Adolfo Bartoli’s Relativity

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    In 1876 Adolfo Bartoli (1851 Mar 19 – 1896 Jul 18), a professor of physics at the University of Bologna, wrote a short paper ("On the movements produced by light and by heat and on the Crookes radiometer") on an odd quirk that some physicists had discovered in the relationship between bodies at various temperatures and the radiation emanating from them. He had conducted an experiment in his imagination and had discovered that in order to preserve the validity of the second law of thermodynamics, the law of entropy, he had to assert that heat radiation exerts pressure upon any surface that absorbs it or reflects it. As I demonstrate below, if Bartoli had pursued one question about his discovery, he could have gone amazingly further and deduced the theory of Special Relativity.

    In 1884 Bartoli returned to the subject in his paper "Radiant Heat and the Second Law of Thermodynamics", which he had published in Il Nuovo Cimento, Italy’s premier scientific journal and which I translated into English for the appendix to this essay. He did so in response to a paper titled "Radiant heat, an exception to the second law of thermodynamics", published in 1882 in the Scientific Proceedings of the Ohio Mechanics’ Institute by Professor Henry T. Eddy. Bartoli quoted at length from his previous paper to show that he had confronted and solved the same problem that Eddy presented and had done so eight years earlier.

    In his deduction Bartoli took as his axioms two basic laws of physics, the first and second laws of thermodynamics. The first law, also known by physicists as the law of conservation of energy, states that no phenomenon can create or destroy energy, only convert it from one manifestation into another. The second law, also known by physicists as the law of entropy, states, in the form in which Bartoli used it, that heat will not of itself flow from a cold body to a hot body, nor can any phenomenon make it do so without the exertion of an appropriate amount of work.

    When he described his imaginary experiment he relied on describing a perfectly spherical system. In reply to some criticism (real or anticipated) that nobody could actually carry out such an experiment, he added a description of the experiment performed with a system comprising a cylindrical tube with two black-faced pistons inserted into its ends and several circular mirrors that an experimenter could insert into the tube and remove from it through slots cut into the tube. I want to use that linear system in what follows.

    In the laboratory of your imagination envision a long, wide cylindrical tube with a perfectly reflecting interior surface and with two black-faced pistons inserted, facing each other, into the ends of the tube. Imagine that we have inserted perfectly reflecting mirrors into the tube so that they rest against the pistons’ faces and that no radiation exists in the tube between the mirrors. Now we remove the mirror lying in front of the colder of the two pistons, which we identify as the piston on our left. Heat radiates into the cavity between the pistons, reflects off the mirror in front of the right-end piston, returns to the left end piston, and gets re-absorbed. When the system reaches equilibrium the piston re-absorbs heat at the same rate at which it radiates heat. We then slide the mirror back in front of the left-end piston.

    How much heat do we have in the cavity? Because we have assumed that the mirrors and the inner wall of the tube reflect radiation perfectly, we can say that the radiation in the cavity does not decay. So we calculate the amount of radiation that the piston emitted into the cavity until it reached equilibrium. The chief complication in this case lies in the fact that the piston radiates heat in all directions on a hemisphere, so we must take into account the fact that the radiation emitted in a given direction will reach equilibrium at a time different from that at which radiation emitted in different directions needs. However, once the radiation emitted in a given direction reaches equilibrium, its contribution to the heat energy in the cavity doesn’t change, so we need only add up the contributions coming from all available directions to get the answer to our question.

    If we let R represent the length of the cavity, then the overall distance that a ray of heat travels in going from the left-end piston to the right-end piston equals L=R/cosθ, in which θ represents the angle that the ray makes with the centerline of the cavity. We assume that the left-end piston radiates heat in a given direction at the rate of K watts per square meter, so from a given element of area dA we get KcosθdA watts radiating into the cavity, the cosine factor representing the fact that an observer viewing the emission directly sees the emitting surface tilted away from them at the angle θ. We now want to calculate the amount of heat radiated into a given direction in the time it takes a ray emitted in that direction to reach the right-end piston. For that we simply integrate the emission rate over the area of the piston and multiply by the time it takes the radiation to travel from the left-end piston to the right-end piston, for which time we have t=L/c, in which c represents the heat’s speed of propagation. We then want to combine the heat radiated into all of the directions that comprise a given angle θ and we do that by multiplying Ldθ by sinθ. Finally to calculate the total amount of heat in the cavity we then integrate that result over the angle θ from zero to ninety degrees and double the result to account for the heat going back from the right-end piston to the left-end piston. We calculate the total energy of the radiant heat in the cavity as

(Eq’n 1)

which conforms to what Bartoli got with his spherically symmetric system.

    Now we slip a third mirror into the tube just in front of the mirror blocking the left-end piston and remove the mirror blocking the right-end piston. Heat will fill the cavity from the right-end piston, but we can ignore it because we can treat it as a problem separate from the one we have at hand. We move our third mirror from the left-end piston to the right-end piston, thereby pushing the heat radiation in the cavity into the right-end piston. In so doing we leave the cavity devoid of any heat radiation whatsoever, which means that, in essence, we have moved heat from a body at absolute zero into a body that has some non-zero temperature. Bartoli did not say that explicitly in his paper, but it was certainly implicit in what he did (and the physicists who read his paper certainly understood what he implied). But moving energy from one body to another necessitates that we do work. And moving heat energy from a body at absolute zero necessitates that the work we do equal the amount of energy that we move. The increment of work that we do on a system equals the product of the force we apply and the increment of distance over which we apply it, so equating that increment of work to the increment of energy that we have changed in the system we get, as Bartoli did,

(Eq’n 2)

Dividing that by the area of the moving mirror and by the increment of the distance over which the force does its work gives us the pressure that the mirror must exert upon the radiation to move it,

(Eq’n 3)

And, finally, Newton’s third law of motion requires that the radiation exert an equal and oppositely directed pressure upon the mirror in order to uphold the law of conservation of linear momentum. Thus we infer, as Bartoli did, that heat radiation and, by extension, light exert pressure.

    Che molto bello! This in itself suffices to make Adolfo Bartoli one of the greatest physicists in history. After all, Ludwig Boltzmann adapted Bartoli’s imaginary experiment directly and used it to deduce the Stefan-Boltzmann law. In 1893 Wilhelm Wien used an imaginary experiment involving the adiabatic expansion of heat radiation inside a cavity to deduce his well-known displacement law. In 1899 Pyotr Nikolayevich Lebedev (1866 Feb 24 – 1912 Mar 01) became the first person to conduct an experiment that demonstrated the existence of radiation pressure. And Bartoli’s use of an imaginary experiment may have inspired Einstein’s use of such experiments in his own work; in 1895, when he lived in Pavia, Einstein’s house stood about half a mile from the university, where Bartoli was professor of physics, and Einstein made emphatic use of the concept of radiation pressure in his famous 1905 papers. But as I demonstrate below, from that point Bartoli, had he conceived the right question, could have gone much further and deduced the theory of Special Relativity himself. That fact comes as a shock, because we don’t think of thermodynamics as having the kind of intimate relationship with the structure of space and time that would enable anyone to do such a thing.

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    Let us suppose that Bartoli had wanted to analyze his cylinder-and-pistons system in motion, to see how motion through the æther would affect it. Bartoli had available to him a still-new technology that could have provided inspiration for the desired experiment; the steam-driven railroad train would do the job nicely. He could imagine conducting his experiment on a fast train moving along a straight section of track, his experiment so set up inside one of the carriages that the axis of the tube pointed in the same direction as that of the train’s motion. In that version of the experiment the observer moving with the cylinder would assume the existence of an æther wind blowing through their apparatus, the wind occurring due to the motion of the apparatus relative to the æther.

    We know that Bartoli used the concept of the æther in his theorizing, because he described that concept in part in the 4th Hypothesis that he wrote at the end of the paper to which I referred earlier. In particular, he believed that the æther can support temperature differences that exert a kind of friction on light and heat radiation passing through it, that friction producing effects that uphold the second law of thermodynamics. However, in all theories light propagates through stationary æther at the same speed in all directions. That fact necessitates that light propagate at different speeds in different directions for any observer moving through the æther. To that observer it would appear that an ætherial wind hindered the motion of light in one direction and aided it in the opposite direction. Given that the cylinder moves in a direction parallel to its axis of symmetry, we assume that the radiation would move in one direction slower than the measured speed of light and in the other direction faster than the measured speed of light, depending upon whether the radiation propagates against the æther wind or with it. The waves propagate upstream and downstream at speeds

(Eq’n 4)

A person moving through the æther would thus observe a strange form of the Doppler effect.

