. 59–82; David Sherry, ‘Thermoscopes, Thermometers, and the Foundations of Measurement’, Studies in History and Philosophy of Science, Part A, 42 (Dec 2011), 509–24; and William F. Wright, ‘Early Evolution of the Thermometer and Application to Clinical Medicine,’ Journal of Thermal Biology, 56 (2016), 18–30.

4 Chang, Inventing Temperature, p. 4.

5 The focus on thermometry instead of temperature carried over into studies spawned by Chang’s book, including those at the conference ‘The Making of Measurement’, held at the University of Cambridge, July 23–24, 2015 and the published as a special issue of Studies in History and Philosophy of Science, Part A, vols. 65–66 (October–December, 2017).

6 Or a multi-word lexeme, such as fire engine, but I will not repeat the exception.

7 Quinn, p. 16.

8 Another statement that will mislead is Chang’s use of the term ‘thermoscope’. In chapter 1, he says he will use it for instruments in which the numbers

9 How the words were in fact used will be described below. Middleton said he would introduce the distinction that ‘a thermometer is simply a thermoscope provided with a scale.’ Chang says he ‘will follow Middleton’ (p. 41) and, in the glossary, says that a thermoscope ‘indicates the relative changes or comparisons of temperatures, without giving numbers’ (p. 258). But Chang also says he gives the term a ‘non-standard’ meaning (p. 251) and that a thermoscope does indeed have a scale, but the scale is ordinal not cardinal. Neither distinction is well maintained in the book. If the second were, ‘thermoscope,’ not ‘thermometer’, would have been the term for the whole period before Thomson’s introduction of an absolute scale. The confusing treatment in Chang is also noticed by David Sherry, ‘Thermoscopes, Thermometers, and the Foundations of Measurement,’ Studies in History and Philosophy of Science, 42 (2011), 511. Middleton’s distinction has unfortunately been too widely and too uncritically adopted. See M. Valleriani, ‘Pneumatics, the Thermoscope, and the New Atomistic Conception of Heat’, Galileo Engineer (Dordrecht: Springer, 2010) for a recent example.

10 Sanctorius, Commentaria in artem medicinalem Galeni (Venice: Somaschus), pt. III, cap. 85, particula X, p. 612 in the 1632 edition.

11 W. E. Knowles Middleton’s translation (p. 9) was ‘I wish to tell you about a marvellous way in which I am accustomed to measure, with a certain glass instrument, the cold and hot temperature of the air . . . . we can measure with the compass the degrees and ultimate limits of heat and cold’. Middleton gets ‘cold and hot temperature’ correct but then seems stumped by ‘ultimas mansions’.

12 Helpful recent roadmaps through the science of the primary sources include works by Peter Weinberger, J. J. Mares, and Wayne Saslow (in addition to Chang and Brush).

13 Categories chapter 8 and Metaphysics book 5(Δ), chapters 13 and 14, 1020a7–b25.

14 For entry to the literature, see Joel Kaye, A History of Balance: 1250–1375 (Cambridge: Cambridge University Press, 2014), especially bibliographical note 107 on p. 213; Edith Sylla, ‘Medieval Quantification of Qualities: The “Merton School”’, Archive for History of Exact Sciences, 8 (1971), 7–39; and Borrelli, p. 32.

15 “For if justice could be more or less justice, certain problems might thereon arise, as is also the case with all qualities which we may call dispositions. And some go so far as to say that these cannot admit of degrees. Health and justice themselves, they contend, are not subject to such variations, but people in varying degrees are possessed of health, justice and so on.” Aristotle, Categories ch. 8, 10b30–35.

16 For the topic in Scholasticism, see Robert Pasnau, ‘Scholastic Qualities, Primary and Secondary’, Primary and Secondary Qualities: The Historical and Ongoing Debate (Oxford: Oxford University Press, 2020).

17 Bernardino Telesio, De rerum natura iuxta propria principia libri IX (Rome: Antonium Bladum, 1586). The final edition of the work included the chapter ‘Calor frigusque uni, eidemque subjecto non erat indendum’ (The heat and cold in something are not one and the same), bk. 3, ch. 31. A nice summary of Telesio’s thinking on this, along with his influencers and influences, and associated literature appear in Michaela Boenke, ‘Bernardino Telesio’, The Stanford Encyclopedia of Philosophy, ed. Edward N. Zalta (Winter 2018 Edition).

18 Bacon, Sylva Sylvarum, century 1, before no. 69.

19 For the late seventeenth to the early nineteenth centuries, including the development of the concept of temperature more broadly, see Hasok Chang, ‘Rumford and the Reflection of Radiant Cold: Historical Reflections and Metaphysical Reflexes’, Physics in Perspective, 4 (2002), 127–69, and Chang, Inventing Temperature, pp. 164–68. Chang’s discussion of Pictet is on pp. 164–67, on Rumford p. 167.

20 Heated clay gives off colours ranging from a deep red at 500°C to white at 1400°C. Joseph V. Noble, ‘The Technique of Attic Vase-Painting’, American Journal of Archaeology, 64 (Oct., 1960), 307–18. ‘Text-box 62: Heat and colour, pyrometric cones and test-pieces’, Ceramics in Archaeology: From Prehistoric to Medieval Times (p. 373).

21 In 1880, William Thomson gave several everyday examples of such indicators and called them ‘discontinuous intrinsic thermoscopes’. Encyclopaedia Britannica, 9th ed. (1880), s.v. ‘heat’.

