Reconstructing the early history of the theory of heat through Fourier’s experiments

We present a project for undergraduate students illustrating the historical path that led to the foundations of the early theory of heat developed by Joseph Fourier. It is based on the reconstruction of 12 key experiments carried out by the French scientist to acquire significant results about heat already known from previous authors, as well as to investigate the relevant parameters ruling heat propagation in media. Upon this empirical evidence, Fourier was able to elaborate his celebrated heat equation that he applied to a number of different problems, whose solution led to predictions and consequences to be tested in further observations. The dominant experimental activity of the project was complemented by a dedicated theoretical one, aimed at illustrating how physical evidences entered into the theoretical framework behind Fourier’s heat equation. The project ended with a dissemination activity, which resulted both in the realization of 12 videos illustrating the experiments performed (available on the YouTube platform) and in the presentation of the whole project to other students and the general public at science fairs and similar events.


Introduction
The physics of heat is usually discussed, in general physics courses, by routinely presenting quite a natural transition from thermology to thermodynamics, leaving just as interesting exercises the problems related to heat transmission. This is, however, not at all the historical route that led people to realize what heat was [1], and historians very well know that thermodynamics was born just from the complex problems of heat conduction [2], which were consistently presented (and solved) just two hundred years ago in the treatise Théorie analytique de la chaleur [3] by the Frenchman Jean Baptiste Joseph Fourier [4,5]. Of course, thermal phenomena were studied since ancient times, and the quantitative study of heat started as early as the beginning of the XVIII century [6]. However what we want here is to highlight the fact, to which few are accustomed, that the first insights into complex thermal phenomena came from the study of concrete problems related to heat transmission, rather than as currently explained in college or university physics courses, resorting rather to later achievements.
As a matter of fact, explicit reference to the effective complexities, revealed by historical studies aimed at reconstructing the actual path followed by thermal science, can greatly help students to overcome the objective difficulties offered by such an intricate topic. Previous works presenting different case studies [7,8], indeed, have shown all the enormous potential of adopting an approach aimed at following how the historical path leading to given results developed, thus unveiling how people have reached those results rather than focusing on the mere results. This is evidently very useful especially for students in their understanding of given topics, when they can usually rely only on college or university textbooks that, even without trivializing the topic, nevertheless simplify it by linearizing a historical path that-as history teaches-is never at all linear.
For this reason, on the occasion of the centenary alluded above, we have developed a project for undergraduate students aimed at reconstructing the historical path that led to the foundations of the theory of heat as epitomized in Fourier's treatise. The project can be easily adapted to different possible uses for different types of students and the public. Indeed, although it is true what is commonly held that Fourier's essential contribution concerns the equation on heat conduction, together with the mathematical methods introduced by him for its solutions, our project is centered neither on the Fourier equation nor on Fourier series and Fourier transform [9]. The theory set out in the 1822 treatise was, indeed, based on a number of key experiments carried out by Fourier himself [4], and including also more general experiences from previous authors since the beginning of the XVIII century [10]. Here we just focus on this experimental path, albeit we do arrive at presenting the key theoretical results (see supplementary material I) while pointing out the unavoidable Fourier's reference to measurable quantities (partially) introduced by himself, and finally checking the theoretical results obtained with further experimental evidence. study and attentive comparison of the facts known up to this time: all these facts I have observed afresh in the course of several years with the most exact instruments that have hitherto been used [3].
The very foundations of Fourier's analytical theory of heat are, then, genuinely experimental. He started his work around 1805 and, for about two years he repeated all relevant experiments carried out previously by other authors, with the intent to become acquainted with the different physical aspects of the phenomenon of heat propagation, adding then experiments of his own on heat transmission in solids and liquids [4,5]. This allowed him to provide the first theoretical formulation of the problem of heat conduction, even formulating his new mathematical technique to solve the equations for different continuous bodies, and then devise further experiments to test various of his theoretical results. All this work conveyed in his Memoir submitted to the Institut de France on December 1807, later revised and extended in 1811, whose publication was however so greatly delayed [11] that Fourier prepared a third version in the form of a book, eventually appeared in the Théorie analytique de la chaleur of 1822. Although practically only the book version is universally known, its very source is the 1807 paper (where experiments are described in some detail): that manuscript [12] is, actually, Fourier's true masterpiece, which can now be appreciated in its entirety thanks to the work by Grattan-Guinness and Ravetz [4]. In the following (especially for experiments) we will refer to this monograph, as well as to the 1822s treatise, for our historical reconstruction of the path that led to the first theory of heat. Such a reconstruction is aimed mainly at a didactic project for students, so that it does not claim to be exhaustive; in particular, we will confine ourselves just to heat conduction, without considering other Fourier's applications to Earth science and related topics (such as the greenhouse effect) [9]. Also, given the great number of relevant scattered contributions about temperature and heat before Fourier, as well as a large number of his own contributions, here we will limit to a minimal though significant number of key experimental observations that are able, nevertheless, to fully understand and appreciate Fourier's magnificent work.

