# A History of Mathematics/Recent Times/Applied Mathematics

APPLIED MATHEMATICS.
Notwithstanding the beautiful developments of celestial mechanics reached by Laplace at the close of the eighteenth century, there was made a discovery on the first day of the present century which presented a problem seemingly beyond the power of that analysis. We refer to the discovery of Ceres by Piazzi in Italy, which became known in Germany just after the philosopher Hegel had published a dissertation proving a priori that such a discovery could not be made. From the positions of the planet observed by Piazzi its orbit could not be satisfactorily calculated by the old methods, and it remained for the genius of Gauss to devise a method of calculating elliptic orbits which was free from the assumption of a small eccentricity and inclination. Gauss' method was developed further in his Theoria Motus. The new planet was re-discovered with aid of Gauss' data by Olbers, an astronomer who promoted science not only by his own astronomical studies, but also by discerning and directing towards astronomical pursuits the genius of Bessel.
Friedrich Wilhelm Bessel (1784–1846) was a native of Minden in Westphalia. Fondness for figures, and a distaste for Latin grammar led him to the choice of a mercantile career. In his fifteenth year he became an apprenticed clerk in Bremen, and for nearly seven years he devoted his days to mastering the details of his business, and part of his nights to study. Hoping some day to become a supercargo on trading expeditions, he became interested in observations at sea. With a sextant constructed by him and an ordinary clock he determined the latitude of Bremen. His success in this inspired him for astronomical study. One work after another was mastered by him, unaided, during the hours snatched from ​sleep. From old observations he calculated the orbit of Halley's comet. Bessel introduced himself to Olbers, and submitted to him the calculation, which Olbers immediately sent for publication. Encouraged by Olbers, Bessel turned his back to the prospect of affluence, chose poverty and the stars, and became assistant in J. H. Schröter's observatory at Lilienthal. Four years later he was chosen to superintend the construction of the new observatory at Königsberg. In the absence of an adequate mathematical teaching force, Bessel was obliged to lecture on mathematics to prepare students for astronomy. He was relieved of this work in 1825 by the arrival of Jacobi. We shall not recount the labours by which Bessel earned the title of founder of modern practical astronomy and geodesy. As an observer he towered far above Gauss, but as a mathematician he reverently bowed before the genius of his great contemporary. Of Bessel's papers, the one of greatest mathematical interest is an "Untersuchung des Theils der planetarischen Störungen, welcher aus der Bewegung der Sonne ensteht" (1824), in which he introduces a class of transcendental functions, $\scriptstyle {J_{n}(x)}$, much used in applied mathematics, and known as "Bessel's functions." He gave their principal properties, and constructed tables for their evaluation. Recently it has been observed that Bessel's functions appear much earlier in mathematical literature. Such functions of the zero order occur in papers of Daniel Bernoulli (1732) and Euler on vibration of heavy strings suspended from one end. All of Bessel's functions of the first kind and of integral orders occur in a paper by Euler (1764) on the vibration of a stretched elastic membrane. In 1878 Lord Rayleigh proved that Bessel's functions are merely particular cases of Laplace's functions. J. W. L. Glaisher illustrates by Bessel's functions his assertion that mathematical branches growing out of physical inquiries as a rule "lack the easy flow ​or homogeneity of form which is characteristic of a mathematical theory properly so called." These functions have been studied by C. Th. Anger of Danzig, 0. Schlömilch of Dresden, R. Lipschitz of Bonn (born 1832), Carl Neumann of Leipzig (born 1832), Eugen Lommel of Leipzig, I. Todhunter of St. John's College, Cambridge.
Prominent among the successors of Laplace are the following: Siméon Denis Poisson (1781–1840), who wrote in 1808 a classic Mémoire sur les inégalités séculaires des moyens mouvements des planètes, Giovanni Antonio Amadeo Plana (1781–1864) of Turin, a nephew of Lagrange, who published in 1811 a Memoria sulla teoria dell' attrazione degli sferoidi ellitici, and contributed to the theory of the moon. Peter Andreas Hansen (1795–1874) of Gotha, at one time a clockmaker in Tondern, then Schumacher's assistant at Altona, and finally director of the observatory at Gotha, wrote on various astronomical subjects, but mainly on the lunar theory, which he elaborated in his work Fundamenta nova investigationes orbitœ verœ quam Luna perlustrat (1838), and in subsequent investigations embracing extensive lunar tables. George Biddel Airy (1801–1892), royal astronomer at Greenwich, published in 1826 his Mathematical Tracts on the Lunar and Planetary Theories. These researches have since been greatly extended by him. August Ferdinand Möbius (1790–1868) of Leipzig wrote, in 1842, Elemente der Mechanik des Himmels. Urbain Jean Joseph Le Verrier (1811–1877) of Paris wrote, the Recherches Astronomiques, constituting in part a new elaboration of celestial mechanics, and is famous for his theoretical discovery of Neptune. John Couch Adams (1819–1892) of Cambridge divided with Le Verrier the honour of the mathematical discovery of Neptune, and pointed out in 1853 that Laplace's explanation of the secular acceleration of the moon's mean motion accounted for only half the observed acceleration. ​Charles Eugène Delaunay (born 1816, and drowned off Cherbourg in 1872), professor of mechanics at the Sorbonne in Paris, explained most of the remaining acceleration of the moon, unaccounted for by Laplace's theory as corrected by Adams, by tracing the effect of tidal friction, a theory previously suggested independently by Kant, Robert Mayer, and William Ferrel of Kentucky. George Howard Darwin of Cambridge (born 1845) made some very remarkable investigations in 1879 on tidal friction, which trace with great certainty the history of the moon from its origin. He has since studied also the effects of tidal friction upon other bodies in the solar system. Criticisms on some parts of his researches have been made by James Nolan of Victoria. Simon Newcomb (born 1835), superintendent of the Nautical Almanac at Washington, and professor of mathematics at the Johns Hopkins University, investigated the errors in Hansen's tables of the moon. For the last twelve years the main work of the U.S. Nautical Almanac office has been to collect and discuss data for new tables of the planets which will supplant the tables of Le Verrier. G. W. Hill of that office has contributed an elegant paper on certain possible abbreviations in the computation of the long-period of the moon's motion due to the direct action of the planets, and has made the most elaborate determination yet undertaken of the inequalities of the moon's motion due to the figure of the earth. He has also computed certain lunar inequalities due to the action of Jupiter.
