Essay: Disquisitiones generales circa superficies curvas

Overview
Carl Friedrich Gauss's Disquisitiones generales circa superficies curvas (1827) is a foundational treatise that established the intrinsic differential geometry of surfaces. It develops a systematic analytic framework for studying the geometry of a surface through quantities defined on the surface itself, rather than through its embedding in three-dimensional space. The work introduced the central concept of Gaussian curvature and demonstrated that curvature can be expressed entirely in terms of the surface metric.
Gauss combined careful coordinate calculations with geometric insight to produce general formulas and theorems applicable to arbitrary smooth surfaces. The results unified prior scattered observations and set rigorous standards for later developments in geometry and analysis.

Definitions and methods
Gauss begins by parametrizing a surface locally with coordinates and deriving the first and second fundamental forms, represented by the metric coefficients E, F, G and the second-form coefficients L, M, N. The first fundamental form encodes lengths and angles on the surface, while the second relates to how the surface bends in space. Gauss showed how to compute all classical curvature measures from these coefficients and their derivatives.
The approach is analytic: partial derivatives of the parametrization produce tangent vectors and normal vectors, and the algebra of these derivatives yields explicit expressions for curvature. Gauss introduced coordinate-invariant combinations of these coefficients, preparing the way for recognizing truly intrinsic quantities.

Theorema Egregium
The central result, the Theorema Egregium, asserts that Gaussian curvature can be calculated solely from the first fundamental form and its first and second derivatives, hence depends only on the intrinsic metric of the surface. Thus curvature is invariant under local isometries and does not change when a surface is bent without stretching. This striking conclusion shows that examples such as a plane and a cylinder, which are locally isometric, share zero Gaussian curvature despite different appearances in space.
Gauss provided multiple formulations and proofs, including an elegant argument via the Gauss map that relates the infinitesimal area distortion of the map sending each surface point to its unit normal on the sphere to the Gaussian curvature. He also derived the classical expression K = (LN - M^2)/(EG - F^2), connecting the two fundamental forms.

Consequences and legacy
The Disquisitiones laid the groundwork for intrinsic differential geometry and influenced Bernhard Riemann's later generalization to higher-dimensional manifolds and Riemannian metrics. The recognition that curvature is an intrinsic property redirected attention from embeddings to the metric, becoming a cornerstone for modern geometric analysis and the mathematical language used in physics, most notably general relativity.
Beyond its theoretical impact, the essay provided calculational tools still taught in geometry courses: the metric tensor viewpoint, coordinate formulas for curvature, and the significance of geodesic curvature and geodesic coordinates. Gauss's clarity in distinguishing intrinsic from extrinsic properties has echoed through two centuries of geometry, making the Disquisitiones a seminal text whose ideas remain central to contemporary mathematics.