Inventing Temperature Read online

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  bound to matter, incapable of being disengaged from the molecules except through changes of state, chemical reactions, or some unusual physical agitation.

  Laplace escaped from this conceptual tangle by taking the extraordinary step of putting free caloric inside molecules, departing considerably from Lavoisier's original picture. The particles of free caloric were bound, but still exerted repulsive forces on each other; this way, free caloric in one molecule was capable of dislodging free caloric from other molecules. On the other hand, latent caloric, also bound in molecules, did not exert repulsive forces and could be ignored in Laplace's force-based derivations.35 Laplace called the free caloric disengaged from molecules the free caloric of space, which was a third state of caloric (very similar to the older notion of radiant caloric), in addition to Lavoisier's latent/combined caloric and free/sensible caloric.36

  Armed with this refined ontology, Laplace proceeded to argue that there would be a definite correlation between the density of free caloric contained in molecules and the density of free caloric tossing about in intermolecular spaces, because the amount of caloric being removed from a given molecule would clearly be a function of the intensity of the cause of the removal. So the density of free caloric of space could be utilized for the measurement of temperature, even its definition. With this concept of temperature, Laplace's argument that the air thermometer was "the true thermometer of nature" consisted in showing that the volume of air under constant pressure would be proportional to the density of the free caloric of space.37

  Laplace gave various demonstrations of this proportionality. The most intuitive one can be paraphrased as follows.38 Laplace took as his basic relations:

  where P is the pressure, K 1and K 2constants, ρ the density of the gas, and c the amount of free caloric contained in each molecule. The first relation follows from regarding the pressure of a gas as resulting from the self-repulsion of caloric contained in it. The repulsive force between any two molecules would be proportional to c2, and the pressure exerted by a molecular layer of density ρ on a layer of the same density proportional to ρ2. In favor of the second relation Laplace argued that temperature, the density of free caloric in intermolecular space, would be

  35. On this count, curiously, Laplace's mature view was more in agreement with De Luc's than with Haüy's.

  36. See Laplace [1821] 1826, 7, for an explanation of this picture. See also Laplace [1823] 1825, 93, 113, for the emphasis that latent caloric did not enter into his calculations. For the term la chaleur libre de l'espace, see Laplace 1821, 335.

  37. Laplace [1821] 1826, 4. The "extreme rarity" of the free caloric of space, due to the high speed at which caloric was transmitted between molecules, guaranteed that its amount would be a negligible fraction of the total amount of free caloric contained in a body. Then the amount of the free caloric of space could serve as a measure of the total free-caloric content, without actually constituting a significant portion of the latter.

  38. This follows the exposition in Laplace [1821] 1826, 3-6, supplemented by insights taken from Laplace [1823] 1825. See Brush 1965, 12-13, for a similar treatment.

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  proportional to the amount of caloric emitted (and absorbed) by each molecule in a given amount of time. This quantity would be proportional to the intensity of its cause, namely the density of caloric present in its environment, ρc, and also to the amount of free caloric in each molecule available for removal, c. By combining equations (1) and (2) Laplace obtained P = KT/V, where V is the volume (inversely proportional to ρ for a given amount of gas), and K is a constant. For fixed P, T is proportional to V; that is, the volume of a gas under constant pressure gives a true measure of temperature.

  It now remained to make a truly quantitative derivation and, in the abstract, what a good "Newtonian" had to do was clear: write down the force between two caloric particles as a function of distance, and then perform the appropriate integrations in order to calculate the aggregate effects. Unfortunately, this was a nonstarter. The fact that Laplace (1821, 332-335) did start this derivation and carried it through is only a testimony to his mathematical ingenuity. Laplace had no idea, and nor did anyone else ever, what the intercaloric force function looked like. It was obviously impossible to infer it by making two-particle experiments, and there were few clues even for speculation. In his derivations Laplace simply wrote f(r) for the unknown aspect of that function and kept writing different symbols for its various integrals; the unknown expression in the final formula, a definite integral, was given the symbol K and treated as a constant for a given type of gas, and turned out not to matter for anything important. The real work in the derivation was all done by various other assumptions he introduced along the way.39 These assumptions make an impressive list. In addition to the basic calorist picture of a gas, Laplace assumed: that the gas would be in thermal equilibrium and uniform in density; that its molecules would be spherical, stationary, and very far away from each other; that each molecule would contain exactly the same amount of caloric; that the force between the caloric particles would be a function of distance and nothing else, and negligible at any sensible distances; that the particles of the free caloric of space moved at a remarkably high speed; and so on.

  Since these assumptions were not theoretically defended or empirically testable, it is perhaps not a great surprise that even most French theorists moved away from Laplacian calculations on caloric. Perhaps the sole exception worth noting is Siméon-Denis Poisson (1781-1840), who continued to elaborate the Laplacian caloric theory even after Laplace's death.40 Not many people bothered to argue against the details of Laplace's caloric theory.41 Rather, its rejection was made wholesale amid the general decline and rejection of the Laplacian research program.

