Jakob Steiner
Encyclopedia
Jakob Steiner was a Swiss
mathematician
who worked primarily in geometry
.
, Canton of 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 tutoring. 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 Entwickelungen he received, through C. G. J. Jacobi, who was then professor at Königsberg University, and earned an honorary degree there; and 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
(1834). This he occupied until his death in Bern on 1 April 1863.
He was described by Thomas Hirst as follows:
. 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. 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.
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 curve
s 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 pedal
s and roulette
s, 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 system
s, a basic kind of block designs.
Switzerland
Switzerland name of one of the Swiss cantons. ; ; ; or ), in its full name the Swiss Confederation , is a federal republic consisting of 26 cantons, with Bern as the seat of the federal authorities. The country is situated in Western Europe,Or Central Europe depending on the definition....
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
who worked primarily in geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
.
Personal and professional life
Steiner was born in the village of UtzenstorfUtzenstorf
Utzenstorf is a municipality in the administrative district of Emmental in the canton of Bern in Switzerland. It is regionally famous for its medieval castle, Schloss Landschut.-Geography:...
, Canton of Bern. At eighteen he became a pupil of Heinrich Pestalozzi, and afterwards studied at Heidelberg
Heidelberg
-Early history:Between 600,000 and 200,000 years ago, "Heidelberg Man" died at nearby Mauer. His jaw bone was discovered in 1907; with scientific dating, his remains were determined to be the earliest evidence of human life in Europe. In the 5th century BC, a Celtic fortress of refuge and place of...
. Thence 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 N. H. Abel, then also staying at Berlin, founded his famous Journal
Crelle's Journal
Crelle's Journal, or just Crelle, is the common name for a mathematics journal, the Journal für die reine und angewandte Mathematik .- History :...
(1826).
After Steiner's publication (1832) of his Systematische Entwickelungen he received, through C. G. J. Jacobi, who was then professor at Königsberg University, and earned an honorary degree there; and through the influence of Carl Gustav Jacob Jacobi and of the brothers Alexander and Wilhelm von Humboldt
Wilhelm von Humboldt
Friedrich Wilhelm Christian Karl Ferdinand Freiherr von Humboldt was a German philosopher, government functionary, diplomat, and founder of Humboldt Universität. He is especially remembered as a linguist who made important contributions to the philosophy of language and to the theory and practice...
a new chair of geometry was founded for him at Berlin
Berlin
Berlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...
(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 geometryGeometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
. 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
Synthetic geometry
Synthetic or axiomatic geometry is the branch of geometry which makes use of axioms, theorems and logical arguments to draw conclusions, as opposed to analytic and algebraic geometries which use analysis and algebra to perform geometric computations and solve problems.-Logical synthesis:The process...
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
Rigour
Rigour or rigor has a number of meanings in relation to intellectual life and discourse. These are separate from public and political applications with their suggestion of laws enforced to the letter, or political absolutism...
in his proofs. He has been considered the greatest pure geometer since Apollonius of Perga
Apollonius of Perga
Apollonius of Perga [Pergaeus] was a Greek geometer and astronomer noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and René Descartes...
.
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
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
. He next gives by aid of these projective rows and pencils a new generation of conics
Conic section
In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...
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
Duality (projective geometry)
A striking feature of projective planes is the "symmetry" of the roles played by points and lines in the definitions and theorems, and duality is the formalization of this metamathematical concept. There are two approaches to the subject of duality, one through language and the other a more...
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
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
is given on the drawing-paper. He also wrote "Vorlesungen über synthetische Geometrie", published posthumously at Leipzig
Leipzig
Leipzig Leipzig has always been a trade city, situated during the time of the Holy Roman Empire at the intersection of the Via Regia and Via Imperii, two important trade routes. At one time, Leipzig was one of the major European centres of learning and culture in fields such as music and publishing...
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 published in Crelle's Journal
Crelle's Journal
Crelle's Journal, or just Crelle, is the common name for a mathematics journal, the Journal für die reine und angewandte Mathematik .- History :...
, the first volume of which contains his first four papers. The most important are those relating to algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
s 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
Pierre de Fermat
Pierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus, including his adequality...
'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
Luigi Cremona
Luigi Cremona was an Italian mathematician. His life was devoted to the study of geometry and reforming advanced mathematical teaching in Italy. His reputation mainly rests on his Introduzione ad una teoria geometrica delle curve piane...
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
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
, 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 pedal
Pedal curve
In the differential geometry of curves, a pedal curve is a curve derived by construction from a given curve ....
s and roulette
Roulette (curve)
In the differential geometry of curves, a roulette is a kind of curve, generalizing cycloids, epicycloids, hypocycloids, trochoids, and involutes....
s, especially of their areas.
Steiner also made a small but important contribution to combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
. In 1853, Steiner published a two pages article in Crelle's Journal
Crelle's Journal
Crelle's Journal, or just Crelle, is the common name for a mathematics journal, the Journal für die reine und angewandte Mathematik .- History :...
on what nowadays is called Steiner system
Steiner system
250px|right|thumbnail|The [[Fano plane]] is an S Steiner triple system. The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line....
s, a basic kind of block designs.
See also
- Steiner triple system
- Steiner surfaceSteiner surfaceIn geometry, a branch of mathematics, the Steiner surfaces, discovered by Jakob Steiner, are mappings of the real projective plane into three-dimensional real projective space...
- Steiner's problemSteiner's problemSteiner's problem is the problem of finding the maximum of the functionIt is named after Jakob Steiner.The maximum is at x=e, where e denotes the base of natural logarithms...
- Steiner treeSteiner treeThe Steiner tree problem, or the minimum Steiner tree problem, named after Jakob Steiner, is a problem in combinatorial optimization, which may be formulated in a number of settings, with the common part being that it is required to find the shortest interconnect for a given set of objects.The...
- Steiner chainSteiner chainIn geometry, a Steiner chain is a set of n circles, all of which are tangent to two given non-intersecting circles , where n is finite and each circle in the chain is tangent to the previous and next circles in the chain...
- Poncelet–Steiner theorem
- Parallel axes rule
- Steiner–Lehmus theorem
- Steiner inellipseSteiner inellipseIn geometry, the Steiner inellipse of a triangle is the unique ellipse inscribed in the triangle and tangent to the sides at their midpoints. It is an example of an inconic. By comparison the inscribed circle of a triangle is another inconic that is tangent to the sides, but not necessarily at the...