22 Per-Gunnar Ottosson, Scholastic Medicine and Philosophy: A Study of Commentaries on Galen’s Tegni (ca. 1300–1450), (Bibliopolis, 1984), ch. 1.

23 For example, ‘Temperatura: . . ., good disposition, temperatnes,’ Thomas, Dictionarium linguae Latinae et Anglicanae (1587). Sometimes also in phrases ‘temperamentum ad pondus’ (mixed to an equilibrium) or Martial’s ‘sequi temperamentum in re aliqua’ (to follow the proper mix in anything).

24 For example, in the Natural History (first century AD), bk. 34, Pliny writes, ‘Sequens temperature statuaria est eademque tabularis hoc modo’ (The proper mixture for statues and tablets is as follows). In De Natura Fossilium (1546), bk. 8, George Agricola writes, ‘Nam incensa Corintho aurum, argentum, aes in unum confluxerant, tribusque aeris Corinthii generibus fortuna dedit temperamentum’ (For in a fire at Corinth gold, silver and copper were melted into one; good luck produced the blend for three kinds of Corinthian copper), and ‘varie permiscentes metalla cum metallis: κράματα Graeci vocant, Latini temperaturas’ (The mixings of metal with metals Greek-writers call krámata, Latin-writers temperaturas). In De Re Metallica (1556), Agricola writes, ‘Temperatura primum adiecto plumbo coquatur in catino cinereo’ (The alloy, with the lead in it, is first heated in a cupel).

25 For example, ‘autoritatis et obsequii temperatura’ (temperature of auctorytie and of obsequy or seruyce), Desiderius Erasmus, ‘Liturgia Virginis Lauretanae’ (1525) in Opera Omnia (Basil: 1540–1541). The English translation was published as a pamphlet by Richard Wyer (London: 1533).

26 For example, Pliny says the ‘caeli temperies’ (temperature of the climate) affects blossoming of a particular rose in Spain. Natural History, bk. 21.

27 English examples are in Oxford English Dictionary, s.v. ‘distemperature’.

28 Joel Kaye, A History of Balance (Cambridge: Cambridge University Press, 2014), chapters 3 and 4; for the examples, pp. 169, 146, 176.

29 William Turner, The Names of Herbs in Greek, Latin, English, Dutch, and French (1548).

30 For the scattered evidence for invention(s) of the thermometer, see Taylor, ‘The Origin of the Thermometer’; Middleton, ch. 1; and now, with previously unnoticed evidence regarding Galileo, Matteo Valleriani, ‘Pneumatics, the Thermoscope and the New Atomistic Conception of Heat’.

31 ‘quo metimur . . . omnes gradus caliditatis, vel frigiditatis’. Sanctorius, Commentaria in artem medicinalem Galeni (Venice: Somaschus), pt. III, cap. 85, particula X, p. 612 in the 1632 edition. Middleton’s translation, ‘measure the degrees of heat and cold’, is potentially misleading.

32 Sagredo says the devices he was making were Sanctorius’s instrument but also, in a later letter, that the original was Galileo’s invention. We do not know how Galileo’s, Sagredo’s, and Sanctorius’s differed.

33 Francis Bacon, Novum Organum, in Instauratio Magna (London: 1620), bk. 2, aphorism 13.

34 ‘Hinc patet ampullam hanc Thermoscopium non inepte appellari posse’ and ‘quod ego Thermoscopium libenter appellarem’. Giuseppe Biancani, Sphaera mundi seu cosmographia demonstratiua (Bologna: Girolamo Tamburini, 1620), pp. 127 and 111.

35 ‘Thermomètre ou instrument pour mésurer les degrez de chalour ou de froidure qui sont en l'air (The thermometer, an instrument to measure the degrees of heat and cold in the air)’. Récréations Mathématiques (1624), p. 75, plate p. 69. The work has long been attributed to the Parisian Jesuit Jean Leurechon but on little evidence. See Albrecht Heeffer, ‘Récréations Mathématiques: A Study of Its Authorship, Sources and Influence,’ Gibecière, vol. 1 (Pont-à-Mousson: Jean Appier Hanzelet, 2010), available at https://www.researchgate.net/publication/266334553.

36 Middleton, History of the Thermometer (1966) proposed to distinguish a thermoscope, a device without a scale, from a thermometer, a device with a scale. This distinction was not sustained in the seventeenth century. In the 1660s, Robert Boyle used ‘thermometer’, ‘thermoscope’, and ‘weather-glass’ interchangeably, as in Robert Boyle, New Thermometrical Experiments and Thoughts. In the same years, Robert Hooke used ‘thermoscope’ for devices with a scale. In the late nineteenth century, William Thomson used ‘thermoscope’ for anything that indicates temperature, including thermometers with scales but also safety devices that trip when a certain temperature is reached.

37 The history of thermometric scales has been traced in Bolton, Evolution of the Thermometer ; Taylor, ‘The Origin of the Thermometer’; Middleton, A History of the Thermometer ; and Chang, Inventing Temperature.

