HOME

TheInfoList



OR:

Karl Georg Christian von Staudt (24 January 1798 – 1 June 1867) was a German
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
who used
synthetic geometry Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry without the use of coordinates or formulae. It relies on the axiomatic method and the tools directly related to them, that is, compass ...
to provide a foundation for arithmetic.


Life and influence

Karl was born in the Free Imperial City of Rothenburg, which is now called
Rothenburg ob der Tauber Rothenburg ob der Tauber () is a town in the district of Ansbach of Mittelfranken (Middle Franconia), the Franconia region of Bavaria, Germany. It is well known for its well-preserved medieval old town, a destination for tourists from around the ...
in Germany. From 1814 he studied in Gymnasium in Ausbach. He attended the
University of Göttingen The University of Göttingen, officially the Georg August University of Göttingen, (german: Georg-August-Universität Göttingen, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany. Founded ...
from 1818 to 1822 where he studied with
Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
who was director of the observatory. Staudt provided an
ephemeris In astronomy and celestial navigation, an ephemeris (pl. ephemerides; ) is a book with tables that gives the trajectory of naturally occurring astronomical objects as well as artificial satellites in the sky, i.e., the position (and possibly ...
for the orbits of
Mars Mars is the fourth planet from the Sun and the second-smallest planet in the Solar System, only being larger than Mercury. In the English language, Mars is named for the Roman god of war. Mars is a terrestrial planet with a thin at ...
and the
asteroid An asteroid is a minor planet of the inner Solar System. Sizes and shapes of asteroids vary significantly, ranging from 1-meter rocks to a dwarf planet almost 1000 km in diameter; they are rocky, metallic or icy bodies with no atmosphere. ...
Pallas Pallas may refer to: Astronomy * 2 Pallas asteroid ** Pallas family, a group of asteroids that includes 2 Pallas * Pallas (crater), a crater on Earth's moon Mythology * Pallas (Giant), a son of Uranus and Gaia, killed and flayed by Athena * Pa ...
. When in 1821
Comet A comet is an icy, small Solar System body that, when passing close to the Sun, warms and begins to release gases, a process that is called outgassing. This produces a visible atmosphere or coma, and sometimes also a tail. These phenomena ...
Nicollet-Pons was observed, he provided the elements of its
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
. These accomplishments in
astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
earned him his doctorate from
University of Erlangen A university () is an institution of higher (or tertiary) education and research which awards academic degrees in several academic disciplines. Universities typically offer both undergraduate and postgraduate programs. In the United States, th ...
in 1822. Staudt's professional career began as a secondary school instructor in
Würzburg Würzburg (; Main-Franconian: ) is a city in the region of Franconia in the north of the German state of Bavaria. Würzburg is the administrative seat of the ''Regierungsbezirk'' Lower Franconia. It spans the banks of the Main River. Würzburg ...
until 1827 and then
Nuremberg Nuremberg ( ; german: link=no, Nürnberg ; in the local East Franconian dialect: ''Nämberch'' ) is the second-largest city of the German state of Bavaria after its capital Munich, and its 518,370 (2019) inhabitants make it the 14th-largest ...
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 In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, ...
. 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 In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and of the six l ...
to "construct the fourth harmonic associated with three points on a straight line", the
projective harmonic conjugate In projective geometry, the harmonic conjugate point of an ordered triple of points on the real projective line is defined by the following construction: :Given three collinear points , let be a point not lying on their join and let any line ...
. Indeed, in 1889
Mario Pieri Mario Pieri (22 June 1860 – 1 March 1913) was an Italian mathematician who is known for his work on foundations of geometry. Biography Pieri was born in Lucca, Italy, the son of Pellegrino Pieri and Ermina Luporini. Pellegrino was a lawyer. Pie ...
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 Bryn Mawr College ( ; Welsh: ) is a women's liberal arts college in Bryn Mawr, Pennsylvania. Founded as a Quaker institution in 1885, Bryn Mawr is one of the Seven Sister colleges, a group of elite, historically women's colleges in the United ...
paraphrased much of von Staudt's work in English for ''The Mathematical Gazette''. When
Wilhelm Blaschke Wilhelm Johann Eugen Blaschke (13 September 1885 – 17 March 1962) was an Austrian mathematician working in the fields of differential and integral geometry. Education and career Blaschke was the son of mathematician Josef Blaschke, who taugh ...
published his
textbook A textbook is a book containing a comprehensive compilation of content in a branch of study with the intention of explaining it. Textbooks are produced to meet the needs of educators, usually at educational institutions. Schoolbooks are 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 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 ...
in his three volumes of ''Beiträge zur Geometrie der Lage'' published from 1856 to 1860. In 1922 H. F. Baker 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 In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a ...
s and the relation of
pole and polar In geometry, a pole and polar are respectively a point and a line that have a unique reciprocal relationship with respect to a given conic section. Polar reciprocation in a given circle is the transformation of each point in the plane into it ...
: :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
quadric In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is de ...
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 (german: Wurftheorie). It is based on
projective range In mathematics, a projective range is a set of points in projective geometry considered in a unified fashion. A projective range may be a projective line or a conic. A projective range is the dual of a pencil of lines on a given point. For inst ...
and the relation of
projective harmonic conjugate In projective geometry, the harmonic conjugate point of an ordered triple of points on the real projective line is defined by the following construction: :Given three collinear points , let be a point not lying on their join and let any line ...
s. Through operations of addition of points and multiplication of points, one obtains an "algebra of points", as in chapter 6 of Veblen & Young's textbook on projective geometry. The usual presentation relies on
cross ratio In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, t ...
(''CA,BD'') of four collinear points. For instance, Coolidge 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'' : ::(CA,BD) + (CA,BD') = (CA,BS) .\ :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 ::(CA,BD) \cdot (CA,DD') = (CA,BD'). :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. A summary statement is given by Veblen & Young as Theorem 10: "The set of points on a line, with P_\infin removed, forms a field with respect to the operations previously defined". As Freudenthal 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. The algebra of throws was described as "projective arithmetic" in ''The Four Pillars of Geometry'' (2005). In a section called "Projective arithmetic", he says :The real difficulty is that the construction of ''a'' + ''b'' , for example, is different from the construction of ''b'' + ''a'', so it is a "coincidence" if ''a'' + ''b'' = ''b'' + ''a''. Similarly it is a "coincidence" if ''ab'' = ''ba'', of any other law of algebra holds. Fortunately, we can show that the required coincidences actually occur, because they are implied by certain geometric coincidences, namely the Pappus and Desargues theorems. If one interprets von Staudt’s work as a
construction of the real numbers In mathematics, there are several equivalent ways of defining the real numbers. One of them is that they form a complete ordered field that does not contain any smaller complete ordered field. Such a definition does not prove that such a complete ...
, then it is incomplete. One of the required properties is that a bounded sequence has a
cluster point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
. As
Hans Freudenthal Hans Freudenthal (17 September 1905 – 13 October 1990) was a Jewish-German-born Dutch mathematician. He made substantial contributions to algebraic topology and also took an interest in literature, philosophy, history and mathematics education ...
observed: :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
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
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 Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
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.
Hans Freudenthal Hans Freudenthal (17 September 1905 – 13 October 1990) was a Jewish-German-born Dutch mathematician. He made substantial contributions to algebraic topology and also took an interest in literature, philosophy, history and mathematics education ...
(1974) "The Impact of Von Staudt's Foundations of Geometry", in ''For Dirk Struik'', R.S. Cohen editor, D. Reidel. Also found in ''Geometry – von Staudt’s Point of View'', Peter Plaumann & Karl Strambach editors, Proceedings of NATO Advanced Study Institute, Bad Windsheim, July/August 1980, D. Reidel,
One of the Italian mathematicians was Giovanni Vailati who studied the circular order property of the real projective line. The science of this order requires a quaternary relation called the
separation relation In mathematics, a separation relation is a formal way to arrange a set of objects in an unoriented circle. It is defined as a quaternary relation ' satisfying certain axioms, which is interpreted as asserting that ''a'' and ''c'' separate ''b'' fro ...
. Using this relation, the concepts of monotone sequence and limit can be addressed, in a cyclic "line". Assuming that every monotone sequence has a limit,
H. S. M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...
(1949) ''The Real Projective Plane'', Chapter 10: Continuity,
McGraw Hill McGraw Hill is an American educational publishing company and one of the "big three" educational publishers that publishes educational content, software, and services for pre-K through postgraduate education. The company also publishes referen ...
the line becomes a
complete space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
. These developments were inspired by von Staudt’s deductions of field axioms as an initiative in the derivation of properties of ℝ from axioms in projective geometry.


