Gauss's Disquisitiones Generales Circa Superficies Curvas, first published in 1827, is one of the foundational documents of differential geometry. One finds here many of the core ideas that still dominate the theory. It was in this paper that the Gauss map, the first fundamental form (and the notations E, F, and G connected to it), Gaussian curvature, the "Theorema Egregium," and a form of the Gauss-Bonnet theorem (for small geodesic triangles) were first introduced. In fact, that one could almost teach a course on the differential geometry of surfaces from it, provided one supplemented Gauss's text with an account in modern notation. This is pretty much what is done, in fact, in the second volume of Michael Spivak's famous book, A Comprehensive Introduction to Differential Geometry.
The translation included here was first published in 1902, and later reprinted a couple of times (most recently in volume 62 of the journal Astérisque, which also has the Latin original on facing pages). Also included are Gauss's original abstract (several pages long, and quite useful), a translation of an unpublished earlier version of the paper (confusingly titled "New Investigations...", which suggests a later paper), and notes, both by the translators (Adam Hiltebeitel and James Morehead, whose names have to be found in the fine print, alas) and by Peter Pesic, who has also provided an overall introduction.
Gauss's article is only forty pages long! These must be some of the richest pages in the history of mathematics. They served as the inspiration for Riemann's famous essay on the foundations of geometry. Much of differential geometry has sprung from these two texts.
Since the older editions are now quite hard to obtain (I have tried!), it is really an all-around Good Thing to have this Dover reprint. If your library wasn't paying attention in 1902 (or 1965, or 1979), here is your chance to add a crucial book to its collection. And, at the price, you might want a copy yourself.
Dombrowski, Peter. 150 Years After Gauss's "Disquisitiones Generales Circa Superficies Curvas". Astérisque 62. Société Mathématique de France, 1979.
Gauss, Karl Friedrich. General Investigations of Curved Surfaces of 1827 and 1825, translated with notes by James Cadall Morehead and Adam Miller Hiltebeitel. Princeton University Library, 1902. Reprinted with an introduction by Richard Courant, Raven Press, 1965.
Spivak, Michael. A Comprehensive Introduction to Differential Geometry, five volumes. Publish or Perish, 1970, 1979, 1999.
Fernando Q. Gouvêa is professor of mathematics at Colby College in Waterville, ME. While he is primarily a number theorist and a historian of mathematics, he is an admirer of differential geometry and has taught elementary introductions to it every few years.
Introduction to the Dover Edition
|Part I.||General Investigations of Curved Surfaces (1827)|
|Part II.||New General Investigations of Curved Surfaces (1825)|
|Part III.||Additional Notes|
|Appendix: Basic Formulas of Spherical Trigonometry|