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The Steiner Prize
The Steiner prize, endowed by Jacob Steiner, was first awarded in 1864 by the Berlin Academy of Sciences (also known at that time as the Prussian Academy of Sciences).
The following information about the Steiner prize is taken from three sources which we have combined and slightly modified: (i) an extract from the article A History of Prizes in Mathematics by Jeremy Gray which was published in the book J Carlson, A Jaffe and A Wiles (eds.), The Millennium Prize Problems (Clay Mathematics Institute/American Mathematical Society, 2006); (ii) Notes, Bulletin of the American Mathematical Society 17 (1) (1910); and (iii) Notes, Bulletin of the American Mathematical Society 29 (3) (1923).
1. The Steiner Prize.
The early years of the Steiner prize from the University of Berlin illustrates the problems of prize competitions only too well. It was endowed in the will of the distinguished exponent of synthetic geometry, the Swiss mathematician Jacob Steiner, who had taught most of his life at Berlin University and died in 1863. Steiner stipulated that the prize, of 8,000 Thaler, be awarded once every two years for a geometric topic treated synthetically. Let us note that value of the prize was roughly equivalent to one years salary for a university professor. The first time the prize was awarded, 1864, it was divided between Luigi Cremona and Rudolf Sturm for their answers to a question set by Karl Weierstrass concerning cubic surfaces, a currently active topic. In 1868 the prize was shared between Hermann Kortum (1832-1904) and Henry J S Smith for works on cubic and quartic curves in the plane.
The competition ran easily enough as long as Steiner's followers were still alive, but soon successors proved hard to find. In 1870 Carl Borchardt proposed the topic of lines of curvature on surfaces, but no essays were forthcoming, and in 1872 the prize went to Otto Hesse for his work in geometry as a whole, and in 1874 to Luigi Cremona, again in recognition of his work in general. In 1874 the judges called for entries on the theory of polyhedra, but none were forthcoming and the topic was withdrawn in 1876. Instead, the prize was awarded to Heinrich Schröter for his work extending and deepening Steiner's geometrical methods. The judges then announced a prize of 1,800 marks for an essay on higher algebraic space curves, but there were no entries. The closing date was extended to 1880, and the prize was awarded to Theodor Reye 'for his distinguished work on pure geometry'. In 1880 there was still no satisfactory entry. The judges awarded the prize to Leonard Lorenz Lindelöf (1827-1908) (father of Ernst Leonard Lindelöf) for a solution to Steiner's problem about the maximum volume of polyhedra of a given type and further extended the closing date for the essay originally set in 1876. Finally, in 1882, they announced two significant essays that were worthy of sharing the prize: Max Noether's and Henri Halphen's. A third essay received an honourable mention, but the author's name was not revealed (most likely it was Rudolf Sturm, who promptly published on that topic). And so it went on. In 1884 and 1888 Wilhelm Fiedler and Hieronymus Zeuthen were rewarded for their distinguished contributions to geometry. Only in 1886 was the prize awarded, to Ernst Kötter (1859-1922), for an essay on the question proposed in 1882 and modified in 1884, which called for a theory of higher curves and surfaces that invoked really existing objects to replace the imaginary points, lines, and planes of contemporary algebraic geometry.
In 1888 Leopold Kronecker, with the support of Lazarus Fuchs, asked that the terms of reference of the prize be changed. This was difficult to achieve, but in late 1889 it was agreed that from 1890 the competition would be announced once every five years, and in the event that no entry was judged satisfactory the prize money could be allocated to significant work, primarily in the field of geometry, written in the previous ten years - a marked relaxation of the original rules. In this spirit, no entries having been received on the set topic (lines of curvature on surfaces, again) Sigmund Gundelfinger (1846-1910) and Friedrich Schottky shared the prize in 1895. A more detailed description follows.
The Academy of Sciences of Berlin at the Leibniz Meeting, 3 July 1890, announced the following problem for the Steiner Prize:-
For the year 1900 the Academy announced the following problem for the Steiner Prize:-
Other winners were Guido Hauck (1845-1905) and Ferdinand von Lindemann. It was the same story in 1905, when the prize went to Gaston Darboux.