Works

* 1831: ''Über die Kurven, 2. Ordnung''. Nürnberg * 1845: ''De numeris Bernoullianis: commentationem alteram pro loco in facultate philosophica rite obtinendo'', Carol. G. Chr. de Staudt. Erlangae: Junge. * 1845: ''De numeris Bernoullianis: loci in senatu academico rite obtinendi causa commentatus est, Carol. G. Chr. de Staudt. Erlangae: Junge. The following links are to
Cornell University Cornell University is a private statutory land-grant research university based in Ithaca, New York. It is a member of the Ivy League. Founded in 1865 by Ezra Cornell and Andrew Dickson White, Cornell was founded with the intention to tea ...
Historical Mathematical Monographs: * 1847
Geometrie der Lage
Nürnberg. * 1856
Beiträge zur Geometrie der Lage, Erstes Heft
Nürnberg. * 1857
Beiträge zur Geometrie der Lage, Zweites Heft
Nürnberg. * 1860
Beiträge zur Geometrie der Lage, Drittes Heft
Nürnberg.


See also

*
W-curve In geometry, a W-curve is a curve in projective ''n''-space that is invariant under a 1-parameter group of projective transformations. W-curves were first investigated by Felix Klein and Sophus Lie in 1871, who also named them. W-curves in the re ...


References

* * * John Wesley Young (1930) ''Projective Geometry'', Chapter 8: Algebra of points and the introduction of analytic methods,
Open Court Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' (YF ...
for
Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure a ...
. {{DEFAULTSORT:Staudt, Karl Georg Christian von 1798 births 1867 deaths Geometers People from Erlangen 19th-century German mathematicians