38 Chang, Inventing Temperature, p. 160.

39 See especially Chang, Inventing Temperature.

40 Robert Boyle, New Experiments Touching Cold (1665) in Michael Hunter and Edward B. Davis, eds, The Works of Robert Boyle, vol. 4 (Pickering & Chatto, 1999), p. 240. The passage Bolton quotes on p. 41 is actually a paraphrase from Peter Shaw, The Philosophical Works of the Honourable Robert Boyle, Esq; Abridged, Methodized, and Disposed . . ., 2nd ed., vol. 3 (London, 1738), p. 579.

41 Ibid.

42 Robert Boyle, A Brief Account of the Utilities of Hygroscopes (1673) in Michael Hunter and Edward B. Davis, eds, The Works of Robert Boyle, vol. 7 (Pickering & Chatto, 1999), p. 433.

43 Two of countless examples: ‘Thermometers [show] . . . the differing temper of Heat and Cold’, in Edmund Halley, ‘An account of several Experiments made to examine the Nature of the Expansion and Contraction of Fluids by Heat and Cold, in order to ascertain the Divisions of the Thermometer’, Philosophical Transactions, 18 (1694), 655. ‘This Liquor perpetually varies its Temperature as to Cold and Heat’, in Robert Boyle, A Free Enquiry into the Vulgarly Received Notion of Nature (1686) in Michael Hunter and Edward B. Davis, eds, The Works of Robert Boyle, vol. 10 (Pickering & Chatto, 1999), p. 505.

44 Robert Boyle, ‘A New Frigorifick Experiment Shewing, How a Considerable Degree of Cold May Be Suddenly Produced Without the Help of Snow, Ice, Haile, Wind, or Niter, and That at Any Time of the Year’, Philosophical Transactions (July 18, 1666), in Michael Hunter and Edward B. Davis, eds, The Works of Robert Boyle, vol. 5 (Pickering & Chatto, 1999), p. 520.

45 Louise Diehl Patterson, ‘Thermometers of the Royal Society, 1663–1768’, American Journal of Physics, 19 (1951), 523–35. Louise Diehl Patterson, ‘The Royal Society’s Standard Thermometer, 1663-1709’, Isis, 44 (Jun., 1953), 51–64. Middleton, pp. 58–62.

46 Felicity Henderson, ‘Unpublished Material from the Memorandum Book of Robert Hooke, Guildhall Library MS 1758’, Notes and Records of the Royal Society of London, 61 (2007), pp. 129–75.

47 ‘L’hiver a été très-modéré, le plus grand froid n’s fait descendre la liqueue du Thermometre ordinaire qu’a 23 degrés ½’. Mémoires de l’Académie royale des sciences (1734).

48 ‘la liqueur du thermomètre est descendue à 5 degrés au dessous de la congélation’. Mémoires de l’Académie royale des sciences (1747), p. 573.

49 ‘Etat du Thermomètre / Degrés’, Louis Cotte, Traité De Météorologie (Imprimerie Royale, 1774), p. 228. A nearby table (p. 230) did have ‘température’ as a heading but entries were not numbers and were not about hot and cold; they were ‘froid & humide’, ‘variable, sec & froid’, etc.

50 His calculations and his procedure for how best to construct a thermometer appear in his working papers, Adversaria, ed. Kirstine Meyer (Copenhagen: B. Lunos bogtrykkeri, 1910), pp. 202–13.

51 Christian Wolff, ‘Relatio de Novo Barometrorum & Thermometrorum Concordantium Genere’, Acta Eruditorum (1714), pp. 380–81. See Bolton, pp. 65–66, and Chang, Inventing Temperature, p. 77.

52 In the 1830s, The Popular Encyclopedia (Glasgow: Blackie & Son, 1835-[41]), s.v. ‘Thermometer’, offered a similar assessment of the history: ‘We now come to mention the greatest improvement made upon the thermometer since the period of its invention.’ The work of Rømer and Fahrenheit is summarized. ‘From this period the thermometer became of scientific utility.’ The Popular Encyclopedia (Glasgow: Blackie & Son, 1835-[41]), s.v. ‘Thermometer’.

53 ‘TEMPERATURE. Ce mot pris pour Degré de chaleur’. Jean-Andre De Luc, Recherches sur les modifications de l’atmosphère, vol. 2 (Geneva, 1772), p. 483.

54 For Réaumur’s thermometer, see Middleton, pp. 79–80, and now especially Jean-Francois Gauvin, ‘The Instrument That Never Was: Inventing, Manufacturing, and Branding Réaumur’s Thermometer During the Enlightenment’, Annals of Science, 69 (Oct., 2012), pp. 515–49.

55 ‘les différents degrés de froid et de chaud’. René-Antoine Ferchault de Réaumur, ‘Observations du Thermometre Faites en MDCCXL a Paris, & dans d’autres endroits, soit du Royaume, soit des Pays étrangers’, Mémoires de l'Académie royale des sciences (1740), second part, pp. 539–66.

56 ‘la température de 12 degrés’, ‘la température qui étoit de neuf degrés’, and ‘celle température étoit de 45 degrés au thermomètre de Fahrenheit’. Abbé Jean-Antoine Nollet, ‘Recherches sur les moyens de suppléer à l’usage de la glace dans les temps & dans les lieux où elle manque’, Mémoires de l'Académie royale des sciences (1756), second part, pp. 85, 90, and 90 respectively.

57 ‘A Method for supplying the Want of Ice-houses for cooling Liquors. By the Abbe Nollet’. The London Chronicle, May 29–June 1, 1762, p. 515.

