Popular Science Monthly/Volume 75/August 1909/Josiah Willard Gibbs and his Relation to Modern Science IV

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The third stage of thermodynamics has for its point of departure Maxwell's observation that the second law is not a mathematical but an empirical or statistical truth, and his prediction that any attempt to deduce it from dynamic principles, such as Hamilton's principle, without introducing some element of probability, is foredoomed to failure.[1] "We have reason to believe of the second law," says Maxwell, "that though true, its truth is not of the same order as that of the first law," being an empirical generalization from the facts of nature in the first instance, while the molecular theory shows it to be "of the nature of a strong probability which, though it falls short of certainty by less than any assignable quantity, is not an absolute certainty." This statement of Maxwell's not only resumes the knowledge of his time, but has not been improved upon by later investigators, whose work shows that the truth of the second law is. certain to the limit of human probability only. The theory of probabilities itself is exact as far as human observation goes. In 6,000 throws of dice, a particular facet will not necessarily turn up 1,000 times, but the probability of its doing so will be more nearly one sixth, the greater the number of throws. In the vital statistics of a great city the data of births, deaths, illegitimacy, etc., will be more nearly the same from week to week, the greater the population of the city; even the introduction of new dynamic factors, as seasonal change, epidemics, vaccination, antitoxin, etc., may alter particular effects but will not change the general tendency towards uniformity. Maxwell has observed that everything irregular, even the motion of a bit of paper falling to the ground, tends, in the long run, to become regular, and this is the rationale of testing the second law with respect to gases. In the kinetic theory of gases, the first scientific statement of which is due to Clausius, we assume a gas to be an assemblage of elastic spheres or molecules, flying in straight lines in all directions, with swift haphazard collisions and repulsions, like so many billiard balls. These, by Maxwell's calculations, will, if enclosed and left to themselves, gradually tend to an ultimate steady condition of perfectly equalized and permanently distributed velocities (i. e., uniform temperature or thermal equilibrium) called "Maxwell's state." "This possible form of the final partition of energies," Maxwell claims, " is also the only form." At this point the work of Boltzmann becomes of central importance, especially on account of its profound influence on the later works of Gibbs. In Boltzmann's application of probabilities to Maxwell's problem, the starting point or initial stage of any sequence of events is called a " highly improbable one," because its certainty decreases the more the events proceed to some final or "most probable" state. For example, the blowing up of the Maine is to us a moral or mathematical certainty, but it may not be so aeons hence, while its predisposing or exciting causes are even now " highly improbable " in that we know nothing positive about them. When a gas is brought into a new physical state, its initial stage is, in Boltzmann's argument, a highly improbable one from which the system of molecules will continually hasten towards successive states of greater probability until it finally attains the most probable one, or Maxwell's state of equilibrated partition of energy and thermal equi- librium. Maxwell's law of final distribution of velocities as determined by Boltzmann's probability coefficient is, therefore, a sufficient condi- tion for thermal equilibrium, and Boltzmann found that the entropy of any state of gas molecules is proportional to the logarithm of the probability of its occurrence; or as Larmor puts it, the principle that the trend of an isolated system is towards states for which the entropy continually increases is analogous to the principle that the general trend of a system of molecules is through a succession of states whose intrinsic probability of occurrence continually increases. As a measure of the degree of variation of the gas molecules from Maxwell's state, Boltzmann introduces a function H such that, as the distribution of molecular velocities constantly tends toward the most probable dis- tribution, H varies with the time and is found to be constantly diminish- ing in value. The necessary condition for thermal equilibrium is, therefore, that H should irreversibly attain a minimum value. Thus Boltzmann's " minimum theorem " becomes, like the Clausius doctrine of maximum entropy, a theorem of extreme probability, 128 or to quote the aphorism of Gibbs which Boltzmann chose as a motto for his Gastheorie: "The impossibility of an uncompensated decrease of entropy seems to be reduced to an improbability." 129 Applying similar reasoning to the material universe, Boltzmann finds that the following assumptions are possible : either the whole universe is in a highly im- probable (i. e.„ initial) state, or, as the facts of physical astronomy would seem to indicate, the part of it known to us is in a state of thermal equilibrium, with certain districts, such as the earth we live

"It can never be proved from the equations of motions alone, that the minimum function H must always decrease. It can only be deduced from the laws of probability, that if the initial state is not specially arranged for a cer- tain purpose, but haphazard governs freely the probability that H decreases is always greater than it increases." Boltzmann, Nature, 1894-5, LI., 414.

