Gauss as a geometer

In an attempt to reveal the breadth of Gauss's interest in geometry, this account is divided into six chapters. The first mentions the fundamental theorem of algebra, which can be proved only with the aid of geometric ideas, and in return, an application of algebra to geometry: the connection between the Fermat primes and the construction of regular polygons. Chapter 2 shows his essentially ‘modern’ approach to quaternions. Chapter 3 is a sample of his work in trigonometry. Chapter 4 deals with his approach to the geometry of numbers. Chapter 5 sketches his differential geometry of surfaces: his use of two parameters, the elements of distance and area, his theorema egregium, and the total curvature of a geodesic polygon. Finally, Chapter 6 shows that he continually returned to the subject of non-Euclidean geometry, which was so precious and personal that he would not publish anything of it during his lifetime, and yet did not wish to let it perish with him.