Across the Rhine – Möbius’s Algebraic Version of Projective Geometry

In 1827 August Möbius published his

Der barycentrische Calcul

(

The barycentric calculus

). This assigns coordinates to points in the plane by regarding each point as the centre of gravity (or barycentre) of three weights attached at the vertices of a fixed but arbitrary triangle. Each choice of weights (other than all three zero) defines a pair of ratios that specifies a point in the plane except for a one-dimensional family of weights that defines a line “at infinity”. Möbius’s barycentric coordinates, which are very simply related to the usual system of homogeneous coordinates that were introduced later by other mathematicians, are thus well suited to describing projective transformations and therefore the projective geometry of conic sections. However, they are even better suited to handling duality, and provide a good algebraic approach to projective geometry. Möbius also described how to introduce a system of projective coordinates in the plane without recourse to Euclidean ideas or methods.

Möbius also extended his notion of duality to higher dimensional spaces, and found a new form of duality that is not a generalisation of pole and polar and applies only in spaces of odd dimension.