At the public Leibniz session of the Royal Prussian Academy of Sciences held in Berlin on 30 June 1910, the academy proposed the following problem for the solution of which the Steiner prize of 7000 marks was to be awarded in 1914:-
2. Steiner Prize winners.
The following list is taken from an article Le prix Steiner by L G Vidiani which was published in Quadrature 56 (April-June 2005), 30-31.
1866 Rudolf Sturm (Bromberg) and Luigi Cremona (Bologna): for the establishment of properties of third-order surfaces.
1868 Hermann Kortum (Bonn) and Henry John Stephen Smith (Oxford): given thirteen points in the plane, they determined, by means of a geometric construction, the three points which, joined to the given points, form a system of sixteen points of intersection of two plane curves of the fourth degree.
The subject proposed for 1870 (determine and discuss all third-order surfaces, whose lines of curvature are algebraic, and these lines of curvature themselves) was proposed a second time for 1872, then finally withdrawn.
1870 Ludwig Schläfli (Bern): for his two works on third-order surfaces.
The subject for 1874 (convex polyhedron of given exterior surface, containing a maximal volume) was renewed for 1876, then withdrawn.
1874 Luigi Cremona (Rome): for his work in the field of geometry.
1876 Heinrich Schröter (Breslau): for his work in the field of geometry.
The subject for 1878 (theory of higher algebraic curves in space) was renewed for 1880 and 1882.
1878 Theodor Reye (Strasbourg): for his work in the field of geometry.
1880 L Lindelöf (Helsingfors): for his work in the field of geometry.
1882 Max Noether (Erlangen) and Georges-Henri Halphen (Paris).
The subject of the prize for 1884 was renewed for 1886.
1884 Wilhelm Fiedler (Zurich): for his recognized work in geometry.
1886 Ernst Kötter (Berlin).
The subject of the prize for 1888 was renewed for 1890 and then withdrawn.
1888 H G Zeuthen (Copenhagen): for his recognized work in geometry.
In 1890, there were no laureates and it was decided that the prize would now be awarded every five years. In 1895, the academy had not received any papers.
1895 Sigmund Gundelfinger (Darmstadt) and Friedrich Schottky (Marburg): for their recognised work in geometry.
The subject for 1900 was renewed for 1905 and 1910 and then withdrawn. The academy received no papers.
1900 Karl Friedrich Geiser (Zurich) and David Hilbert (Göttingen) and Ferdinand Lindemann (Munich): for their recognised work in geometry.
1905 Guido Hauck (Charlottenburg): for his recognised work in geometry.
1910 Gaston Darboux (perpetual secretary of the Paris Academy of Sciences): for his recognised work in geometry.
Due to the First World War, it was not possible to award the Steiner Prize in 1915 to one of the seven papers received. It was finally awarded in 1922.
The subject for 1920 was renewed for 1925 but, due to inflation, the Academy was no longer in a position to award the prize, which was discontinued.
1922 Eugenio Giuseppe Togliatti: determination of all surfaces of the fifth and sixth order, containing an infinity of conics.
The following information about the Steiner prize is taken from three sources which we have combined and slightly modified: (i) an extract from the article A History of Prizes in Mathematics by Jeremy Gray which was published in the book J Carlson, A Jaffe and A Wiles (eds.), The Millennium Prize Problems (Clay Mathematics Institute/American Mathematical Society, 2006); (ii) Notes, Bulletin of the American Mathematical Society 17 (1) (1910); and (iii) Notes, Bulletin of the American Mathematical Society 29 (3) (1923).
1. The Steiner Prize.
The early years of the Steiner prize from the University of Berlin illustrates the problems of prize competitions only too well. It was endowed in the will of the distinguished exponent of synthetic geometry, the Swiss mathematician Jacob Steiner, who had taught most of his life at Berlin University and died in 1863. Steiner stipulated that the prize, of 8,000 Thaler, be awarded once every two years for a geometric topic treated synthetically. Let us note that value of the prize was roughly equivalent to one years salary for a university professor. The first time the prize was awarded, 1864, it was divided between Luigi Cremona and Rudolf Sturm for their answers to a question set by Karl Weierstrass concerning cubic surfaces, a currently active topic. In 1868 the prize was shared between Hermann Kortum (1832-1904) and Henry J S Smith for works on cubic and quartic curves in the plane.