W 7V. Connect. Acad., III., 229. on, noticeably removed from this condition. The probability of the latter state of affairs is smaller, the further such a state is removed from thermal equilibrium, but it can be made as great as we please to assume the universe to be great. But there is necessary and sufficient prob- ability that our earth as we know it is in its present state. By the second law (irreversible increase of entropy in natural processes) there is still greater probability that it tends to a final state of thermal equi- librium or death; and since the universe itself is so great, there is sufficient probability that other worlds than ours may deviate from thermal equilibrium. As a graphic exposition of this theory, which shows the vast scope of the second law of thermodynamics, a curve can be plotted with the variables H and the time as coordinates, to visualize what takes place in the universe. The // curve is shaped like a suc- cession of inverted trees, the summits of which represent " the worlds where visible motion and life exist/' 130 Physicists have found that the Maxwell-Boltzmann distribution of velocities is satisfactory for gases whose molecules move independently and at random; but when the molecules are supposed to be subject to one another's influence, it does not account for certain facts of nature such as the measured specific heats of gases or individual peculiarities of their spectra. In monatomic gases like argon, helium and mercury, the ratio of the specific heats will account for the three degrees of molecular freedom ascribed to them by the mathematical theory, but in the case of dia- tomic gases, like hydrogen or oxygen, the theory calls for six degrees of freedom, while experiment will account for only five. Boltzmann met these objections with frank or ironical admissions as to the ultimate inadequacy of all human hypotheses, 181 and although his theory is to some extent invalidated by facts like the above, 182 his subtle handling of molecular thermodynamics gives the physicist deeper insight into "° " Almost all these trees are extremely low, and have branches very nearly horizontal. Here H has nearly the minimum value. Only very few trees are higher, and have branches inclined to the axis of abscissa?, and the improbability of such a tree increases enormously with its height." Boltzmann, Nature, 1894-5, LI., 581.

m « Neither the theory of Gases nor any other physical theory can be quite a congruent account of facts, and I can not hope with Mr. Burbury that Mr. Bryan will be able to deduce all the phenomena of spectroscopy from the electro- magnetic theory of light. Certainly, therefore, Hertz is right when he says: ' The rigour of science requires that we distinguish well the undraped figure of Nature itself from the gay-coloured vesture with which we clothe it at our plea8ure. , But I think the predilection for nudity would be carried too far if we were to forego every hypothesis. Only we must not demand too much from hypotheses." Boltzmann, Ibid., 413.

The principal opponent of the Maxwell-Boltzmann partition of energies was Lord Kelvin in his " Nineteenth Century Clouds over the Dynamical Theory of Heat and Light." When asked what he had against it, he replied point-blank : "I don't think there is a single thing about it that is right" (Science, Jan. 3 r 1908, p. 6). such unusual phases of matter as radiation in rarefied gases, where the system has no temperature at all, because its internal motions have not settled down to a definite average. Helmholtz's dynamic proof of the second law assumes the existence of cyclic systems with reversible circular motions, like those of the gyroscope or the governor of a steam engine, in other words it assumes matter to be made of rotational or gyrostatic stresses in the ether. Gibbs's " Elementary Principles of Statical Mechanics" (1903) 18S is based upon no assumptions whatever except that the systems involved are mechanical, obeying the equations of motion of Lagrange and Hamilton. " One is building on insecure foundations," he says, "who rests his work on hypotheses concerning the constitution of matter," and his statistics deal, not with the behavior of gas molecules in isolated systems, but with large averages of vast ensembles of systems of the same kind (solid, liquid or gas), " differing in the configurations and velocities which they have at any given instant, and differing not merely infinitesimally, but it may be so as to embrace every conceivable combination of configuration and veloci- ties." The problem is, given the distribution of these ensembles in phase (i. e., in regard to configuration and velocities) at some one time, to find their distribution at any required time. To solve this problem Gibbs establishes a fundamental equation of statistical mechanics, which gives the rate of change of the systems in regard to distribution in phase. A particular case of this equation gives the condition for sta- tistical equilibrium or permanent distribution in phase. Integration of the equation in the general case gives certain constants relating to the extent, density and probability of distribution of the systems in phase, which Gibbs interprets as the principles of conservation of "extension in phase," of "density in phase," and of "probability in phase." Boltzmann found that when the gas molecules have more than two degrees of freedom, the equations can not be integrated and further progress is impossible. He got around this difficulty by using Jacobi's " method of the last multiplier," which integrates the equations of mo- tion. Gibbs found that the principle of " conservation of extension-in- phase," supplies such a Jacobian multiplier, " if we have the skill or good fortune (he says) to .perceive that the multiplier will make the first member of the equation an exact differential." Boltzmann's prob- ability coefficient is used as the index of the canonical distribution of ensembles, and when the exponent of this coefficient is zero, the latter becomes unity, producing a distribution in phase called " micro- canonical," in which all the systems in the ensemble have the same energy, as in Maxwell's " state." After demonstrating the possibility of irreversible phenomena in the various ensembles, and after a careful study of their behavior when isolated, subjected to external forces or to