Prehistory
As early as in 1701, Isaac Newton published anonymously [13] a short note concerning the cooling in the air of a heated iron block, by proposing a first-rate equation for heat transfer obtained from the assumption that the driving force of the phenomenon is the temperature difference between iron and air. According to him, indeed, in an iron plate that cools in a stream of air at a constant temperature flowing uniformly, equal quantities of air in contact with the plate transport, in equal times, quantities of heat proportional to the temperature difference between the plate and the air (whatever heat and temperature meant to him: neither the terms were already well defined). More specifically, he assumed that the cooling rate of a given heated body is proportional to the temperature difference with the environment in which it is immersed, and, by solving the resultant differential equation, the well-known Newton's law of cooling is obtained: the body temperature decreases exponentially with time until it reaches (asymptotically) room temperature. However, at least up to Fourier, this was known as the logarithmic law for cooling, since Newton originally expressed it in terms of the logarithm of the excess of temperature to be proportional to the time elapsed, and later scholars variously recognized it is valid approximately only for small excesses [14].
The first experiment of the project just refers to this simple cooling mechanism but, in order not to distract too much attention from the main topic of heat transfer, we decided to propose only introductory, qualitative observations as follows.
• Exp. N.1: Newton's experiment on an iron plate cooling An aluminum 1 plate is heated by a candle flame (or even on an electric stove or a gas burner) and then placed on a wooden tripod (see figure 1(a)). A fan blows air parallel to the plate, in order to transport the heated air away from the plate. Small pieces of wax, beeswax and sealing wax (or even rosin) are placed on the hot plate, which initially liquefies them, and then the times of cooling are measured (with a stopwatch) until the different wax pieces lose their fluidity and harden.
Originally, Newton 'found' that 'the excess of the heat of the iron and of the hardening bodies above the heat of the atmosphere, found by the thermometer, were in geometrical (e) Biot's experiment (Exp. N.5). See text. 1 Of course, Newton's experiment as well as the subsequent experiments originally made use of iron, rather than aluminum (the metal of such element was not yet produced at those times). However, for practical time reasons, we have preferred to adopt such a metal in most of our experimental reconstructions.
progression when the times were in arithmetical progression', which is a direct consequence of the exponential cooling law. 2 However, since the verification of such a result would have required different temperature measurements (quite easy to perform in a dedicated activity, but here inopportune), we preferred to resort to another observation. The difference between the melting temperatures of sealing wax (100°÷ 110°C, depending on the sample used) and beeswax (62°C) is approximately the double of that between the beeswax and common wax (45°C): such materials were chosen precisely for this property. For small times, by approximating the exponential cooling curve between the considered melting temperatures with a straight line (which is quite a crude approximation for the values considered), the time interval between the hardening of sealing wax and beeswax is approximately double that between the hardening of beeswax and common wax. The rough verification of such a property illustrates well the spirit of Newton's cooling law, its exponential (or 'logarithmic') nature along with the kind of approximations (linearization) employed; at the same time, it does not resort to anachronistic temperature measurements (note that the intent of Newton's paper was just the definition of a temperature scale). The next relevant step was an intriguing experiment performed by G Amontons shortly after Newton, in 1703 [17]. By depositing a given substance in several points of an iron rod heated at one end, he observed that it melted in succession, so that he realized that heat flows in solids in the direction of decreasing temperatures. As a result, he was able to guess that the temperature of the rod varies with distance in a predictable way (he actually assumed that such variation was linear) [10].
• Exp. N.2: Amontons' experiment on heat propagation in a metal bar Drops of wax are dropped onto an aluminum rectangular bar at approximately the same distance from each other. The flame of a gas burner (or of a candle) is applied to one end of the bar, and the successive liquefaction of the various pieces of hardened wax is observed. A second bar is prepared (after several attempts) where the different pieces of wax are not equidistant from each other, but rather placed in such a regular way that the time elapsed between the melting of one piece of wax and another is approximately the same for different distances from the heat source (see figure 1(b)).
The second part of the experiment is quite laborious, requiring some calibration of the distance between the heat source and the first wax marker (setting the general time evolution), as well as the distance between the first and the second marker (the time elapsed between the first two meltings serves as a reference for the position of the other markers).

