Principia generalia theoriae figurae fluidorum in statu aequilibrii

GAUSS, Carl Friedrich

Published by Dieterich, Göttingen, 1830

About this Item

RARE OFFPRINT OF GAUSS ON THE CALCULUS OF VARIATIONS. First edition, the rare separately-paginated offprint, and a fine copy in original state, of this early contribution of Gauss (1777-1855) to mathematical physics, which contains a fundamental and groundbreaking contribution to the calculus of variations. "Gauss ranks, together with Archimedes and Newton, as one of the greatest geniuses in the history of mathematics" (Printing & the Mind of Man, p. 155). He is best known today for giving the first satisfactory proof of the fundamental theorem of algebra (1799), for Disquisitiones arithmeticae (1801), which revolutionized number theory, and for Disquisitiones generales circa superficies curvas (1828), which laid the foundations of differential geometry. The present work deals with the theory of capillary action, the rise or fall of liquids in narrow tubes, the foundations of which were laid by Thomas Young in 1805, who provided a qualitative theory for surface tension, and by Pierre-Simon Laplace, who mathematically formalized the relationship described by Young a year later. Gauss wrote that Laplace s "investigations, which have found their confirmation in striking agreement with careful experiments, are among the most beautiful enrichments of natural science that we owe to the great mathematician" (translation from p. 2). However, Laplace s theory depended on hypotheses about molecular forces, which were little known at the time, and so open to criticism, by Young among others. "The principle which [Gauss] adopted is that of virtual velocities, a principle which under his hands was gradually transforming itself into what is now known as the principle of the conservation of energy. Instead of calculating the direction and magnitude of the resultant force on each particle arising from the action of neighbouring particles, he formed a single expression which is the aggregate of all the potentials arising from the mutual action between pairs of particles … With its sign reversed it is now called the potential energy of the system … The condition of equilibrium is that this expression shall be a minimum. This condition when worked out gives not only the equation of the free surface in the form already established by Laplace, but the conditions of the angle of contact of this surface with the surface of a solid. Gauss thus supplied the principal defect in the great work of Laplace" (Britannica). To find the conditions under which the potential energy is a minimum, Gauss used the calculus of variations and his own generalization of d Alembert s principle of virtual work .In the process he created the method for varying double integrals with variable limits, a problem that both Euler and Jacobi had failed to solve. "In 1830 appeared Principia generalia theoriae figurae fluidorum in statu aequilibrii, his one contribution to capillarity and an important paper in the calculus of variations, since it was the first solution of a variational problem involving double integrals, boundary conditions, and variable limits" (DSB). "Gauss had always been interested in physics …[This] is a theoretical paper, not connected to any experiment … one of Gauss s objectives appeared to be to show how much mathematics could contribute to the elucidation and explanation of nature … Principia generalia should be counted among Gauss s work in the calculus of variations. It also contains interesting results in potential theory … it is closely connected to Gauss s other work in this area" (Bühler, pp. 121-3). This is an offprint from Commentationes Societatis Regiae Scientiarum Göttingensis. It was published in Vol. VII (1832), pp. 39-88 of the Commentationes Classis Mathematicae. As was often the case with the Commentationes, articles appeared in offprint form before the publication of the journal, in this case two years before. Gauss offprints are rarely found in original state as here. ABPC/RBH list only two other copies, both in modern bindings (one wi. Seller Inventory # 5927

Bibliographic Details

Title: Principia generalia theoriae figurae ...

Publisher: Dieterich, Göttingen

Publication Date: 1830

Edition: First edition.