"Yale Bicentennial Publications," 1903. Translated into German by Ernst Zermelo, Leipzig, 1905. JO SI AH WILLARD GIBBS 195 the spheres of one another's influence, Gibbs finds that the processes of statistical mechanics are to all human perception analogous to those of thermodynamics, the familiar formulae of which appear, as Bumstead puts it, " almost spontaneously, as it seems from the consideration of purely mechanical systems.". The differential equation relating to average values in the ensemble is found to correspond with the funda- mental equation of thermodynamics; the modulus of distribution of ensembles turns out to be analogous to the temperature, while the average index of probability in phase is the analogue of the entropy with reversed sign, and being a minus quantity, is found to decrease just as entropy increases. Most of the objections filed against Gibbs's statistical demonstration, turn upon the fact that it is difficult, perhaps impossible, to apply the reversible dynamics of ideal, frictionless systems to the spontaneous irreversible phenomena of nature without making some physical assumptions. " Entropy/' Burbury objects, 184 " may, for all that appears, either increase or diminish in a system which is dynamically reversible. This then can not be strictly applied to an irreversible process." Gibbs has met these objections fairly. " Our mathematical fictions," 135 he says, to quote Burbury's paraphrase of his argument, " give us no information whether the distribution of phases is towards uniformity or away from it. Our experience with the real world, however, teaches us that it is towards uniformity." All actual mechanical systems are, as Gibbs pointed out long before, in reality thermodynamic, 136 and it seems odd that the critics who rejected Boltz- mann's proof, because it did not agree with the facts of nature, should now, for a logical quibble, take exception to Gibbs's because it does. It has been predicted that future truth in physical science will often be found in the sixth place of decimals, for not everything in nature works out according to specifications. We can, if we choose, regard mathe- matics as a metaphysical diversion or employ it practically as a means of interpreting the physical facts of nature, empirically ascertained by man. In these matters, says Gibbs elsewhere, " Nature herself takes us by the hand and leads us along by easy steps as a mother teaches her child to walk," 137 and he would have agreed with Langley that man may put questions to nature if -he -will, but is in no position to dictate her answers to them. 138 Nature seems tres femme in this re- spect, especially in regard to mathematical fictions, that is, ideal or lim- iting cases devised by the finite mind of man. 130 Like any other human

Phil. Mag., 1904, 6. s., VIII., 44. 