58 Joseph Black, Lectures on the Elements of Chemistry, ed. John Robison, vol. 1 (Edinburgh, 1803), p. 77.

59 Ibid., p. 76.

60 Alexander Law, Notes of Black's Lectures, vol. 1, p. 5, cited in Charles Coulston Gillispie, Dictionary of Scientific Biography: Volumes 1–2 (1981), p. 178. Compare Black, Lectures on the Elements of Chemistry, p. 78.

61 Black, Lectures on the Elements of Chemistry, pp. 79–81.

62 See Robison’s account in ‘Preface’, Black, Lectures on the Elements of Chemistry, pp. xxxiv–xxxix.

63 Chang, Inventing Temperature, p. 14.

64 Charles Mason and Jeremiah Dixon, ‘Observations for Determining the Length of a Degree of Latitude in the Provinces of Maryland and Pennsylvania, in North America’, Philosophical Transactions, 58 (1768), p. 313.

65 Alexander Wilson, ‘An Account of the Remarkable Cold Observed at Glasgow, in the Month of January, 1768’, Philosophical Transactions, 61 (1771), p. 328.

66 Nevil Maskelyne, ‘M. De Luc’s Rule for Measuring Heights by the Barometer, Reduced to the English Measure of Length, and Adapted to Fahrenheit’s Thermometer, and Other Scales of Heat, and Reduced to a more Convenient Expression’, Philosophical Transactions, 64 (1774), pp. 158–70; Samuel Horsley, ‘M. de Luc’s Rules, for the Measurement of Heights by the Barometer, Compared with Theory, and Reduced to English Measures of Length, and Adapted to Fahrenheit’s Scale of the Thermometer: With Tables and Precepts, for Expediting the Practical Application of Them’, Philosophical Transactions, 64 (1774), 214–301.

67 Henry Cavendish et al., ‘The Report of the Committee Appointed by the Royal Society to Consider of the Best Method of Adjusting the Fixed Points of Thermometers; and of the Precautions Necessary to be Used in Making Experiments with Those Instruments’, Philosophical Transactions, 67 (1777), pp. 816–57.

68 George Augustus William Shuckburgh, ‘Observations Made in Savoy, in order to Ascertain the Height of Mountains by Means of the Barometer; being an Examination of Mr. De Luc’s Rules, Delivered in his Recherches sur les Modifications de l’atmosphere’, Philosophical Transactions, 67 (1777), p. 518.

69 In 1740, Reaumur spelled out degrés or abbreviated deg or degr. In 1768, Mason and Dixon, trained in surveying, a profession that was already using ° for angular degrees, used the symbol for thermometric degrees as well. De Luc spelled out dégrés in 1772. Use of ° became common in the 1780s. I have not noticed any attempt to use a sexagesimal system for temperature.

70 In America, the transition can be witnessed in issues of the Pennsylvania Gazette. In the 1750s and ’60s, reports would say, for example, ‘the liquor in the thermometer rose thirty degrees’ (Oct. 11, 1753) but not that the temperature did. Similarly for a report on unusually cold weather (Jan. 8, 1767). But in the 1780s, one reads of ‘temperature’ being reduced (Jun. 29, 1785) or of sixty or seventy degrees being a ‘proper temperature’ for adding rennet when making cheese (Jun. 15, 1785). Benjamin Franklin (born 1706), in ‘Physical and Meteorological Observations, Conjectures, and Suppositions,’ Philosophical Transactions of the Royal Society, 55 (1765), p. 187, used ‘temperature’ the old way; Thomas Jefferson (born 1743), in Notes on the State of Virginia, used it the new way.

71 ‘Température: Nom que l’on donne au degré de chaleur’. Mathurin-Jacques Brisson, Dictionnaire raisonné de physique (Paris).

72 ‘Il se dit aussi Du degré de chaleur.’ Le Dictionnaire de l'Académie française, 6th ed. (Paris, 1835).

73 Hasok Chang, ‘Rumford and the Reflection of Radiant Cold’.

74 De Luc, Recherches sur les modifications de l’atmosphère (Geneva: 1772), part 4, chapter 3. William Roy, ‘Experiments and Observations Made in Britain, in order to Obtain a Rule for Measuring Heights with the Barometer’, Philosophical Transactions, 67 (1777), p. 704. Horace-Bénédict de Saussure, Essais sur l’Hygrométrie (Neuchatel: 1783), p. 108.

75 Reviews of the efforts appear in Gay-Lussac, ‘Enquiries concerning the Dilatation of the Gases and Vapors’, A Journal of Natural Philosophy, Chemistry, and the Arts, 3 (London: 1801), 212–13; and John Dalton, ‘Essay IV: On the Expansion of Elastic Fluids by Heat’, Memoirs and Proceedings of the Manchester Literary and Philosophical Society, 5 (1802), pp. 595–96.

76 Gay-Lussac, ‘Enquiries Concerning the Dilatation of the Gases and Vapors’, p. 208, translation of ‘Je conclus que tous les gaz, en général, se dilatent également par les mêmes degrés de chaleur; pourvu qu’on les mette tous dans les mêmes conditions’, Annales de Chimie, vol. 43 (Sept., 1802), p. 172.