    Look again at Bartoli’s perfectly reflecting tube capped at both ends with perfectly black pistons and assume that it moves through the æther in a direction parallel to its axis. Assuming further that we move with the tube, pick a single frequency and observe how heat radiation of that frequency comes off the pistons. At the rear piston the radiation gets emitted against the ætherial current and thus, because it moves away from the piston slower than the standard speed of light, each part of the wave gets partly held back and the wave comes out with a wavelength shorter than it would have if it got emitted from a body at rest. At the forward piston the radiation gets emitted with the ætherial current and thus, because it moves away from the piston faster than the standard speed of light, each part of the wave gets pulled away from the piston and the wave comes out with a wavelength longer than it would have if it got emitted from a body at rest.

    In the case in which no æther wind blows through the tube we have a simple relation among frequency, wavelength, and speed of propagation; to wit, λν=c. If we now have an æther wind blowing through the tube at a speed V and the radiation coming straight off the pistons (that is, parallel to the tube’s axis) at the same frequency, the wavelengths come out as

(Eq’n 5)

Because we refer this calculation to an absolute space, via the æther, we may call that equation a representation of the Newtonian Doppler shift.

    Both the experimenter on the train and an observer standing on a platform beside the track expect to see that same change of wavelengths. The train observer attributes the change to the effect of the æther wind blowing through his apparatus and the platform observer attributes the change to the motion of the pistons as they emit the radiation. But the platform observer also expects the radiation to display the Bartoli effect, the moving pistons doing work on or gaining work from the radiation as they emit it.

    Assume that Bartoli himself stands on the trackside platform and conducts experiments with an array of apparatus identical to what his assistant has on the train. In his tube the pistons each emit heat in pulses of radiation of a single wavelength that propagates parallel to the tube’s axis, each pulse carrying energy in the amount Q. Kirchhoff’s radiation law of 1859 tells Bartoli that he can make his apparatus do this, if only in concept.

    Then Bartoli asks what will happen in his assistant’s apparatus. To calculate the amount of work done in the emission of a pulse, he uses Equation 2 in the form

(Eq’n 6)

in which the integral of KA over the time taken to emit the pulse equals the energy Q that the generator puts into the pulse. The energy that he would expect to measure then conforms to

(Eq’n 7)

the plus sign applying to the radiation coming from the piston moving in the direction of the radiation and the minus sign applying to the radiation coming from the piston moving away from the radiation.

    The absorption of each pulse works like emission in reverse. Again work gets done on or by the piston as it absorbs the radiation, depending on whether the piston moves away from or toward the radiation. The amount of energy in a pulse conforms to Equation 7, so for the energy coming from one piston going into to the opposite piston we have

(Eq’n 8)

But that gives us a net decrease in the amount of energy in the radiation.

    We can calculate how long our imaginary experiment takes to diminish the energy in a parcel of heat radiation by a certain proportion. If we make the distance between the pistons equal five meters and cover the piston faces with perfectly reflecting mirrors, the radiation will make the round trip from and back to one of the piston faces 30,000,000 times every second. We also assert that the pistons move at a speed of 1000 meters per second, a speed achievable by an artillery shell of the 1870's. Thus we conceive an apparatus that Bartoli and his colleagues could imagine building and using, though they would not actually attempt such a feat. In that device a parcel of heat radiation carrying energy Q would have its energy content diminished after N round trips in the device to

(Eq’n 9)

Because the square of V/c gives us a very small number, we can exploit the binomial theorem and approximate the result as

(Eq’n 10)

We can thus calculate approximately how long it takes the energy content of the parcel to decline to half its original value. It would take about 75 seconds for the parcel’s energy to decline by fifty percent, so we can take that time as a kind of half-life of the heat radiation in the device.

    Of course, that diminution does not violate the first law of thermodynamics (the conservation of energy), because the energy lost by the radiation appears in the pistons as a net increase in their directed kinetic energy. Bouncing back and forth between the mirrors, the pulse exerts more force on the leading mirror than it exerts upon the following mirror. The combination of the net force and the motion of the tube-and-pistons apparatus does work while diminishing the radiant heat in the pulse. However, this process of converting heat, at any temperature, directly into net work without exploiting a temperature difference violates the second law of thermodynamics.

    In this case we must use a form of the second law that appears to differ from Clausius’s version. Rudolf Clausius told us that heat will not of itself flow from a cold body to a hot body. To move heat from a cold body to a hot body we must put the bodies into thermal contact with an engine (called a refrigerator) and then we must do a certain minimum amount of work upon that engine to move a given amount of heat from the cold body to the hot body. We now imagine running that apparatus in reverse and infer that we can extract work from the heat in a hot body by making that heat go through the engine to the cold body, on the proviso that only a certain amount of the heat can become work in that circumstance and that the rest must go into the cold body. Thus we infer that no process can convert heat into work with perfect efficiency: some of the heat must go into a body colder than the body from which we drew the heat. The process described above clearly violates that law.

    One may argue that the pressure of the heat radiation on the rear of each piston will solve the problem, but we can easily refute that argument. Simply put, the work done on the following piston by radiation coming from outside the cylinder equals the work that the leading piston does on the radiation coming from outside the cylinder and striking its front. Thus we infer that the external radiation field will not interfere with the process described above.

    Bartoli can only make Equation 8 conform to the law of entropy by multiplying it by the square of what we now call the Lorentz factor,

(Eq’n 11)

He knows that he must make that correction in a way that does not negate the augmentation or diminution of energy as expressed in the factors in Equation 8. He also knows that, because the Lorentz factor depends only upon the velocity of the system in his laboratory, that factor must have the same value for augmentation that it has for diminution. Thus he writes for the change in the radiant energy in the system

(Eq’n 12)

in which the plus sign applies to augmentation of energy by work done on the radiation and the minus sign applies to the diminution of energy by work done by the radiation.

    We can see that the law of entropy requires the Lorentz factor in that equation, but what does it mean in physical terms? Bartoli would have recognized that Equation 12 resembles the equation describing the Doppler shift and that would make perfect sense to him because he knows that radiation bouncing off moving mirrors gets its wavelengths shifted. But Doppler never devised an equation like Equation 12, with a gamma factor in it, so Bartoli has to figure out the meaning of the Lorentz factor on his own deductive power.

    He notices that Equation 12 resembles Equation 5. He also notices that the radiation coming off the leading piston has elongated wavelength and diminished energy and that the radiation coming off the following piston has shortened wavelength and augmented energy, so he writes

(Eq’n 13)

in which K represents a constant that converts units of length into units of reciprocal energy. But in any wave phenomenon there exists a simple relationship among wavelength, frequency, and the speed of propagation: Bartoli knows that λν=c, so he modifies Equation 13 to read

(Eq’n 14)

in which t represents the time elapsed between two consecutive emissions of two identical parts of the wave, the crests for example. Also c represents a constant, the radiation’s speed of propagation, as measured by Bartoli, whether the radiation emanates from a stationary object or from a moving one. Thus t remains as the only variable factor which the gamma factor can modify.

    Bartoli writes

(Eq’n 15)

in which the primed tee represents the period of the wave emitted from the moving pistons in his assistant’s apparatus and the unprimed tee represents the period of the wave emitted from the stationary pistons in his apparatus. Because gamma always takes values equal to or greater than one, Equation 15 tells Bartoli that a moving body emits radiation more slowly than an identical stationary body does.

    So Bartoli can thus assert that a moving generator pulsates more slowly than does an identical stationary generator. Because we can use any pulsating source to drive a clock, Bartoli infers that moving clocks count time more slowly than stationary clocks do. He has discovered the relativistic effect that we call time dilation. But he would have done so only in the sense that Hendrik Antoon Lorentz did, by assuming that motion through the æther makes objects more ponderous and thus makes clocks tick slower. We must note that Lorentz devised his theory as a correction of the æther theory to explain the Michelson-Morley experiment, which occurred a decade after Bartoli would have done his theorizing. Bartoli does not yet have a perfectly relativistic system.

