Jacob Steiner (1796–1863), "the greatest geometrician since the time of Euclid," was born in Utzendorf in the Canton of Bern. He did not learn to write till he was fourteen. At eighteen he became a pupil of Pestalozzi. Later he studied at Heidelberg and Berlin. When Crelle started, in 1826, the celebrated mathematical journal bearing his name, Steiner and Abel became leading contributors. In 1832 Steiner published his Systematische Entwickelung der Abhängigkeit geometrischer Gestalten von einander, "in which is uncovered the organism by which the most diverse phenomena (Erscheinungen) in the world of space are united to each other." Through the influence of Jacobi and others, the chair of geometry was founded for him at Berlin in 1834. This position he occupied until his death, which occurred after years of bad health. In his Systematische Entwickelungen, for the first time, is the principle of duality introduced at the outset. This book and von Staudt's lay the foundation on which synthetic geometry in its present form rests. Not only did he fairly complete the theory of curves and surfaces of the second degree, but he made great advances in the theory of those of higher degrees. In his hands synthetic geometry made prodigious progress. New discoveries followed each other so rapidly that he often did not take time to record their demonstrations. In an article in Crelle's Journal on Allgemeine Eigenschaften Algebraischer Curven he gives without proof theorems which were declared by Hesse to be "like Fermat's theorems, riddles to the present and future generations." Analytical proofs of some of them have been given since by others, but Cremona finally proved them all by a synthetic method. Steiner discovered synthetically the two prominent properties of a surface of the third order; viz. that it contains twenty-seven straight lines and a pentahedron which has the double points for its vertices and the lines of the Hessian of the given sur-