The competition ran easily enough as long as Steiner's followers were still alive, but soon successors proved hard to find. In 1870 Carl Borchardt proposed the topic of lines of curvature on surfaces, but no essays were forthcoming, and in 1872 the prize went to Otto Hesse for his work in geometry as a whole, and in 1874 to Luigi Cremona, again in recognition of his work in general. In 1874 the judges called for entries on the theory of polyhedra, but none were forthcoming and the topic was withdrawn in 1876. Instead, the prize was awarded to Heinrich Schröter for his work extending and deepening Steiner's geometrical methods. The judges then announced a prize of 1,800 marks for an essay on higher algebraic space curves, but there were no entries. The closing date was extended to 1880, and the prize was awarded to Theodor Reye 'for his distinguished work on pure geometry'. In 1880 there was still no satisfactory entry. The judges awarded the prize to Leonard Lorenz Lindelöf (1827-1908) (father of Ernst Leonard Lindelöf) for a solution to Steiner's problem about the maximum volume of polyhedra of a given type and further extended the closing date for the essay originally set in 1876. Finally, in 1882, they announced two significant essays that were worthy of sharing the prize: Max Noether's and Henri Halphen's. A third essay received an honourable mention, but the author's name was not revealed (most likely it was Rudolf Sturm, who promptly published on that topic). And so it went on. In 1884 and 1888 Wilhelm Fiedler and Hieronymus Zeuthen were rewarded for their distinguished contributions to geometry. Only in 1886 was the prize awarded, to Ernst Kötter (1859-1922), for an essay on the question proposed in 1882 and modified in 1884, which called for a theory of higher curves and surfaces that invoked really existing objects to replace the imaginary points, lines, and planes of contemporary algebraic geometry.
In 1888 Leopold Kronecker, with the support of Lazarus Fuchs, asked that the terms of reference of the prize be changed. This was difficult to achieve, but in late 1889 it was agreed that from 1890 the competition would be announced once every five years, and in the event that no entry was judged satisfactory the prize money could be allocated to significant work, primarily in the field of geometry, written in the previous ten years - a marked relaxation of the original rules. In this spirit, no entries having been received on the set topic (lines of curvature on surfaces, again) Sigmund Gundelfinger (1846-1910) and Friedrich Schottky shared the prize in 1895. A more detailed description follows.
The Academy of Sciences of Berlin at the Leibniz Meeting, 3 July 1890, announced the following problem for the Steiner Prize:-
The solution of an important problem in the theory of lines of curvature of surfaces; preferably, the determination of the conditions under which the lines of curvature of algebraic surfaces are algebraic curves.No paper on this subject was received. In accordance with the terms of the Steiner foundation, the Berlin Academy of Sciences then utilised the prize, thus unawarded, for the purpose of recognising certain important geometrical contributions published during the previous few years. One half of it was awarded to Dr Sigmund Gundelfinger, professor at the polytechnic school at Darmstadt, for his remarkable work leading to the fundamental investigation and development of the methods introduced into geometry by Otto Hesse; the other half was awarded to Dr Friedrich Schottky, professor at the University of Marburg, who had afforded valuable assistance in a series of most important special problems of geometry by demonstrating their relations to the theory of abelian functions of two, three, and four variables.
For the year 1900 the Academy announced the following problem for the Steiner Prize:-
To completely solve any important, hitherto unsolved problem relating to the theory of curved surfaces, taking into account so far as possible the methods and principles evolved by Steiner. It is required that sufficient analytical explanations shall accompany the geometrical investigations to verify the correctness and completeness of the solution. Without wishing to limit the choice of subject the Academy takes the opportunity to call attention to the special problems to which Steiner has referred in his general remark at the end of his second paper on maximum and minimum in figures in a plane, on a sphere, and in space. For the solution of the problem a prize of 4000 marks is offered, with an additional sum of 2000 marks. Papers offered in competition may be written in German, French, English, Italian, or Latin, and must be submitted before 31 December 1899. The result will be announced at the Leibniz Meeting of 1900.In 1900 David Hilbert was awarded a one-third part of the Steiner prize for his work on the foundations of geometry (Grundlagen der Geometrie, 1899).
Other winners were Guido Hauck (1845-1905) and Ferdinand von Lindemann. It was the same story in 1905, when the prize went to Gaston Darboux.
