The Mathematician in Modern Physics

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"The Mathematician in Modern Physics" is an article by Carl Barus and published in Science Vol. 40, Jul-Dec 1914, pp. 721-727.


  • When I began my work... in 1876 the theory of Weber "Das electrodynamische Grundgesetz" of 1846, was rife in... continental Europe. Electrodynamics through the genius of Ampère (1821-22) had already definitely captured magnetism. Weber embraced the whole of electromagnetics in a single equation, consistent with the law of the conservation of energy. It was a beautiful theory; but it was action at a distance gone mad. ...[E]ven Gauss and Riemann did not escape temptation, while Clausius revised and modified the argument throughout, bringing out a new theory of his own. ...[I]t is a superb piece of vigorous mathematical reasoning ...[W]hat these men did was to postulate a force which depended upon the states or motion of the point where force originates; but any phase of the force hammers away at any distant point co-temporaneously with the time of its origin [instantaneously]. These electrical forces... did what gravitational forces still persist in doing. ...[W]e can not but marvel how perilously near they came to the state... to-day. If they had only retarded their potentials! ...[T]hey suspected nothing as the 3 X 1010 velocity which characterizes the relation of Weber's electrostatic to his electromagnetic system of units, measured by Weber and Kohlrausch in 1856, is the velocity of light.
  • In England... [Weber genre theories] were vigorously condemned. Thomson and Tate, T and T'... anathematized them as all the more pernicious in proportion as they were beautiful. They were completely swept away by the profound originality and incisiveness of the Faraday-Maxwell hypothesis (1854). Maxwell's great book (1873)... was looked at askance in Germany, Helmholtz alone excepted.
  • The aim of the earlier thinkers, to reduce the whole of electrical science to one equation was now to be realized in a way that marks one of the most important epochs in the history of physical science; an epoch comparable only to that of Newton...
  • [A]t a single stroke of the wand... the whole domain of light and heat was annexed to electricity. It interpreted the meaning of the transparent and the opaque body of reflection and refraction. It introduced a new cosmical force, the light pressure long after found by Lebedew (1900) and... Nichols and Hull. It harmonized the divergent views of Fresnel and Neumann... and it gave to optics a new lease of life by lifting it over the obstructions of the elastic theory.
  • Maxwell's best friends were apprehensive, since the theory predicted even more than was believed to exist, until in 1877 the new Maxwellian light dawned upon the mind of Hertz. The theory endowed the world medium, the ether with new potencies, in insisting on its continuity, on the point to point transfer of electric force, so that ether stress became one of its familiar images, a veritable charm to conjure by.
  • Hydrodynamics... profited and such beautiful researches as those of the Bjerknes (1863 et. seq.) father and son, were stimulated in proportion as they fitted into the electromagnetic scheme.
  • It was inevitable... that in the further treatment of Maxwell's equation the use of vector methods of computation should become indispensable in physics.
  • The ether electrically ignored heretofore has become all embracing. Woe to him that lisps, action at a distance!
  • Maxwell's theory, which according to Hertz means Maxwell's equations, thus includes the whole of physics, dynamics alone excepted, and the world equation has advanced another step.
  • Maxwell indeed, following the established custom, endeavored to call dynamics to his aid; but... mechanics had no counsel to give.
  • Naturally the theory so revolutionary gained headway but slowly... unfortunately Kelvin, and (I believe) Rayleigh long remained unconvinced.
  • When therefore the theory was universally accepted, it was already ripe for the modification, which Hertz himself actually began.
    The ether as Maxwell left it has two independent properties, specific inductive capacity and permeability, which may be regarded as associated in the velocity of the electromagnetic wave passing through it. But the equations apply only for a medium at rest or at least approximately at rest, to a quasi-stationary medium. It is fortunate that a very coarse approximation to rest suffices; otherwise the early workers would have lacked encouragement.
  • The new epoch, now about to dawn, thus found its point of departure in the motion of electrical systems. It has been in the main an era of confusion and bewilderment and one was to learn the hopelessness of any fundamental proof in physics. Instead of any fundamental proof in physics. Instead of subjecting physics to the arbitrament of dynamics, we see dynamics pleading at the gates of electrical science, when electricity, distraught within itself, has no fundamental interpretations to offer.
  • The troubles begin with the study of the first-order effects of moving optical systems, in the researches of Fizeau (1851); they become grave in the famous experiment of Michelson (1881) where the effects to be observed are of the second order.
  • The speed of the earth, regarded optically from axes fixed in the ether, is zero. The ether and the earth have no relative velocity. This is tantamount to a rejection of the ether. Judge the consternation!
  • As Maxwell's equation contained no direct reference to the motion of the charged body, a first attempt... was made by Hertz (1890) to supply this deficiency; but it was not of permanent value.
  • The real interpretative advance came from Lorentz, in 1892. Although he fully realized and had endeavored to explain away the Michelson difficulties, Lorentz none the less boldly put his coordinates in an absolutely fixed ether, penetrating all bodies, even the atoms. He then went back to the methods of Weber, but with this essential difference that he included the whole dictum of the Maxwellian electro-magnetics in his postulates.