'» Ibid., 45. lu Tr. Connect. Acad., III., 108. ™ Proc. Am. Ass. Adv. Sc, 1886, Salem, 1887, XXXV., 62. m "Let us read Bacon again, and agree with him that we understand only what we have observed." S. P. Langley, Science, 1902, XV., p. 927. m " Physical chemistry is not yet a quantitative science; it is a pseudo- quantitative science. There are all the outward signs of a quantitative science. 196 THE POPULAR SCIENCE MONTHLY instrument of precision, our mathematical methods are but an approxi- mation to the subtler aspects of nature, and it is only by eternal vigi- lance in regard to sources of human error that workers in physical sci- ence have put aside personal equation and infallibility and thus avoided what Rowland calls the " discontinuity " of the ordinary legal or culti- vated mind. 140 " Gibbs has: not sought to give a mechanical explana- tion of heat," says Professor Bumstead, "but has limited his task to demonstrating that such an explanation is possible. And this achieve- ment forms a fitting culmination of his life's work." 141 The naturalist Haeckel has explicitly denied the doctrine of uni- versal increase of entropy 142 because, pointing as it does to the ultimate thermal death of different worlds, it conflicts with his monistic con- ception of the universe as a perpetuum mobile, consisting of infinite substance in eternal motion, without beginning and without end. Yet the cosmogony of Kant and Laplace, which Haeckel accepts, points to the same conclusion as well as to formative periods in the history of the solar and sidereal systems, in which entropy decreases, and energy, instead of dissipating; tends, after a maxitfvum of degradation, to con- centrate. Even possibilities of this kind put the second law on a lower plane of probability than the first as far as man is concerned, unless it be that the irreversible processes of nature are in reality cyclic, in which case we should have Nietzsche's " eternal return " of all things. But as Bumstead has so admirably said, " It is nearer the truth to base the doctrine of entropy upon the finite character of our perceptions than upon infinity of time." In connection with the validity of the second law arises the impor- tant question of the extent of its application to animate nature and whether it is capable of reversal in vital processes. " The first law (conservation of energy) has been proven," says Otwald, " with an exactness of 1 : 1,000 even for physiological combustion (including mechanical and psychical work performed)." The second law, whether in the Clausius form of increase of entropy, the Kelvin form of dissi- We have formulas and tables; we make use of thermodynamics and the differ- ential calculus; but this is for the most part a vain show. Long before we reach the point where the formula is to be tested experimentally we slip in a simplifying assumption: that the concentration of one component may be considered as a constant; that the heat of dilution is zero; that the solute may be treated in all cases as though it. were an indifferent gas; that the concentration of the dissociated portion of a salt may be substituted for the total concentration; etc., etc. The result is that our calculations apply at best only to limiting or ideal cases, where an error in deducing the formula may be masked by errors in observation. Helmholtz did not do this, but ITelmholtz is considered old-fashioned." W. D. Bancroft, ./. Phys. Chem. y 1899, III., 604.