77 Dalton, ‘Essay IV’, p. 600. Dalton’s emphasis.

78 ‘Le thermomètre, tel qu'il est aujourd’hui construit, ne peut servir à indiquer des rapports exacts de chaleur, parce que l'on ne sait pas encore quel rapport il y a entre les degrés du thermomètre et les quantités de chaleur qu'ils peuvent indiquer. On croit, il est vrai, généralement, que des divisions égales de son échelle représentent des tensions égales de calorique; mais cette opinion n'est fondée sur aucun fait bien positif.’ Joseph Louis Gay-Lussac, ‘Enquiries concerning the Dilatation of the Gases and Vapors’, Annales de Chimie, vol. 43 (Sept., 1802), pp. 138–39. A translation, used here, was published in English the same year, ‘Enquiries concerning the Dilatation of the Gases and Vapors’, A Journal of Natural Philosophy, Chemistry, and the Arts, 3 (1802), pp. 207–16, 257–67. An adjacent article in that issue of the journal was William Henry, ‘A Review of Experiments, Which Have Been Supposed to Disprove the Materiality of Heat’.

79 John Dalton, New System of Chemical Philosophy, pt. 1 (Manchester: R. Bickerstaff, 1808), p. 9.

80 Louis-Jacques Thenard, Traité de chemie élémentaire, théoretique et pratique, vol. 1 (Paris: Crochard, 1813), p. 37. Translation from Chang, Inventing Temperature, p. 69.

81 Here the most relevant papers by Thomson, or Thomson and Joule, are the following, gathered in – and cited here with the page numbers in – Sir William Thomson, Mathematical and Physical Papers, vol. I (Cambridge University Press, 1882), hereafter MPP-I. ‘On an Absolute Thermometric Scale Founded on Carnot’s Theory of the Motive Power of Heat, and Calculated from Regnault’s Observations’, Cambridge Philosophical Society Proceedings (1848), art. 38 in MPP-I, pp. 100–06. ‘An Account of Carnot’s Theory of the Motive Power of Heat; With Numerical Results Deduced from Regnault’s Experiments on Steam’, Transactions of the Royal Society of Edinburgh (1849), art. 41 in MPP-I, pp. 113–54. ‘On the Dynamical Theory of Heat, with Numerical Results from Mr. Joule’s Equivalent of a Thermal Unit, and M. Regnault’s Observations on Steam’, Transactions of the Royal Society of Edinburgh (1851), art. 48 in MPP-I, pp. 174–332. ‘Thermo-Electric Currents’, Transactions of the Royal Society of Edinburgh, vol. 21 (1854), added as pt. 6 of art. 48 in MPP-I, pp. 232–91.

82 Thomson, ‘On the Absolute Thermometric Scale’ (1848), MPP-I, p. 100.

83 Thomson, ‘On the Absolute Thermometric Scale’ (1848), MPP-I, p. 102.

84 Emile Clapeyron, ‘Mémoire sur la puissance mortice de la chaleur,’ Journal de l’École Polytechnique, 14, 13–190. Richard Taylor, trans., ‘Memoir on the Motive Power of Heat’, Scientific Memoirs, Selected from the Transactions of Foreign Academies of Science and Learned Societies and from Foreign Journals, 1, 347–76.

85 Sadi Carnot, Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance [Reflections on the motive power of fire and on machines capable of developing that power] (Paris: Bachelier, 1824). R. H. Thurston, trans., Reflections on the Motive Power of Heat, second, revised, edition (New York: John Wiley & Sons and London: Chapman & Hall, 1897). Robert Fox, trans., Reflections on the Motive Power of Heat (Manchester: Manchester University Press, 1986).

86 The only problem was that Regnault had latent heat by weight and Thomson needed it by volume. Thomson expected Regnault would gather that data eventually. In the interim Thomson would derive one from the other by assuming steam obeyed the gas laws of Boyle and Dalton. He figured he could refine his results later.

87 Thomson, ‘On the Absolute Thermometric Scale’ (1848), MPP-I, pp. 100–06.

88 Thomson, ‘On the Absolute Thermometric Scale’ (1848), MPP-I, p. 104.

89 Thomson, ‘An Account of Carnot’s Theory’ (1849), MPP-I, pp. 113–54.

90 For a history of this μ, see William H. Cropper, ‘Carnot’s Function: Origins of the Thermodynamic Concept of Temperature’, American Journal of Physics, 55 (1987), 120–29; the detailed studies cited there; and Chang, Inventing Temperature, pp. 182–86.

91 Values for the new scale are the mechanical effects given in Table II in ‘Account of Carnot’s Theory’ (1849), p. 140, linearly scaled so that the temperatures at 1° and 100° are the same on both old and new scales. Chang, Inventing Temperature, p. 191, presents the results of this calculation, mapping degrees on an air thermometer centigrade scale to degrees on Thomson’s proposed scale.

92 Thomson, ‘Thermo-Electric Currents’, MPP-I, p. 233.

93 Thomas S. Kuhn examined the work of a dozen men working on the topic. ‘Energy Conservation as an Example of Simultaneous Discovery’, Critical Problems in the History of Science, ed. Marshall Clagett (Madison, WI: University of Wisconsin Press, 1959), pp. 321–56.

94 Antoine Lavoisier, ‘Réflexions sur le phlogistique,’ in Jean-Baptiste Dumas, ed., Œuvres de Lavoisier, vol. 2 (Paris: Imprimerie Impériale, 1862), p. 641. Nicholas W. Best, trans., ‘Lavoisier’s “Reflections on Phlogiston” II: On the Nature of Heat’, Foundations of Chemistry, vol. 18 (2016). Emphasis in original.