The mathematical discussion of Saturn's rings was taken up first by Laplace, who demonstrated that a homogeneous solid ring could not be in equilibrium, and in 1851 by B. Peirce, who proved their non-solidity by showing that even an irregular solid ring could not be in equilibrium about Saturn. The mechanism of these rings was investigated by James Clerk Maxwell in an essay to which the Adams prize was awarded. ​He concluded that they consisted of an aggregate of unconnected particles.
The problem of three bodies has been treated in various ways since the time of Lagrange, but no decided advance towards a more complete algebraic solution has been made, and the problem stands substantially where it was left by him. He had made a reduction in the differential equations to the seventh order. This was elegantly accomplished in a different way by Jacobi in 1843. R, Radau (Comptes Rendus, LXVII., 1868, p. 841) and Allégret (Journal de Mathématiques, 1875, p. 277) showed that the reduction can be performed on the equations in their original form. Noteworthy transformations and discussions of the problem have been given by J. L. F. Bertrand, by Émile Bour (1831–1866) of the Polytechnic School in Paris, by Mathieu, Hesse, J. A. Serret. H. Bruns of Leipzig has shown that no advance in the problem of three or of $\scriptstyle {n}$ bodies may be expected by algebraic integrals, and that we must look to the modern theory of functions for a complete solution (Acta Math., XI., p. 43).
Among valuable text-books on mathematical astronomy rank the following works: Manual of Spherical and Practical Astronomy by Chauvenet (1863), Practical and Spherical Astronomy by Robert Main of Cambridge, Theoretical Astronomy by James C. Watson of Ann Arbor (1868), Traité élémentaire de Mécanique Céleste of H. Resal of the Polytechnic School in Paris, Cours d' Astronomie de l'École Polytechnique by Faye, Traité de Mécanique Céleste by Tisserand, Lehrbuch der Bahnbestimmung by T. Oppolzer, Mathematische Theorien der Planetenbewegung by O. Dziobek, translated into English by M. W. Harrington and W. J. Hussey.
During the present century we have come to recognise the advantages frequently arising from a geometrical treatment of mechanical problems. To Poinsot, Chasles, and Möbius we ​owe the most important developments made in geometrical mechanics. Louis Poinsot (1777–1859), a graduate of the Polytechnic School in Paris, and for many years member of the superior council of public instruction, published in 1804 his Éléments de Statique. This work is remarkable not only as being the earliest introduction to synthetic mechanics, but also as containing for the first time the idea of couples, which was applied by Poinsot in a publication of 1834 to the theory of rotation. A clear conception of the nature of rotary motion was conveyed by Poinsot's elegant geometrical representation by means of an ellipsoid rolling on a certain fixed plane. This construction was extended by Sylvester so as to measure the rate of rotation of the ellipsoid on the plane.
A particular class of dynamical problems has recently been treated geometrically by Sir Robert Stawell Ball, formerly astronomer royal of Ireland, now Lowndean Professor of Astronomy and Geometry at Cambridge. His method is given in a work entitled Theory of Screws, Dublin, 1876, and in subsequent articles. Modern geometry is here drawn upon, as was done also by Clifford in the related subject of Bi-quaternions. Arthur Buchheim of Manchester (1859–1888), showed that Grassmann's Ausdehnungslehre supplies all the necessary materials for a simple calculus of screws in elliptic space. Horace Lamb applied the theory of screws to the question of the steady motion of any solid in a fluid.
Advances in theoretical mechanics, bearing on the integration and the alteration in form of dynamical equations, were made since Lagrange by Poisson, William Rowan Hamilton, Jacobi, Madame Kowalevski, and others. Lagrange had established the "Lagrangian form" of the equations of motion. He had given a theory of the variation of the arbitrary constants which, however, turned out to be less fruitful in results than a theory advanced by Poisson. ​Poisson's theory of the variation of the arbitrary constants and the method of integration thereby afforded marked the first onward step since Lagrange. Then came the researches of Sir William Rowan Hamilton. His discovery that the integration of the dynamic differential equations is connected with the integration of a certain partial differential equation of the first order and second degree, grew out of an attempt to deduce, by the undulatory theory, results in geometrical optics previously based on the conceptions of the emission theory. The Philosophical Transactions of 1833 and 1834 contain Hamilton's papers, in which appear the first applications to mechanics of the principle of varying action and the characteristic function, established by him some years previously. The object which Hamilton proposed to himself is indicated by the title of his first paper, viz. the discovery of a function by means of which all integral equations can be actually represented. The new form obtained by him for the equation of motion is a result of no less importance than that which was the professed object of the memoir. Hamilton's method of integration was freed by Jacobi of an unnecessary complication, and was then applied by him to the determination of a geodetic line on the general ellipsoid. With aid of elliptic coordinates Jacobi integrated the partial differential equation and expressed the equation of the geodetic in form of a relation between two Abelian integrals. Jacobi applied to differential equations of dynamics the theory of the ultimate multiplier. The differential equations of dynamics are only one of the classes of differential equations considered by Jacobi. Dynamic investigations along the lines of Lagrange, Hamilton, and Jacobi were made by Liouville, A. Desboves, Serret, J. C. F. Sturm, Ostrogradsky, J. Bertrand, Donkin, Brioschi, leading up to the development of the theory of a system of canonical integrals.
​An important addition to the theory of the motion of a solid body about a fixed point was made by Madame Sophie de Kowalevski (1853–1891), who discovered a new case in which the differential equations of motion can be integrated. By the use of theta-functions of two independent variables she furnished a remarkable example of how the modern theory of functions may become useful in mechanical problems. She was a native of Moscow, studied under Weierstrass, obtained the doctor's degree at Göttingen, and from 1884 until her death was professor of higher mathematics at the University of Stockholm. The research above mentioned received the Bordin prize of the French Academy in 1888, which was doubled on account of the exceptional merit of the paper.