H. A. Rowland, Am. J. 8c. t 1899, 4. s.. VIII., 409. 
Bumstead, Am. J. 8c, 1903, 4. s M XLI., 199. 

m Haeckel, "The Riddle of The Universe," New York, 1900, 24G-248. JOSIAH WILLARD GIBBS 197 pation of available energy, or the Gibbs-Helmholtz form of decrease of free energy, is assumed by recent physiologists to be characteristic of all spontaneous or metabolic processes, but both Helmholtz 143 and Kelvin 144 have doubted whether it is either necessary or sufficient for their production, while Maxwell 145 and Boltzmann 146 have asserted, what Gibbs's statistical researches seem to prove, that it is sometimes possible for entropy to decrease, that is for small isolated temporary violations of the second law to occur in any real body. Has animal or vegetable protoplasm ever the power ascribed to Maxwell's demon of reversing the thermodynamic order of nature, and directing physico- chemical forces? Such a demon, according to Lord Kelvin, might, through his superior intelligence or motor activity, render one half of a bar of metal glowing hot, while the other half remained icy cold. We have something analogous to this in certain diseases, as gangrene, aphasia, various forms of paralysis, the curious vasomotor and trophic disorders of the nervous system. Are these phenomena then of a thermodynamic nature? The animal body, Lord Kelvin thought, does not act like a thermodynamic engine, but " in a manner more nearly analogous to that of an electric motor working in virtue of energy supplied to it by a voltaic battery/' Here, as Gibbs has shown in his theory of the chemical cells, the electromotive force would be identical with the free energy upon which the surface energies of the body must ultimately depend. Beyond these speculations we know nothing. Gibbs himself avowed his express disinclination to "explain the mys- teries of nature," while Lord Kelvin, although affirming that physicists are bound "by the everlasting law of honor," to explain everything material upon physical principles, mystified friends and opponents alike by falling back upon a " vital principle " with " creative power " behind it as the causa causans of biological happenings. But the busi- ness of physics is with the material facts of the universe, and the invo- cation of creative power explains nothing and is subversive of deter- minism, or the relation of cause and effect in science. It may be that " man was born too late to ascertain final causes " : he can only inter- pret the physical facts of his experience as he finds them and with the means at his disposal. An interesting attempt to explain the relation of life and mind to matter is found in the energetische Weltanschauung ,ia Helmholtz, J. f. Math., v. 100, 137. Auerback, " Kanon der Physrik.," 414. m Kelvin, "Pop. Lect.," II., 190, 463, 404. See, also, the discussion in Science, 1903, N. S., XVIII., 138-146. "• Maxwell, Nature, 1877-8, XVII., 280. "* Der grosse Meister, dem auch diese Zeilen huldigen mochten, hat einst den Gedanken ausgesprochen, dass es in der Welt vielleicht Stellen giebt, wo die Entropie nicht wHchst, sondern zunimmt," O. Chwolson. Boltzmann, " Festschr.," 1904, 33. 198 THE POPULAR SCIENCE MONTHLY or energetic philosophy of Ostwald, which confessedly derives 147 from the thermodynamic argument of Gibbs, but should not be confused with the latter. Gibbs was concerned only with applying the laws of mechanics to physical chemistry. Compared with the case of nature, he says, thermodynamic systems are "of an ideal simplicity." To Ostwald, however, mind and matter are but forms of energy, which is the only thing eternal and immortal. " We can deal with measurable things, never with the unknown heart of nature/' says Ostwald, yet his basic principle, energy, is to all intents and purposes identical with the eternal infinite substance of Spinoza, Goethe and Haeckel, " sive Deus, sive Natura naturans, sive Anima mundi appelletur." Matter, in Ostwald's scheme, is a group of energies in space; thought becomes a mode of energy involving evolution of heat, and " the problem of the connection between body and spirit belongs to the same series as the connection between chemical and electrical energy, which is treated in the theoiy of voltaic chains." 148 Falling in love, listening to a Beethoven symphony, identifying oneself with nature, are to Ostwald instances of dissipation of energy like any other. 149 Philosophy of this kind does not clear up the mystery of the relation of mind and matter. Descartes assumed that mind and matter exist apart as parallels, hav- ing no causal connection with each other. Spinoza held that neither can exist apart; indeed, he sometimes asserts their practical identity as different modes of the same eternal substance. But however inti- mately they may be associated, no scientist or philosopher has yet proven, whether .in the body of man or in the origin of the universe, that one is either the cause or the effect of the other. Assuming matter in mass to be ultimately made up of rotational, vortical or gyrostatic stresses or of energies, whether kinetic or poten- tial, we encounter the formidable objection of Boltzmann, that it seems illogical, not to say unmechanical, to postulate motion as the primary idea with the moving thing as the derived one. Motion of what? we have a right to ask, since Ostwald disdains the ether of the physicists. 150 Matter, in the words of Sir Oliver Lodge, may be physically resolved i«r it YVi r wollen daher den Versuch wagen, eine Weltansicht ohne die Benutzung des Begriffs der Materie ausschliesslich aus energetischem Material aufzubauen ... In der ftir die neuere Chemie grundlegenden Abhandlung von. Willard Gibbs ist sogar dies Postulat praktisch in weitestem Umfange durch- gefiihrt worden, allerdings ohne dass es ausdriicklich aufgestellt worden ware." W. Ostwald, " Vorles. tiber Naturphilosophie," 165. u *Monist, 1907.