Protohistory
The turning points in the history of thermal studies were essentially two. First, in 1714 D Fahrenheit perfected a mercury thermometer capable of obtaining reproducible measurements (the main problem that plagued temperature measuring instruments before Fahrenheit) [15]. Then, in 1761, Black convincingly clarified the difference between temperature and heat, by observing that when ice melts, it absorbs heat without changing its temperature [16]. In addition to introducing, therefore, the concept of latent heat, with his experiments he also 2 From the exponential cooling law ΔT = ΔT 0 e −αt , where ΔT is the excess temperature at time t with respect to room temperature (while ΔT 0 corresponds to the initial plate temperature) and α is a constant, it is easy to prove that if times t follow an arithmetic progression t n = n t 1 , then the excess temperatures follow a geometric one: introduced that of specific heat, just observing that equal masses of different substances require different amounts of heat to increase their temperatures by the same value: he then firstly realized that heat can be 'accumulated' in bodies. A major advancement in the understanding of the phenomenon was introduced in 1776 by Lambert, who provided the first definitive treatment of it, by realizing that in the Amontons' experiment heat does propagate inside the bar, but it is also dispersed from its surface. In other words, two different phenomena (conduction and radiation) contribute to what is observed in the experiment [17].
By following Newton, Lambert also realized that a 'logarithmic' law (rather than a linear one, as assumed by Amontons) rules the phenomenon: in a long metal bar heated at one end, and left to cool in the air, the temperature exponentially decreases with distance along the bar. Notably, he ascribed 'Newton's law' only to radiation, and further realized that the temperature profile depends on the bar geometry (whether the bar section is rectangular, circular or other shape).

The existence of heat conduction is recognized
Although during the XVIII century, it was widely accepted that metals were the best substances for heat transmission, a practical problem arose in deciding which metals were the best conductors of heat: in fact, in many factories and foundries, indeed, the (expensive) dispersion of large quantities of heat could be avoided or at least limited with an appropriate choice of the materials used. A first quantitative, and not subjective, answer to the problem came from B Franklin, who in 1780 suggested to the Dutch-born scholar Ingenhousz to measure at what distance the heat penetrates inside wires of different metals heated at one end, by observing the melting of the thin layer of wax with which they were coated [18].
• Exp. N.3: Franklin's experiment with the Ingenhousz apparatus In a metal box (for example a baking pan for plumcake) six rods made of wood, iron, steel, copper, brass, aluminum, respectively, are opportunely inserted (at a given distance of 4 cm) and fixed; each rod has a length of 25 cm and a diameter of 5 mm. Common wax is melted in a small pot and poured over the rods. Water is then heated in a saucepan until it boils, after which it is poured into the box (placed on wood) until it abundantly covers the terminals of the rods inside it. After some time (1-3 min, if the water is almost at its boiling temperature) it is observed that the wax on the rods melts up to different distances from the connection to the box, the greater corresponding to the copper and the smaller to the wood (which practically does not melt). It can also be appreciated that, with respect to copper, the melting distance is about 2/3 in aluminum, 1/2 in brass, 1/5 or so in iron and somewhat less in steel: metals can be ordered according to their ability to conduct heat (see figure 1(c)).
Ingenhousz's experiment was instrumental to recognize the existence of conduction of heat, distinguished from the radiation of heat and, in this direction, at the end of the century the important figure of Count Rumford emerged with his experiments (performed around 1798) about radiant heat, revealing first that the amount of heat that one body receives from another is proportional to the temperature difference between the two bodies (following Newton's law), and then that the heat radiated from the surface of a body depends on the state of that surface. [19] • Exp. N.4: Rumford's experiment on the radiative cooling of a metal cylinder A small hole is made on the lid of a metal can (for example a can for soft drinks, to which a suitable lid made from the base of another can is adapted) to insert a bulb thermometer; the can is placed stably on a wooden tripod. Boiling water (or so) is poured into the (open) can, which is then immediately closed with the lid, into which the thermometer is inserted. With a stopwatch, the times it takes for the temperature of the water in the can to drop every 2 or 3 degrees are noted, acknowledging that they increase considerably as the temperature drops and approaches that of the room (with an appropriate table reporting temperature/time data a graph may be built, which can better illustrate the Newton property). The experiment is repeated with the can covered (tightly) with a thin white cloth, noting a faster cooling (but exhibiting the same property), which therefore depends on the surface state of the radiant body (see figure 1(d)).

