Karl Georg Christian von Staudt
Encyclopedia
Karl Georg Christian von Staudt (January 24, 1798 – June 1, 1867) was a German
mathematician
born in the Free Imperial City of Rothenburg, which is now called Rothenburg ob der Tauber
in Germany. From 1814 he studied in Gymnasium in Ausbach. He attended the University of Göttingen from 1818 to 1822 where he studied with Gauss
who was director of the observatory. Staudt provided an ephemeris
for the orbits of Mars
and the asteroid
Pallas
. When in 1821 Comet
Nicollet-Pons was observed, he provided the elements of its orbit
. These accomplishments in astronomy
earned him his doctorate from University of Erlangen in 1822.
Staudt's professional career began as a secondary school instructor in Würzburg
until 1827 and then Nuremberg
until 1835. He married Jeanette Dreschler in 1832. They had a son Eduard and daughter Mathilda, but Jeanette died in 1848.
The book Geometrie der Lage (1847) was a landmark in projective geometry
. As Burau (1976) wrote:
Furthermore, this book (page 43) uses the complete quadrangle to "construct the fourth harmonic associated with three points on a straight line", the projective harmonic conjugate.
Hans Freudenthal
(1974) has provided this picture of von Staudt's contribution to the foundations of projective geometry:
Indeed, in 1889 Mario Pieri
translated von Staudt, before writing his I Principii della Geometrie di Posizione Composti in un Systema Logico-deduttivo (1898). In 1900 Charlotte Scott
of Bryn Mawr College
paraphased much of von Staudt's work in English for The Mathematical Gazette.
When Wilhelm Blaschke
published his textbook
Projective Geometry in 1948, a portrait of the young Karl was placed opposite the Vorwort.
Staudt went beyond real projective geometry and into complex projective space
in his three volumes of Beiträge zur Geometrie der Lage published from 1856 to 1860.
The Von Staudt-Clausen theorem is partially named after him.
In 1922 H. F. Baker
wrote of von Staudt's work:
Von Staudt is also remembered for his view of conic section
s and the relation of pole and polar
:
For example, in the development by Veblen & Young (page 141) the result is stated as Theorem 10: "The set of points on a line, with
removed, forms a field
with respect to the operations previously defined". As Freudenthal (1974, page 199) notes
Another affirmation of von Staudt's work with the harmonic conjugates comes in the form of a theorem:
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
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....
born in the Free Imperial City of Rothenburg, which is now called Rothenburg ob der Tauber
Rothenburg ob der Tauber
Rothenburg ob der Tauber is a town in the district of Ansbach of Mittelfranken , the Franconia region of Bavaria, Germany, well known for its well-preserved medieval old town, a destination for tourists from around the world. In the Middle Ages, it was an Imperial Free City...
in Germany. From 1814 he studied in Gymnasium in Ausbach. He attended the University of Göttingen from 1818 to 1822 where he studied with Gauss
Gauss
Gauss may refer to:*Carl Friedrich Gauss, German mathematician and physicist*Gauss , a unit of magnetic flux density or magnetic induction*GAUSS , a software package*Gauss , a crater on the moon...
who was director of the observatory. Staudt provided an ephemeris
Ephemeris
An ephemeris is a table of values that gives the positions of astronomical objects in the sky at a given time or times. Different kinds of ephemerides are used for astronomy and astrology...
for the orbits of Mars
Mars
Mars is the fourth planet from the Sun in the Solar System. The planet is named after the Roman god of war, Mars. It is often described as the "Red Planet", as the iron oxide prevalent on its surface gives it a reddish appearance...
and the asteroid
Asteroid
Asteroids are a class of small Solar System bodies in orbit around the Sun. They have also been called planetoids, especially the larger ones...
Pallas
2 Pallas
Pallas, formally designated 2 Pallas, is the second asteroid to have been discovered , and one of the largest. It is estimated to constitute 7% of the mass of the asteroid belt, and its diameter of 530–565 km is comparable to, or slightly larger than, that of 4 Vesta. It is however 20%...
. When in 1821 Comet
Comet
A comet is an icy small Solar System body that, when close enough to the Sun, displays a visible coma and sometimes also a tail. These phenomena are both due to the effects of solar radiation and the solar wind upon the nucleus of the comet...
Nicollet-Pons was observed, he provided the elements of its orbit
Orbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...
. These accomplishments in astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
earned him his doctorate from University of Erlangen in 1822.
Staudt's professional career began as a secondary school instructor in Würzburg
Würzburg
Würzburg is a city in the region of Franconia which lies in the northern tip of Bavaria, Germany. Located at the Main River, it is the capital of the Regierungsbezirk Lower Franconia. The regional dialect is Franconian....
until 1827 and then Nuremberg
Nuremberg
Nuremberg[p] is a city in the German state of Bavaria, in the administrative region of Middle Franconia. Situated on the Pegnitz river and the Rhine–Main–Danube Canal, it is located about north of Munich and is Franconia's largest city. The population is 505,664...
until 1835. He married Jeanette Dreschler in 1832. They had a son Eduard and daughter Mathilda, but Jeanette died in 1848.
The book Geometrie der Lage (1847) was a landmark in projective geometry
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...
. As Burau (1976) wrote:
- Staudt was the first to adopt a fully rigorous approach. Without exception his predecessors still spoke of distances, perpendiculars, angles and other entities that play no role in projective geometry.
Furthermore, this book (page 43) uses the complete quadrangle to "construct the fourth harmonic associated with three points on a straight line", the projective harmonic conjugate.
Hans Freudenthal
Hans Freudenthal
Hans Freudenthal was a Dutch mathematician. He made substantial contributions to algebraic topology and also took an interest in literature, philosophy, history and mathematics education....