W. Ostwald, "Individuality and Im mortality," 44-46. 

wo tt what the atom of each element is, whether it is a movement or a thing, or a vortex, or a point having inertia, all these questions are surrounded by profound darkness. I dare not use any less pedantic word than entity to designate the ether, for it would be an exaggeration of our knowledge to speak of it as a body, or even a substance," Lord Salisbury, " Rep. Brit. Ass. Adv. Sc," 1894, 8. * JO SI AH WILLARD GIBBS 199 "perhaps, into electricity, and that into some hitherto unimagined mode of motion of the ether," but no dynamic theory of the ether can resolve the ether into nothing. Assuming thought to be a mode of energy, the metaphysical argument that mind is at the bottom of motion seems more likely, in the last analysis, than that motion should be the cause of mind, for we can not conceive of a thing moving unless something moves it. Mind seems almost like an assemblage or com- plex of causes in itself, and is probably related to the brain as music to the violin. Destroy the violin and there will be an end of its music, but it needs other coefficients than the violin itself to get music out of it. Ostwald has himself admitted the force of Leibnitz's argument, that no mechanical explanation of cerebral action will ever account for the genesis of thought or the nature of consciousness : " Nihil in intel- lectu quod non prius in sensu, nisi intellectus ipse." Individual think- ing may be the result of physico-chemical differences of structure or substance in the brain, but apart from the evidence of mind in the evolution and structure of the universe, different aspects of mind, as ideas, sensations and sentiments, seem to have an individual life of their own so far as man is concerned, and are " things " in the sense that, like external forces, they have profoundly influenced and deter- mined the actions of individuals and of entire races. Human thought as a function of the human brain may disappear with man himself, but this does not annul the possibility of mind existing in manifold ways elsewhere in the universe. The electric waves of wireless telegraphy undoubtedly existed as motions in the air before man discovered and labeled them and may continue to exist and be apprehended in other spheres of thought when man is gone. Man's capacity for error in these matters is determined by his anthropomorphic tendencies and by the fact that his intelligence is finite. Of the possibly infinite number of attributes of eternal sub- stance postulated by Spinoza, the human mind can apprehend only two — thought and extension, and even here thought and sensation are the fundamental facts, while "all else is an inference and is probably essentially unlike what it appears to our senses." It seems impossible to break down the fact that there is no absolute causal connection between the two primary categories of Spinoza, who has anticipated most of modern psychology. For this reason such subjects as spirit- ualism, phrenology, faith-healing, telepathy have remained in the limbo of pseudo-science, although each has undoubtedly a shadowy reason for existence. It is as fair as any other hypothesis, then, to assume that man, in his higher mental or psychical activities, may, under certain conditions, be " freed from the galling yoke of space and time," or, in other words, released from the thraldom of the second law. Yet such an assumption, even if made by a Kelvin, would be, in our present state of knowledge, an expression of individual personal belief, a literary or humane analogy, a leaning in the direction of the "fair humanities of old religion/' but not a scientific fact. To fix our ideas for the material world we may accept the expanded statement of the second law which Ostwald gave in his Ingersoll lecture in 1906 : lB1 i( Every known physical fact leads to the conclusion that diffusion or a homogeneous distribution of energy is the general aim of all happen- ings. ... A partial concentration may be brought in a system, but only at the expense of greater dissipation, and the sum total is always an increase in dissipation." 152 Through the labors of Joule and Kel- vin, Maxwell and Boltzmann, Gibbs and Helmholtz, Carnot's simple generalization about heat engines has been elevated to the dignity of an irrevocable law of nature, a principle of scientific determinism, giving one of the most complete and satisfactory answers that man can furnish to the great question: How does any event in the material universe come to pass? In Darwin's picture of nature the quiet woods and waters, so calm and peaceful on the surface, are in reality centers of " strange and cruel life," the struggle and turmoil of creatures con- tinually preying upon each other, even trees and plants and the tiniest particles of animate bodies taking part in a definite, never-ending war for existence. But the stern law of life, whereby the strong war down the weak, loses all moral, or human significance when seen as due, in the last analysis, to an inevitable tendency to dissipation of energy or as the resultant of a play of complex forces, which, through some prin- ciple of " least action," must inexorably flow from higher to lower potentials. As Spinoza pointed out long ago, Nature could 'not change these laws which flow from its very being, without ceasing to be itself, and the conclusion of physics and biology that Nature is never on the side of the weak becomes, as far as man is related to the material uni- verse, identical with Spinoza's denial of final causes.

