General Investigations of Curved Surfaces: Edited with an Introduction and Notes by Peter Pesic

Copyright

Copyright © 2005 by Dover Publications, Inc. All rights reserved.

Bibliographical Note

This Dover edition, first published in 2005, is an unabridged republication of the work first published by the Princeton University Library, Princeton, New Jersey in 1902. It was translated from the Latin and German by Adam Hiltebeitel and James Morehead. A new Introduction, Additional Notes on the text, Appendix, and Bibliography have been prepared by Peter Pesic.

The editor wishes to thank Roger Cooke, Grant Franks, and Jeremy Gray for their helpful comments.

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INTRODUCTION TO THE DOVER EDITION

Thou, Nature, art my goddess; to thy law my services are bound. By choosing this motto, Carl Friedrich Gauss (1777-1855) indicated his deep commitment to natural science.¹ His profound discoveries earned him the title of prince of mathematicians; his multifaceted achievements challenge the common distinction between pure and applied mathematics. James Clerk Maxwell thought Gauss’s work on magnetism may be taken as models of physical research by all those who are engaged in the measurement of any of the forces in nature.² Gauss was also deeply involved in astronomy, surveying, and geodetic mapping. These intensely practical activities converged with his pure concerns in his seminal formulation of the theory of curved surfaces. Gauss’s work, generalized by his student Georg Friedrich Bernhard Riemann, was part of the essential groundwork for Albert Einstein’s general theory of relativity. According to Einstein, the importance of Gauss for the development of modern physical theory and especially for the mathematical fundamentals of the theory of relativity is overwhelming indeed; ... If he had not created his geometry of surfaces, which served Riemann as a basis, it is scarcely conceivable that anyone else would have discovered it.³

From the beginning, Gauss’s mind spanned practice and speculation. The son of a day laborer and an almost illiterate mother, he was a calculating prodigy who at age ten grasped how to sum the first hundred integers without tedious addition. At age fifteen, he calculated the number of primes less than a given number (going up into the hundred thousands), from which he conjectured the celebrated prime number theorem.⁴ In early youth he also began considering the consequences for geometry were the parallel postulate of Euclid not true.⁵ At age eighteen he used algebra in order to accomplish a geometrical tour de force, the construction of a regular seventeen-sided polygon using only straightedge and compass, which decided him to pursue mathematics rather than philology.

Gauss’s involvement with geometry was also practical. He remained an active astronomical observer over the course of fifty years, using sextant and telescope and making lengthy calculations. His determination of the orbit of the planetoid Ceres led to his appointment as professor of astronomy and director of the Göttingen observatory. He brought together his knowledge of mathematical and practical astronomy in his Theory of the Motion of the Heavenly Bodies (1809).⁶

Gauss’s involvement in surveying and geodesy was no less intense.⁷ In post-Napoleonic Europe, economic and military considerations called for greatly improved maps, which required much more accurate surveys. For more than a decade (1818-1832), most of Gauss’s time was devoted to directing the surveying of the Electorate of Hanover, a major project that involved extensive fieldwork to make triangulations, followed by endless calculations. Gauss took charge of the arduous work personally. In summer, he spent most of his time away from home, going from village to village, arranging for sightlines by removing trees, and directing his military assistants. Once he was thrown from his horse, another time his carriage overturned and his theodolite fell on him; he suffered from injuries, heat, and lack of sleep. During the same time, his wife was seriously ill (she died in 1831) and he became estranged from one of his sons, who left for America. Gauss struggled with every aspect of the observations as well as the exacting calculations they entailed. He was in fact the founder of modern geodesy. Gauss was especially proud of having invented the heliotrope, an instrument whose movable mirror could reflect light over longer distances than ever before practicable, which he devised after being bothered by the reflection of sunlight from a distant window.

Though one of his biographers deplored that so much of Gauss’s valuable time was taken up by the survey, its practical needs led Gauss to several important mathematical discoveries. ⁸ At age seventeen, he had already devised the method of least squares to deal with astronomical data; later, he applied this fundamental tool for analyzing data to geodetic measurements. ⁹ The exigencies of map-making required projecting the curved surface of the earth onto a flat plane. Several projections were already in use, such as the Mercator or stereographic projections, but their varying advantages and disadvantages led Gauss to reconsider the question mathematically. In an important paper of 1822, he asked: What system of mapping would give the most faithful possible projection, in the sense that the image is similar to the original in the smallest parts, as he put it?¹⁰ Here emerges a deep theme: the search for invariants, in this case quantities that test the similarity of image to original. Because surveying involves the measurement of angles between sightlines to different points of reference, Gauss naturally considered those projections that would preserve angles, so that an angle surveyed on earth would be the same as the corresponding angle shown on the plane map (a condition obeyed by Mercator projections, for instance). In his 1822 paper, Gauss set forth the properties of these conformal mappings.¹¹

The mapping problem raises the question of the relation between different sorts of maps and the curved surface they represent because different projections can render startlingly different versions of the same earth. For instance, a Mercator projection makes the polar regions seem huge in comparison to comparable areas at lower latitudes. Gauss then inquired whether curved surfaces have any properties that reflect intrinsic properties of their curvature. The surprising results form the climax of his General Investigations of Curved Surfaces (Disquisitiones generales circa superficies curvas) of 1827.