95 Decades later, in 1871, James Clerk Maxwell wrote, ‘Heat, therefore, may pass out of one body into another just as water may be poured from one vessel into another, and it may be retained in a body for any time, just as water may be kept in a vessel. We have a right therefore to speak of heat as a measurable quantity, and to treat it mathematically . . . . [The] word Caloric was introduced to signify heat as a measurable quantity. So long as the word denoted nothing more than this, it might be usefully employed.’ Theory of Heat, p. 7.

96 Printed as Appendix A in Thurston, trans., Reflections on the Motive Power of Heat (1897), p. 219.

97 Benjamin [Thompson] Count of Rumford, ‘An Inquiry concerning the Source of the Heat which is excited by Friction’, Philosophical Transactions of the Royal Society, 88 (1798), p. 99. Emphasis in original. Advocates of the caloric theory saw nothing insurmountable in the second reason, just as nowadays the ability to generate apparently limitless amounts of static electricity by rubbing materials together weighs little against the theory of electrons. Thompson’s first reason was more important. For engagements with his conclusions at the time, see, for example, William Henry, ‘A Review of Experiments, Which Have Been Supposed to Disprove the Materiality of Heat’, cited above; Claude-Louis Berthollet, ‘Note VI’, Essai de Statique Chimique, 2 (Paris: 1803), pp. 247–50; Charles Haldat, ‘Inquiries Concerning the Heat Produced by Friction’, Journal de Physique, 65, p. 213, reprinted in Nicholson’s Journal of Natural Philosophy, 26 (1810); and ‘Caloric’, The Cyclopaedia; or, Universal Dictionary of Arts, Sciences, and Literature, ed. Abraham Rees (London: 1819).

98 James P. Joule, ‘On the Mechanical Equivalent of Heat’, Philosophical Transactions of the Royal Society of London, 140 (1850), 62.

99 ‘Que des volumes égaux de tous les fluides élastiques pris à une même température et sous une même pression, étant comprimés ou dilatés subitement d’une même fraction de leur volume, dégagent ou absorbent la même quantité absolue de chaleur.’ Pierre Louis Dulong, ‘Recherches sur la chaleur spécifique des fluides élastiques’, Mémoires de l’Académie des Sciences, 10 (1831), 188. The translation is given by Joule in ‘Mechanical Equivalent of Heat’, p. 62. The emphasis is in both Dulong and Joule.

100 Julius Robert Mayer, ‘Bemerkungen über die Kräfte de unbelebten Natur’, Annalen de Chemie und Pharmacie, 42 (1842), 233–40. Translated into English by G. C. Foster as ‘Remarks on the Forces of Inorganic Nature’, Philosophical Magazine, 24 (1862), 371–77. Reprinted in Stephen G. Brush, ed, The Kinetic Theory Of Gases: An Anthology Of Classic Papers With Historical Commentary (World Scientific, 2003), pp. 71–77.

101 Foster, trans., ‘Remarks on the Forces’, p. 374; Brush, Kinetic Theory, p. 74.

102 Foster, trans., ‘Remarks on the Forces’, p. 377; Brush, Kinetic Theory, p. 77. Foster (in 1862) added this footnote: ‘When the corrected specific heat of air is introduced into the calculation this number is increased, and agrees then with the experimental determinations of Mr. Joule’.

103 Donald S. L. Cardwell, James Joule: A Biography (Manchester: Manchester University Press, 1989), p. 57.

104 There are conflicting reports about what numbers Joule reported at this meeting, and the uncertainty may be telling us something about the situation. Joule, young and not an academic, had had a hard time getting respect for his proposal – based on simple experiments with a brewer’s vat – that there is one universal conversion factor. He now had here an audience of Britain’s scientific elite at its most prestigious event. It would have been important that he speak with conviction. Cardwell, Joule: A Biography, p. 82, reports that the meeting was running late and Joule was forced to summarize. We do not have a record of exactly what he said. ‘On the Mechanical Equivalent of Heat, as determined by the Heat evolved by the Friction of Fluids, . . . Read before the Mathematical and Physical Section of the British Association at Oxford, June 1847,’ Philosophical Magazine, vol. 31, p. 173, and the abstract published in the Association’s own Proceedings, report 781.5 for water and 782.1 for sperm oil. But the report printed in The London Literary Gazette and Journal of Belles Lettres, Arts, Sciences, etc., Saturday, June 26, 1847, p. 459, written by someone at the meeting, says Joule gave 775.4 for water and 775.9 for sperm oil. That same issue of the Philosophical Magazine, p. 114, printed a letter, ‘On the Theoretical Velocity of Sound,’ from Joule, written three weeks after the meeting, in which he wrote, ‘The equivalent of a degree of heat per lb. of water, determined by careful experiments brought before the British Association at Oxford, is 775 lb. through a foot.’ That would be consistent with the reporter’s account. Then, oddly, Joule’s Scientific Papers, vol. 1, p. 282, reprints that letter, but adds as follows: ‘by careful experiments made since those brought before the British Association’ (emphasis added). I wonder if just before the meeting Joule realized a problem with his numbers, if this as much as delays in the agenda explains why he merely summarized, and if a lack of resolve partly explains the lukewarm reception he received.