Apart from his work in mathematical physics, Gibbs made several important contributions to pure mathematics, notably in his theory of " dyadics," a variety of the multiple or matricular algebras which Ben- jamin Peirce classified as " linear associative." The tendency of his mind was always toward broad, general views and the simplifications that go with such an outlook, and here mention should be made of his charming address on multiple algebra and his innovation of vector analysis, a calculus designed to give the student of physics a clearer

W. Ostwald, " Individuality and Immortality," Boston, 1906, 42.

As a fundamental formula for all material happenings, analogous to the " world-formula " of Laplace, J G. Vogt proposes the following (Polit. Anthrop. Rev., Leipzig, 1907-8, VI., 573) : If Pe represent the positive or dissipation^ potential {emissives Potential) and Pr the negative or concentrational potential (rezeptive8 Potential) of any given set of forces, then Pe -f- Pr = or f d Pe + / dPr = 0. This is, however, only another restatement of Newton's Third Law of Motion, that action and reaction are equal and in opposite directions. insight into such space relations as strains, twists, spins and rotational or irrotational movements in general. Maxwell, who once declared that he had been striving all his life to be freed from the yoke of the Carte- sian coordinates, had already found such an instrument in the Hamil- tonian quaternions, the application of which he brilliantly demon- strated in his great treatise on electricity and magnetism. Quater- nions are elegant, consistent, concise and uniquely adapted to Euclidean space, but physicists have latterly found them artificial and unnatural to their science, because the square of the quaternionic vector becomes a negative quantity. 153 The Gibbsian vectors obviate this difficulty, and while seemingly uncouth, furnish' a mode of attack more simple and direct and adaptable to space of any dimensions. Their capacity for interpreting space relations was amply tested by Gibbs in his five papers on the electromagnetic theory of light and his application of vectors to the calculation of orbits, since incorporated in recent German treatises on astronomy. The fact that vectors tend to displace the quaternionic analysis of Sir William Rowan Hamilton involved our author in a lengthy controversy with Hamilton's best interpreter, the ingenious and versatile Tait, 154 who looked upon Gibbs as " one of the retarders of quaternionic progress," defining his system as " a sort of hermaphrodite monster compounded of the notations of Hamilton and Grassmann." But Gibbs did not regard his method as strictly orig- inal ; he was only concerned with its application in the task of teaching students ; and when, after testing it by twenty years' experience in the class-room, he reluctantly consented to the publication of his lectures in full, the task was confided to one of his pupils, our author declining, with a characteristic touch of conscience, to have the work appear under his name or even to read the proof. In the controversy with Tait there is, as in most controversies, an amusing element of human nature. The name of Hamilton is undoubtedly one of the most illustrious in the history of science, and Tait and his adherents seemed to regard it as an impertinence and a desecration of his memory that any other

"I have the highest admiration for the notion of a quaternion; but . . . as I consider the full moon far more beautiful than any moonlit view, so I regard the notion of a quaternion as far more beautiful than any of its appli- cations. ... I compare a quaternion formula to a pocket-map — a capital thing to put in one's pocket, but which for use must be unfolded: The formula, to be understood, must be translated into coordinates," Arthur Cayley, Proc. Roy. 80c. Edinb., 1892-5, XX., 271. At the Southport meeting of the British Asso- ciation in 1903, Professor Larmor, while admitting the extreme usefulness of the different methods of vector analysis, argued that their slow progress in physics was due to the lack of uniformity in definitions and notations, requir- ing that each system must be mastered separately before it can be applied. To which Professor Boltzmann not inaptly replied that the confusion might have been avoided, if Hamilton had adopted the notations of Grassmann in the first instance.

Nature, 1891-3, passim. system than quaternions should be proposed. "The ideas which flashed into the mind of Hamilton at the classic Brougham Bridge " became the occasion of a joined battle between the perfervid clan-loyalty of the Celt and the cool individualism of the Saxon ; on one side,