  39. A similar view is given by Heilbron 1993, 178-180, and also Truesdell 1979, 32-33. There was probably a good deal of continuity between this situation and Laplace's earlier, more acclaimed treatments of capillary action and optical refraction, in which he demonstrated that the particular form of the force function was unimportant; see Fox 1974, 101.

  40. See, for instance, Poisson 1835. According to Fox (1974, 127 and 120-121), Poisson "seems to have pursued the [Laplacian] program with even greater zeal than the master himself."

  41. One of those who did was the Scottish mining engineer Henry Meikle, who attacked Laplace's treatment of thermometry directly with a cogent technical argument. See Meikle 1826 and Meikle 1842.

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  Although generally discredited, the Laplacian treatment remained the only viable theoretical account of the thermal physics of gases until the revival and further development of Sadi Carnot's work in the 1840s and 1850s (see "William Thomson's Move to the Abstract" in chapter 4), and the only viable microphysical account until the maturity of the molecular-kinetic theory in the latter half of the century.

  Regnault: Austerity and Comparability

  The principles of thermometry thus endured "the rise and fall of Laplacian physics" and returned to almost exactly where they began. The two decades following Laplace's work discussed earlier seem to be mostly characterized by a continuing erosion in the confidence in all theories of heat. The consequence was widespread skepticism and agnosticism about all doctrines going beyond straightforward observations. The loss of confidence also resulted in a loss of theoretical interest and sophistication, with both pedagogic and professional treatments retreating into simpler theoretical conceptions.42 (I will give a further analysis of post-Laplacian empiricism in "Regnault and Post-Laplacian Empiricism.") An emblematic figure for this period is Gabriel Lamé (1795-1870), renowned mathematician, physicist, and engineer. Lamé was a disciple of Fourier's and also modeled himself after Pierre Dulong and Alexis-Thérèse Petit, who were his predecessors in the chair of physics at the Paris École Polytechnique. He stated his position in no uncertain terms in the preface of his physics textbook for the École: Petit and Dulong constantly sought to
free teaching from those doubtful and metaphysical theories, those vague and thenceforth sterile hypotheses which used to make up almost the whole of science before the art of experimenting was perfected to the point where it could serve as a reliable guide. … [After their work] it could be imagined that at some time in the future it would be possible to make the teaching of physics consist simply of the exposition of the experiments and observations which lead to the laws governing natural phenomena, without it being necessary to state any hypothesis concerning the first cause of these phenomena that would be premature and often harmful. It is important that science should be brought to this positive and rational state.

  For this kind of attitude he won the admiration of Auguste Comte, the originator of "positivism," who had been his classmate at the École Polytechnique.43

  In his discussion of the choice of thermometric fluids Lamé agreed that gases seemed to reveal, better than other substances, the pure action of heat unadulterated by the effects of intermolecular forces. However, like Dalton and Gay-Lussac (and Haüy before his Laplacian indoctrination), Lamé clearly recognized the limits to the conclusions one could derive from that assumption:44

  42. See Fox 1971, 261-262, 276-279.

  43. See Fox 1971, 268-270; the quoted passage is from Lamé 1836, 1:ii-iii, in Fox's translation on pp. 269-270.

  44. Lamé 1836, 1:256-258; cf. Haüy [1803] 1807, 1:263-264.

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  Although the indications of the air thermometer could be regarded as exclusively due to the action of heat, from that it does not necessarily follow that their numerical values measure the energy of that action in an absolute manner. That would be to suppose without demonstrating that the quantity of heat possessed by a gas under a constant pressure increases proportionally to the variation of its volume. If there were an instrument for which such a proportionality actually held, its indications would furnish an absolute measure of temperatures; however, as long as it is not proven that the air thermometer has that property, one must regard its reading as an as yet unknown function of the natural temperature. (Lamé 1836, 1:258)

  The blitheness of simply assuming linearity here might have been obvious to Lamé, who is mainly remembered now as the man who introduced the use of curvilinear coordinates in mathematical and physical analysis.

  Into this state of resignation entered Henri Victor Regnault (1810-1878), with a solution forged in a most austere version of post-Laplacian empiricism. Regnault's career is worth examining in some detail, since the style of research it shaped is directly relevant to the scientific and philosophical issues at hand. Regnault may be virtually forgotten now, perhaps nearly as much as De Luc, but in his prime he was easily regarded as the most formidable experimental physicist in all of Europe. Regnault's rise was so triumphant that Paul Langevin (1911, 44), though critical of him, drew a parallel with the glory days of Napoleon. Orphaned at the age of 2 and growing up without means, Regnault benefited enormously from the meritocratic educational system that was a legacy of the French Revolution. With ability and determination alone he was able to gain his entry to the École Polytechnique, and by 1840, at the age of 30, succeeded Gay-Lussac as professor of chemistry there. In that same year he was elected to the chemistry section of the Académie des Sciences, and in the following year became professor of experimental physics at the Collège de France. By then he was an obvious choice for a renewed commission by the minister of public works to carry out experimental studies to determine all the data and empirical laws relevant to the study and operation of steam engines.

