STEINER, JAKOB

born March 18, 1796, Utzenstorf, Switz. died April 1, 1863, Bern Swiss mathematician, one of the founders of modern synthetic, or projective, geometry. Steiner had no early schooling and did not learn to read or write until he was 14. Against the wishes of his parents, at 18 he went to the Pestalozzi School at Yverdon, Switz., where his extraordinary geometric intuition was discovered. Later, he went to Heidelberg and Berlin to study, supporting himself precariously as a tutor. By 1824 he had discovered the geometric transformation known as inversive geometry, but he did not publish his work. The founding of August Crelle's Journal (1826) gave him an opportunity to publish some of his other geometric discoveries, and in 1832 he received an honorary doctorate from the University of Knigsberg. Two years later he occupied the chair of geometry established for him at the University of Berlin, a post he held until his death. Steiner's Systematische Entwickelung der Abhngigkeit geometrischer Gestalten von Einander (1832; Systematic Development of the Dependency of Geometric Forms on One Another) includes a thorough discussion of the principle of duality, which states that, in projective geometry, if one of two dual theories is true, the other is also true. He had an extreme dislike for the use of algebra and analysis and often expressed the opinion that calculation replaces thinking, whereas pure geometry stimulates thinking. Using only synthetic methods, he proved that a surface of the third order contains only 27 lines, a theorem that would appear to depend on analysis for proof. Steiner was considered the greatest geometer since Apollonius of Perga (c. 262190 BC), and he produced works that are considered the classical authority on synthetic geometry. He contributed many basic concepts and results in projective geometry and discovered the Steiner surface (also known as the Roman surface), which has a double infinity of conic sections on it. He also developed the Steiner theorem, which holds that the two pencils (collections of geometric objects) by which a conic is projected from two of its points are projectively related, and the Poncelet-Steiner theorem, which demonstrates that only one given circle with its centre and a straightedge are needed for any Euclidean construction. His other work was primarily on the properties of algebraic curves and surfaces and maxima and minima. His collected writings were published as Gesammelte Werke, 2 vol. (188182; Collected Works).