" The broad Scots tongue that flatters, scolds defies,
The thick Scots wit that fells you like a mace,"

and on the other, the overconscientious, ethical arguments of a super-sensitive spirit, obviously nettled at certain rough pleasantries which were understood but not appreciated. In 1893 Heaviside, an English vectorist, reports " confusion in the quaternionic citadel : alarms and excursions and hurling of stones and pouring of water upon the invading hosts."[2] The vectorists were denounced as a " clique " and ridiculed especially for their lack of elegance, their alleged intellectual dishonesty and the fact that their pupils were " spoon-fed " upon mathematico-physical pap. But some of the notations held up to ridicule turned out to be things like Poisson's theorem or the difficult hydrodynamic problem "given the spin in a case of liquid motion to find the motion," which Helmholtz solved with one of his strokes of genius, and which Gibbs showed could be understood and interpreted by the average student without genius by a simple application of vectorial methods. The real point at issue in the controversy, the fundamental difference in the ideals of European and American education, lies here. Both have their relative advantages and defects, but the object of one has been to bring the best to the highest development, while the other is concerned with increasing the efficiency of the average man. One has been exclusive, aiming at the survival of the fittest; the other is democratic and inclusive, and aims, in Huxley's words, to make the greatest number fit to survive. The merits of the case are well summed up in Gibbs's final statement: "The notions which we use in vector analysis are those which he who reads between the lines will meet on every page of the greatest masters of analysis, or of those who have probed deepest the secrets of nature, the only difference being that the vector analyst, having regard for the weakness of the human intellect, does as the early painters who wrote beneath their pictures " This is a tree." "This is a horse."[3] This view is in perfect accord with the recent trend of mathematical teaching, European or American, which is to emphasize the meaning and interpretation of equations and formulae rather than their demonstrations or manipulation; in short, to substitute visualizing methods, the art of thinking straight and seeing clear, for what is conventional and scholastic. A Harvard professor is said to have told his students that the demonstration of a theorem is no evidence that it is understood, but the intelligent use of it is; and the object of such teaching as Gibbs's was to enable the JOSIAH WILLARD GIBBS 203 student to see physical phenomena with the " clarity of vision " which Tait himaelf thought characteristic of the truly mathematical mind, and of which a good criterion is afforded in Helmholtz's unforgetable state- ment about Michael Faraday: "With wonderful sagacity and intel- lectual precision, Faraday performed on his brain the work of a great mathematician without using a single mathematical formula/' 157 At Yale Gibbs was esteemed an ideal teacher of physics, cordial, quick, helpful, willing to devote unlimited time to assist plodders and giving his students ample opportunity to learn " what may be regarded as known, what is guessed at, what a proof is and how far it goes." Of the qualities that make for distinction of mind and character he had the impersonal gift, "le don d'etre n6 essentiellement imper- sonnel," which Renan thought highest of all, and which, fortunately for the advance of real knowledge, has been characteristic of most of the great leaders of science. He could build no wall of personal ego- tism between himself and the external facts, and " few could come in contact with this serene and impartial mind without feeling profoundly its influence in all his future studies of nature." 158 We know little of his life beyond the fact that he was a man of stoic fiber, who lived and worked alone. The countenance in the portraits expresses the Puritan austerity with lines that tell of mental stress and struggles with illness, but the man himself was " unassuming in manner, genial and kindly in his intercourse with his fellow men." " In the minds of those who knew him," concludes his biographer, " the greatness of his intellectual achievements will never overshadow the beauty and dignity of his life." 159 American contributions to physics, from Franklin to Michelson, have been characterized by originality of invention and experiment. The work of Gibbs has a place apart as that of a mathematical theorist whose ideas have found wide application in the main current of modern thought, and his true position is best described in his own often-quoted •estimate of his great predecessor, Clausius. " Such work as that of Clausius," he says, " is not measured by counting titles or pages. His true monument lies not on the shelves of libraries, but in the thoughts of men and the history of more than one science." 160 The general scientific reputation of Gibbs is of this kind, while in his chosen field of activity, the austere region of physics in which Newton and La- grange, Hamilton and Jacobi are the leaders, his is assuredly the most •distinguished American name. ^Helmholtz, Faraday Lecture, 1881. ,M Bumstead, Am. J. Sc., 1903, 4. s., XLL, 201. "• Bumstead, loc. cit. M Gibbs, Proc. Am. Acad. Arts and Sc, 1889, N. S., XVI., 465.

  1. Nature, 1877–8, XVII., 280.
  2. Ibid., 1892-3, XLVII., 534.
  3. Ibid., 464.