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Jakob Steiner
Jakob Steiner (18 March 1796 – 1 April 1863) was a Swiss mathematician who worked primarily in geometry.
Steiner was born in the village of Utzenstorf, Canton of Bern. At 18, he became a pupil of Heinrich Pestalozzi and afterwards studied at Heidelberg. Then, he went to Berlin, earning a livelihood there, as in Heidelberg, by tutoring. Here he became acquainted with A. L. Crelle, who, encouraged by his ability and by that of Niels Henrik Abel, then also staying at Berlin, founded his famous Journal (1826).
After Steiner's publication (1832) of his Systematische Entwickelungen he received, through Carl Gustav Jacob Jacobi, who was then professor at Königsberg University, and earned an honorary degree there; and through the influence of 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 until his death in Bern on 1 April 1863.
He was described by Thomas Hirst as follows:
Steiner's mathematical work was mainly 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 synthetic geometry if equal or higher results were obtained by analytical geometry 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 the rigour in his proofs. He has been considered the greatest pure geometer since Apollonius of Perga.
In his Systematische Entwickelung der Abhängigkeit geometrischer Gestalten von einander he laid the foundation of modern synthetic geometry. In projective geometry even parallel lines have a point in common: a point at infinity. Thus two points determine a line and two lines determine a point. The symmetry of point and line is expressed as projective duality. Starting with perspectivities, the transformations of projective geometry are formed by composition, producing projectivities. Steiner identified sets preserved by projectivities such as a projective range and pencils. He is particularly remembered for his approach to a conic section by way of projectivity called the Steiner conic.
In a second little volume, Die geometrischen Constructionen ausgeführt mittels der geraden Linie und 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 über synthetische Geometrie", published posthumously at Leipzig by C. F. Geiser and H. Schroeter in 1867; a third edition by R. Sturm was published in 1887–1898.
Other geometric results by Steiner include development of a formula for the partitioning of space by planes (the maximal number of parts created by n planes), several theorems about the famous Steiner's chain of tangential circles, and a proof of the isoperimetric theorem (later a flaw was found in the proof, but was corrected by Weierstrass).
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Jakob Steiner
Jakob Steiner (18 March 1796 – 1 April 1863) was a Swiss mathematician who worked primarily in geometry.
Steiner was born in the village of Utzenstorf, Canton of Bern. At 18, he became a pupil of Heinrich Pestalozzi and afterwards studied at Heidelberg. Then, he went to Berlin, earning a livelihood there, as in Heidelberg, by tutoring. Here he became acquainted with A. L. Crelle, who, encouraged by his ability and by that of Niels Henrik Abel, then also staying at Berlin, founded his famous Journal (1826).
After Steiner's publication (1832) of his Systematische Entwickelungen he received, through Carl Gustav Jacob Jacobi, who was then professor at Königsberg University, and earned an honorary degree there; and through the influence of 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 until his death in Bern on 1 April 1863.
He was described by Thomas Hirst as follows:
Steiner's mathematical work was mainly 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 synthetic geometry if equal or higher results were obtained by analytical geometry 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 the rigour in his proofs. He has been considered the greatest pure geometer since Apollonius of Perga.
In his Systematische Entwickelung der Abhängigkeit geometrischer Gestalten von einander he laid the foundation of modern synthetic geometry. In projective geometry even parallel lines have a point in common: a point at infinity. Thus two points determine a line and two lines determine a point. The symmetry of point and line is expressed as projective duality. Starting with perspectivities, the transformations of projective geometry are formed by composition, producing projectivities. Steiner identified sets preserved by projectivities such as a projective range and pencils. He is particularly remembered for his approach to a conic section by way of projectivity called the Steiner conic.
In a second little volume, Die geometrischen Constructionen ausgeführt mittels der geraden Linie und 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 über synthetische Geometrie", published posthumously at Leipzig by C. F. Geiser and H. Schroeter in 1867; a third edition by R. Sturm was published in 1887–1898.
Other geometric results by Steiner include development of a formula for the partitioning of space by planes (the maximal number of parts created by n planes), several theorems about the famous Steiner's chain of tangential circles, and a proof of the isoperimetric theorem (later a flaw was found in the proof, but was corrected by Weierstrass).