First attempts at a theoretical systematization
With the increase of experimental evidence, although scattered, at the beginning of the XIX century the time was ripe for the search for a theoretical systematization of the phenomenon of heat propagation. Among the first to try his hand at this undertaking was JB Biot, who addressed the problem of heat conduction in a thin bar heated at one end, just as Lambert did [20]. He first repeated some crucial experiments himself, confirming Lambert's results according to which propagation in a straight bar follows a 'logarithmic' law, but then, since 'it was not enough to conclude these results by experiment; it was necessary to find them by theory, for experiment alone shows only some isolated facts, while it is theory that makes us perceive the relations between them'.
Differently from Lambert, however, he assumed that the underlying Newton's law applies not only to the radiation from the surface but to conduction as well, since (according to him) it refers in general to the surface of contact of two bodies. This resulted into an unsuccessful theoretical derivation of a (differential) equation describing heat conduction (Newton's law is inadequate for conductive heat transfer). His approach based on Newtonian action at a distance, where heat propagation in the bar depends only on the temperature differences between the various parts of a body, while does not involve the distance between the points, was questioned by Laplace [21], who suggested Biot to adopt a new concept of heat transfer ruled bygradients of temperature within an undivided body, just as in Euler's hydrodynamics where the accelerating force is not due to a pressure difference effected by a piston, but rather as the result of a pressure gradient within the fluid. Such a 'potential theory' point of view was precisely what Fourier would later adopt, as we will see below.
Here, instead, we will focus on Biot's experimental activity, focusing on the result he obtained regarding the validity (explicitly shown) of Newton's law in the Amontons-Lambert experiment, that is, how temperature exponentially decreases with distance from the source along a heated straight metal bar. Biot emphasizes that this result is obtained only when the bar has reached a stationary state (to which both bar heating and air cooling due to surface radiation contribute), recognizable when the different temperatures along the bar no longer change as time goes by (obviously with the flame still on).
• Exp. N.5: Biot's experiment on the exponential drop in temperature Six wells (15 mm deep) are drilled in a 25 × 25 × 750 mm aluminum bar at 40 mm distance from each other, starting from one end of the bar. The other end is heated with a candle flame. A bulb thermometer is inserted in the olive oil poured in each well, measuring the temperature in different places inside the bar, as the distance from the source increases. For a certain time (about an hour), the temperatures recorded by each thermometer rise with time, after which they stabilize on the maximum values reached, clearly revealing the presence of a stationary state.
Temperatures are recorded at regular intervals of time and, once such a state is reached, the simultaneous look at the heights reached by the liquid in the different thermometers shows an approximately exponential decrease (confirmed by the acquired numerical data reported in a graph). Then the candle flame is extinguished, and the temperatures are recorded (as above) along the bar during its cooling, showing their exponential decrease with time. A cooling rate (in different places) comparable to the heating rate can be as well appreciated (see figure 1(e)).

The tradition of the theory of heat begins
Biot's work on heat propagation revived Fourier's interest in the subject and, soon after Biot presented his results to the Académie, Fourier started his own analysis of the problem. His initial approach was basically mathematical in nature, following the action at a distance reasoning of Biot, but he soon (and suddenly) abandoned it, also taking an empirical, observational approach [10].
Fourier firstly repeated a number of well-known key experiments, such as for example about the cooling of a thermometer placed in a liquid [4,10], and then launched his own 'experimental campaign' on heat conduction in solids, upon which his theoretical masterpiece is based. The main results he obtained, explicitly reported earlier in his 1807s manuscript [12], concerned the temperature distribution and cooling rate in a heated iron annulus, spheres and cubes. The apparently unusual choice of a ring as a 'starting point' was done in order to test Biot's suggestion about the establishment of a stationary state in the bar, according to which it resulted from the balance of the heat supplied by the source at one end of the bar and that lost through the section of the bar at the opposite end. By 'closing the bar on itself' (in a sense), deforming it appropriately into a ring, Fourier aimed at both understanding the problem properly (differently from Biot) and testing the theoretical results he obtained already in 1807, and later refined. For clarity of explanation, we postpone to the following section the appropriate theoretical description of the present project, while reporting here the whole experimental part, including that directly devoted to testing the results obtained mathematically; moreover, when necessary we will refer also to 1822s Théorie in order to follow Fourier's reasoning and presentation of results, although it omitted explicit experimental results. The reaching of a steady (or 'permanent') state during heating is viewed, in Fourier's reasoning, as due to the fact that the source finally provides (instant by instant) the amount of heat that the ring surface dissipates into the air, thus allowing the temperature to stabilize. In part B, Fourier evidently tested a property of the solution of the heat equation he got for the steady state in a ring. 3 The interesting conclusion he draws about the steady state temperature distribution was that 'when the circumference is divided into equal parts, the temperatures at the points of division, included between two consecutive sources of heat, are represented by the terms of a recurring series' described by the relation T 3 = q T 2 − T 1 (for any three equidistant points 1, 2, 3). depending on the ring geometric (length ℓ of the average circumference and section S) and physical (external h and specific k thermal conductivity; see below) properties. From this law, it is immediate to deduce that three consecutive points x 1 , x 2 , x 3 separated by a common distance λ verify the relation (T 1 + T 3 )/T 2 = α λ + α − λ ≡ q, where q does not depend neither on the x-position nor on source parameters (note, however, that this is true if the succession of the three mentioned points does not cross the point where the heat source is placed). The same applies when the separation λ is (for example) doubled; in such a case, the constant value α 2 λ + α −2 λ ≡ r satisfies the relation q 2 = r + 2 or, as pointed out by Fourier, = + q r 2 .