105 Letter from Joule to Thomson, October 6, 1848. Quoted in Chang, Inventing Temperature, pp. 182–83.

106 Letter from Thomson to Joule, October 27, 1848. Quoted in Donald S. L. Cardwell, Springs of Scientific Creativity: Essays on Founders of Modern Science, ed. Rutherford Aris, Howard Ted Davis, and Roger H. Stuewer (University of Minnesota Press, 1983), p. 59.

107 Thomson, ‘An Account of Carnot’s Theory’ (1849), MPP-I, p. 119.

108 Thomson, ‘On the Dynamical Theory of Heat’ (1851), MPP-I, p. 174.

109 Thomson, ‘On the Dynamical Theory of Heat’ (1851), MPP-I, pp. 174–315.

110 For the first equation, see Thomson, ‘Account of Carnot’s Theory’ (1849), MPP-I, p. 134; for the second, Thomson, ‘On the Dynamical Theory of Heat’ (1851), MPP-I, p. 190.

111 Thomson, ‘On the Dynamical Theory of Heat’ (1851), MPP-I, p. 198. The scale can be gotten from Table 38, ‘Table of the Motive Power of Heat’, column IV.

112 Cardwell, James Joule, p. 99.

113 ‘Carnot’s function . . ., or any arbitrary function of Carnot’s function, may be defined as temperature’. Joule and Thomson, ‘On the Thermal Effects of Fluids in Motion’, part 2, section 4 (1854), MPP-I, p. 393.

114 See the footnote that Thomson added in 1881 to a reprint of the 1848 paper. MPP-I, p. 106. The math works because Joule was correct about the μ values; in an ideal gas they fall off reciprocally. Thomson was summing those values, and the integral of 1 ∕ t is log_e(t). The reason his result was not closer than 0.6° is that steam, on which Thomson’s μ values were based, is not an ideal gas. I disagree with Chang, Inventing Temperature (p. 183) that the footnote is retrospective bravado. Thomson is referring in the note to three attempts at a scale, not just two. The note helpfully explains how the three were in fact related.

115 Thomson, ‘On the Dynamical Theory of Heat’ (1851), MPP-I, note on p. 233.

116 Keith Hutchison, ‘Mayer’s Hypothesis: A Study of the Early Years of Thermodynamics’, Centaurus, 20 (1976), 288 and 290.

117 The hypothesis, in fact, had several interchangeable corollaries. Hutchison, ‘Mayer’s Hypothesis’, p. 279 lists seven.

118 Thomson, ‘On the Dynamical Theory of Heat’ (1851), MPP-I, p. 220.

119 Joule and Thomson, ‘On the Thermal Effects of Fluids in Motion’, part 4 (1862), MPP-I, p. 393. Emphasis added. Cited by Chang, Inventing Temperature, p. 185.

120 Joule and Thomson, ‘On the Thermal Effects of Fluids in Motion’, part 4 (1862), MPP-I, p. 430. Cited by Chang, Inventing Temperature, p. 195.

121 Where we now use 273.15°, by definition, they used 273.7°, based on their measurements.

122 Hauptsatz in German. ‘Law’ would eventually prevail, but ‘main principle’ was a good rendering at the time. Browne used it in his translation of 1879. Preston, The Theory of Heat, (1894, 1904, and 1919) used ‘fundamental principle’.

123 In The Mechanical Theory of Heat (1875), p. 90, Clausius claimed to have been the first to publish this equation, in Poggendorfs Annalen, vol. 93 (1854), p 500.

124 It is, for example, what remains constant in the other two phases of a four-phase Carnot cycle. Two are isothermal, the other two isentropic.

125 The fraction attracted the attention of Rudolph Clausius in Germany but also William Rankine in Scotland. While Clausius used the symbol S and the name ‘entropy’, Rankine used $\varphi$ and labeled the fraction as simply a thermodynamic function. Well into the twentieth century, those writing in English normally used Clausius’s ‘entropy’ and Rankine’s $\varphi$. Of course, what – if any – physical macroscopic physical property corresponds to entropy has troubled physicists and countless students since. In 1910, Hugh L. Callendar, ‘The Caloric Theory of Heat and Carnot’s Principle’, Proceedings of the Physical Society of London, 23 (1910), 153–89, proposed that entropy is nothing but the old caloric. The suggestion has received insufficient consideration.

126 Rudolph Clausius, Die Mechanische Wärmetheorie (Braunschweig: Vieweg, 1876). Walter R. Brown, trans., The Mechanical Theory of Heat (London: Macmillan and Co., 1879).

127 The identity was anticipated in Rankine, ‘On the Geometrical Representation of the Expansive Action of Heat, and the Theory of Thermodynamic Engines’, Transactions of the Royal Society, 144 (1854), 115–75, but it was Clausius’s approach that was more influential.

128 The importance of this is noted by Clausius, ‘Formation of the Two Fundamental Equations’, The Mechanical Theory of Heat (1879), chapter 5; Tait, ‘Elements of Thermodynamics’, Heat (1884, 1892, 1895, 1904), chapter 21; and Preston, ‘Thermodynamic Formulae’, The Theory of Heat (1894, 1904, 1919), section 5. Like Thomson before them, Tait and Preston use Rankine’s ϕ instead of Clausius’s S.

129 In 1880, Thomson had a way to use entropy in an equation for temperature, but the formula worked by reducing entropy to a constant that could then be eliminated from the computation, so it never had to be measured. Thomson, Heat (1880), §48. The article was also published the same year as the entry for ‘Heat’ in the ninth edition of the Encyclopaedia Britannica.

