Gauss’ basic theorem in this memoir had already been set forth in his 1813 memoir on the attraction of homogeneous spheroids, and he probably knew it even before 1810. In Paragraph 24 of this paper he set up another important theorem of the potential: “If a closed surface is a surface of equilibrium for the attracting and repelling forces of masses which are totally in the outer space, then the resultant of forces at every point of that surface as well as at every point of all the inner space is equal to zero.” (Collected Works, V, 307.) From a letter to Bessel dated December 31, 1831, we know that Gauss discovered this theorem several months earlier that year.
Another theorem concerns the case where the attracting and repelling masses are inside space bounded by a closed surface. At every point of the surface, if it is a surface of equilibrium, the resulting force will be directed to one and the same side; according to whether the aggregate of the former or the latter is larger, the resultant at all points will be directed inward or outward. If, however, the aggregate of the attracting masses is equal to the repelling, if there is a closed and inclusive equilibrium, then the resultant of forces at every point of the same and simultaneously in all outer space will be equal’ to zero. (Collected Works, V, 307.)
The theorem which Gauss regarded as the most important part of his memoir is a conclusion from the previous theorem that there is always one and only one distribution of given mass over a surface, so that the potential of this mass at all points of that surface assumes prescribed values (Collected Works, V, 240).
Gauss made electromagnetic measurements for the first time on October 22, 1832, and continued them until 1836. At first he was testing Ohm’s law under the most varied conditions. In 1833 he discovered Kirchhoff’s laws of branched circuits, discovered by the latter in 1845. He established the principle of minimum heat, later set forth by Kirchhoff in 1848, according to which the heat produced by the current is a minimum for the actual current distribution. In addition. Gauss gave exact proof of the identity of frictional electricity and that produced by galvanic elements and thermoelectric forces, a fact of which all physicists at that time were not convinced. These were preliminary studies.
Gauss’ main interest lay in the study of the laws of induction. In 1834 he constructed an induction coil, and was enabled to recognize the damping of a magnet vibrating in a coil as a result of induced current; this led him to the construction of a copper damper for his magnetic apparatus, and to the discovery of “sympathetic vibrations.” January 23, 1835, marked the acme of this work, when he formulated41 the law of induction known now as the “Franz Neumann Law of Potential,” although Neumann did not enunciate it until 1845.
By means of certain transformations on Ampère’s law. Gauss arrived at a law formulated by Grassmann in 1845, which differs quite substantially from the fundamental law of Ampère. Grassmann did not realize that his law was equivalent to that of Ampère, and he proposed experiments to decide between the two. Gauss realized that this was impossible if only closed currents are available, and that Grassmann’s law contradicts the law of action and reaction for current elements, whereas Ampère’s law obeys the last-named law for the elements.
The peak of Gauss’ work in physics was concerned with the magnitude of electrodynamic forces. He discovered the first part of Neumann’s Law of Potential and then made use of the known fact that an induced current has a direction such that the electrodynamic forces resist the motion, so that in consequence positive work is performed during this motion. Thus he deduced from the law of the electrodynamic forces the law of the phenomena of induction. It is the same train of thought followed later by Franz Neumann and then much later by Helmholtz in his celebrated memoir on the conservation of force, and ties up with Clerk Maxwell’s equations.
In those days the general fundamental law was sought in a generalization of Coulomb’s law, which agreed in form with Newton’s law of gravitation. Newton’s work enjoyed unique authority. Gauss was not satisfied with what he had done. The problem was not regarded as having been solved until it was possible to trace it back to the position and motion of electrical charges. In 1835 Gauss attempted to formulate such fundamental laws, and one of them, called “Gauss’ Fundamental Law,” is examined in detail in Clerk Maxwell’s Treatise. Gauss had proved that the basic law represents correctly the electromagnetic phenomena, but Maxwell showed that it fails for the phenomena of induction. Gauss probably saw this failure, for he did not publish this law.
In later years Gauss approached Faraday’s view on field action. In 1845 he stated in a letter to Weber that the generalizations of Coulomb’s law had not satisfied him, because they assumed an instantaneous propagation, whereas his real aim had been the derivation of the forces from an action which is not instantaneous, but propagated in time in a manner similar to light. He stated frankly that, in view of the fact he had not solved this problem, he saw no justification for publishing anything about these electrodynamic investigations. In 1858 Gauss’ pupil Bernhard Riemann was the first to attempt to replace Poisson’s equation for the potential by one which results in propagation of the potential with the velocity of light, c.
XIV
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Surface Theory, Crystallography, and Optics
The general theory of curved surfaces, developed in the nineteenth century, owed its origin to geodesy. When Gauss visited von Zach at the Seeberg observatory near Gotha in 1812, he found his solution to the problem of finding the attraction of an elliptic spheroid, which he published in 1813. (Collected Works, V, 1.) In the time from 1812 to 1816 he was busy with the theory of the shortest lines on the elliptic spheroid, and conceived of the cartesian coordinates of a point of a curved surface as functions of two auxiliary magnitudes.