• Exp. N.7: Fourier's experiments on the cooling of a metal ring (heat distribution and mean temperature) (A) With the same apparatus as above, when the flame is blown out, it is observed that all temperatures decrease (again, exponentially) with time, until they all become equal to the ambient temperature (see figure 2(b)) [3].
(B) After some time from the start of cooling (that is, since the flame is blown out), the numerical data show that half the sums of the temperatures registered by thermometers placed in diametrically opposing points rapidly converge to a common value (T 1 + T 4 )/2 = (T 2 + T 5 )/2 = (T 3 + T 6 )/2. Such a property remains valid for the whole subsequent duration of the cooling, the mean value decreasing exponentially with time [3, 12].
(C) By repeating the experiments with the candle placed at a different point, with two candles (in different places) as heat sources, or covering the ring with a layer of carbon black (obtained from a burning candle), the previous conclusions do not change [12].
Fourier promptly noted that the cooling of the ring through its surface does not alter the heat distribution along the ring, but only lowers the temperature at each point (and this explains, for example, its exponential decrease with time). Moreover, the property of the mean temperatures (part B) in the cooling regime, following the initial fluctuations after the removal of the heat source, again deduced mathematically 4 reveals an interesting symmetric state shown by the ring. Indeed, by denoting with A and B, the two opposite points on the ring where the temperature effectively equals the mean temperature, such points divide the ring into two symmetrical halves, one with temperatures higher and one with temperatures lower than the mean value, and the thermal evolution of the system is just due to the heat fluxes at the two points that tend to bring each half toward the mean temperature.
Inspired by what is already known about a straight bar, Fourier realized that the thermal state of a body can be represented at given times by the temperatures of its different points. During the heating process in a fluid (such as air), after some time the heated body tends to reach a steady state, while the cooling process following the removal of the heat source results in a final temperature equal to that of the fluid (air). In later experiments, he also proved that, in analogous conditions, heating and cooling processes are similar. If a body is heated (from the outside) by immersing it into a fluid, its final temperature at any point of the body is approximately equal to that of the fluid, when the steady state is reached. Similarly, in the cooling process in air, the body starts to cool from its surface, finally reaching approximately the air temperature at all its points. shows a lower value with respect to the other, until both register the common higher value corresponding to the boiling water, reached with some appreciable time delay inside the sphere.
(B) The sphere is then removed from the boiling water, along with its corresponding thermometer inside it, and quickly suspended in the air by means of a flat Teflon support (from which a disk with a radius smaller than that of the sphere has been removed); the same is done with the other thermometer, exposed to air (and thus rapidly cooling down to the room temperature). Similarly, as above, the temperature inside the sphere is observed to decrease only after some time from the beginning of its cooling to air, again denoting some delay in the outward propagation of heat.
(C) The same is done with an iron cube, with an edge of 50 mm and a 30 mm deep welldrilled perpendicular to one of its faces at its center. In all tests, the inner temperature always approximately reaches the boiling water value at the body's surface after some time. Even just after the immersion of the two bodies in the air, their inner temperature does not change appreciably (for short times), roughly denoting that heat begins to dissipate at body surfaces (see figure 3(a)) [3].
