Mathematicians From Germany

"The equation of Clausius to which I must now call your attention is of the following form:pV=\frac{2}{3}T-\frac{2}{3}\sum\sum(\frac{1}{2}Rr).Here p denotes the pressure of a fluid, and V the volume of the vessel which contains it. The product pV, in the case of gases at constant temperature, remains, as Boyle's Law tells us, nearly constant for different volumes and pressures. ...The other member of the equation consists of two terms, the first depending on the motion of the particles, and the second on the forces with which they act on each other. The quantity T is the kinetic energy of the system... that part of the energy which is due to the motion of the parts of the system. ...In the second term, r is the distance between any two particles, and R is the attraction between them. ...The quantity ½Rr or half the product of the attraction into the distance across which the attraction is exerted is defined by Clausius as the virial of the attraction. ∑∑(½Rr)... indicates that the value of ½Rr is to be found for every pair of particles and the results added together. Clausius has established this equation by a very simple mathematical process... it indicates two causes which may affect the pressure of the fluid on the vessel which contains it... We may therefore attribute the pressure of a fluid either to the motion of its particles or to a repulsion between them."

"If for the entire universe we conceive the same magnitude to be determined, consistently and with due regard to all circumstances, which for a single body I have called entropy, and if at the same time we introduce the other and simpler conception of energy, we may express in the following manner the fundamental laws of the universe which correspond to the two fundamental theorems of the mechanical theory of heat. 1. The energy of the universe is constant. 2. The entropy of the universe tends to a maximum."

"There is no doubt that Clausius with this paper created classical thermodynamics. Compared with his work here, all preceding except Carnot's is of small moment. Clausius exhibits here the quality of a great discoverer: to retain from his predecessors major and minor—in this case, from LaPlace, Poisson, Carnot, Mayer, Holtzmann, Helmholtz, and Kelvin—what is sound while frankly discarding the rest, to unite previously disparate theories, and by one simple if drastic change to construct a complete theory that is new yet firmly based upon previous partial success. ...By no means disregarding the results of experiment, Clausius was the first theorist of thermodynamics who was not enslaved to them... those which to him seemed dubious were to be rejected... Clausius had another handle... his kinetic theory of gases... Both Rankine's model and Clausius' model... led to a theory... "dynamical"... [F]aith... gave... Rankine and Clausius... confidence... while Kelvin, not yet an atomist, wavered. ...[I]n the molecular theory Clausius was not only the wiser man but also the better physicist."

"In the present memoir I... develope... principles of the calculation of the work of the steam-engine. I have... limited myself to the steam-engines now in use, without... consideration of... recent... interesting attempts to employ vapour in a superheated state."

"In The Kind of Motion We Call Heat, Clausius had shown how to relate the temperature and pressure of a volume of gas to the motion of the atoms, and was able to deduce their average speed. ...That calculation drew a quick response from the Dutch meteorologist Christopher Buys Ballot. ...It atoms were really flying through the air at hundreds of meters per second, shouldn't the fragrant vapors of a hot dinner race through the room...? In figuring out the answer... Clausias added a fundamentally new innovation to gas theory. Atoms... banged into each other a good deal. ...battling through all the other atoms ...What mattered was the average distance between collisions. This turned out to be an all-important quantity... and Clausius gave it the name mean free path."

"In their calculations, Clausius (and Waterston, for that matter) had imagined all atoms in a gas moving at the same speed. They knew this wasn't true... but they didn't have the mathematical sophistication to tackle the full problem. Maxwell... defined a mathematical function called the distribution of velocities, which kept track of how many atoms were moving at any particular speed relative to the average, and by dealing in terms of this distribution... was able to give his calculations a precision that those of Clausius lacked."