There are in vogue three forms for the expression of the kinetic energy of a dynamical system: the Lagrangian, the Hamiltonian, and a modified form of Lagrange's equations in which certain velocities are omitted. The kinetic energy is expressed in the first form as a homogeneous quadratic function of the velocities, which are the time-variations of the co-ordinates of the system; in the second form, as a homogeneous quadratic function of the momenta of the system; the third form, elaborated recently by Edward John Routh of Cambridge, in connection with his theory of "ignoration of co-ordinates," and by A. B. Basset, is of importance in hydro-dynamical problems relating to the motion of perforated solids in a liquid, and in other branches of physics.
In recent time great practical importance has come to be attached to the principle of mechanical similitude. By it one can determine from the performance of a model the action of the machine constructed on a larger scale. The principle was first enunciated by Newton (Principia, Bk. II., Sec. VIII., Prop. 32), and was derived by Bertrand from the principle of virtual velocities. A corollary to it, applied in ship​building, goes by the name of William Froude's law, but was enunciated also by Reech.
The present problems of dynamics differ materially from those of the last century. The explanation of the orbital and axial motions of the heavenly bodies by the law of universal gravitation was the great problem solved by Clairaut, Euler, D'Alembert, Lagrange, and Laplace. It did not involve the consideration of frictional resistances. In the present time the aid of dynamics has been invoked by the physical sciences. The problems there arising are often complicated by the presence of friction. Unlike astronomical problems of a century ago, they refer to phenomena of matter and motion that are usually concealed from direct observation. The great pioneer in such problems is Lord Kelvin. While yet an undergraduate at Cambridge, during holidays spent at the seaside, he entered upon researches of this kind by working out the theory of spinning tops, which previously had been only partially explained by Jellet in his Treatise on the Theory of Friction (1872), and by Archibald Smith.
Among standard works on mechanics are Jacobi's Vorlesungen über Dynamik, edited by Clebsch, 1866; Kirchhoff's Vorlesungen über mathematische Physik, 1876; Benjamin Peirce's Analytic Mechanics, 1855; Somoff's Theoretische Mechanik, 1879; Tait and Steele's Dynamics of a Particle, 1856; Minchin's Treatise on Statics; Routh's Dynamics of a System of Rigid Bodies; Sturm's Cours de Mécanique de l'École Polytechnique.
The equations which constitute the foundation of the theory of fluid motion were fully laid down at the time of Lagrange, but the solutions actually worked out were few and mainly of the irrotational type. A powerful method of attacking problems in fluid motion is that of images, introduced in 1843 by George Gabriel Stokes of Pembroke College, Cambridge. It received little attention until Sir William Thomson's ​discovery of electrical images, whereupon the theory was extended by Stokes, Hicks, and Lewis. In 1849, Thomson gave the maximum and minimum theorem peculiar to hydrodynamics, which was afterwards extended to dynamical problems in general.
A new epoch in the progress of hydrodynamics was created, in 1856, by Helmholtz, who worked out remarkable properties of rotational motion in a homogeneous, incompressible fluid, devoid of viscosity. He showed that the vortex filaments in such a medium may possess any number of knottings and twistings, but are either endless or the ends are in the free surface of the medium; they are indivisible. These results suggested to Sir William Thomson the possibility of founding on them a new form of the atomic theory, according to which every atom is a vortex ring in a non-frictional ether, and as such must be absolutely permanent in substance and duration. The vortex-atom theory is discussed by J. J. Thomson of Cambridge (born 1856) in his classical treatise on the Motion of Vortex Rings, to which the Adams Prize was awarded in 1882. Papers on vortex motion have been published also by Horace Lamb, Thomas Craig, Henry A. Rowland, and Charles Chree.
The subject of jets was investigated by Helmholtz, Kirchhoff. Plateau, and Rayleigh; the motion of fluids in a fluid by Stokes, Sir W. Thomson, Köpcke, Greenhill, and Lamb; the theory of viscous fluids by Navier, Poisson, Saint-Venant, Stokes, O. E. Meyer, Stefano, Maxwell, Lipschitz, Craig, Helmholtz, and A. B. Basset. Viscous fluids present great difficulties, because the equations of motion have not the same degree of certainty as in perfect fluids, on account of a deficient theory of friction, and of the difficulty of connecting oblique pressures on a small area with the differentials of the velocities.
Waves in liquids have been a favourite subject with ​English mathematicians. The early inquiries of Poisson and Cauchy were directed to the investigation of waves produced by disturbing causes acting arbitrarily on a small portion of the fluid. The velocity of the long wave was given approximately by Lagrange in 1786 in case of a channel of rectangular cross-section, by Green in 1839 for a channel of triangular section, and by P. Kelland for a channel of any uniform section. Sir George B. Airy, in his treatise on Tides and Waves, discarded mere approximations, and gave the exact equation on which the theory of the long wave in a channel of uniform rectangular section depends. But he gave no general solutions. J. McGowan of University College at Dundee discusses this topic more fully, and arrives at exact and complete solutions for certain cases. The most important application of the theory of the long wave is to the explanation of tidal phenomena in rivers and estuaries.
The mathematical treatment of solitary waves was first taken up by S. Earnshaw in 1845, then by Stokes; but the first sound approximate theory was given by J. Boussinesq in 1871, who obtained an equation for their form, and a value for the velocity in agreement with experiment. Other methods of approximation were given by Rayleigh and J. McCowan. In connection with deep-water waves, Osborne Reynolds gave in 1877 the dynamical explanation for the fact that a group of such waves advances with only half the rapidity of the individual waves.
The solution of the problem of the general motion of an ellipsoid in a fluid is due to the successive labours of Green (1833), Clebsch (1856), and Bjerknes (1873). The free motion of a solid in a liquid has been investigated by W. Thomson, Kirchhoff, and Horace Lamb. By these labours, the motion of a single solid in a fluid has come to be pretty well understood, but the case of two solids in a fluid is not ​developed so fully. The problem has been attacked by W. M. Hicks.
The determination of the period of oscillation of a rotating liquid spheroid has important bearings on the question of the origin of the moon. G. H. Darwin's investigations thereon, viewed in the light of Riemann's and Poincaré's researches, seem to disprove Laplace's hypothesis that the moon separated from the earth as a ring, because the angular velocity was too great for stability; Darwin finds no instability.