From now on, Fourier's original experimental activity focused almost exclusively on the propagation of heat in spheres, even explicitly considering small and large radius limits. Apart from obvious theoretical reasons concerning the symmetry and simplicity of this problem, according to his own statements [3], two more relevant motivations emerged. On one hand, indeed, what is deduced on the heating or cooling of a small sphere 'applies to the movement of heat in a thermometer surrounded by air or fluid', so the corresponding study is important for understanding the operation of a thermometer. On the other hand, 'the problem of the movement of heat in a sphere includes that of the terrestrial temperatures', on which Fourier was also particularly interested. Here, however, we limit to the general study of heat propagation, without entering into such applications, which are beyond the scope of the present project. The first important result was that the cooling (in air) of a heated sphere again follows a 'logarithmic law', as in a bar or a ring, thus revealing the common nature of the radiation cooling phenomenon (which, as we will see below, will be described by an appropriate parameter).
• Exp. N.9: Fourier's experiment on the exponential cooling of a metal sphere The iron sphere used in Exp. N.8 is heated as described above, by immersing it in boiling water. It is then removed and suspended in the air by means of a flat Teflon support as in the previous experiment. With a thermometer placed inside the well, the inner temperature of the sphere is measured at definite time intervals (for example, each 5 minutes); from these numerical data, an exponential decrease is revealed. By repeating the experiment with a different heating mechanism, i.e. using the flame of a candle rather than boiling water, the result for the cooling law remains unchanged (see figure 3(b)) [11,12].
The exponential decrease can be displayed, as done originally by Fourier, by showing the constancy of the ratio ( ) ( ) --T T t t log log 2 1 2 1 for any couple of time-temperature (t 1 , T 1 ) and (t 2 , T 2 ) values registered (note that T denotes the excess temperature of the sphere with respect to air), 5 or-more simply-by the time-temperature plot built from the numerical data collected.
This ubiquitous 'logarithmic law', effectively ruling the cooling of differently shaped and sized bodies, further induced Fourier to investigate the radiation process through the surface of those heated bodies. On a qualitative ground, it was already known that the actual state of the body surface strongly affected its cooling, and Fourier experimentally realized that, when the bodies were covered with a black coating, the cooling rate 'almost doubled' [12]. From his heat equation (and the relevant solution), however, he intriguingly deduced also a specific testable property connecting the cooling rate with geometric properties of the heated body; namely, that the time for a small sphere to halve (or reduce by any given ratio) its temperature scales as the radius of the sphere, while for large spheres the time increases as the squared radius.
• Exp. N.10: Fourier's experiments on the dependence of the cooling rate of a metal sphere on its coating and size (A) Two iron spheres as above, one of which covered with the carbon black of a candle, are heated by a gas burner (or in the flame of a candle). After some time, they are removed from the flame and placed on Teflon supports (with thermometers placed inside their wells) and allowed to cool in the air. Successive temperatures of the spheres are registered at given time intervals, revealing the different cooling rates of the two spheres, quantitatively expressed by the time they take to halve their initial temperature [12].
The same experiment is repeated with two iron spheres (without any coating) of diameters 50 mm and 35 mm, respectively, revealing that the ratio of the times for halving the initial temperatures is approximately equal to the ratio of the corresponding diameters of the two spheres (see figure 3(c)) [3]. 5 By assuming an exponential decrease y = Aα t for the excess temperature y with respect to air, with A, α two constants (0 < α < 1), we have that Fourier's basic reasoning about cooling was that the heat dissipating from the surface of a body propagates by successive layers parallel to the surface, from the innermost to the outermost. As a consequence, if differently shaped bodies are very small-sized, we should expect them to cool equally quickly. This was just the result he obtained when comparing the theoretical expressions for the cooling rates of a cube and a sphere, having the same sizes (diameter of the sphere equal to the side of the cube), in the limit of small dimensions. It was evidenced by the approximately equal times for halving (for example) a given temperature in a cube and in a sphere. Fourier also considered the opposite limit of large dimensions, obviously obtaining that the above result no more applied: the final duration of the cooling resulted to be greater for the cube than for the sphere in the ratio 4 to 3. Also, its scaling with the cube side was as for the scaling law for the sphere considered above, that is a linear/ quadratic scaling with side or radius in the small/large size limit. Such a result was explicitly tested as early as 1807 [12], revealing some discrepancy whose explanation was searched in the possible sources of experimental inaccuracy. (B) The same experiment is repeated with a sphere and a cube with diameter/side equal to 35 mm, revealing a better agreement between the cooling rates (or between the times for a given temperature drop) [12].