In the spring of 1816 Gauss had proposed as a prize problem for a new astronomical journal, founded by von Lindenau and Bohnenberger, the projecting of two curved surfaces on each other, yet preserving similarity in the smallest parts.42 At the time he had the solution and had thought out a projection by means of parallel normals on the sphere of radius unity. In his concept the development or bending of curved surfaces was a special case of projection. At this time he had worked out the concept of the total curvature of a portion of a surface, and the concept of the measure of curvature at one point of a surface, as well as preservation of the measure of curvature in spite of bendings.
Schumacher stimulated the Copenhagen Society of Sciences in 1820 to set up as a prize question for 1821 the above-mentioned problem, which Gauss had proposed to Lindenau and Bohnenberger. There were no applicants, hence the problem was renewed for 1822. Gauss sent in his solution and received the prize. The work did not appear until 1825 in the third and last number of Schumacher’s Astronomische Abhandlungen.
The concept of projection is at the center of Gauss’ theory of curved surfaces. For projections in which complete similarity is preserved, he coined the word “conform” in 1843. This type of projection had a long history, particularly in the field of cartography; it reaches back to the time of the Greeks. It is thought that Gauss was led to this work by the study of shortest lines on curved surfaces. For years he planned to write a major work on the theory and practice of higher geodesy.
In generalizing for geodetic triangles of any curved surface, the first step was to find out the sum of the angles of such a triangle. Gauss proved in his Copenhagen prize essay that for every point of a shortest line the osculating plane contains within itself the subject surface normal. He then showed that the sum of the angles of a geodetic triangle of two right angles deviates by an amount which is given by the area of the corresponding triangle on the sphere of radius unity, if one equates its surface to eight right angles.
After a long, hard struggle Gauss succeeded by the end of 1825 in generalizing his theory of curved surfaces. The Disquisitiones generales circa superficies curvas, one of Gauss’ major works, was presented to the Royal Society of Sciences in Göttingen on October 8, 1827, and published i
n the Commentationes recentiores of the Society in 1828. It was reprinted in Gauss’ Collected Works (IV, 217), and has been translated into four languages. In it he established the famous theorem that in whatever way a flexible and inextensible surface may be deformed, the sum of the principle curvatures at each point will always be the same. The so-called Gauss Theorem states that the measure of curvature of a surface depends only on the expression of the square of a linear element in terms of two parameters and their differential coefficients. Modern progress in the theory of surfaces begins with this work of Gauss. Two things in it profoundly affected subsequent development in the theory. The first was the systematic employment of curvilinear coordinates, and therewith a demonstration of the great advantages which could be derived from their use; the second was the conception of a surface as a two-way extension, not rigid but flexible, which could be made to assume new shapes by bending without stretching. All surfaces derived from a given surface by bending are said to be applicable or developable upon each other. The analytical criteria of whether two given surfaces are applicable upon each other constitute one of the interesting chapters in the general theory.
Gauss worked out the general formula for measure of curvature in 1826. It is not known how he arrived at the general concept of bending of curved surfaces. In December, 1822, he worked out the question of surfaces developable on a plane.
In Gauss’ generalization of Legendre’s theorem of 1789 on the reduction of a small spherical triangle to a plane triangle with sides of the same length, the theory of the shortest lines is connected with the theory of measure of curvature. Thus Gauss’ theorem on the reduction of small geodetic triangles to plane triangles appears as the pinnacle of the structure of Gauss’ general theory of curved surfaces.
Only a portion of his results in the field of curved surfaces was presented in the Disquisitiones generales, and a second memoir on the subject was planned. Soon after completing this work, however. Gauss was led to research on the foundations of geometry, and at the same time was studying surfaces of constant negative curvature. At the instigation of Gauss, the philosophical faculty of the University of Göttingen in 1830 set up the prize question: the determination of a minimum surface by rotation of a curve joining two given points around a given axis. It was solved by his fellow countryman from Brunswick, later his pupil and assistant, C. W. B. Goldschmidt, who received the prize.
Euler was the only geometer mentioned by name in the Disquisitiones generales; Gauss evidently knew Euler’s Recherches sur la courbure des surfaces (1763), but it is uncertain to what extent he knew Euler’s other work on surface theory. Gauss’ remark that the partial differential equation of the second order for the surfaces developable on the plane “was not yet proved with requisite rigor” was a reference to Monge. The French geometer’s investigations of special classes of surfaces had no influence on Gauss; the same is true of his descriptive geometry, which Gauss praised in a review (1813).
Gauss’ work in surface theory was pioneer research for the later nineteenth century in two respects. First, Gauss moved on to the use of an infinite group, in the sense of Sophus Lie, while up until his time in geometry only finite groups of transformations had been considered. Second, he treated the theory of curved surfaces as the geometry of a two-fold extended manifold in a manner which paved the way for the general theory of the multiply extended manifolds, or n-dimensional space.