In this work, Gauss begins with the perspective of a surveyor concerned with the directions of various straight lines in space he specifies through the use of an auxiliary sphere of unit radius, similar to the celestial sphere in astronomy. A parallel radius of the auxiliary sphere can represent a given directed line, so that the direction of that line corresponds to a certain point on the surface of the sphere. Gauss thus opens an extended comparison between spherical geometry and the actual curved surface being mapped. He then uses a two-dimensional coordinate grid on the curved surface, a construction introduced by Leonhard Euler, whose work was fundamental for all that came after.¹² These Gaussian coordinates (as they are now called) are far more convenient for depicting distance relations than the Cartesian x, y, and z coordinates. Here again we recall a surveyor tracking terrestrial curvature using the two dimensions of latitude and longitude. For instance, consider a point on a convex surface touched by an osculating sphere, a sphere that kisses the inside of the surface, matching the local curvature near that point. The radius of the osculating sphere is called the radius of curvature of the surface at that point, so that a flat plane corresponds to an infinite radius of curvature. Yet the earth is not a perfect sphere (as Newton pointed out) because rotation makes its equatorial region bulge slightly, so that the radii of curvature in the polar and equatorial directions differ over the earth’s surface. In 1760, Euler showed that in this case the curvature at some point can be expressed as the reciprocal of the products of the maximum and minimum radii of curvature (signed positively or negatively to denote convexity or concavity), a definition of curvature that came to be standard.

In his quest for a truly exact geodesy, Gauss had to determine the degree to which the earth’s double curvature was observable and could affect his surveys. In addressing this, he came upon far more general and surprising results. He defines the measure of curvature (now called the Gaussian curvature) as the ratio between an infinitesimal area on the curved surface and the corresponding area mapped on a sphere, which is closely related to Euler’s standard definition. Now Gauss took a crucial step forward. He considers all curved surfaces that can be developed on each other, meaning that to each point of the original surface there corresponds a single point on the map, as if the original surface were allowed to be arbitrarily bent but without stretching. The modern term isometry emphasizes that Gauss’s development requires equal distance be preserved. From this, Gauss uses his surface coordinates and in a couple of lines deduces what he calls his remarkable theorem (theorema egregium): "If a curved surface is developed upon any other surface whatever, the measure of curvature in each point remains unchanged."

Gauss’s theorem shows that a surface can have an intrinsic curvature that does not change even if that surface is extrinsically bent (always leaving distances invariant). Thus, no plane map can be faithful (isometric) to the intrinsically curved earth, even in a small region, which may have long seemed intuitive to map-makers. Yet the beauty of Gauss’s theorem is also its counterintuitive surprise; one might have thought that there is no such thing as intrinsic curvature, as opposed to how a surface is deployed in its ambient space. For instance, the intrinsic curvature of a flat plane and of a cylindrical surface are both zero, so that both are intrinsically flat, though intuitively it might seem the cylinder is curved. Here Gauss shows the irrelevance of the three-dimensional space within which we intuitively immerse the cylinder or plane.

Gauss’s theorem led to a new formulation of differential geometry, the study of surfaces in the small. He introduced the use of a metric, an expression for the infinitesimal distance between points that generalizes the Pythagorean formula to curved surfaces expressed in arbitrary coordinates. This was the starting point Riemann took in 1854 to generalize Gauss’s work to a space of n dimensions and determine its intrinsic curvature; indeed, Riemann thought of reformulating all geometry in intrinsic terms.¹³ Gauss also developed Euler’s work on geodesics, the shortest possible lines on a curved surface, which are the natural generalizations of straight lines in the Euclidean plane. Thus, the Cartesian grid of orthogonal straight lines on the plane can be generalized to grids of geodesics on a curved surface.

Gauss then returns to his surveying and calculates the curvature of the surface in terms of the metric (the Gauss equation, as it is now called). He also constructs a triangle whose sides are geodesics. He shows that the excess (or deficit) of the sum of its angles compared to 180° is measured by the triangle’s corresponding area mapped