"[A] fact proved by Rankine and myself... when a quantity of vapour, at its maximum density... enclosed by a surface impenetrable to heat, expands and thereby displaces... e, g. a , with its full force of expansion, a part of the vapour must undergo condensation; whereas in most works on the steam-engine, amongst others in the excellent work of De Pambour, Watt's theorem, that under these circumstances the vapour remains... at... maximum density, is assumed... fundamental..."

"To him we are indebted for the conception of the mean length of the path of a molecule of a gas between its successive encounters with other molecules. As soon as it was seen how each molecule, after describing an exceedingly short path, encounters another, and then describes a new path in a quite different direction, it became evident that the rate of diffusion of gases depends not merely on the velocity of the molecules, but on the distance they travel between each encounter."

"The expression "a machine is driven by heat" is not... strictly accurate. ...[I]n consequence of the changes produced by heat upon ...matter in the machine, the parts ...are set in motion. ...[T]his matter ...[is] that which manifests the action of heat."

"Carnot's annunciation of his theory was defective in that it took no notice of the fact that the hot body gives out more heat than the cold one receives from it, and that it regarded as equal the amount of heat received upon one isothermal side of a cycle and that emitted from the other side; a principle that may hold good for infinitely small cycles, but not for larger ones, in which a difference exists between the thermic quantities proportioned to the size of the cycle. This error and the true condition as pointed out by Clausius are defined by Prof. Rankine, who says, in his paper "On the Economy of Heat in Expansive Machines": "Carnot was the first to assert the law that the ratio of the maximum mechanical effect to the whole heat expended in an expansive machine is a function solely of the two temperatures at which the heat is respectively received and emitted, and is independent of the nature of the working substance. But his investigations, not being based on the principle of the dynamic convertibility of heat, involve the fallacy that power can be produced out of nothing. The merit of combining Carnot's law, as it is termed, with that of the convertibility of heat and power, belongs to Mr. Clausius and Prof. William Thomson; and, in the shape in which they have brought it, it may be stated thus: The maximum proportion of heat converted into expansive power by any machine is a function solely of the temperatures at which heat is received and emitted by the working substance, which function for each pair of temperatures is the same for all substances in nature." The law as thus modified and newly expressed might, as M. Langlois remarks, be designated as the equation of Clausius. But Clausius himself, acknowledging the influence which the Frenchman's ideas had exercised upon him, called it the theorem of Carnot."

"[T]he mechanical theory of heat, had... origin in the well-known fact that heat may be employed for producing mechanical work... [W]e may naturally anticipate that the theory... will in its turn help to place this application... in a clearer light."

"Besides these reasons, which apply to all thermo-dynamic machines, there are others, applicable... particularly to the... steam-engine... [W]ith respect to vapour at a maximum ... this new theory has led... to laws which differ... from those formerly accepted as true..."

"[I]t was formerly assumed, in determining the volumes of the unit of weight of saturated vapour at different temperatures, that vapour even at... maximum density... obeys Mariotte's and GayLussac's laws. ...I have ...shown in my first memoir... the volumes in question can be calculated... under the assumption, that a permanent gas when it expands at a constant temperature only absorbs so much heat as is consumed in the external work thereby performed, and that these calculations lead to values which, at least at high temperatures, differ considerably from Mariotte's and Gay-Lussac's laws."

"[T]he matter... must at... regularly-recurring periods be present in the machine in equal quantity, and in the same state."

"[T]he mechanical theory of heat... render[s] a new investigation of... [the former theory of steam-engines] necessary."

"[W]e may apply the theorems concerning cyclical processes to all thermo-dynamic machines, and thereby arrive at conclusions... independent of the nature of the processes executed by the several machines."

"William Thomson... in March 1851... regarded this result as a proof of the improbability of the above assumption which I had employed. Since then, however, he and J. P. Joule have together undertaken to test experimentally the accuracy... [and] have... shown... with... permanent gases, atmospheric air and hydrogen, the assumption is so nearly true... deviations from exactitude may be disregarded. With [non-permanent gas,] ... deviations were greater... in... accordance with...[my] remark... that the latter would probably be... be accurate for each gas in the same measure as Mariotte's and Gay-Lussac's laws were applicable... Thomson now calculates the volumes of saturated vapours in the same manner as myself."