Although the experimental activity performed by Fourier, as documented in manuscript and published papers [11,12], did not report further dedicated observations, his Théorie did discuss and propose several possible experiments, aimed at testing the theoretical predictions deduced from the different solutions of his heat equation specialized to various systems. Some specific tests (as Exps. N. 6B, 7B, 7C or 9, alluded above) were performed by Fourier himself, while several others were only suggested by him (as Exp. N. 10B). We conclude our experimental path with just one more of these last tests, concerning the verification of the steady state exponential distribution of temperatures in a metal rod with the distance from the heat source, which is then a Fourier-style alternative to Biot's experiment (Exp. N. 5). Indeed, in his treatise Fourier showed that, by heating two differently sized bars of the same material at one of their ends, when the steady state is finally reached, the distances from the source to the points where the two bars attain the same temperature are to each other as the square roots of their thicknesses. 6 6 According to Fourier's analysis, the temperature distribution with distance x from the heat source in a bar with size

Classroom implementation and learning assessment
The complete project concerning the early history of the theory of heat was originally devised for second and third-year university students (in Physics and Engineering); in addition to the reconstruction of the experimental path just described above in some detail, it included also a second part devoted to Fourier's better-known theoretical masterpiece concerning heat propagation [3]. Aimed at illustrating the basic physics (rather than mathematical) ingredients of the theory elaborated by Fourier, how it was generated and how experimentally testable results could be deduced from it, this second part is described in the supplementary material I, and evidently requires that students do know the basic mathematical tools of calculus that are employed in general physics courses discussing thermodynamics at the university level. The first, experimental part described above, instead, does not require such mathematical tools, so that, it can be fruitfully implemented also with final-year college or high school students; what reported below can, then, be easily adapted to these groups of students. The method used for the implementation of the project in the (university) classroom was approximately the same employed in previous successful works (see [7,8] for more details), and can be briefly summarized as follows. An initial selection of a restricted number of interested students was devoted to choose outstanding students with just appropriate abilities in physical reasoning, and not necessarily requiring a good level of scientific knowledge. This was achieved through a simple test asking few Fermi questions [7]. Then the project effectively started, developed in three successive phases: it lasted a total of approximately five months, with scheduled two-hour (or more) weekly meetings.
In the first phase, lasting just five two-hour meetings, the tutor introduced the subject and read, commented and discussed with the students the original texts chosen (see supplementary material II). The students were asked to address a given topic, brought to their attention from reading some original passage where a given problem was actually posed, the instructor playing the role of a 'master' of physics reasoning, enabling students to think like the authors of the texts. In addition to introducing the given problem, the reading of these texts proved crucial to allow the students to fully appreciate how 'philosophical' reasoning was developing, as well as how it was presented to an educated readership.
In the second phase, the longest and most important one covering a time period of more than three months, the students were asked to autonomously reproduce the 12 experiments discussed in the first phase and reported above, by using the resources available or procuring the necessary material. They worked alone or barely supervised and guided by their tutor (who, however, constantly followed their work), in order to let them to realize and solve all the different, practical difficulties encountered in the experimental realizations, without external help. As a matter of fact, indeed, experiments were set up by the students (though assisted by the tutor) and performed by themselves; historical inspiration was always a primary concern, but students were allowed to set up appropriate (for current times) reconstructions of the given apparatuses (for example, as noted above, aluminum was used rather than iron in most experiments to speed up heating or cooling; etc). The students also provided their own interpretation of the results of the experiments (sometimes stimulated by the instructor), and then compared them to the original ones, as reported in the original texts. At the end of this phase, when the experiments were completed, a cumulative discussion of the different results achieved was also included.
This (larger) phase devoted to experimental activity was then complemented by a theoretical one, developed in just a couple of meetings. It was aimed at illustrating just how the physical ingredients deduced from experiments entered into the theoretical framework that led to the celebrated heat equation; or even how predictions of the theory could be directly tested by experiments, again according to Fourier's original approach. The students were thus introduced to Fourier's theoretical analysis of the problem of heat propagation, and, although here the main active character was the tutor explaining to the students the basic theoretical passages performed by Fourier (see supplementary material I), nevertheless constant interaction with the students was always in order, given the strict recourse to the experimental findings already obtained by them during the second phase (just following, again, original Fourier's reasoning).
Finally, the students were asked to present the results acquired, with the filming of the whole series of experiments realized [22], neatly repeated (and with further acquaintance with difficulties related to communication problems), as well as with the creation of an illustrative panel reporting the basic Fourier's theoretical results. Especially the filming of the experiments was an important part of the project, since a number of unexpected technical topics emerged, so the students were further challenged to clarify them. The whole activity was then presented by the students to other students (not involved into the project) and to the general (non-educated) public during science fairs, where experiments were reproduced and the whole historical path explained to visitors, even with the help of the videos and the panel previously realized.