  • The peculiar feature of the ether, its permittance and permeability, were abolished and in their place appears the velocity and density of the electron, or charged particle. Electric fluid exists; magnetic fluid does not.
  • Lorentz then showed with consummate skill that the equations of the classic electromagnetics of Maxwell could be retained, that both the scaler potential and the vector-potential would retain their original form, would be invariant, so to speak, if the time-variable were belated by the interval consumed by light in passing from the source to the point of application in question.
  • The profound originality and power of this and the earlier Lorentz transformation would perhaps not have been detected so soon, but for the unexampled abundance of new resources accruing to experimental physics at this time. In 1892 Lenard had isolated the cathode ray; Röntgen in 1895 discovered the X-ray. As a sort of corollary of the X-ray came the Becquerel-ray in 1896; the radium of the Curies in 1897, soon to be interpreted as to radiation by Thomson and Rutherford. The year 1896 brought the Zeeman effect, virtually predicted by Lorentz. The year 1898 brought Thomson's electron. In these and similar researches, bodies moving with a speed approximating that of light (easily exceeding c/10) were for the first time in history, at the disposal of the investigator.
  • The new bodies, showing an inertia or virtual mass depending in a pronounced way on their speed, made havoc with Newton's laws and swept the classic dynamics mercilessly out of the field, as an arbiter of world phenomena.
  • Theories such as those of Lorentz, 1892, or of Larmor, 1894, were now the only refuge. What could they do, was the ardent question, to replace dynamics?
  • Following the suggestion of Lorentz that the moving system contracts in the direction of motion, or at least apparently contracts to the fixed witness, Einstein in 1905 was the first to clearly perceive the iron logic of the situation; and the logic of a desperate situation is all there is in the theory of relativity.
  • Einstein saw that if systems were to be interconsistent, time periods in the moving system would have to expand in the same second-order ratio to the ken of the fixed observer, so that time specifications and time frequencies may proportionately contract; or that identical clocks in the moving system must go slower. In such a case, any natural phenomenon, preferably a vacuum phenomenon like the velocity of light, is the same in all systems, moving or at rest. One system is as good as another. All observation is relative.
  • In its original form, the principle is as yet rather a detached statement, adapted to definite purposes but lacking in mathematical elegance. It was left to the genius of Minkowski (1908) to mould this flotsam of ideas into a philosophical system of extraordinary symmetry and breadth, the promise of which it is as yet too soon to adequately appreciate.
  • Gravity still acts at a distance, as did the electrical vector in the days of Weber. Nor is the most generalized electromagnetic field able to account for the spectrum distribution of radiation, in the development of which energy threatens to pursue, if it has not already entered, the route of atomistic physics occupied by chemistry.
  • The first step came from W. Wien, whose displacement law of 1893 is embodied in the shift of the maximum of spectrum energy density, from red to violet, with increasing temperatures. Wien showed that a universal function of the ratio of temperature to frequency must here be in question. The determination of this universal function was the culmination of the insight and consistent labors of Planck (1900), who by postulating the energy quantum, became the creator of modern thermodynamics; for this energy element is a saucy reality, whose purpose is to stay. It not only tells us all we know of the distribution of energy in the black body spectrum in its thermal relations, but it gives us, indirectly, perhaps the most accurate data at hand of the number of molecules per normal cubic centimeter of the gas, of the mean translational energy of its molecules, of the molecular mass, of the Boltzmann entropy constant, even of the charge of the electron or electric atom itself.
  • Of the Planck molecular oscillators... If operating continuously under the established electromagnetic laws they lead to the impossible distributions of energy in the spectrum investigated by Rayleigh and Jeans. But if emitting only, when their energy content is a whole number of energy elements, a case thus involving the entropy probability of Boltzmann, Wien's law and the numerical data referred to are deducible with astounding precision.

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Quotes about The Mathematician in Modern Physics[edit]

  • The twenty-first summer meeting of the American Mathematical Society was held at Providence, R.I. ...September 8, 9, 1914, as a part of the ceremonies connected with the celebration of the one hundred and fiftieth anniversary of the founding of Brown University. ...The entertainment of the members given by President Faunce, Chancellor Chace, Professor Davis, and other members of the mathematical faculty of Brown University, was so generous and whole hearted that every one present will carry the most pleasant recollections of it for years to come. ...President Faunce gave a notable address in his inimitable style at the dinner on Tuesday evening and Professor Carl Barus, of the department of physics in Brown University, gave an illuminating talk on "The relations of mathematics to physics." The mere enumeration of all these good things, not to mention the friendly discussions of the scientific papers presented and the general good fellowship always found on these occasions, should be sufficient to make every member of the Society resolve to attend the next meeting...
    • W. DeW. Cairns, "Notes and News" (Nov. 1914) The American Mathematical Monthly (Jan-Dec, 1914) Vol. 21, pp. 313-314, No. 9.

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