At the public Leibniz session of the Royal Prussian Academy of Sciences held in Berlin on 30 June 1910, the academy proposed the following problem for the solution of which the Steiner prize of 7000 marks was to be awarded in 1914:-
To determine all those non-degenerate surfaces of order five on which one or more series of conics lie, and to investigate their properties. It is required to confirm the correctness and completeness of the solution by furnishing an analytic commentary on the results of the geometric investigation.Eugenio Togliatti submitted his entry The determination of all non-degenerate surfaces of the fifth order on which lie one or more series of conics and was deemed to have won the Steiner Prize. The outbreak of World War I in July 1914 led to Germany entering the war on 1 August 1914 and the award of the Steiner Prize to Togliatti had to be delayed. In fact it was not until 1923 that the award could be made:-
... competing memoirs should be written in German, French, Latin, English, or Italian, and be submitted to the secretary of the academy under the usual conditions on or before 31 December 1913.
The Berlin Academy of Sciences has awarded its Steiner prize for a memoir on 'The determination of all non-degenerate surfaces of the fifth order on which lie one or more series of conics' to Dr E G Togliatti, of the University of Turin. This subject was announced in 1910, but the award was delayed by war conditions.German hyperinflation during the 1920s had reduced the value of the prize, making it essentially worthless by the time it was awarded to Togliatti. Because of this hyperinflation, it was decided to discontinue awarded the Steiner prize, so Togliatti became the last to receive this honour.
2. Steiner Prize winners.
The following list is taken from an article Le prix Steiner by L G Vidiani which was published in Quadrature 56 (April-June 2005), 30-31.
1866 Rudolf Sturm (Bromberg) and Luigi Cremona (Bologna): for the establishment of properties of third-order surfaces.
1868 Hermann Kortum (Bonn) and Henry John Stephen Smith (Oxford): given thirteen points in the plane, they determined, by means of a geometric construction, the three points which, joined to the given points, form a system of sixteen points of intersection of two plane curves of the fourth degree.
The subject proposed for 1870 (determine and discuss all third-order surfaces, whose lines of curvature are algebraic, and these lines of curvature themselves) was proposed a second time for 1872, then finally withdrawn.
1870 Ludwig Schläfli (Bern): for his two works on third-order surfaces.
The subject for 1874 (convex polyhedron of given exterior surface, containing a maximal volume) was renewed for 1876, then withdrawn.
1874 Luigi Cremona (Rome): for his work in the field of geometry.
1876 Heinrich Schröter (Breslau): for his work in the field of geometry.
The subject for 1878 (theory of higher algebraic curves in space) was renewed for 1880 and 1882.
1878 Theodor Reye (Strasbourg): for his work in the field of geometry.
1880 L Lindelöf (Helsingfors): for his work in the field of geometry.
1882 Max Noether (Erlangen) and Georges-Henri Halphen (Paris).
The subject of the prize for 1884 was renewed for 1886.
1884 Wilhelm Fiedler (Zurich): for his recognized work in geometry.
1886 Ernst Kötter (Berlin).
The subject of the prize for 1888 was renewed for 1890 and then withdrawn.
1888 H G Zeuthen (Copenhagen): for his recognized work in geometry.
In 1890, there were no laureates and it was decided that the prize would now be awarded every five years. In 1895, the academy had not received any papers.
1895 Sigmund Gundelfinger (Darmstadt) and Friedrich Schottky (Marburg): for their recognised work in geometry.
The subject for 1900 was renewed for 1905 and 1910 and then withdrawn. The academy received no papers.
1900 Karl Friedrich Geiser (Zurich) and David Hilbert (Göttingen) and Ferdinand Lindemann (Munich): for their recognised work in geometry.
1905 Guido Hauck (Charlottenburg): for his recognised work in geometry.
1910 Gaston Darboux (perpetual secretary of the Paris Academy of Sciences): for his recognised work in geometry.
Due to the First World War, it was not possible to award the Steiner Prize in 1915 to one of the seven papers received. It was finally awarded in 1922.
The subject for 1920 was renewed for 1925 but, due to inflation, the Academy was no longer in a position to award the prize, which was discontinued.
1922 Eugenio Giuseppe Togliatti: determination of all surfaces of the fifth and sixth order, containing an infinity of conics.