Carl Friedrich Gauss

Overview

Gauss published the Disquisitiones Arithmeticae in 1801, at twenty-four, and it made number theory a coherent discipline rather than a collection of isolated results; the congruence notation used for modular arithmetic today is introduced in it, together with the first complete proof of quadratic reciprocity. His doctoral dissertation of 1799 argued the fundamental theorem of algebra, that every non-constant polynomial has a root among the complex numbers. In astronomy he developed the method of least squares and the error distribution that carries his name, work driven by the problem of recovering the orbit of a newly discovered asteroid from a handful of observations. In 1827 he showed that the curvature of a surface can be determined from measurements made entirely within the surface, without reference to any surrounding space, which is the starting point of modern differential geometry. He reached non-Euclidean geometry privately and declined to publish. Start with the memoir on curved surfaces in translation rather than with the Disquisitiones.

Where to start reading

• Disquisitiones Arithmeticae 1801

  The book that organised number theory: modular arithmetic, quadratic reciprocity, and the theory of quadratic forms.

• General Investigations of Curved Surfaces 1827

  Shows that curvature is intrinsic to a surface and measurable from inside it, which is where differential geometry begins.

  Read it on Project Gutenberg

• Theoria motus corporum coelestium 1809

  On determining the orbits of celestial bodies from limited observations, including his treatment of least squares and the normal distribution of errors.

Part of

• Number theory
• Non-Euclidean geometry
• Probability theory

Sources

1. Carl Friedrich Gauss MacTutor History of Mathematics
2. Carl Friedrich Gauss Wikipedia