    But Bartoli wants to make sure that he and his assistant can coordinate their measurements of the elapse of time between events occurring in their experiments. For an imaginary experiment that will clarify that issue Bartoli might have obtained inspiration from a couple of still relatively-new technologies.

    In Bartoli’s time those people who could afford to travel (and, as a university professor, Bartoli could) traveled by train, that still-newfangled device that made long-distance travel so convenient. So, perhaps some evening while waiting on a station platform for a train, Professor Bartoli happened to notice the flickering light from the locomotive’s firebox pulsating on the track as his train arrived. He comes to understand that, if his assistant were to project a pulsating light onto the track, he could use the pulsations as the equivalent of a clock and that he could use that fact to compare the measurements of time that he and his assistant make with their more conventional clocks. But how would the assistant generate the pulsations in a uniform manner?

    To answer that question Bartoli might have recalled to mind the concept of a toy that still enjoyed some popularity in his time. Originally invented in 1834 by the British mathematician William George Horner (1786 – 1837 Sep 22), it enjoyed a resurgence of popularity in America in the 1860's under the guidance of William F. Lincoln (no dates found), who called it a zoetrope (roughly meaning wheel of life in Greek). A zoetrope consists of a wide, short cylinder mounted on a spindle that lies on the cylinder’s axis but does not enter the cylinder. Vertical slits cut into the upper half of the cylinder allow a viewer to see the lower half of the inside of the cylinder, where a band displaying pictures lies. As the zoetrope spins, the viewer sees the pictures appear to move. An understanding of how that illusion works led, later in the century, to the invention of motion pictures.

    But Bartoli would more likely have used the Austrian version, called a stroboscope (from the Greek strobos [whirlpool]+skopein [to look at]). Invented by Simon Ritter von Stampfer (1792 Oct 26 – 1864 Nov 10), a stroboscope consisted of an axle passing through the center of a disc in which radial slots were cut. A viewer would hold the stroboscope in front of a mirror and look through the slots at the reflection of the pictures drawn on the opposite side of the disc. The stroboscope would likely have reminded Bartoli of the toothed wheel that Armand Hippolyte Louis Fizeau (1819 Sep 23 – 1896 Sep 18) used in 1849 in his apparatus to measure the speed of light.

    So Bartoli and his assistant set up a stroboscope in the rail carriage where the assistant will perform his experiments, putting it up in such a way that its axle lies parallel to the direction of the train’s motion. With a light source on one side of the disc and a tilted mirror on the other side, the spinning stroboscope projects pulses of light onto the ground beside the track. Along one section of track Bartoli has laid a long mirror whose white frame scatters some of each pulse into one or another of the cameras that Bartoli has set up alongside the track (we assume that Bartoli intends to conduct at least some of his experiments on an especially dark night).

    The assistant spins up the stroboscope until the disc rotates with an angular speed of ω=2π/t, where t represents the time that the disc requires to make one complete rotation. Then the train accelerates to the speed with which it will pass Bartoli’s experimental equipment. Floating on perfectly frictionless bearings, the stroboscope continues to spin at the set rate, doing so at least until the experiment has come to its end.

    Bartoli knows that his assistant gave the disc a certain amount of angular momentum when he spun it up and he knows that the acceleration of the train exerted no torque upon the disc, so the angular momentum before the train accelerated must equal the angular momentum in the disc after the train has gotten itself up to speed. To simplify matters Bartoli has made the disc of a thin, heavy ring with a light membrane stretched over it: almost all of the disc’s mass m resides in the ring a distance r from the centerline of the disc’s axle. Thus, for the before and after angular momenta Bartoli has

(Eq’n 16)

Using Equation 15, he transforms that equation into

(Eq’n 17)

    Bartoli can’t solve that equation by itself: he needs more information. So he imagines that his assistant has laid down a small track across the rail carriage, in a direction perpendicular to the direction of the train’s motion, and has a small object of mass m slide on it (without friction, of course) with a velocity of dy/dt. The acceleration of the train exerts no forces perpendicular to the train’s direction of motion, so the small body’s linear momentum in the y-direction does not change. Thus Bartoli can write

(Eq’n 18)

Again invoking Equation 15, he transforms that equation into

(Eq’n 19)

    Although it seems illegitimate on first impression, Bartoli can, in fact, solve Equations 17 and 19. He reasons that, because nothing distinguishes one lateral direction from another, then whatever transformation applies to dy must also apply in the same way to r. He thus gets

(Eq’n 20)

which necessitates that

(Eq’n 21)

Bartoli accepts Equation 20 because it seems to offer an explanation of the slowing of moving clocks: it seems reasonable to claim that clocks would move more slowly when they become more ponderous. Likewise, Equations 21 express the proposition that distances measured perpendicular to the direction of motion in a system do not change: Bartoli also accepts that result and uses it in his next imaginary experiment.

    Bartoli directs his assistant to make his stroboscope send its pulses of light straight down onto the mirror laid alongside the track. The assistant reports that each pulse travels a distance z’ at the speed of light, taking a time interval t’ to cross that distance. Bartoli knows that in his frame of observation each pulse follows a slanted course whose length he can calculate by way of the Pythogorean theorem,

(Eq’n 22)

The x-coordinate represents the distance the train moves between the instants in which the pulse comes off the tilted mirror in the assistant’s stroboscope and in which the pulse strikes Bartoli’s trackside mirror, equal to the product of the train’s speed V and the time interval t that Bartoli would measure with his clocks. Equation 22 thus becomes

(Eq’n 23)

Equation 15 came into play in the move from the second line of that equation to the third. That equation tells Bartoli that the pulse travels along the slanted path in his frame at the speed of light, thereby confirming what he has discovered so far.

    One assumption that Bartoli and his assistant made, that the pulses move at the speed of light in any direction perpendicular to the motion of the train in the assistant’s frame of observation leads Bartoli to modify Equation 4 to read

(Eq’n 24)

in which θ represents the angle between the direction of the train’s velocity and the direction in which the light gets emitted. That equation would bring Bartoli up short, because it shows him a serious problem with the æther theory of radiation propagation. He sees that radiation emitted from a moving body will encounter a strange form of refraction caused by the æther wind blowing through the body. Knowing that refraction makes radiation turn in to a medium where the radiation propagates more slowly than it does in the medium from which it came, Bartoli understands that radiation emanating from a moving body turns into the æther wind and away from the direction in which that wind blows.

    Bartoli imagines two bodies, one warmer than the other, moving together through the æther. If the warmer body lies directly in front of the cooler body, ætherially upwind of it, then it becomes possible for the ætherial refraction to concentrate radiation from the cooler body onto the warmer body while dispersing the radiation emitted from the warmer body toward the cooler body. He conceives a means of creating a purely passive device that would make net heat flow from the cooler body to the warmer one, in blatant violation of the second law of thermodynamics.

    Bartoli has only one way to save the second law, only one way to ensure that the theories of physics give us a correct description of the workings of Reality. He must dismiss Equation 24 (and Equation 4) as invalid as a description of the real world. He must assert that in all directions a ray of light must travel at the same speed, whether measured by Bartoli or by his moving assistant.

    Thus he deduces what we now call Einstein’s second postulate of Relativity. Note that while Einstein had to introduce the proposition as a postulate, Bartoli would have deduced it from the first and second laws of thermodynamics.

    This goes beyond strange and Bartoli wonders whether he has made an error somewhere. He rechecks his derivations and calculations and finds that, except for Equations 4 and 24, they do not change except for the fact that the assistant no longer sees what Bartoli sees. In the radiation emanating from the moving pistons Bartoli sees a Doppler shift; his assistant sees none. Bartoli detects an increased mass in objects aboard the moving rail carriage; his assistant detects none. What could make such a thing possible?

    Bartoli tries to imagine what his assistant sees and he gets another shock. If light moves at the same speed for both him and his assistant, then the assistant would see Bartoli’s clocks ticking more slowly than his do. Bartoli understands that if he were to project pulses of light vertically toward the trackside mirror, the assistant would see those pulses tracing slanted paths, so he could use a modified version of the analysis that led Bartoli to Equation 23 to show that, indeed, Bartoli’s clocks appear to him to count time more slowly than his clocks do.

    Bartoli examines his experiment in his mind, trying to see where he went wrong. He notices one feature that might have something to do with the presumed error. If he lets his pulse travel up and down, he will need only one clock to time its traverse, but his assistant will need two clocks separated from each other by the distance the train travels as the pulse makes its up and down movement. Could the motion of the train somehow prevent the assistant from properly synchronizing those clocks?