The explanation of the contracted vein has been a point of much controversy, but has been put in a much better light by the application of the principle of momentum, originated by Froude and Rayleigh. Rayleigh considered also the reflection of waves, not at the surface of separation of two uniform media, where the transition is abrupt, but at the confines of two media between which the transition is gradual.
The first serious study of the circulation of winds on the earth's surface was instituted at the beginning of the second quarter of this century by H. W. Dové, William C. Redfield, and James P. Espy, followed by researches of W. Reid, Piddington, and Elias Loomis. But the deepest insight into the wonderful correlations that exist among the varied motions of the atmosphere was obtained by William Ferrel (1817–1891). He was born in Fulton County, Pa., and brought up on a farm. Though in unfavourable surroundings, a burning thirst for knowledge spurred the boy to the mastery of one branch after another. He attended Marshall College, Pa., and graduated in 1844 from Bethany College. While teaching school he became interested in meteorology and in the subject of tides. In 1856 he wrote an article on "the winds and currents of the ocean." The following year he became connected with the Nautical Almanac. A mathematical paper followed in 1858 on "the motion of fluids and solids relative to the earth's ​surface." The subject was extended afterwards so as to embrace the mathematical theory of cyclones, tornadoes, water-spouts, etc. In 1885 appeared his Recent Advances in Meteorology. In the opinion of a leading European meteorologist (Julius Hann of Vienna), Ferrel has "contributed more to the advance of the physics of the atmosphere than any other living physicist or meteorologist."
Ferrel teaches that the air flows in great spirals toward the poles, both in the upper strata of the atmosphere and on the earth's surface beyond the 30th degree of latitude; while the return current blows at nearly right angles to the above spirals, in the middle strata as well as on the earth's surface, in a zone comprised between the parallels 30° N. and 30° S. The idea of three superposed currents blowing spirals was first advanced by James Thomson, but was published in very meagre abstract.
Ferrel's views have given a strong impulse to theoretical research in America, Austria, and Germany. Several objections raised against his argument have been abandoned, or have been answered by W. M. Davis of Harvard. The mathematical analysis of F. Waldo of Washington, and of others, has further confirmed the accuracy of the theory. The transport of Krakatoa dust and observations made on clouds point toward the existence of an upper east current on the equator, and Pernter has mathematically deduced from Ferrel's theory the existence of such a current.
Another theory of the general circulation of the atmosphere was propounded by Werner Siemens of Berlin, in which an attempt is made to apply thermodynamics to aërial currents. Important new points of view have been introduced recently by Helmholtz, who concludes that when two air currents blow one above the other in different directions, a system of air waves must arise in the same way as waves are formed on the ​sea. He and A. Oberbeck showed that when the waves on the sea attain lengths of from 16 to 33 feet, the air waves must attain lengths of from 10 to 20 miles, and proportional depths. Superposed strata would thus mix more thoroughly, and their energy would be partly dissipated. From hydrodynamical equations of rotation Helmholtz established the reason why the observed velocity from equatorial regions is much less in a latitude of, say, 20° or 30°, than it would be were the movements unchecked.
About 1860 acoustics began to be studied with renewed zeal. The mathematical theory of pipes and vibrating strings had been elaborated in the eighteenth century by Daniel Bernoulli, D'Alembert, Euler, and Lagrange. In the first part of the present century Laplace corrected Newton's theory on the velocity of sound in gases, Poisson gave a mathematical discussion of torsional vibrations; Poisson, Sophie Germain, and Wheatstone studied Chladni's figures; Thomas Young and the brothers Weber developed the wave-theory of sound. Sir J. F. W. Herschel wrote on the mathematical theory of sound for the Encyclopædia Metropolitana, 1845. Epoch-making were Helmholtz's experimental and mathematical researches. In his hands and Rayleigh's, Fourier's series received due attention. Helmholtz gave the mathematical theory of beats, difference tones, and summation tones. Lord Rayleigh (John William Strutt) of Cambridge (born 1842) made extensive mathematical researches in acoustics as a part of the theory of vibration in general. Particular mention may be made of his discussion of the disturbance produced by a spherical obstacle on the waves of sound, and of phenomena, such as sensitive flames, connected with the instability of jets of fluid. In 1877 and 1878 he published in two volumes a treatise on The Theory of Sound. Other mathematical researches on this subject have been made in England by Donkin and Stokes.
​The theory of elasticity belongs to this century. Before 1800 no attempt had been made to form general equations for the motion or equilibrium of an elastic solid. Particular problems had been solved by special hypotheses. Thus, James Bernoulli considered elastic laminæ; Daniel Bernoulli and Euler investigated vibrating rods; Lagrange and Euler, the equilibrium of springs and columns. The earliest investigations of this century, by Thomas Young ("Young's modulus of elasticity") in England, J. Binet in France, and G. A. A. Plana in Italy, were chiefly occupied in extending and correcting the earlier labours. Between 1830 and 1840 the broad outline of the modern theory of elasticity was established. This was accomplished almost exclusively by French writers,—Louis-Marie-Henri Navier (1785–1836), Poisson, Cauchy, Mademoiselle Sophie Germain (1776–1831), Félix Savart (1791–1841).
Siméon Denis Poisson (1781–1840) was born at Pithiviers. The boy was put out to a nurse, and he used to tell that when his father (a common soldier) came to see him one day, the nurse had gone out and left him suspended by a thin cord to a nail in the wall in order to protect him from perishing under the teeth of the carnivorous and unclean animals that roamed on the floor. Poisson used to add that his gymnastic efforts when thus suspended caused him to swing back and forth, and thus to gain an early familiarity with the pendulum, the study of which occupied him much in his maturer life. His father destined him for the medical profession, but so repugnant was this to him that he was permitted to enter the Polytechnic School at the age of seventeen. His talents excited the interest of Lagrange and Laplace. At eighteen he wrote a memoir on finite differences which was printed on the recommendation of Legendre. He soon became a lecturer at the school, and continued through life to hold various government scientific posts and professorships. He prepared some 400 publications, ​mainly on applied mathematics. His Traité de Mécanique, 2 vols., 1811 and 1833, was long a standard work. He wrote on the mathematical theory of heat, capillary action, probability of judgment, the mathematical theory of electricity and magnetism, physical astronomy, the attraction of ellipsoids, definite integrals, series, and the theory of elasticity. He was considered one of the leading analysts of his time.