  Thus ensconced in a prestigious institution with ample funds and few other duties, Regnault not only supplied the government with the needed information but also in the course of that work established himself as an undisputed master of precision measurement. Marcelin Berthelot later recalled the strong impression he had received on meeting Regnault in 1849: "It seemed that the very spirit of precision had been incarnated in his person" (Langevin 1911, 44). Young scientists from all over Europe, ranging from William Thomson (later Lord Kelvin) to Dmitri Mendeléeff, visited his fabled laboratory, and many stayed for a while to work and learn as his assistants.45 Regnault may well have frightened the European scientific community into accepting the authority of his results. Matthias Dörries (1998a, 258) notes that it was difficult for other physicists to challenge Regnault's results because they could not afford the apparatus needed to repeat his experiments. The

  45. A list of visitors to Regnault's lab is given by Dumas (1885), 178. On Mendeléeff, see Jaffe 1976, 153.

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  size of his equipment alone might have been enough to overpower potential detractors! Regnault describes in one place a 24-meter tall manometer that he constructed for the measurement of pressure up to 30 atmospheres, later a famous attraction in the old tour of the Collège.46 The sheer volume and thoroughness of his output would have had the same effect. Regnault's reports relating to the steam engine took up three entire volumes of the Mémoires of the Paris Academy, each one numbering 700 to 900 pages, bursting with tables of precise data and interminable descriptions of experimental procedures. In describing the first of these volumes, James David Forbes (1860, 958) spoke of "an amount of minute and assiduous labor almost fearful to contemplate."

  But, as I will discuss further in "Minimalism against Duhemian Holism" and "Regnault and Post-Laplacian Empiricism" in the analysis part, it was not mere diligence or affluence that set Regnault apart from the rest. Jean-Baptiste Dumas (1885, 169) asserted that Regnault had introduced a significant new principle to experimental physics, which he regarded as Regnault's service to science that would never be forgotten. To explain this point, Dumas drew a contrast to the methodology exhibited in the classic treatise of physics by Jean-Baptiste Biot. Whereas Biot would employ a simple apparatus to make observations, and then reason clearly through all the necessary corrections, Regnault realized (as Dumas put it): "In the art of experimenting by way of corrections, the only sure procedure is that which does not require any."47 Dumas summed up Regnault's distinctive style as follows: A severe critic, he allows no causes of error to escape him; an ingenious spirit, he discovers the art of avoiding all of them; an upright scholar, he publishes all the elements relevant to the discussion, rather than merely giving mean values of his results. For each question he introduces some characteristic method; he multiplies and varies the tests until no doubts remain about the identity of the results. (Dumas 1885, 174)

  Regnault aspired to test all assumptions by measurements. This implied that the measurements would need to be made without relying on any theoretical assumptions: "In establishing the fundamental data of physics one must, as far as possible, only make use of direct methods" (Regnault, quoted in Langevin 1911, 49). Regnault aimed at a puritanical removal of theoretical assumptions in the design of all basic measurement methods. This was, however, easier said than done. It is fine to say that all assumptions should be checked by measurements, but how

  46. On that instrument, see Regnault 1847, 349, and Langevin 1911, 53.

  47. For instance, consider the weighing of a given volume of gas, as discussed by Dumas (1885, 174-175). If one puts a sizeable glass balloon containing the gas on one side of the balance and small metal weights on the other, it is necessary to correct the apparent measured weight by estimating exactly the effect of the buoyancy of the surrounding air, for which it is necessary to know the exact pressure and temperature of the air, the exact density and volume of the glass (and its metal frame), etc. Instead of trying to improve that complex and uncertain procedure of correction, Regnault eliminated the need for the correction altogether: he hung an identical glass balloon, only evacuated, on the opposite side of the one containing the gas to be weighed, and thereafter the balance behaved as if it were in a perfect vacuum. In that procedure, the only buoyancy correction to worry about was for the metallic weights balancing the weight of the gas, which would have been quite a negligible effect.

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.76

  can any measurement instruments be designed, if one can make no assumptions about how the material substances that constitute them would behave? Coming back to thermometry: we have seen that all investigators before Regnault were forced to adopt some contentious assumptions in their attempts to test thermometers. In contrast, Regnault managed to avoid all assumptions regarding the nature of caloric, the constancy or variation of specific heats, or even the conservation of heat.48 How did he pull off such a feat?

  Regnault's secret was the idea of "comparability." If a thermometer is to give us the true temperatures, it must at least always give us the same reading under the same circumstance; similarly, if a type of thermometer is to be an accurate instrument, all thermometers of that type must at least agree with each other in their readings. Regnault (1847, 164) considered this "an essential condition that all measuring apparatuses must satisfy." Comparability was a very minimalist kind of criterion, exactly suited to his mistrustful metrology. All that he assumed was that a real physical quantity should have one unique value in a given situation; an instrument that gave varying values for one situation could not be trusted, since at least some of its indications had to be incorrect. (See "Comparability and the Ontological Principle of Single Value" for further discussion of this "principle of single value.")