"The name and fame of Professor Clausius stand as high in this country as in his own. ...his writings ...fell into my hands at a time when I knew but little of the Mechanical Theory of Heat. In those days their author was my teacher; and in many respects I am proud to acknowledge him as my teacher still."

"In a similar manner... passage of the quantity of heat Q, from... temperature t1 to the temperature t2 must be proportional to the quantity Q, and... can only depend upon the two temperatures.... expressed bywherein F(t1,t2) is...[the same] function of... temperatures... for all cases, and... must change... sign when... temperatures are interchanged; so..."

"[I]n every reversible cyclical process... the two transformations... must be equal in magnitude, but opposite in sign; so that their algebraical sum must be zero."

"We proceed now to the consideration of non-reversible cyclical processes. ...[W]e obtain the following theorem, which applies generally to all cyclical processes, those that are reversible forming the limit:—The algebraical sum of all [non-reversible] transformations occurring in a cyclical process can only be positive."

"so that the temperatures t and t′ being arbitrary, the function of two temperatures which applies to the second kind of transformation is reduced... to the function of one temperature which applies to the first kind."

"[T]he equivalence-value of the transformation of work into the quantity of heat Q, of the temperature t, may be represented... wherein f(t) is...[the same] function of the temperature... for all cases. When Q is negative... it will indicate that the quantity... transformed... from heat into work."

"[T]he equationis the analytical expression, for all reversible cyclical processes, of the second fundamental theorem in the mechanical theory of heat."

"Carnot's theorem... brought into agreement with the first fundamental theorem, expresses a relation between... the transformation of heat into work, and the passage of heat from a warmer to a colder body... regarded as... heat at a higher, into heat at a lower temperature. The theorem... may be enunciated... as:—In all cases where a quantity of heat is converted into work, and where the body effecting this transformation... returns to its original condition, another quantity of heat must necessarily be transferred from a warmer to a colder body; and the magnitude of the last quantity of heat, in relation to the first, depends only upon the temperatures of the bodies between which heat passes, and not upon the nature of the body effecting the transformation."

"[W]e may consider the as well as the whole condition of the body... as determined so soon as its t and v are given. We... make these two magnitudes... independent variables, and... consider the pressure p as well as the quantity U... as functions of these. If... t and v receive the increments dt and dv, the corresponding quantity of exterior work done... during an increment of volume dv will be pdv. Hence... and... we obtain"

"This principle, upon which the whole of the following development rests, is... Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time. Everything we know the interchange of heat between two bodies of different temperatures confirms this; for heat everywhere manifests a tendency to equalize existing differences of temperature..."

"[W]e... consider the conversion of work into heat and... the passage of heat from a higher to a lower temperature as positive transformations."

"When the several changes are... such... that... the body returns to its original condition... these changes form a cyclical process, we haveand..."

"[A] simpler symbol for the last function, or rather for its reciprocal... will... be... more convenient... Let...so that T is now the unknown function of the temperature... T1, T2, &c. shall represent... values of this function, corresponding to... t1, t2, &c."

"[T]he function T... hitherto... undetermined; ...by means of a very probable hypothesis it will be possible ...to do. I refer to... my former memoir... that a permanent gas, when it expands at a constant temperature, absorbs only so much heat as is consumed by the exterior work thereby performed. This assumption has been verified by... experiments of Regnault, and in... probability is accurate for all gases to the same degree as Mariotte and Gay-Lussac's law, so that for an ', for which the latter law is perfectly accurate, the above assumption will also be perfectly accurate."

"In my memoir "On the Moving Force of Heat, &c." I have shown that the theorem of the equivalence of heat and work, and Carnot's theorem, are not mutually exclusive, but that, by a small modification... they can be brought into accordance."