    The assistant can demonstrate easily enough that he has his clocks properly synchronized. He finds the point exactly midway between the clocks, places a small quantity of flash powder at that point, and ignites it. The flash expands away from the ignition point at the same speed in all directions and so takes the same amount of time to reach the clocks, illuminating their faces and showing that, indeed, they display precisely the same time; the clocks are truly synchronized.

    Bartoli conducts the experiment in his mind and sees an amazing phenomenon. He imagines having photographed the rail carriage as it passed his position and notes that the flash would have illuminated his clocks in sequence, progressing away from the instant of the flash powder’s ignition. When the flash illuminated the assistant’s clocks it would also have illuminated those of Bartoli’s clocks that stood next to them at that instant, thereby putting a time stamp on Bartoli’s photographs. Bartoli sees that the assistant’s clocks display the same time but that his clocks don’t: according to Bartoli’s clocks the flash illuminated the clock at the rear of the rail carriage before it illuminated the clock at the carriage’s front end. That fact means that, although the two clocks appear synchronized to the assistant, for Bartoli the rear clock actually runs fast relative to the front clock.

    It doesn’t take long for Bartoli to figure out what happened. The flash propagated in all directions at the speed of light and at the same time the rail carriage moved in the positive x-direction at the speed V. Thus, the rear clock ran into its part of the flash before the flash could overtake and illuminate the front clock. To calculate the time difference, as his clocks measure it, Bartoli needs to calculate first the time that the flash takes in going from its origin to each of the assistant’s clocks. The assistant measures the distance between his clocks as x’, so Bartoli divides half that distance by the relative speed between the light and the clocks. A simple geometric analysis tells him that he can actually use Equation 4, even in the absence of an æther, to calculate that speed, so he gets

(Eq’n 25)

The difference between the two times comes out as

(Eq’n 26)

    In that equation the gamma factor from Equation 11 appears as its square. Bartoli knows that the gamma factor applies once to account for the fact that the assistant’s clocks count dilated time. That means that the second gamma factor must multiply the assistant’s measurement of length, making it extend longer for Bartoli than it does for the assistant. He sets that idea aside, intending to come back to it later, because he has just noticed another feature of this distortion of time.

    The fact that the assistant’s well-synchronized clocks appear out of synchrony to Bartoli (as they would do if he photographed both of them simultaneously) necessitates that Bartoli’s well-synchronized clocks appear out of synchrony to the assistant and do so in exactly the same way. By correcting the propagation law for light in order to preserve the validity of the second law of thermodynamics, Bartoli seems to have gotten himself lost. He can’t seem to devise an experiment that will give different results to him and to his assistant in a way that lets them determine whether Bartoli and his apparatus truly lie at rest. Bartoli has discovered a strange, yet appealing, symmetry. He appears to have extended the Copernican Revolution: not only does Earth not occupy the center of space, apparently it also does not occupy the Universe’s center of velocity, the state of absolute rest.

    "There must be an absolute state of rest around here somewhere," he mutters. (Actually it would have sounded more like, "Deve essere un stato assoluto di riposo in qualche posto di qui", because il professore Bartoli would not likely have done his muttering in English.) But no matter what he tries, he just can’t seem to devise an experiment that will enable two observers to find a difference in the laws of physics as they apply to this particular situation. He gains the distinct impression that neither observer can determine whether he moves or lies at rest. A strange vertigo sweeps over him. He feels as if someone had set him adrift on a calm sea with no stars to guide him, only the occasional contact with another drifter.

    At this point, instead of feeling stymied, he would have remembered an odd imaginary experiment that Galileo Galilei described in his famous book, Dialogue on the Two Chief World Systems. Or would he?

    It now becomes necessary to ask whether Bartoli could have read Galileo’s book at all. Due to certain statements that they considered errors the leaders of the Catholic Church put the book on their Index of Prohibited Books in 1633, almost immediately after the book was published. Though people in Protestant countries could read the book openly, no respectable person in a Catholic country would want to risk his reputation by being associated with such a thing. However, Dialogue on the Two Chief World Systems was taken off the Index in 1824, so Bartoli could have read it with a clear conscience.

    In that book Galileo defended the Copernican world system, in which Earth revolves about the sun, in part by addressing and answering one of the more common objections to Koppernigk’s hypothesis. People usually expressed that objection by asking Why can we not feel Earth moving? Galileo responded by having one of his characters (it was a dialogue, after all) suggest that another of the characters;

    "Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some flies, butterflies, and other small flying animals. Have a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. The fish swim indifferently in all directions; the drops fall into the vessel beneath; and, in throwing something to your friend, you need throw it no more strongly in one direction than another, the distances being equal; jumping with your feet together, you pass equal spaces in every direction. When you have observed all these things carefully (though there is no doubt that when the ship is standing still everything must happen in this way), have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still."

That last sentence gives us what we call the Galilean principle of Relativity.

    Of course, we can detect Earth’s motion around the sun, but only by measuring Earth’s motion relative to other objects, such as the stars, just as the sailor could determine whether his ship moved on the sea or sat in the harbor by looking out a porthole. Only absolute motion, motion that we could detect without reference to objects outside our laboratory, eludes us. However, Galileo only suggested experiments involving mechanical effects. Bartoli would have had to extend the principle to cover heat radiation and, by extension, all electromagnetic effects.

    Thus he would say that no manifestation of the laws of physics can reveal any state of absolute motion. That statement necessitates the further statement that the laws of physics have the same form for all observers, regardless of how they might move relative to each other. So Bartoli has inferred what we know as Einstein’s first postulate of Relativity.

    But Bartoli retains a certain skepticism about this result. He turns his attention back to the experiments he wants to perform with the tube-and-pistons apparatus his assistant has erected inside the rail carriage. In accordance with his analysis of Equation 26, he expects that, as measured from his frame, the tube will elongate to

(Eq’n 27)

He thus figures that the radiation in the tube has an energy density lower in his frame than it has in the assistant’s frame. That seems reasonable to him because he knows that from his point of view the pistons emit radiation more slowly, due to time dilation, than they do as seen by the assistant. Then he notices that the pistons themselves give him a new problem to solve.

    To gain a clearer understanding of that problem, he conceives a black cylinder suspended inside a black tube such that the two bodies’ centerlines coincide. Both objects have the same temperature, so each receives heat across the narrow gap between them at a rate equal to the rate at which it emits heat to the other body. Accelerating the cylinder to some velocity V does not change the heat in the cylinder or change its other thermodynamic properties, but time dilation reduces the rate at which the cylinder radiates heat. The moving cylinder thus absorbs heat from the tube faster than it radiates heat back to the tube, so when the system regains equilibrium the cylinder can stop and it will have a temperature greater than that of the tube. We can use that temperature difference to drive a heat engine until that difference declines to negligibility, then we can repeat that procedure. Through that process we can convert heat into work in a closed system, in violation of the second law of thermodynamics. Again Bartoli needs to conceive a means of "saving the appearances" regarding the law of entropy: he needs to figure out what his analysis missed in its description of the system and its behavior.

    If time dilation reducing the flux of radiation from the cylinder causes the problem, then whatever increases the flux to its original value will solve the problem. Bartoli has no dynamic options available to him because the acceleration of the cylinder doesn’t change the thermodynamics of the system. He has only a geometric option to exploit. Recalling how a lens concentrates the heat coming from the sun, he asserts that the moving cylinder must shrink in the direction of motion, doing so in the same proportion in which time dilates on moving clocks. That shrinkage restores thermal equilibrium to his description of the system, but it also gives Bartoli yet another puzzle to solve.

    He has already inferred that moving bodies expand, as described in Equation 27, so why does he now infer that moving bodies contract? (We call that effect the Lorentz-FitzGerald contraction, but in this alternate history people would likely call it the Clausius-Bartoli contraction.) To answer that question he asks how he would measure the length of a moving body. The obvious method consists of photographing the body at a single instant as it passes a long ruler. He understands that he must make one feature of the method explicit: his cameras must photograph the ends of the moving body simultaneously, which means that between the measurements of the locations of the body’s ends by the ruler he has t=0.

