The birth of non-Euclidean geometry -- Riemann's vision of a new approach to geometry -- Poincaré and Klein — groups and geometries -- Klein, Lie, and the “Erlanger programm” -- Apparent contours from Monge to Todd -- L'Espace : Concept abstrait et/ou physique ; la géométrie entre formalisation mathématique et etude de la nature -- Geometrie und Erfahrung -- The geometric challenge of Riemann and Clifford -- Poincaré et Enriques : deux points de vue différents sur les relations entre géométrie, mécanique et physique -- Physical geometry and special relativity. Einstein et Poincaré -- Transport parallèle et connexions en Géométrie et en Physique -- De la Géométrie Formelle à l'Algèbre Abstraite -- Le Principe de Dualité : sa Signification Historique et Epistémologique -- The formal and the transcendental in mathematics -- Un Panorama des Mathématiques -- Mathematical progress as synthesis of intuition and calculus -- What is space? -- La “lineale ausdehnungslehre” (1844) de Hermann Günther Grassmann -- La capture de l'extension comme dialectique géométrique : Dimension et puissance selon l'ausdehnung de Grassmann (1844) -- Helmholtz and Poincaré's considerations on the genesis of geometry -- Le continu contre l'espace -- Geometrical concepts in quantum physics -- Physics and differential geometry -- Actuality of transcendental æsthetics for modern physics.

In the first half of the 19th century geometry changed radically, and withina century it helped to revolutionize both mathematics and physics. It also put the epistemology and the philosophy of science on a new footing. In this volume a sound overview of this development is given by leading mathematicians, physicists, philosophers, and historians of science. This interdisciplinary approach gives this collection a unique character. It can be used by scientists and students, but it also addresses a general readership.