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Jakob Steiner

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JAKOB STEINER (1796-1863), Swiss mathematician, was born on the 18th of March 1796 at the village of Utzendorf (canton Bern). At eighteen he became a pupil of Heinrich Pestalozzi, and afterwards studied at Heidelberg. Thence he went to Berlin, earning a livelihood there, as in Heidelberg, by giving private lessons. Here he became acquainted with A. L. Crelle, who, encouraged by his ability and by that of N. H. Abel, then also staying at Berlin, founded his famous Journal (1826). After Steiner's publication (1832) of his Systematische Entwickelungen he received, through Jacobi's exertions, who was then professor at Konigsberg, an honorary degree of that university; and through the influence of G. J. Jacobi and of the brothers Alexander and Wilhelm von Humboldt a new chair of geometry was founded for him at Berlin (1834). This he occupied till his death, which took place in Bern on the 1st of April 1863.

Steiner's mathematical work was confined to geometry. This he treated synthetically, to the total exclusion of analysis, which he hated, and he is said to have considered it a disgrace to synthetical geometry if equal or higher results were obtained by analytical methods. In his own field he surpassed all his contemporaries. His investigations are distinguished by their great generality, by the fertility of his resources, and by such a rigour in his proofs that he has been considered the greatest geometrical genius since the time of Apollonius.

In his Systematische Entwickelung der Abhangigkeit geometrischer Gestalten von einander he laid the foundation of modern synthetic geometry. He introduces what are now called the geometrical forms (the row, flat pencil, &c.), and establishes between their elements a one-one correspondence, or, as he calls it, makes them projective. He next gives by aid of these projective rows and pencils a new generation of conics and ruled quadric surfaces, "which leads quicker and more directly than former methods into the inner nature of conics and reveals to us the organic connexion of their innumerable properties and mysteries." In this work also, of which unfortunately only one volume appeared instead of the projected five, we see for the first time the principle of duality introduced from the very beginning as an immediate outflow of the most fundamental properties of the plane, the line and the point.

In a second little volume, Die geometrischen Constructionen ausgefuhrt mittelst der geraden Linie and eines festen Kreises (1833), republished in 1895 by Ottingen, he shows, what had been already suggested by J. V. Poncelet, how all problems of the second order can be solved by aid of the straight-edge alone without the use of compasses, as soon as one circle is given on the drawing-paper. He also wrote Vorlesungen uber synthetische Geometric, published posthumously at Leipzig by C. F. Geiser and H. Schroeter in 1867; a third edition by R. Sturm was published in 1887-1898.

The rest of Steiner's writings are found in numerous papers mostly published in Crelle's Journal, the first volume of which contains his first four papers. The most important are those relating to algebraical curves and surfaces, especially the short paper Allgemeine Eigenschaften algebraischer Curven. This contains only results, and there is no indication of the method by which they were obtained, so that, according to L. O. Hesse, "they are, like P. Fermat's theorems, riddles to the present and future generations." Eminent analysts succeeded in proving some of the theorems, but it was reserved to L. Cremona to prove them all, and that by a xxv. 28 a uniform synthetic method, in his book on algebraical curves. Other important investigations relate to maxima and minima. Starting from simple elementary propositions, Steiner advances to the solution of problems which analytically require the calculus of variation, but which at the time altogether surpassed the powers of that calculus. Connected with this is the paper Vom Kriimmungsschwerpuncte ebener Curven, which contains numerous properties of pedals and roulettes, especially of their areas.

Steiner's papers were collected and published in two volumes (Gesammelte Werke, 1881-1882) by the Berlin Academy.

See C. F. Geiser's pamphlet Zur Erinnerung an Steiner (Zurich, 1874).