130 P. G. Tait, Heat (London: Macmillan and Co.), §385 in all editions.

131 Thomas Preston, The Theory of Heat (London: Macmillan and Co., 1894), chap. 8, art. 307, p. 642.

132 For example, James Clerk Maxwell, Theory of Heat (London: Longmans, Green, and Co., 1871), chapter 22, pp. 281–312.

133 Preston, Theory of Heat (1894), chap. 1, art. 55, p. 70.

134 Much of the following on the specific heat anomaly is drawn from Henk W. de Regt, ‘Philosophy and the Kinetic Theory of Gases’, The British Journal for the Philosophy of Science, 47 (1996), 31–62.

135 James Clerk Maxwell, ‘On the Results of Bernoulli’s Theory of Gases as Applied to their Internal Friction, their Diffusion, and their Conductivity for Heat’, in ‘Notes and Abstracts’, Report of the Thirtieth Meeting of the British Association for the Advancement of Science (1861), pp. 15–16.

136 Henry William Watson, Treatise on the Kinetic Theory of Gases (Oxford: Clarendon Press, 1876), pp. 23–25.

137 Ludwig Boltzmann, Lectures on Gas Theory, trans. Stephen G. Brush (University of California, 1964 [1896–98]), p. 23.

138 J. H. Jeans, The Dynamical Theory of Gases (1904), ch. 6, art. 124, p. 108, under heading ‘Temperature: Definition’.

139 J. H. Poynting and J. J. Thomson, A Textbook of Physics: Heat, 3rd ed. (London: 1904), p. 140.

140 The Collected Papers of Albert Einstein, vol. 2, The Swiss Years: Writings, 1900–1909 (English translation supplement), trans. Anna Beck (Princeton, NJ: Princeton University Press), pp. 214–24.

141 Thomas Preston and J. Rogerson Cotter, The Theory of Heat, 3rd ed. (London: Macmillan and Co., 1894), chap. 9, art. 374, p. 803. Emphasis on ‘defined’ added; the rest is in the original.

142 Ludwig Boltzmann, ‘Über die beziehung dem zweiten Haubtsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung respektive den Sätzen über das Wärmegleichgewicht,’ Wiener Berichte, 76 (1877), 373–435. Kim Sharp and Franz Matschinsky, trans. ‘Translation of Ludwig Boltzmann’s Paper “On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium”’, Entropy, 17 (2015), 1971–2009.

143 Sydney Chapman and Thomas George Cowling, The Mathematical Theory of Non-Uniform Gases (Cambridge: Cambridge University Press, 1939), ch. 2, art. 2.41, p. 37.

144 Ibid.

145 J. Casas-Vázquez and D. Jou, ‘Nonequilibrium Temperature Versus Local-Equilibrium Temperature’, Physical Review E, 49 (Feb. 1994); and J. Casas-Vázquez and D. Jou, ‘Temperature in Non-Equilibrium States: A Review of Open Problems and Current Proposals’, Reports on Progress in Physics, 66 (2003) provide entry into the literature. The first article reports, ‘Out of equilibrium, . . . definition (1) [that is, 1 ∕ T = dS ∕ dU ] is not directly operative and relation (2) [that is, (³⁄₂) k T = ⟨(¹⁄₂) m v ²⟩] is used as a definition of temperature . . . in molecular-dynamics simulations’ (p. 1042); little was said about these simulations. The later progress report studies them in some depth and also explores cases where both kinetic and thermodynamic temperatures can be calculated but differ slightly.

146 Josiah Willard Gibbs, Elementary Principles in Statistical Mechanics (New York: Charles Scribner’s Sons, 1902), pp. 169–76.

147 L. Onsager, ‘Statistical Hydrodynamics,’ Il Nuovo Cimento, 6 (1949), p. 281.

148 V. Berdichevsky, I. Kunin, and F. Hussain, ‘Negative Temperature of Vortex Motion,’ Physical Review A, 43 (Feb 15, 1991), 2050.

149 See Ramsay’s reflections (including the stories about Simon and Townes) in the prefatory abstract to ‘Paper 6.1: “Thermodynamics and Statistical Mechanics at Negative Absolute Temperatures,” N. F. Ramsey, Phys. Rev. 103, 20–28 (1956)’, Spectroscopy With Coherent Radiation: Selected Papers of Norman F. Ramsey (With Commentary) (World Scientific, 1998), p. 389.

150 Mark W. Zemansky, Heat and Thermodynamics, 5th ed. (McGraw-Hill, 1968), pp. 487–92.

151 E.g. Henning Struchtrup, ‘Work Storage in States of Apparent Negative Thermodynamic Temperature,’ Physical Review Letters, 120 (2018); and Quanmin Guo, ‘Negative Absolute Temperatures,’ preprint, submitted Oct. 4, 2019, https://arxiv.org/ftp/arxiv/papers/1910/1910.01915.pdf.

152 Maxwell, Theory of Heat, p. 2.

153 In recent decades, discussions about temperature as a case study in incommensurability and conceptual change – cited in a note at the beginning of this essay – have died down some. But for background, revival and essays relevant here, see Corinne Bloch-Mullins and Theodore Arabatzis, eds, Concepts, Inducti,on, and the Growth of Scientific Knowledge.