"[T]he entire quantity of heat, Q, absorbed by the gas during a change of volume and temperature may be decomposed into two portions. One of these, U, which comprises the sensible heat and the heat necessary for interior work, if... present... determined by the state of the gas at the beginning and at the end of the alteration; while the other portion... the heat expended on exterior work, depends, not only upon the state of the gas at these two limits but also upon the manner in which the alterations have been effected..."

"Theorem of the equivalence of Heat and Work. ...Mechanical work may be transformed into heat, and conversely heat into work, the magnitude of the of the one being always proportional to that of the other."

"[T]he new theory is opposed, not to the real fundamental principle of Carnot, but to the addition "no heat is lost;" for it is... possible that in the production of work... a certain portion of heat may be consumed, and a further portion transmitted from a warm body to a cold one; and both portions may stand in a certain definite relation to the quantity of work produced.<!--p. 17->"

"[M]any facts have lately transpired which tend to overthrow the hypothesis that heat is itself a body, and to prove that it consists in a motion of the ultimate particles of bodies. If this be so, the general principles of mechanics may be applied to heat; this motion may be converted into work, the loss of ' in each particular case being proportional to the quantity of work produced. These circumstances, of which Carnot was also well aware, and the importance of which he expressly admitted, pressingly demand a comparison between heat and work, to be undertaken with reference to the divergent assumption that the production of work is not only due to an alteration in the distribution of heat, but to an actual consumption thereof; and inversely, that by the expenditure of work, heat may be produced."

"I. Deductions from the principle of the equivalence of heat and work. We shall forbear entering... on the nature of the motion... supposed... within a body, and shall assume... a motion of the particles... and that heat is the measure of their '. Or... more generally... lay down one maxim... founded on the above assumption: — In all cases where work is produced by heat, a quantity of heat proportional to the work done is consumed; and inversely, by the expenditure of a like quantity of work, the same amount of heat may be produced."

"The s... may be divided into two classes: those which the atoms of a body exert upon each other... which depend... upon the nature of the body, and those which arise from the foreign influences to which the body may be exposed. According to these two classes of forces... I have divided the work done by heat into interior and exterior work."

"Carnot proves that whenever work is produced by heat... a... quantity of heat passes from a warm body to... cold... [e.g.,] the vapour... generated in the of a steam-engine... passes... to the condenser where it is precipitated... This transmission Carnot regards as the change of heat corresponding to the work... He says... no heat is lost in the process, that... [its] quantity remains unchanged; and he adds, "This is a fact... never... disputed... confirmed by various calorimetric experiments. To deny it, would be to reject the entire theory of heat, of which it forms the principal foundation.""

"The steam-engine having furnished us with a means of converting heat into a motive power, and our thoughts being thereby led to regard a certain quantity of work as an equivalent for the amount of heat expended in its production, the idea of establishing theoretically some fixed relation between a quantity of heat and the quantity of work which it can possibly produce, from which relation conclusions regarding the nature of heat itself might be deduced, naturally presents itself. Already, indeed, have many successful efforts been made with this view; I believe, however, that they have not exhausted the subject, but that, on the contrary, it merits the continued attention of physicists... The most important investigation in connexion with this subject is that of S. Carnot. Later... represented analytically... by Clapeyron"

"The careful experiments of Joule, who developed heat... by the application of mechanical force, establish... not only the possibility of increasing the quantity of heat, but also the fact that the newly-produced heat is proportional to the work expended in its production.<!--p. 15->"

"[M]any facts.. lately transpired... tend to overthrow the hypothesis that heat is... a body, and to prove that it consists in a motion of the... particles of bodies. If... so... principles of mechanics may be applied to heat; this motion may be converted into work, the loss of '... being proportional to... work produced.<!--p. 15->"

"My memoirs "On the Mechanical Theory of Heat" are of different kinds. Some are devoted to the development of the general theory and to the application thereof to those properties of bodies which are usually treated of in the doctrine of heat. Others have reference to the application of the mechanical theory of heat to electricity. ...Other memoirs... have reference to the conceptions I have formed of the molecular motions which we call heat. These conceptions, however, have no necessary connexion with the general theory, the latter being based solely on certain principles which may be accepted without adopting any particular view as to the nature of molecular motions. I have therefore kept the consideration of molecular motions quite distinct from the exposition of the general theory."