    Now he combines Equation 15 with a form of Equation 26 that calculates the difference that he would see in a photograph of the assistant’s clocks;

(Eq’n 28)

That equation tells him, via his t=0, that

(Eq’n 29)

He knows that if a time t had elapsed between his measurements of the locations of the body’s endpoints, he would have to calculate the distance as the usual x+Vt. Making appropriate substitution from Equations 15 and 27 transforms that calculation into

(Eq’n 30)

Substituting into that equation from Equation 29 gives him

(Eq’n 31)

which expresses the Clausius-Bartoli contraction.

    Now Bartoli understands how he could have two different descriptions of length. Instead of representing the amount by which one of a pair of moving clocks runs fast relative to the other, Equation 29 represents one clock existing so far into the future relative to the other clock. That means that the moving clock at the rear of the rail carriage partly overtakes the front clock: because that effect gets prorated over the length of the carriage, it results in a uniform contraction of the vehicle.

    Bartoli writes Equations 28 and 30 along with y=y’ and z=z’ and he presents them as a complete set of equations for transforming the measurements made by one observer into the equivalent measurements made by another observer. Thus Bartoli would have obtained the four equations of what physicists would call the Bartoli Transformation, which equations we actually call the Lorentz Transformation. He understands that he has done something remarkable. Isaac Newton added the element of time to the geometry of Euclid and Apollonius to create his dynamic geometry, which formed the basis for modern physics. Now Bartoli sees that he has added the element of time to geometry in a much more intimate way, one that obscures the distinction between space and time. Moreover, he does not understand the equations as Lorentz did, as indicating the effect of an æther wind, but rather as Einstein did, as indicating a distorting of space and time between two inertial frames. He might even notice, as Hermann Minkowski actually did some thirty years later, that he can get from his transformation equations a four-dimensional analogue of the Pythagorean theorem,

(Eq’n 32)

As an excellent geometer, Bartoli would recognize that calculation as expressing the mathematical fact that his transformation equations represent something analogous to a rotation of one coordinate grid relative to another. Thus he might have made the discovery that Minkowski made in 1908.

    Would Bartoli have taken his analysis further? We cannot doubt, as I will demonstrate, that he could have done so. And, given that it involves another application of his theory of radiation pressure, we may think it likely that he would have taken at least one more step in the development of Relativity.

    Bartoli’s assistant has established in the rail carriage a short straight track oriented in the direction in which the train moves. On that track he sets a perfectly black disc that can slide freely on the track and at the opposite ends of the track he mounts a pair of spotlights aimed at the disc. At one given instant he makes the spotlights emit pulses of light at the disc, each pulse carrying the same amount of energy E, and the disc absorbs the radiation completely. Each pulse, as it gets absorbed, exerts a force on the disc, but the two pulses exert equal and opposite forces on the disc: the forces cancel each other and, so, the disc does not change its movement. The disc does not acquire any momentum from the pulses, but gains a total energy Q=2E.

    Standing by the track as the rail carriage carrying the experiment passes, Bartoli observes the light pulse coming from the forward spotlight diminished by the fact that it has done work on the spotlight during its emission, so the forward face of the disc absorbs an amount of energy equal to

(Eq’n 33)

Likewise, the light pulse coming from the aft spotlight gains energy from the spotlight doing work on it as it gets emitted, so the rearward face of the disc absorbs an amount of light that conforms to the description

(Eq’n 34)

We then have E1+E2=γQ describing the total energy that the disc absorbs as seen from Bartoli’s frame.

    In Bartoli’s observations and analysis the disc has gained net momentum, because more energy strikes the rear of the disc than strikes the front. For a beam of light (in other words, light all propagating in one direction) falling onto a surface with a certain area, Bartoli can convert pressure into force and then use Equation 3 in the form

(Eq’n 35)

If Bartoli integrates that equation over an interval of time, then he will calculate from Equations 33 and 34 the net impulse that the slug acquires as

(Eq’n 36)

The disc has not changed its velocity for the assistant, so it must have an unchanged velocity for Bartoli. That fact necessitates, because linear momentum equals the product of a body’s mass and that body’s velocity, that the mass of the disc increase as the disc gains linear momentum. We have

(Eq’n 37)

and that gives us, for ΔV=0,

(Eq’n 38)

which corresponds to Einstein’s famous equation E=mc2.

    Of course Bartoli and his colleagues would have assumed that this formula means only that energy confers extra mass on bodies containing it (or, putting it another way, that a hot body carries slightly more mass than does a colder identical body). They would not have suspected that mass itself could turn into energy, but they would have nonetheless created the tool that Antoine Henri Becquerel (1852 Dec 15 – 1908 Aug 25), the Curies, and others would have used to understand radioactivity when Becquerel discovered it in 1896: they would not have had to wait the additional nine years for Einstein to create that tool for them.

    Such an act of creativity, the deduction of Relativity from a science originating in the study of steam engines, seems too good to come true. But we can understand it if we bring a reasonable aphorism into play, Louis Pasteur’s "Chance favors only the prepared mind." The prepared mind in this case came from Bartoli’s theory of radiation pressure, which prepared Bartoli’s mind to take advantage of a chance meditation on the motion of radiation in a system subject to an æther wind.

    Thus one year after publishing his paper on radiation pressure, Bartoli, like Einstein on our timeline, at age 26 would have published his work laying out the theory of Special Relativity. Twenty-eight years before a minor clerk in the Swiss Patent Office shook the foundations of modern physics an Italian professor of physics could have done it and simply astonished the world.

    But Bartoli did not make that wonderful discovery, though in fairness I point out that neither Ludwig Boltzmann nor Wilhelm Wien, who both used modified versions of Bartoli’s imaginary experiment, made it either. Had the time simply not come for physicists to contemplate the thermodynamics of moving systems? That seems a strange question in light of the fact that Bartoli made the heart of his imaginary experiment a moving mirror. So what stopped him or, more to the point, failed to start him?

    We can see readily one reason that explains why Bartoli did not discover Relativity. In "Adolfo Bartoli and the problem of radiant heat" (Annals of Science, Vol. 46, Issue 2 March 1989, pp 183 - 194), Bruno Carazza and Helge Kragh note that Bartoli felt some reluctance to accept the existence of radiation pressure and that he eventually dismissed the hypothesis as false to Reality. When someone asked Isaac Newton how he made his wonderful discoveries, he replied, "By thinking ever unto the problem." We can infer, then, that Bartoli did not put enough thought into the discovery that he had made. He did not come to know it with sufficient intimacy to notice opportunities that it might present. A child that one does not love will not develop its full potential.

    I will leave to others the task of discovering why neither Boltzmann nor Wien discovered the Theory of Relativity in the laws of thermodynamics.

hghg

    So how would physics have developed differently if Professor Bartoli had written "Sulla termodinamica dei corpi in movimento" (On the Thermodynamics of Moving Bodies) before Einstein was born and more than a quarter century before Einstein wrote "Zur Elektrodynamik bewegter Körper" (On the Electrodynamics of Moving Bodies)?

    Almost immediately it would have struck an almost, if not completely, fatal blow against the concept of the æther. In addition to producing violations of the laws of thermodynamics, the refraction of light emitted from moving bodies by the æther wind would yield effects that astronomers would have noticed by 1876. The æther might still exist, certainly, and support the propagation of light, but it can only do so on the condition that it have absolutely no effect upon that propagation, especially with respect to the velocity of propagation. Thus the æther becomes irrelevant to physics and might as well not exist at all.

    As I noted above, one subtle change would have involved the discovery of radioactivity in 1896. Instead of waiting nine years for the mathematical theory necessary to account for the phenomenon, physicists and chemists would have found that theory waiting for them. They would have known about the special relationship between mass and energy.

    Accustomed to thinking of mass as a form of energy a generation earlier than on our timeline, physicists at the beginning of the Twentieth Century might have discerned the relationship between mass-energy and frequency implied by Planck’s quantum hypothesis of 1900 and then added the relationship between linear momentum and wave vector that Louis de Broglie inferred in 1924. With the wave mechanics version of the quantum theory available a generation earlier our modern understanding of the structure of matter would have come to us earlier as well. The theory of electrical conduction offers one example and, in particular, the theory of semiconductors. It may seem trivial, but consider how history might have gone differently if people had listened to Franklin Roosevelt on portable, transistor-based radios instead of the huge sets actually available in the 1930's.