In 1831 Gauss suddenly manifested a great predilection for crystallography, probably as a hobby or diversion. After a few weeks he had fully mastered the subject as it was then known. He measured crystals with a twelve-inch Reichenbach theodolite, and calculated and sketched their most difficult forms. On June 30, 1831, he wrote to his friend Gerling:
Recently I have begun to busy myself with the study of crystallography, which was formerly quite foreign to me. At first I found it very difficult to orient myself somewhat in it; it seemed to me as though the memoir of your colleague Hessel43 in the physics dictionary is the model of a confused lecture, to follow which a greater patience than mine is necessary. The reflecting goniometers, as they are equipped up to now according to Wollaston, seem to me to be rather incomplete instruments; I have thought out a very simple apparatus and had one made, by means of which the crystal is fastened at the telescope of a theodolite and thereby can keep its correct position with the greatest sharpness, and I am quite curious as to how experiments begun with it will turn out. I thus hope to be able to determine with ease and without repetition the angles between two surfaces as sharply as only the plane quality of the surfaces allows, and shall then go through a series of crystals.
Gauss was interested in the question of the rationality or irrationality of the ratios of the crystallographic coefficients. His system of crystallographic notation was essentially the one later devised by Professor William Hallows Miller (1808–1880) of Cambridge University. As a matter of fact, this system’s use of the indices was first devised in 1825 by Whewell. In Germany crystallography had been developed at that time by Franz Ernst Neumann and Grassmann. Miller was the pupil and successor of Whewell at Cambridge, and did not publish his system until 1838. His works have a certain laconicism which is reminiscent of Gauss’ style. Miller’s crystal-notation system represented the face by a symbol composed of three numerals, or indices, which are the denominators of three fractions with unity for their numerator and in the ratio of the multiples of the parameters. He asserted the principle that his axes must be parallel to possible edges of the crystal. This system brought the symbols of the crystallographer into a form similar to that employed in algebraic geometry, and obtained expressions suitable for logarithmic computation. Gauss and Miller followed Whewell, Neumann, and Grassmann in representing the faces of a crystal by normals to the faces, which are conceived as all passing through a common point. This point is taken as the center of an imaginary sphere, the sphere of projection. The points, or poles, in which the sphere is met by these normals, and which therefore give the relative directions in space of the faces of the crystal, can have their positions on the sphere determined by the methods of spherical trigonometry. A great circle (zone circle) traversing the poles of any two faces will traverse all the poles corresponding to faces in a zone with them. Miller, Gauss, and Neumann used the stereographic projection, and thus were able at once to project any of these great circles on a sheet of paper with ruler and compasses. Thus elaborate edge drawings of crystals became of comparatively little importance. Their system gave expressions for working all the problems that a crystal can present, and it gave them in a form that appealed at once to the sense of symmetry and appropriateness of the mathematician.
Gauss complimented Miller with having “exactly hit the nail on the head” in his crystallography; after a short time he laid aside all his papers, observations, calculations, and sketches. Strangely, Gauss never published anything on crystallography and never talked about it again.
Work in astronomy forced Gauss into the problems of optics in the early period of his life. More accurately expressed, it was the problems of dioptrics, which were caused by the insufficiency of telescopes of that day. The main problem was the calculation of achromatic and spherically correct telescope objectives. As early as 1807 J. G. Repsold, owner of the well-known Hamburg optical works, turned to Gauss with questions on the construction of an achromatic double objective. Gauss wrote to Repsold on September 2, 1809, asking for exact values of refraction and dispersion of two kinds of glass, and gave him some advice on carrying out exact measurement. Twice in 1810 Schumacher in the name of Repsold asked Gauss for the calculation of a new, double objective of eight-foot focal length.
The first two objectives which Repsold produced according to Gauss’ formulas yielded poor results. The thickness of the glasses was so slight that the glass lost its spherical form in polishing. On October 6, 1810, Gauss wrote Schumacher that he was ready to repeat the calculation for somewhat greater thickness of lenses. Schumacher soon wro
te Gauss that a new experiment by Repsold was crowned with full success. As a matter of fact, up until 1810 Gauss had not paid special attention to the theory of achromatic objectives. His calculations went back essentially to Euler’s Dioptrik. The double objective calculated for Repsold was corrected spherically for the marginal rays and chromatically in the axis.
In 1817 Gauss published an article on achromatic double objectives, especially with reference to complete removal of chromatic dispersion. He came to the following result: Complete removal of chromatic dispersion among the marginal rays and rays next to the axis is of course possible, or more definitely, an objective can be calculated, which unites at one and the same point all rays of two definite colors, those which impinge at a definite distance from the axis, as well as those which impinge infinitely near it (and indeed, as is always presupposed here, parallel to it). In this result Gauss dropped the old condition that the convex lens would have to be biconvex.
Carl Friedrich Gauss, Titan of Science_A Study of His Life and Work Page 20