His work on elasticity is hardly excelled by that of Cauchy, and second only to that of Saint-Venant. There is hardly a problem in elasticity to which he has not contributed, while many of his inquiries were new. The equilibrium and motion of a circular plate was first successfully treated by him. Instead of the definite integrals of earlier writers, he used preferably finite summations. Poisson's contour conditions for elastic plates were objected to by Gustav Kirchhoff of Berlin, who established new conditions. But Thomson and Tait in their Treatise on Natural Philosophy have explained the discrepancy between Poisson's and Kirchhoff's boundary conditions, and established a reconciliation between them.
Important contributions to the theory of elasticity were made by Cauchy. To him we owe the origin of the theory of stress, and the transition from the consideration of the force upon a molecule exerted by its neighbours to the consideration of the stress upon a small plane at a point. He anticipated Green and Stokes in giving the equations of isotropic elasticity with two constants. The theory of elasticity was presented by Gabrio Piola of Italy according to the principles of Lagrange's Mécanique Analytique, but the superiority of this method over that of Poisson and Cauchy is far from evident. The influence of temperature on stress was first investigated experimentally by Wilhelm Weber of Göttingen, and afterwards mathematically by Duhamel, who, assuming Poisson's theory of elasticity, examined the alterations of ​form which the formulæ undergo when we allow for changes of temperature. Weber was also the first to experiment on elastic after-strain. Other important experiments were made by different scientists, which disclosed a wider range of phenomena, and demanded a more comprehensive theory. Set was investigated by Gerstner (1756–1832) and Eaton Hodgkinson, while the latter physicist in England and Vicat (1786–1861) in France experimented extensively on absolute strength. Vicat boldly attacked the mathematical theories of flexure because they failed to consider shear and the time-element. As a result, a truer theory of flexure was soon propounded by Saint-Venant. Poncelet advanced the theories of resilience and cohesion.
Gabriel Lamé (1795–1870) was born at Tours, and graduated at the Polytechnic School. He was called to Russia with Clapeyron and others to superintend the construction of bridges and roads. On his return, in 1832, he was elected professor of physics at the Polytechnic School. Subsequently he held various engineering posts and professorships in Paris. As engineer he took an active part in the construction of the first railroads in France. Lamé devoted his fine mathematical talents mainly to mathematical physics. In four works: Leçons sur les fonctions inverses des transcendantes et les surfaces isothermes; Sur les coordonnées curvilignes et leurs diverses applications; Sur la théorie analytique de la chaleur; Sur la théorie mathématique de l'élasticité des corps solides (1852), and in various memoirs he displays fine analytical powers; but a certain want of physical touch sometimes reduces the value of his contributions to elasticity and other physical subjects. In considering the temperature in the interior of an ellipsoid under certain conditions, he employed functions analogous to Laplace's functions, and known by the name of "Lamé's functions." A problem in elasticity called by Lamé's name, viz. ​to investigate the conditions for equilibrium of a spherical elastic envelope subject to a given distribution of load on the bounding spherical surfaces, and the determination of the resulting shifts is the only completely general problem on elasticity which can be said to be completely solved. He deserves much credit for his derivation and transformation of the general elastic equations, and for his application of them to double refraction. Rectangular and triangular membranes were shown by him to be connected with questions in the theory of numbers. The field of photo-elasticity was entered upon by Lamé, F. E. Neumann, Clerk Maxwell. Stokes, Wertheim, R. Clausius, Jellett, threw new light upon the subject of "rari-constancy" and "multi-constancy," which has long divided elasticians into two opposing factions. The uni-constant isotropy of Navier and Poisson had been questioned by Cauchy, and was now severely criticised by Green and Stokes.
Barré de Saint-Venant (1797–1886), ingénieur des ponts et chaussées, made it his life-work to render the theory of elasticity of practical value. The charge brought by practical engineers, like Vicat, against the theorists led Saint-Venant to place the theory in its true place as a guide to the practical man. Numerous errors committed by his predecessors were removed. He corrected the theory of flexure by the consideration of slide, the theory of elastic rods of double curvature by the introduction of the third moment, and the theory of torsion by the discovery of the distortion of the primitively plane section. His results on torsion abound in beautiful graphic illustrations. In case of a rod, upon the side surfaces of which no forces act, he showed that the problems of flexure and torsion can be solved, if the end-forces are distributed over the end-surfaces by a definite law. Clebsch, in his Lehrbuch der Elasticität, 1862, showed that this problem is ​reversible to the case of side-forces without end-forces. Clebsch extended the research to very thin rods and to very thin plates. Saint-Venant considered problems arising in the scientific design of built-up artillery, and his solution of them differs considerably from Lamé's solution, which was popularised by Rankine, and much used by gun-designers. In Saint-Venant's translation into French of Clebsch's Elasticität, he develops extensively a double-suffix notation for strain and stresses. Though often advantageous, this notation is cumbrous, and has not been generally adopted. Karl Pearson, professor in University College, London, has recently examined mathematically the permissible limits of the application of the ordinary theory of flexure of a beam.
The mathematical theory of elasticity is still in an unsettled condition. Not only are scientists still divided into two schools of "rari-constancy" and "multi-constancy," but difference of opinion exists on other vital questions. Among the numerous modern writers on elasticity may be mentioned Émile Mathieu (1835–1891), professor at Besançon, Maurice Levy of Paris, Charles Chree, superintendent of the Kew Observatory, A. B. Basset, Sir William Thomson (Lord Kelvin) of Glasgow, J. Boussinesq of Paris, and others. Sir William Thomson applied the laws of elasticity of solids to the investigation of the earth's elasticity, which is an important element in the theory of ocean-tides. If the earth is a solid, then its elasticity co-operates with gravity in opposing deformation due to the attraction of the sun and moon. Laplace had shown how the earth would behave if it resisted deformation only by gravity. Lamé had investigated how a solid sphere would change if its elasticity only came into play. Sir William Thomson combined the two results, and compared them with the actual deformation. Thomson, and afterwards G. H. Darwin, computed that the resistance of the earth to ​tidal deformation is nearly as great as though it were of steel. This conclusion has been confirmed recently by Simon Newcomb, from the study of the observed periodic changes in latitude. For an ideally rigid earth the period would be 360 days, but if as rigid as steel, it would be 441, the observed period being 430 days.
