# 1911 Encyclopædia Britannica/Steiner, Jakob

**STEINER, JAKOB** (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
Entwickehingen 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.

*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, &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 Conslruclionen*
*ausgeführt mittelst der geraden Linie und eines festen Kreises* (1833),
republished in 1895 by Öttingen, 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.

The rest of Steiner's writings are found in numerous papers mostly
blished in *Crelle's Journal*, the first volume of which contains
is 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
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*
Krümmungsschwerpuncte 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 J. Steiner*