Since we were also interested in measuring the learning outcomes of the project, we devised some surveys aimed at investigating both the students' impressions and project's effectiveness. The first one was addressed just to the students involved (given to them at the very end of the project), and concerned their previous knowledge (A), their engagement into the project (B), project setting (C) and its outcome (D); entries were as follows: Your overall satisfaction about the project.
This survey revealed that, despite quite a poor knowledge of the subject at the start of the project (especially for Fourier's experiments), the activities undertaken clearly aroused a strong interest in it, the proven strength of the project being the continued stimulus of students' curiosity, according to the favourable reception of the method employed too. The second survey was instead addressed to a small number of teachers evaluating students' activities during their public performances, and concerned the knowledge acquired (A), the competencies developed (B) and the abilities mastered (C) by the students; entries were as follows: A. Knowledge about the general topic.
Knowledge about the specific topics. B. Behavioural competencies (communication, initiative, etc).
Abilities in demonstrating science.
The external evaluators (not involved in the project at all), with working interests ranging from general physics, applied physics, theoretical and experimental physics, expressed an overall extremely favourable view regarding the students involved in the project, not limited to an appreciation of the high level of knowledge they had acquired, but also referred to their skills in scientific communication and demonstration.

Discussion and conclusions
The project discussed in the present paper was aimed at illustrating, to undergraduate students, the early historical path resulted into the foundations of the theory of heat developed by Joseph Fourier two centuries ago. Such a path was strongly based on key experiments, which were at first carried out by the French scientist in order both to acquire the significant known results about heat and to investigate the relevant parameters ruling heat propagation. When sufficient empirical evidence was gained, Fourier was then able to elaborate the basic features of his theoretical work on heat propagation upon that firm physical ingredients, according to his own approach at variance with the fashioned Laplace one based on action at a distance. This point was particularly emphasized to the students, being a common way of proceeding in scientific research in general, which, however, is not always clear or obvious to them. Indeed, while undergraduate students plainly accept that the scientific discourse is constructed with arguments based on experimental evidence, they do not crucially acknowledge the role of hypotheses and their need to be tested by experiments. In this respect, the comparison between the Biot theoretical approach recalled above and the early Fourier one, both based practically on the same starting experimental evidence, is particularly illuminating. They indeed led to different theoretical descriptions of heat propagation, the 'correct' choice between them being decided only by resorting to further experimental tests. Also, the intriguing Fourier's idea of using a closed ring to experimentally test Biot's assumption on the establishment of a stationary state in a linear bar, further highlights to students the strict duality between theories and experimentation. The resulting heat equation deduced by Fourier, with the relevant boundary conditions required by the given problem at hand to be worked out, was then solved by introducing appropriate mathematical tools (based on trigonometric expansion), and the relevant predictions and consequences were again subjected to further experimental test. In the present project, we just focussed mainly on this complex experimental path, rather than on the betterknown theoretical work, to allow the students involved to realize how our significant knowledge on heat (not limited to heat propagation) was effectively acquired. In the supplementary material, however, we discuss also the second part of the project, devoted rather to Fourier's theoretical work, so that we urge the interested reader to refer to such part. Note that, while the experimental activity of the project presented here can be easily addressed also to college or high school students, the theoretical one is instead aimed at second or third-year university students (in Physics, Engineering or in general, STEM students).
The method adopted to develop our project was the same employed in recent years to present other relevant historical cases [7,8], based on direct reference to original texts (here centered about-but not limited to-Fourier's papers [3, 11,12]; see supplementary material II), from which we have reconstructed a series of 12 key experiments.
As in previous projects, what was acquired and realized by students was subsequently 'put into practice' by asking them to disseminate to others their achievements. This resulted firstly in the realization of 12 videos illustrating the experiments performed (uploaded to the YouTube platform [22]), and then the whole project was presented to other students and to the general public in science fairs and similar events (such as the European Researchers' Night, Futuro Remoto Science Fair in Italy, or even in high schools), mainly focussing again on experimental activities. This part actually allowed the students involved in the project to effectively test different skills: preparation of videos and their filming, as well as public presentation, required a total time comparable to that required for the first part of the project on the realization of the experiments.
The final result, as can be appreciated from the videos available on YouTube and from the general satisfaction of the students involved (as well as the public participating in the events), is totally encouraging, and certainly urges to further explore the interesting connections suggested by the history of physics in advanced education and dissemination projects.

Acknowledgments
We gratefully acknowledge the kind assistance by Gianluca Amato and Matteo Olimpo during the preparation of the experiments (and their filming) discussed in the present work, as well as technical assistance by Stefano Marrazzo in preparing some of the metal devices employed in the experimental activity.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).