# Jakob Steiner

**Jakob Steiner** (18 March 1796 – 1 April 1863) was a Swiss mathematician who worked primarily in geometry.

Jakob Steiner | |
---|---|

Born | |

Died | 1 April 1863 67) | (aged

Citizenship | Swiss |

Known for | Poncelet–Steiner theorem |

Scientific career | |

Fields | Mathematics |

Influences | Fritz Bützberger |

## Life

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:

*"He is a middle-aged man, of pretty stout proportions, has a long intellectual face, with beard and moustache and a fine prominent forehead, hair dark rather inclining to turn grey. The first thing that strikes you on his face is a dash of care and anxiety, almost pain, as if arising from physical suffering—he has rheumatism. He never prepares his lectures beforehand. He thus often stumbles or fails to prove what he wishes at the moment, and at every such failure he is sure to make some characteristic remark."*

## Mathematical contributions

Steiner's mathematical work was mainly confined to geometry. This he treated synthetically, to the total exclusion of analysis, which he hated,[1] 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. He introduces what are now called the geometrical forms (the *row*, *flat pencil*, etc.), and establishes between their elements a one-to-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 connection of their innumerable properties and mysteries. In this work also, of which 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 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).

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 algebraic 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. Hosse, they are, like Fermat's theorems, riddles to the present and future generations. Eminent analysts succeeded in proving some of the theorems, but it was reserved to Luigi Cremona to prove them all, and that by a uniform synthetic method, in his book on algebraic 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 variations, but which at the time altogether surpassed the powers of that calculus. Connected with this is the paper *Vom Krümmungsschwerpuncte ebener Curven*, which contains numerous properties of pedals and roulettes, especially of their areas.

Steiner also made a small but important contribution to combinatorics. In 1853, Steiner published a two pages article in *Crelle's Journal* on what nowadays is called Steiner systems, a basic kind of block design.

His oldest papers and manuscripts (1823-1826) were published by his admirer Fritz Bützberger on the request of the Bernese Society for Natural Scientists.[2]

## See also

## Notes

- "Steiner (print-only)". History.mcs.st-and.ac.uk. Retrieved 2012-09-20.
- O'Connor & Robertson. "Fritz Bützberger".
*MacTutor History of Mathematics*. University of St. Andrews. Retrieved October 14, 2018.

## References

- Viktor Blåsjö (2009) "Jakob Steiner’s Systematische Entwickelung: The Culmination of Classical Geometry", Mathematical Intelligencer 31(1): 21–9.

## External links

- Steiner, J. (1796-1863)
- O'Connor, John J.; Robertson, Edmund F., "Jakob Steiner",
*MacTutor History of Mathematics archive*, University of St Andrews. - Jacob Steiner's work on the Isoperimetric Problem at
*Convergence*(by*Jennifer Wiegert*) *Encyclopædia Britannica*.**25**(11th ed.). 1911.
. *New International Encyclopedia*. 1905.
. Media related to Jakob Steiner at Wikimedia Commons