    Or consider a more horrible possibility. If an Italian Catholic, rather than a German Jew, had discovered Relativity, then Adolf Hitler and his Nazis would have had less cause to dismiss modern physics as what they derided as mere "Jewish physics". Would their greater openness to modern physics have led any people in Nazi Germany to invent the nuclear bomb?

    An understanding of Relativity and of the quantum theory is certainly necessary to the development of a nuclear weapon, but the key piece of knowledge that enabled the invention of the nuclear bomb didn’t come to us until 1938 when Otto Hahn and his team of chemists in Germany, with theoretical help from Lise Meitner, discovered nuclear fission. That discovery would likely have occurred at about the same time on our alternate timeline and, thus, there would have been no German nuclear bomb because there would have been insufficient time to carry out the necessary research and development work.

    On our timeline Werner Heisenberg actually tried to instigate a German nuclear bomb project but abandoned the attempt when he understood that the Nazi way of doing business made a crash program, similar to the Manhattan Project, too dangerous to him and his team. That fact would likely have been true on our alternate timeline. Thus history would likely have gone pretty much as it has done on our timeline.

    Comparing this imaginary timeline with our real, as-lived timeline, I can discern a difference in the emphasis in modern physics regarding thermodynamics. When I was at UCLA in 1968 we learned the statistical version of thermodynamics, the version initiated by the studies of Maxwell and Boltzmann of gases as collections of vast numbers of particles. The classical version, the version of Kirchhoff, Clausius, and Bartoli, seemed old-fashioned, more suited for engineering than for physics. Classical thermo felt more like the gritty world of iron, coal, and steam, of clanking factories and chuffing locomotives, while statistical physics put us into the sparkling modern world of plutonium and titanium, of sleek spaceships and humming electricity. But had Bartoli used the law of entropy to deduce important facts about the nature of space and time, that attitude would not have developed.

    We would have had, instead of the required course in statistical thermodynamics, likely a two-part course in classical thermodynamics and radiation thermodynamics. The course in statistical dynamics would have followed that preliminary course and grown out of it. The law of entropy certainly would have received much more attention in its classical form that it does now.

Appendix:

Il Calorico Raggiante e il Secondo Principio di Termodinamica;

nota di

ADOLFO BARTOLI

    In this appendix I present the text, which I have translated from the original Italian into English, of Bartoli’s paper. I have put Bartoli’s footnotes, referenced in the text by numbers in parentheses, at the bottom of the text. I have included my own comments in red. I have also put Bartoli’s quotes from his previous paper in a different font rather that use the continuing << at the beginning of every line.

    Let me note here that translate does not accurately denote what happens when we convert a text from one language to another: the task does not consist of simply making a word-for-word replacement. Even between closely related languages, such as Norwegian and English, that simplistic process will not yield a good result that accurately conveys the meaning that the author wrote into the original text; that fact stands even more strongly true in regard to languages, such as Italian and English, that come from different families (Latin-based and Germanic in this example). We would better use the word transformation to denote what we do, to convey the idea that one cannot merely pick up a Beep-to-Boop dictionary and begin translating. In taking on this task I have read a short teach-yourself book on Italian to gain some familiarity with the grammar. I will still stumble over idiomatic constructs, but in this case I have assumed that my knowledge of physics will compensate for my deficiencies in Italian and thus enable me to produce a comprehensible, if not perfect, transformation of Bartoli’s Nineteenth-Century Italian text into its Twenty-First-Century English equivalent.

    This paper appeared in Il Nuovo Cimento, which was founded in 1844 and became the official organ of the Italian Physical Society when that society was founded in 1897.

Il Nuovo Cimento

Journal founded for Physics and Chemistry

3rd Series, Vol. 15

Pisa, 1884

pages 193 - 202

Radiant Heat and the Second Law of Thermodynamics

by

Adolfo Bartoli

    A recent publication by Professor H.T. Eddy of the University of Cincinnati [Henry T. Eddy (1844 - 1921; Yale, 1867)], which bears the attractive title <<Radiant heat, an exception to the second law of thermodynamics>>, printed in July 1882 in the Scientific Proceedings of the Ohio Mechanics’ Institute (pg 105 to pg 114), has quickly brought itself to the attention of the scientific public. A long summary by Boltzmann appears in Beiblatter in 1883, no. 4, pg. 251 and a very brief item from Prof. Bazzi appeared in Nuovo Cimento, the issue of January 1884; to say nothing of many other appearances in other journals.

    Eddy’s paper has raised many objections, all of which Eddy has answered. It is likely what many other papers by skillful physicists, of both the Old and the New Worlds, would have to be in order to succeed in the first rank on a subject that seems to be of great interest.

    In this paper I permit the full reproduction of any statements from one of my previous papers, titled: The Hypothesis of Movable Equilibrium of Temperature and The Second Law of Thermodynamics (1), a paper of which one part, which is reproduced here now, was inserted into another paper titled <<On the Movements Produced by Light and by Heat and on the Radiometer of Crookes>>, which was published in Florence in July 1876, on the press of the Successors of Le Monnier (13), therefore eight years ago now.(2).

    I reproduce here, in full, the sections reading from page 22 (the second part) to page 27 of my paper, without taking away or adding a single syllable:

    <<I display here briefly (saying then) some of the reasons that made me determined to undertake this research (3).

    A and B would be two extremely thin concentric spherical shells [Bartoli used the Italian word involucro, which means wrapping] that are perfectly reflecting on their interiors as well as on their exteriors and that have radii RA and RB respectively, with RB>RA.

    A spherical body C, of radius ξ<RA, completely black (in the sense given in the words of Kirchhoff, Pogg. Ann. [Poggendorff’s Annalen. See footnote 14 for more] Ed CIX, 1275) would also have a common center with the two shells: to the exterior [of body C] the completely black surface b of a sphere of radius ρ>RB concentric with the previous objects, would contain inside itself the system formed from the sphere C and from the two perfectly reflecting shells. The body C is found (in principle) in thermal equilibrium [presumably with the heat radiation surrounding it]: at a given instant we suppose that the reflector B is destroyed; the body b radiates heat into the whole space included between the surface b and the reflector A. From then, when the body b comes into thermal equilibrium [again, presumably with the heat radiation surrounding it], we suppose that at a given instant shell B is reconstructed and shell A disappears. Thereafter we let the radius of shell B diminish until that radius becomes equal to RA (that same shell remaining always spherical during the operation). With this cycle of operation, a cycle which we are able to imagine repeating any number of times, one can take a certain quantity of heat from body b and transport it onto body C. On the temperature of the bodies b and C we have not made a true hypothesis [i.e. we have made no assumptions]; we are able, therefore, to suppose that the temperature of C is greater than that of b. If we suppose that for any single cause the initial temperatures of body b and of body C remain constant, however much heat the one has lost and the other has acquired, it would be possible to pass as large a quantity of heat as we want from a colder body to a warmer body.

    The mechanism of the operation I have described depends properly on this fact: that if a body C of completely black surface s is found within a perfectly reflecting shell Σ, if this shell comes to shrink, always remaining closed, until coming to adhere to the surface of the body C, the body C comes to gain a quantity of heat

where

K is the quantity of heat emitted in one second from a unit of the surface of s;

ω is the surface of the hemisphere of radius 1;

t, the time spent by a ray of heat which leaves the element ds, to travel its own path, from the point of departure to that which, after a certain number of reflections, comes to encounter anew the surface of C;

θ the angle that the same ray that leaves from ds makes with the normal to the same element.

    If the body C is a sphere of radius ν [it's hard to tell, but this is the Greek letter nu, different from the letter vee used to represent the velocity of propagation], the surface Σ is that of a sphere concentric to the first and of radius R very large with respect to ν, the preceding integral reduces to

with s=4πν2, and v the velocity of propagation of a ray of heat. Then the body C, after this operation, gains a quantity of heat Θ.

    It can be that someone would have objected to the previous reasonings [on the basis of] the physical impossibility of destroying, or re-establishing in an imperceptible time, those shells, or among other things to vary the dimensions, maintain always closed, etc., etc. I select a case in which those objections can no more be made.