Among text-books on elasticity may be mentioned the works of Lamé, Clebsch, Winkler, Beer, Mathieu, W. J. Ibbetson, and F. Neumann, edited by O. E. Meyer.
Riemann's opinion that a science of physics only exists since the invention of differential equations finds corroboration even in this brief and fragmentary outline of the progress of mathematical physics. The undulatory theory of light, first advanced by Huygens, owes much to the power of mathematics: by mathematical analysis its assumptions were worked out to their last consequences. Thomas Young (1773–1829) was the first to explain the principle of interference, both of light and sound, and the first to bring forward the idea of transverse vibrations in light waves. Young's explanations, not being verified by him by extensive numerical calculations, attracted little notice, and it was not until Augustin Fresnel (1788–1827) applied mathematical analysis to a much greater extent than Young had done, that the undulatory theory began to carry conviction. Some of Fresnel's mathematical assumptions were not satisfactory; hence Laplace, Poisson, and others belonging to the strictly mathematical school, at first disdained to consider the theory. By their opposition Fresnel was spurred to greater exertion. Arago was the first great convert made by Fresnel. When polarisation and double refraction were explained by Young and Fresnel, then Laplace was at last won over. Poisson drew from Fresnel's formulæ the seemingly paradoxical deduction that a small circular disc, illuminated by a luminous point, ​must cast a shadow with a bright spot in the centre. But this was found to be in accordance with fact. The theory was taken up by another great mathematician, Hamilton, who from his formulæ predicted conical refraction, verified experimentally by Lloyd. These predictions do not prove, however, that Fresnel's formulæ are correct, for these prophecies might have been made by other forms of the wave-theory. The theory was placed on a sounder dynamical basis by the writings of Cauchy, Biot, Green, C. Neumann, Kirchhoff, McCullagh, Stokes, Saint-Venant, Sarrau, Lorenz, and Sir William Thomson. In the wave-theory, as taught by Green and others, the luminiferous ether was an incompressible elastic solid, for the reason that fluids could not propagate transverse vibrations. But, according to Green, such an elastic solid would transmit a longitudinal disturbance with infinite velocity. Stokes remarked, however, that the ether might act like a fluid in case of finite disturbances, and like an elastic solid in case of the infinitesimal disturbances in light propagation.
Fresnel postulated the density of ether to be different in different media, but the elasticity the same, while C. Neumann and McCullagh assume the density uniform and the elasticity different in all substances. On the latter assumption the direction of vibration lies in the plane of polarisation, and not perpendicular to it, as in the theory of Fresnel.
While the above writers endeavoured to explain all optical properties of a medium on the supposition that they arise entirely from difference in rigidity or density of the ether in the medium, there is another school advancing theories in which the mutual action between the molecules of the body and the ether is considered the main cause of refraction and dispersion. The chief workers in this field are J. Boussinesq, W. Sellmeyer, Helmholtz, E. Lommel, E. Ketteler, W. Voigt, and Sir William Thomson in his lectures delivered at the ​Johns Hopkins University in 1884. Neither this nor the first-named school succeeded in explaining all the phenomena. A third school was founded by Maxwell. He proposed the electro-magnetic theory, which has received extensive development recently. It will be mentioned again later. According to Maxwell's theory, the direction of vibration does not lie exclusively in the plane of polarisation, nor in a plane perpendicular to it, but something occurs in both planes—a magnetic vibration in one, and an electric in the other. Fitzgerald and Trouton in Dublin verified this conclusion of Maxwell by experiments on electro-magnetic waves.
Of recent mathematical and experimental contributions to optics, mention must be made of H. A. Rowland's theory of concave gratings, and of A. A. Michelson's work on interference, and his application of interference methods to astronomical measurements.
In electricity the mathematical theory and the measurements of Henry Cavendish (1731–1810), and in magnetism the measurements of Charles Augustin Coulomb (1736–1806), became the foundations for a system of measurement. For electro-magnetism the same thing was done by Andrè Marie Ampère (1775–1836). The first complete method of measurement was the system of absolute measurements of terrestrial magnetism introduced by Gauss and Wilhelm Weber (1804–1891) and afterwards extended by Wilhelm Weber and F. Kohlrausch to electro-magnetism and electro-statics. In 1861 the British Association and the Royal Society appointed a special commission with Sir William Thomson at the head, to consider the unit of electrical resistance. The commission recommended a unit in principle like W. Weber's, but greater than Weber's by a factor of $\scriptstyle {10^{7}}$. The discussions and labours on this subject continued for twenty years, until in 1881 a general agreement was reached at an electrical congress in Paris.
​A function of fundamental importance in the mathematical theories of electricity and magnetism is the "potential." It was first used by Lagrange in the determination of gravitational attractions in 1773. Soon after, Laplace gave the celebrated differential equation,
$\scriptstyle {{\frac {\partial ^{2}V}{dx^{2}}}+{\frac {\partial ^{2}V}{dy^{2}}}+{\frac {\partial ^{2}V}{dz^{2}}}=0}$,
which was extended by Poisson by writing $\scriptstyle {-4\pi k}$ in place of zero in the right-hand member of the equation, so that it applies not only to a point external to the attracting mass, but to any point whatever. The first to apply the potential function to other than gravitation problems was George Green (1793–1841). He introduced it into the mathematical theory of electricity and magnetism. Green was a self-educated man who started out as a baker, and at his death was fellow of Caius College, Cambridge. In 1828 he published by subscription at Nottingham a paper entitled Essay on the application of mathematical analysis to the theory of electricity and magnetism. It escaped the notice even of English mathematicians until 1846, when Sir William Thomson had it reprinted in Crelle's Journal, vols. xliv. and xlv. It contained what is now known as "Green's theorem" for the treatment of potential. Meanwhile all of Green's general theorems had been re-discovered by Sir William Thomson, Chasles, Sturm, and Gauss. The term potential function is due to Green. Hamilton used the word force-function, while Gauss, who about 1840 secured the general adoption of the function, called it simply potential.
