b. Schulpforta, Saxony, 1790; d. Leipzig, Germany, 1868
Astronomy; geometry; statics
Möbius was briefly a student of GAUSS [VI.26], and worked as an astronomer at Leipzig University for almost all of his life. His finest mathematical work was his Der barycentrische Calcul (1829), in which he introduced algebraic methods into the study of projective geometry. He showed in this way how points can be described by a homogeneous triple of coordinates, lines can be described by linear equations, the concept of cross-ratio can be introduced, and the duality of points and lines in the plane can be handled algebraically. He also introduced a Möbius net, which is the projective equivalent of squared paper in Cartesian geometry. His work is all the more remarkable because Möbius knew very little of Poncelet’s radical reinvention of projective geometry only a few years before. In its turn his work was for a time overshadowed by Jakob Steiner’s synthetic treatment of projective geometry of 1832, and then Plücker’s two books on algebraic curves in the 1830s, but the simplicity and generality of Möbius’s methods were important in establishing projective geometry as a rigorous mainstream subject.
In the 1830s Möbius developed a geometrical theory of statics and the composition of forces, and it was in this connection that he showed that whereas duality in plane geometry necessarily gives rise to a conic, duality in space need not. Möbius’s study of duality in space, which pairs points with planes, led him to consider the set of all lines in space, which is a four-dimensional space. It pleased the educator Rudolf Steiner very much that the ordinary three-dimensional space may also be thought of as a four-dimensional space, because Steiner’s philosophy was directed against breaking what he saw as a stranglehold of orthodox teaching.
Möbius is also remembered for the “Möbius band” (or MÖBIUS STRIP [IV.7 §2.3]), a one-sided or nonorientable surface, but the first mathematician to describe such a surface was his compatriot J. B. Listing, in July 1858 (published in 1861). Möbius discovered it only in September 1858 (publishing it in 1865). He is also one of the most important mathematicians to study inversion in circles, and his account of it in 1855 is one reason that such transformations are often called Möbius transformations.
Fauvel, J., R. Flood, and R. J. Wilson, eds. 1993. Möbius and His Band. Oxford: Oxford University Press.
Möbius, A. 1885-87. Gesammelte Werke, edited by R. Baltzer (except volume 4, edited by W. Scheibner and F. Klein), four volumes. Leipzig: Hirzel.