Tuesday, March 18, 2014

Jakob Steiner and Analytical Geometry

Jacob Steiner
(1796 – 1863)
On March 18, 1796, Swiss mathematician Jakob Steiner was born. Steiner's work was mainly confined to geometry. Moreover, he has been considered the greatest pure geometer since Apollonius of Perga.

Steiner became acquainted with several influential scientists, who supported his career like A. L. Crelle or N. H. Abel. In 1832, he published his Systematische Entwickelungen and earned himself a great reputation. Through the influence of Carl Gustav Jacob Jacobi and of the brothers Alexander and Wilhelm von Humboldt a new chair of geometry was founded for him at Berlin two years later. He held this position for the rest of his life.

Steiner devoted most of his mathematical work to geometry and tried to avoid analysis as much as he could, since he hated it. His research became known for its great generality, the fertility of his resources, and for the rigour in his proofs. Jakob Steiner was soon widely considered as the greatest pure geometer since Apollonius of Perga. In his work "Systematische Entwickelung der Abhängigkeit geometrischer Gestalten von einander", he laid the foundation of modern synthetic geometry and introduced what are now called the geometrical forms. Between their elements, Steiner managed to establish a one-to-one correspondence, or, made them projective, as he referred to it. Furtherly, he gave 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 the organic connection of their innumerable properties and mysteries. Also, 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 can be seen.

Another important work by Steiner was "Die geometrischen Constructionen ausgeführt mittels der geraden Linie und eines festen Kreises", which was published in 1833. The work was influenced by J V. Poncelet and in it, Steiner showed 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. A posthumously published work was his "Vorlesungen über synthetische Geometrie". However, next to his scientific books, Steiner authored numerous papers, which were later published in Crelle's Journal. Especially the one's concerning algebraic curves and surfaces were the most influential to later mathematicians. Futher very important research, published by the Swiss mathematician was his work on maxima and minima, which surpassed contemporary scientists significantly.

At yovisto you may be interested in a video lecture on "Steiner's regions of space problem" by n. J. Wildberger.

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