Large contributions to electricity and magnetism have been made by William Thomson. He was born in 1824 at Belfast, Ireland, but is of Scotch descent. He and his brother James studied in Glasgow. From there he entered Cambridge, and was graduated as Second Wrangler in 1845. William ​Thomson, Sylvester, Maxwell, Clifford, and J. J. Thomson are a group of great men who were Second Wranglers at Cambridge. At the age of twenty-two W. Thomson was elected professor of natural philosophy in the University of Glasgow, a position which he has held ever since. For his brilliant mathematical and physical achievements he was knighted, and in 1892 was made Lord Kelvin. His researches on the theory of potential are epoch-making. What is called "Dirichlet's principle" was discovered by him in 1848, somewhat earlier than by Dirichlet. We owe to Sir William Thomson new synthetical methods of great elegance, viz. the theory of electric images and the method of electric inversion founded thereon. By them he determined the distribution of electricity on a bowl, a problem previously considered insolvable. The distribution of static electricity on conductors had been studied before this mainly by Poisson and Plana. In 1845 F. E. Neumann of Königsberg developed from the experimental laws of Lenz the mathematical theory of magneto-electric induction. In 1855 W. Thomson predicted by mathematical analysis that the discharge of a Leyden jar through a linear conductor would in certain cases consist of a series of decaying oscillations. This was first established experimentally by Joseph Henry of Washington. William Thomson worked out the electro-static induction in submarine cables. The subject of the screening effect against induction, due to sheets of different metals, was worked out mathematically by Horace Lamb and also by Charles Niven. W. Weber's chief researches were on electrodynamics. Helmholtz in 1851 gave the mathematical theory of the course of induced currents in various cases. Gustav Robert Kirchhoff (1824–1887) investigated the distribution of a current over a flat conductor, and also the strength of current in each branch of a network of linear conductors.
The entire subject of electro-magnetism was revolutionised ​by James Clerk Maxwell (1831–1879). He was born near Edinburgh, entered the University of Edinburgh, and became a pupil of Kelland and Forbes. In 1850 he went to Trinity College, Cambridge, and came out Second Wrangler, E. Routh being Senior Wrangler. Maxwell then became lecturer at Cambridge, in 1856 professor at Aberdeen, and in 1860 professor at King's College, London. In 1865 he retired to private life until 1871, when he became professor of physics at Cambridge. Maxwell not only translated into mathematical language the experimental results of Faraday, but established the electro-magnetic theory of light, since verified experimentally by Hertz. His first researches thereon were published in 1864. In 1871 appeared his great Treatise on Electricity and Magnetism. He constructed the electro-magnetic theory from general equations, which are established upon purely dynamical principles, and which determine the state of the electric field. It is a mathematical discussion of the stresses and strains in a dielectric medium subjected to electro-magnetic forces. The electro-magnetic theory has received developments from Lord Rayleigh, J. J. Thomson, H. A. Rowland, R. T. Glazebrook, H. Helmholtz, L. Boltzmann, O. Heaviside, J. H. Poynting, and others. Hermann von Helmholtz turned his attention to this part of the subject in 1871. He was born in 1821 at Potsdam, studied at the University of Berlin, and published in 1847 his pamphlet Ueber die Erhaltung der Kraft. He became teacher of anatomy in the Academy of Art in Berlin. He was elected professor of physiology at Königsberg in 1849, at Bonn in 1855, at Heidelberg in 1858. It was at Heidelberg that he produced his work on Tonempfindung. In 1871 he accepted the chair of physics at the University of Berlin. From this time on he has been engaged chiefly on inquiries in electricity and hydrodynamics. Helmholtz aimed to determine in what direction experiments should be made to ​decide between the theories of W. Weber, F. E. Neumann, Riemann, and Clausius, who had attempted to explain electrodynamic phenomena by the assumption of forces acting at a distance between two portions of the hypothetical electrical fluid,—the intensity being dependent not only on the distance, but also on the velocity and acceleration,—and the theory of Faraday and Maxwell, which discarded action at a distance and assumed stresses and strains in the dielectric. His experiments favoured the British theory. He wrote on abnormal dispersion, and created analogies between electro-dynamics and hydrodynamics. Lord Rayleigh compared electro-magnetic problems with their mechanical analogues, gave a dynamical theory of diffraction, and applied Laplace's coefficients to the theory of radiation. Rowland made some emendations on Stokes' paper on diffraction and considered the propagation of an arbitrary electro-magnetic disturbance and spherical waves of light. Electro-magnetic induction has been investigated mathematically by Oliver Heaviside, and he showed that in a cable it is an actual benefit. Heaviside and Poynting have reached remarkable mathematical results in their interpretation and development of Maxwell's theory. Most of Heaviside's papers have been published since 1882; they cover a wide field.
One part of the theory of capillary attraction, left defective by Laplace, namely, the action of a solid upon a liquid, and the mutual action between two liquids, was made dynamically perfect by Gauss. He stated the rule for angles of contact between liquids and solids. A similar rule for liquids was established by Ernst Franz Neumann. Chief among recent workers on the mathematical theory of capillarity are Lord Rayleigh and E. Mathieu.