    Let F be a perfectly reflective [internally] circular cylinder; (5) and A and B would be two moveable pistons entering the cylinder and terminating in a completely black surface. On the surfaces A and B there are two diaphragms a, b which are perfectly reflecting and movable perpendicular to the axis of the cylinder (by means of an aperture made in the external wall of the cylinder). In this case the physical possibility of carrying out the operation described in principle in this second part, will not be able to come [about, in] contrast with any [other case]; [it tells us] only that it is possible to make to pass from the body A onto the body B a quantity of heat sensible to the most delicate instruments of physics, though it would be necessary to employ a cylinder of very great length.

    We return now to the case of the perfectly reflective spherical shell concentric to a perfectly black spherical body of radius ν much smaller than the radius R of the shell: if it is found that the shell, remaining always closed, would have come to adhere to the surface of the body, this [the body] would gain a quantity of heat

retaining the same notation from the preceding page: if, instead, the shell, remaining also always spherical, varies its radius from R to R’ (with R>R’), the black body gains a quantity of heat q,

(6)

    The preceding considerations and the second law of thermodynamics demand that in this case with deformation of the shell it is necessary to spend a quantity of work = Eq, E being the mechanical equivalent of heat; in such a case in explaining the expense of such work, the simplest hypothesis is that, even when the shell is not deformed, each element of the surface found a pressure, a repulsion as the effect of heat rays emitted from the black body that is found at the center of the shell (7). In this hypothesis it is easy to see that if we indicate with Q the quantity of heat that a square meter of the surface of the shell receives in one second, when the radius of the shell is R, and with p the repulsive force exerted by the beam of heat over that same square meter of the surface, we will have by a minuscule diminution δR, of the radius R,

so that

    We calculate behind this formula the repulsion that a solar beam would exert upon a square meter of a perfectly reflective plane surface oriented normal to the direction of the beam, situated on the surface of Earth (it is understood that allowance is made for absorption of the beam due to the atmosphere). By the observations of Pouillet [Claude Servais Mathias Pouillet (1791 Feb 16 – 1868 Jun 14), who measured the heat coming from the sun in 1837 and 1838]:

Q=0.293833 calorie.

[the modern value is 0.327 kilocalorie per second per square meter = 1.366 kilowatts per square meter]

    By the experiments of Joule:

E=428 kilogram-meters,

[the modern value is 4186 Joules per kilocalorie or 427 kilogram-meters per kilocalorie if we divide out the acceleration of gravity]

and accepting

v=298 000 000 meters,

[Bartoli means meters per second; this is the speed of light, now valued at 299,792,458 meters per second.]

there is obtained

p=0.84 milligrams,

[9.2 milli-Newtons per square meter or 0.939 milligrams weight per square meter is the modern value for fully reflected sunlight.]

per square meter: that is

p=0.0084 milligram,

per square decimeter.>>

    The other mechanisms which I indicated on page 25 of the paper herein cited, are four and I have set them out in the manuscript of the paper recalled at the beginning of this work: I believe it is unnecessary to describe them at least for now: here we are also examining the case of an imperfect reflector which would have necessarily absorbed so much percent of the incident radiation, and it is found that also in this case it is always possible to make heat pass from a very cold body to one which is very hot without this passage being compensated by an equivalent passage of inverse heat (equivalent understood in the sense of the thermodynamic principle of the equivalence of the transformations).

    To put then the preceding results into harmony with the second law of thermodynamics we make diverse hypotheses of which one alone is published in my paper from then; that is, that of the pressure exerted from the radiation onto the reflective bodies (seen through the other hypotheses in the note at the end of this writing).

    This hypothesis of a pressure due to the radiation was then put by me to a series of tests with many experiments which were carried out in the Department of Physics at the University of Bologna in 1874, experiments that are found printed from page 49 to page 53 of my paper edited by Le Monnier: I will report here one of those experiments, because it is connected a little with the argument of this paper: the same words that I employed before serving me in the reporting:

    <<If it would be true that light and heat would have been able directly to produce a movement, either attractive or repulsive, it is clear that this movement would yet be sensible not only through normal incidence but also through sensibly different angles.

    So, in the usual globe (8) [Bartoli used the word pallone, which now means ball or balloon, to denote a glass globe that had the shape of a party balloon] I have fastened at the extremity of a lever made from a thin blade of aluminum [Bartoli used the word lastrina, which translates as little sheet, but blade works better] about six millimeters long and that is terminating at an extremity with a circle of about 6 centimeters in diameter. The lever has balanced from the other side a large bullet [Bartoli uses pallino da caccia, which I translate as little ball (bullet) from hunting]; it has been stretched exactly horizontal in the middle of four wires which we suspended in the same manner of the plate in a set of lever scales, the four wires would then be suspended from an extremely thin wire of silver. Another time in a globe of equal dimensions I have introduced, I suspend upon a needle of steel, an apparatus formed of a horizontal blade of aluminum that terminates in two horizontal discs also of aluminum. The blade is 7 centimeters long; and the diskettes have 3 centimeters of diameter. Finally in a third globe a little smaller than the preceding I have suspended in its center on a needle of steel a horizontal disc of aluminum 10 centimeters in diameter: upon the disc were drawn with a point two perfectly visible, mutually perpendicular diameters.

    In the globes then have come to be made a vacuum as complete as possible.

    A solar beam, directed and concentrated by means of a lens on the blade or on the discs and making an angle of 30˚ or 40˚ with the normal to the disc, does not produce sensible deviations in the movable system.

    I have experimented also with a beam of solar light polarized both with a black mirror and with a large Nicol prism [15] (a prism of the apparatus of Ruhmkorff [16] for its action of magnetism on transparent bodies) in this case, but the beam that is obtained being necessarily very weakened.

    I have pushed, concentrating with a great Fresnel lens, the intensity of the heat to such a point to twist one of the discs, only in this case is happening oscillations of the disc, but not a proper movement of rotation.

    In this experiment the system of suspension is so delicate, (an extremely thin silver wire or a plane of glass upon a steel point), the heat beam so concentrated, that truly the directed impulse of the light would make in the Crookes radiometer [17] the movement of the little mill [the rotor]; the lever in these last two experiments would have to have had in the instant in which the solar beam came striking, received a knock so strong from doing it to make hundreds of turns before it would have stopped.

    I think, then, I demonstrate victoriously that it is not possible to attribute to ordinary light, as to polarized light, any perceptible impulsive power upon the bodies on which it falls (9).>>

    And lower, on page 54, of the same paper adding:

    <<These results, however, not implying an extremely weak impulsive action (so small as not able to be felt by the means I have employed) not possible to be exerted by light or by incident heat.>>

    Florence 1876 Jul 01.

    The conclusion of my experiments being therefore clear and being able to express like this << also admits that radiation producing a pressure on surfaces, this to be so small and of such a meaning as not able to explain the phenomena observed first by Fresnel (10) and then by Crookes and by me.

    I end here this brief note of claim, my aim being principally that of establishing my priority (by eight years) over Eddy: keeping me to return a little on the subject that forms the title of this writing.

    Florence 1884 May 01.

NOTE TO THE PRECEDING PAPER – THE VARIED HYPOTHESES PUT FORTH IN ACCORDANCE WITH THE RESULTS OF THE THEORY OF RADIATION WITH THE SECOND LAW OF THERMODYNAMICS.

    In that first work that has the same title of the preceding note, displaying to such an aim varied hypotheses, that I want to indicate here; although any of this not offering a great probability and more interest that other relative to the history of the tentative facts in discovering the truth, in the field of experimental science.

    1st Hypothesis. – Of a pressure exerted by the radiation on surfaces on which it is striking. It is the hypothesis the consequences of which you will study experimentally in my paper on Movements, etc. cited above (18).

    2nd Hypothesis. – Considering well the mechanisms of the operations with which is obtained in such a manner to pass heat from a very cold body to a very hot one, it shows that in all it is necessary that a surface is come to move perpendicularly to the radiation: so the spherical shell that is destroyed: or the spherical mirror of large diameter that comes gradually to contract: or the mirrored piston that slides within an indefinite cylinder, there is something about it offered all examples: it would be enough to suppose in all of these cases that would necessitate a work to move a mirrored surface perpendicularly to the radiation to that which is exposed, work with the minimum will be easy to determine in harmony with the second law.