"Let Q... be the quantity of heat which must be imparted to a body during its passage... from one condition to another, any heat withdrawn from the body being... [a] negative quantity... Q may be divided into three parts... the first... in increasing the heat... in the body, the second in producing the interior [work]... the third in producing the exterior work. ...[T]he second ...[and] first... together... represented by... function U... [are] completely determined by the initial and final states of the body. The third part... the equivalent of exterior work, can, like this work itself, only be determined when the precise manner in which the changes of condition took place is known. If W be the quantity of exterior work, and A the equivalent of heat for the unit of work, the value of the third part will be A · W, and the first fundamental theorem will be..."

"[T]he second fundamental theorem in the mechanical theory of heat... appropriately... called the theorem of the equivalence of transformations..: If two transformations which, without necessitating any other permanent change, can mutually replace one another, be called equivalent, then the generation of the quantity of heat Q of the temperature t from work, has the equivalence-value'and the passage of the quantity of heat Q from the temperature t1 to the temperature t2, has the equivalence-value'wherein T is a function of the temperature, independent of the nature of the process by which the transformation is effected."

"[A]ccording to Mariotte and Gay-Lussac's law,{{center|1=p = \frac{a+t}{v} \cdot \text{const}.}}"

"If to the last expression we give the form...the passage of the quantity of heat Q, from the temperature t1 to the temperature t2, has the same equivalence value as... the transformation of the quantity Q from heat at the temperature t1 into work, and from work into heat at the temperature t2."

"T is nothing more than the temperature counted from a°, or about 273° C. below the freezing-point; and, considering the point... as the absolute zero... T is simply the absolute temperature. ...For this reason I introduced, at the commencement, ...T for the reciprocal value of the function f(t)."

"Riemann himself modestly apologizes for the philosophical shortcomings of his essay on the ground of his inexperience in philosophical matters. But the crudeness of his speculations affords a very striking illustration... of the well-known fact that exclusive devotion to the labors of the mathematical analyst has a tendency to develop certain special powers of the intellect at the expense of its general grasp and strength. Although Sir William Hamilton, no doubt, overstated the case against the mathematicians, I believe that his suggestions are not wholly unworthy of attention, and that there is force in the words of D'Alembert (referred to by Sir William Hamilton)... We have here five distinct propositions, which... may be stated in distinct form as follows: 1. That the nature of space is to be deduced from its concept. 2. That the concept of space can be formed and determined only by its subsumption under a higher concept. 3. That our space is a "triply extended Multiple or Aggregate," the higher concept under which its concept is to be subsumed being that of an "n-fold extended Multiple" or a "multiply extended Aggregate" (eine n-fach ausgedehnte Mannigfaltigkeit), and that—translating Riemann's phraseology into its plain logical import—the (logical) extension of this higher concept determines the number of the possible kinds of space. 4. That the conceptual possibility of space is coextensive with its empirical possibility, though not with its empirical reality. 5. That continuous quantities are coördinate with discrete quantities, i.e., are species of the same genus, both being in their nature multiples or aggregates."

"To imagine that conclusions respecting the nature of space and the origin of its concept can be drawn from the mere fact that space is a function of three variables, and may thus in a manner be classified with similar functions, is a mockery of all reasoning from which an old scholastic would have turned with the scornful reminder that coördination and subsumption, for the purpose of effectually aiding in the formation of a particular concept, must not only be under a genus, but under a genus proximum."