The great principle of the conservation of energy was established by Robert Mayer (1814–1878), a physician in Heilbronn, and again independently by Colding of ​Copenhagen, Joule, and Helmholtz. James Prescott Joule (1818–1889) determined experimentally the mechanical equivalent of heat. Helmholtz in 1847 applied the conceptions of the transformation and conservation of energy to the various branches of physics, and thereby linked together many well-known phenomena. These labours led to the abandonment of the corpuscular theory of heat. The mathematical treatment of thermic problems was demanded by practical considerations. Thermodynamics grew out of the attempt to determine mathematically how much work can be gotten out of a steam engine. Sadi-Carnot, an adherent of the corpuscular theory, gave the first impulse to this. The principle known by his name was published in 1824. Though the importance of his work was emphasised by B. P. E. Clapeyron, it did not meet with general recognition until it was brought forward by William Thomson. The latter pointed out the necessity of modifying Carnot's reasoning so as to bring it into accord with the new theory of heat. William Thomson showed in 1848 that Carnot's principle led to the conception of an absolute scale of temperature. In 1849 he published "an account of Carnot's theory of the motive power of heat, with numerical results deduced from Regnault's experiments." In February, 1850, Rudolph Clausius (1822–1888), then in Zürich (afterwards professor in Bonn), communicated to the Berlin Academy a paper on the same subject which contains the Protean second law of thermodynamics. In the same month William John M. Rankine (1820–1872), professor of engineering and mechanics at Glasgow, read before the Royal Society of Edinburgh a paper in which he declares the nature of heat to consist in the rotational motion of molecules, and arrives at some of the results reached previously by Clausius. He does not mention the second law of thermodynamics, but in a subsequent paper he declares that it could be derived ​from equations contained in his first paper. His proof of the second law is not free from objections. In March, 1851, appeared a paper of William Thomson which contained a perfectly rigorous proof of the second law. He obtained it before he had seen the researches of Clausius. The statement of this law, as given by Clausius, has been much criticised, particularly by Rankine, Theodor Wand, P. G. Tait, and Tolver Preston. Repeated efforts to deduce it from general mechanical principles have remained fruitless. The science of thermodynamics was developed with great success by Thomson, Clausius, and Rankine. As early as 1852 Thomson discovered the law of the dissipation of energy, deduced at a later period also by Clausius. The latter designated the non-transformable energy by the name entropy, and then stated that the entropy of the universe tends toward a maximum. For entropy Rankine used the term thermodynamic function. Thermodynamic investigations have been carried on also by G. Ad. Hirn of Colmar, and Helmholtz (monocyclic and polycyclic systems). Valuable graphic methods for the study of thermodynamic relations were devised in 1873–1878 by J. Willard Gibbs of Yale College. Gibbs first gives an account of the advantages of using various pairs of the five fundamental thermodynamic quantities for graphical representation, then discusses the entropy-temperature and entropy-volume diagrams, and the volume-energy-entropy surface (described in Maxwell's Theory of Heat). Gibbs formulated the energy-entropy criterion of equilibrium and stability, and expressed it in a form applicable to complicated problems of dissociation. Important works on thermodynamics have been prepared by Clausius in 1875, by R. Rühlmann in 1875, and by Poincaré in 1892.
In the study of the law of dissipation of energy and the principle of least action, mathematics and metaphysics met on ​common ground. The doctrine of least action was first propounded by Maupertius in 1744. Two years later he proclaimed it to be a universal law of nature, and the first scientific proof of the existence of God. It was weakly supported by him, violently attacked by König of Leipzig, and keenly defended by Euler. Lagrange's conception of the principle of least action became the mother of analytic mechanics, but his statement of it was inaccurate, as has been remarked by Josef Bertrand in the third edition of the Mécanique Analytique. The form of the principle of least action, as it now exists, was given by Hamilton, and was extended to electrodynamics by F. E. Neumann, Clausius, Maxwell, and Helmholtz. To subordinate the principle to all reversible processes, Helmholtz introduced into it the conception of the "kinetic potential." In this form the principle has universal validity.
An offshoot of the mechanical theory of heat is the modern kinetic theory of gases, developed mathematically by Clausius, Maxwell, Ludwig Boltzmann of Munich, and others. The first suggestions of a kinetic theory of matter go back as far as the time of the Greeks. The earliest work to be mentioned here is that of Daniel Bernoulli, 1738. He attributed to gas-molecules great velocity, explained the pressure of a gas by molecular bombardment, and deduced Boyle's law as a consequence of his assumptions. Over a century later his ideas were taken up by Joule (in 1846), A. K. Krönig (in 1856), and Clausius (in 1857). Joule dropped his speculations on this subject when he began his experimental work on heat. Krönig explained by the kinetic theory the fact determined experimentally by Joule that the internal energy of a gas is not altered by expansion when no external work is done. Clausius took an important step in supposing that molecules may have rotary motion, and that atoms in a molecule may move relatively to each other. He assumed that the force acting ​between molecules is a function of their distances, that temperature depends solely upon the kinetic energy of molecular motions, and that the number of molecules which at any moment are so near to each other that they perceptibly influence each other is comparatively so small that it may be neglected. He calculated the average velocities of molecules, and explained evaporation. Objections to his theory, raised by Buy's-Ballot and by Jochmann, were satisfactorily answered by Clausius and Maxwell, except in one case where an additional hypothesis had to be made. Maxwell proposed to himself the problem to determine the average number of molecules, the velocities of which lie between given limits. His expression therefor constitutes the important law of distribution of velocities named after him. By this law the distribution of molecules according to their velocities is determined by the same formula (given in the theory of probability) as the distribution of empirical observations according to the magnitude of their errors. The average molecular velocity as deduced by Maxwell differs from that of Clausius by a constant factor. Maxwell's first deduction of this average from his law of distribution was not rigorous. A sound derivation was given by O. E. Meyer in 1866. Maxwell predicted that so long as Boyle's law is true, the coefficient of viscosity and the coefficient of thermal conductivity remain independent of the pressure. His deduction that the coefficient of viscosity should be proportional to the square root of the absolute temperature appeared to be at variance with results obtained from pendulum experiments. This induced him to alter the very foundation of his kinetic theory of gases by assuming between the molecules a repelling force varying inversely as the fifth power of their distances. The founders of the kinetic theory had assumed the molecules of a gas to be hard elastic spheres; but Maxwell, in his second presentation of the theory in 1866, ​went on the assumption that the molecules behave like centres of forces. He demonstrated anew the law of distribution of velocities; but the proof had a flaw in argument, pointed out by Boltzmann, and recognised by Maxwell, who adopted a somewhat different form of the distributive function in a paper of 1879, intended to explain mathematically the effects observed in Crookes' radiometer. Boltzmann gave a rigorous general proof of Maxwell's law of the distribution of velocities.
None of the fundamental assumptions in the kinetic theory of gases leads by the laws of probability to results in very close agreement with observation. Boltzmann tried to establish kinetic theories of gases by assuming the forces between molecules to act according to different laws from those previously assumed. Clausius, Maxwell, and their predecessors took the mutual action of molecules in collision as repulsive, but Boltzmann assumed that they may be attractive. Experiment of Joule and Lord Kelvin seem to support the latter assumption.
Among the latest researches on the kinetic theory is Lord Kelvin's disproof of a general theorem of Maxwell and Boltzmann, asserting that the average kinetic energy of two given portions of a system must be in the ratio of the number of degrees of freedom of those portions.