(1974) has provided this picture of von Staudt's contribution to the foundations of projective geometry:
- To be able to consider von Staudt's approach as a rigorous foundation of projective geometry, one need only add explicitly the topological axioms which are tacitly used by von Staudt. ... how can one formulate the topologyTopologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
of projective space without the support of a metric? Von Staudt was still far from raising this question, which a quarter of a century later would become urgent. ... Felix KleinFelix KleinChristian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
noticed the gap in von Staudt's approach; he was aware of the need to formulate the topology of projective space independently of Euclidean space.... the Italians were the first to find truly satisfactory solutions for the problem of a purely projective foundation of projective geometry, which von Staudt had tried to solve.
Indeed, in 1889 Mario Pieri
Mario Pieri
Mario Pieri was an Italian mathematician who is known for his work on foundations of geometry.Pieri was born in Lucca, Italy, the son of Pellegrino Pieri and Ermina Luporini. Pellegrino was a lawyer. Pieri began his higher education at University of Bologna where he drew the attention of Salvatore...
translated von Staudt, before writing his I Principii della Geometrie di Posizione Composti in un Systema Logico-deduttivo (1898). In 1900 Charlotte Scott
Charlotte Scott
Charlotte Angas Scott D.Sc. was a British mathematician who spent the later part of her career in the United States and was influential in the development of American mathematics, including the mathematical education of women.Scott played an important role in Cambridge changing its rules for the...
of Bryn Mawr College
Bryn Mawr College
Bryn Mawr College is a women's liberal arts college located in Bryn Mawr, a community in Lower Merion Township, Pennsylvania, ten miles west of Philadelphia. The name "Bryn Mawr" means "big hill" in Welsh....
paraphased much of von Staudt's work in English for The Mathematical Gazette.
When Wilhelm Blaschke
Wilhelm Blaschke
Wilhelm Johann Eugen Blaschke was an Austro-Hungarian differential and integral geometer.His students included Shiing-Shen Chern, Luis Santaló, and Emanuel Sperner....
published his textbook
Textbook
A textbook or coursebook is a manual of instruction in any branch of study. Textbooks are produced according to the demands of educational institutions...
Projective Geometry in 1948, a portrait of the young Karl was placed opposite the Vorwort.
Staudt went beyond real projective geometry and into complex projective space
Complex projective space
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines...
in his three volumes of Beiträge zur Geometrie der Lage published from 1856 to 1860.
The Von Staudt-Clausen theorem is partially named after him.
In 1922 H. F. Baker
H. F. Baker
Henry Frederick Baker was a British mathematician, working mainly in algebraic geometry, but also remembered for contributions to partial differential equations , and Lie groups....
wrote of von Staudt's work:
- It was von Staudt to whom the elimination of the ideas of distance and congruence was a conscious aim, if, also, the recognition of the importance of this might have been much delayed save for the work of Cayley and Klein upon the projective theory of distance. Generalised, and combined with the subsequent Dissertation of Riemann, v. Staudt's volumes must be held to be the foundation of what, on its geometrical side, the Theory of Relativity, in Physics, may yet become.
Von Staudt is also remembered for his view of conic section
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...
s and the relation of pole and polar
Pole and polar
In geometry, the terms pole and polar are used to describe a point and a line that have a unique reciprocal relationship with respect to a given conic section...
:
- Von Staudt made the important discovery that the relation which a conic establishes between poles and polars is really more fundamental than the conic itself, and can be set up independently. This "polarity" can then be used to define the conic, in a manner that is perfectly symmetrical and immediately self-dual: a conic is simply the locus of points which lie on their polars, or the envelope of lines which pass through their poles. Von Staudt’s treatment of quadricQuadricIn mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial...
s is analogous, in three dimensions.
Algebra of throws
In 1857, in the second Beiträge, von Staudt contributed a route to number through geometry called the Algebra of Throws (or Algebra of Jets; ). It is based on the production of the fourth harmonic point from three given on a line. Though the algebra can be exhibited purely as an "algebra of points", as in chapter 6 of Veblen & Young's textbook on projective geometry, the usual presentation relies on cross ratio (CA,BD) of four collinear points. For instance, Coolidge (1940) wrote:- How do we add two distances together? We give them the same starting point, find the point midway between their terminal points, that is to say, the harmonic conjugate of infinity with regard to their terminal points, and then find the harmonic conjugate of the initial point with regard to this mid-point and infinity. Generalizing this, if we wish to add throws (CA,BD) and (CA,BD' ), we find M the harmonic conjugate of C with regard to D and D' , and then S the harmonic conjugate of A with regard to C and M :
- In the same way we may find a definition of the product of two throws. As the product of two numbers bears the same ratio to one of them as the other bears to unity, the ratio of two numbers is the cross ratio which they as a pair bear to infinity and zero, so Von Staudt, in the previous notation, defines the product of two throws by
- These definitions involve a long series of steps to show that the algebra so defined obeys the usual commutative, associative, and distributive laws, and that there are no divisors of zero.
For example, in the development by Veblen & Young (page 141) the result is stated as Theorem 10: "The set of points on a line, with
removed, forms a fieldField (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
with respect to the operations previously defined". As Freudenthal (1974, page 199) notes
- ...up to Hilbert, there is no other example for such a direct derivation of the algebraic laws from geometric axioms as found in von Staudt's Beiträge.
Another affirmation of von Staudt's work with the harmonic conjugates comes in the form of a theorem:
- The only one-to-one correspondence between the real points on a line which preserves the harmonic relation between four points is a non-singular projectivity.