    The attempted experiments in this sense were made by me at Bologna in the summer of 1874 with the means having been put at my disposal by my teacher and friend Prof. Emilio Villari, I was making to oscillate in vacuum a very long horizontal lever bearing at one end a light vertical mirror of 40 square centimeters of surface fixed perpendicularly to the lever: it was made at great length to oscillate the first system in darkness thereafter with a solar beam maintained with a heliostat perpendicular to the mirror. The logarithmic decrement of the scale (calculated after hundreds of oscillations) that being in the first case 0.35984 is reduced to 0.34439 in the second case (the first and the last scales of oscillation being in two cases sensibly equal): but it remains uncertain that this decrease would have been attributed to the perturbation that the solar beam introduced in the apparatus produced on the moving system: perturbations that were for me impossible to avoid. I have resumed then the experiment under diverse form and the results although by now incomplete I have delivered it in a sealed packet, that was entrusted in March 1882 to a distinguished Academy.

    3rd Hypothesis. – that the emissive power of a body, that as Clausius has demonstrated, also has to depend (in consistency with the 2nd law of thermodynamics) on the quality of the medium that surrounded the body, having to depend also on the conditions and the circumstances of the bodies to these which it sent or with which it exchanged radiations (11): in the way that the power became null in the case of a body surrounded by a perfectly reflective shell and so small in the case of a shell that is not completely reflecting, that in this last case the direct thermal transformation, crossing, that is, of heat to the mirror being equivalent to that inverse that was reflected with one of the operations from me indicated: and this hypothesis would go back to reject the other of the movable equilibrium or of the radiation now from all supposed/admitted. But this hypothesis, little probable in itself, is lent little good to an attempt that allowed from considerations.

    4th Hypothesis. – to consider the aether or il quid [this may be a misprint for il liquido, the liquid] that transmits radiation in the vacuum by [means of] a ponderable material like a material medium (although extremely light) and [that is] able in certain ways to have degrees of diverse temperature; and [to consider also] that the radiations that were themselves being transmitted proving in the propagation a kind of friction which would absorb one part: and this part would have to be such as to compensate the inverse thermal transformation that it makes in the operations that I imagined for making heat to pass from a very hot body to a very cold one. It would be this hypothesis not conflicting with the consequences at which Muller (12) arrived in a remarkable paper in which [he describes using] the most delicate means [of] measuring the variations of velocities by testing the radiations against the variation of the intensity of the radiation and of the number of the corresponding vibrations.

    These are the hypotheses that I have discussed then: that I have transcribed here, as if with the same words.

A. BARTOLI

    P.S. While this manuscript was sent for the press there arrived for me No. 5 of Annalen der Physik of Wiedemann [19]. Therein on page 31 is a paper by Boltzmann having the title <<Ueber eine von Hrn. Bartoli entdeckte Beziehung der Wärmestrahlung zum zweiten Hauptsatze>> [On one of Mr. Bartoli’s discovered Relations of the Heat Radiation to the Second Fundamental Law] that I have not yet had the time to examine attentively, but where my priority over Eddy is indubitably recognized.

Bartoli’s footnotes

    (1) The manuscript of this work was sent in 1875 for the press to the manager of Il Nuovo Cimento the which would kindly be immediately published; but I asked that they would stop the press because correction in some points I would have been able (it was) sent in time for the competition for the Chair of Physics at the University of Catania, it’s not a mistake, that I vacate a little time after (1875) in which occasion was that paper registered among the qualifications to the competition: my manuscript however was not ever in full published, and only one part was instilled in my paper printed by Le Monnier in 1876.

    (2) It was said my paper was found registered in Comptes Rendus T. LXXXIII, No 9, pg 518, year (1876) in the Bulletin Bibliographique des ouvrages reçus dans la sèance du 21 Aout 1876, and so also in the bibliographical bulletins of almost all the principal Italian and foreign academies. A summary of this paper appeared immediately in the Mondes of Moigno, T. XL, pp 685 - 686; and the appendix that is found at the end of this paper was reproduced in whole in the Rivista scientifico-industriale di Firenze [Scientific-Industrial Review of Florence] of 1876. A mention of the same paper is also made incidentally by Lippmann, Journal de Physique, 1876, T. 5, pg 871 (at the foot of the note). And a briefest summary (it is yet possible to say summary), extracted from Mondes of Moigno, appeared in Fortschritte der Physik im Jahre 1876 (Berlin 1881) on pg 888 and 1541 with the following name and title little recognizable;

    C. Bartbi. Sopra i movimenti prodotti dalla luce e dal calore e sopralti radiometri di Crookes, Firenze 1876. br. 8e. (sic). [apparently Fortschritte’s typesetter misread Bartoli’s name, read sopra il as sopralti, and radiometro as radiometri.]

    (3) Researches were implied, as the title of my paper, On the movements produced by light and by heat etc., said.

    (4) The reader is asked to make the figure.

    (5) The reader is asked to make the figure.

    (6) One may notice the analogy of this formula with the alternative

that expresses the work carried out against the electrostatic pressure, necessary to contract a spherical conducting shell, making the radius to come down from R’‘ ending at R’‘’<R’‘, and maintaining constant the level of the potential L’ with the gradual subtraction of the electricity in the necessary quantity.

    This last formula has been established by the distinguished Prof. Beltrami, in his paper On the theory of systems of electrified conductors – Reports of the Lombardy Institute, Series 2, Vol. XV, issue XII-XIII – Nuovo Cimento, 3. s. T. XII, pg 5.

    (7) Very soon in another paper I will discuss the various hypotheses that are able to be made, and I will demonstrate how with another mechanism different from that already indicated it would succeed in obtaining the same result (note of the paper of 1876).

    (8) A glass globe of three decimeters diameter, in which is made a vacuum at a leveled barometric test.

    (9) I don’t know, because in the summaries that were made in this paper of mine it asserts of this experiment, by secure result, and that is more important than how much they are described by me; perhaps, it seems to me, one of the more decisive also among the thousand other radiometric experiments that are being published.

    (10) Fresnel, Ann. de Ch. et de Physique 2. S. T. XXIX. pg 57 and pg 107: and also Oeuvres complètes de A. Fresnel (published by De Senarmont, Verdet, etc.) Paris, 1868, T. 2. pg 567 - 672.

    (11) Clausius, Pogg. Ann. Bd. CXXI. s. l. year 1864 and Vülner, Physik, Leipzig 1875 BdIII. S. 216 - 217

    (12) Pogg. Ann. CXV, 86; year 1872. [Über die Fortpflanzung des Lichtes (On the Propagation of Light), J. J. Müller, Vol 221, Issue 1, Pages 86-132 in Annalen der Physik. The author is Johann Jakob Müller (1846 Mar 04 -- 1875 Jan 14), a professor of physics at the Swiss Federal Institute of Technology in Zurich.]

My footnotes

    (13) successors of Le Monnier. Il signore Felice Le Monnier (1806 Dec 01 – 1884 Jun 27) founded his publishing house in Florence in 1840 and transferred it to the Società Successori Le Monnier (Society of the Successors of Le Monnier) in 1859.

    (14) Poggendorff’s Annalen refers to the journal properly called Annalen der Physik und Chemie and now called Annalen der Physik (the journal in which Einstein first published his masterworks of 1905). Founded in 1790 as Gilbert’s Annalen der Physik by Friedrich Albrecht and Carl Gren, it was edited from 1824 to 1876 by Johan Christian Poggendorff (1796 Dec 29 - 1877 Jan 24) and edited by Gustav Heinrich Wiedemann from 1877 to 1899.

    (15) The Nicol prism consists of two pieces of birefringent material (such as calcite, commonly called Icelandic spar) glued together at a slant by a suitable transparent adhesive (such as Canadian balsam). The prism splits an incident ray of light into two polarized rays. Bartoli reports that he used one of these devices to create a ray of plane-polarized sunlight for his experiments.

    (16) Heinrich Daniel Ruhmkorff (1803 Jan 15 – 1877 Dec 20), who is best known for inventing a potent induction coil that many experimenters used throughout the last half of the Nineteenth Century and well into the Twentieth.

    (17) notes on the Crookes radiometer.

    (18) Sopra i movimenti prodotti dalla luce e dal calore e sopra il radiometro di Crookes (On the Movements produced by Light and by Heat and On the Radiometer of Crookes).

    (19) Gustav Heinrich Wiedemann (1826 Oct 02 – 1899 Mar 24), succeeded Johan Christian Poggendorff as editor of Annalen der Physik and edited the journal from 1877 to 1899.

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