Karl Georg Christian von Staudt

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Anne Nies M340 Paper #3 29 Nov 2011 Karl Georg Christian von Staudt Although he was not initially a mathematician, Karl Georg Christian von Staudt’s principal contribution was to geometry, in particular he was working to remove the dependence of geometry upon metrics. His work, primarily done in the middle of the 19th century, was the first since classical times to return to more abstract ideas about how to approach geometry, providing the world with “the first purely synthetic theory of imaginary points, lines and planes in projective geometry [2].” This was a product of his “idea of developing projective geometry without the use of numbers [4].” His most influential works, the books Geometrie der Lage (1847; “The Geometry of Position 1 ”) and Bieträge zur Geometrie der Lage (1856-60; “Contributions to the Geometry of Position”), provided support for the work of Hilbert and Klein. Although influential, Geometrie der Lage, was not widely read as it was difficult to understand; primarily because there are no illustrations, applications, sketches or references to other work, nor is a foundation laid down on axioms; and many new technical terms are not used, not always with explanation [3]. When he was twenty-one (in 1819) von Staudt graduated from the University of Gottingen in Germany. He then began studying astronomy under Gauss. Three years later he received his doctorate degree in astronomy from the University of Erlangen, after publishing some astronomical articles. He then taught and gave lectures for thirteen years, until in 1835 he was appointed professor of mathematics at the University of Erlangen. Twelve years later, at age 49 he published Geometrie der Lage. Von Staudt spent his life in central Germany. The University of Gottingen is only about 300 kilometers north of his birth place, Rothenburg ob der Tauber, and the University of Erlagen is only about 75 kilometers east. Nies 1 1 In older literature, projective geometry is sometimes called "higher geometry," "geometry of position," or "descriptive geometry" (Cremona 1960, pp. v-vi) [4].

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Paper about Karl Georg Christian von Staudt's contributions to geometry.

Transcript of Karl Georg Christian von Staudt

Page 1: Karl Georg Christian von Staudt

Anne NiesM340

Paper #329 Nov 2011

Karl Georg Christian von Staudt

Although he was not initially a mathematician, Karl Georg Christian von Staudt’s

principal contribution was to geometry, in particular he was working to remove the

dependence of geometry upon metrics. His work, primarily done in the middle of the

19th century, was the first since classical times to return to more abstract ideas about

how to approach geometry, providing the world with “the first purely synthetic theory of

imaginary points, lines and planes in projective geometry [2].” This was a product of his

“idea of developing projective geometry without the use of numbers [4].” His most

influential works, the books Geometrie der Lage (1847; “The Geometry of Position1”)

and Bieträge zur Geometrie der Lage (1856-60; “Contributions to the Geometry of

Position”), provided support for the work of Hilbert and Klein. Although influential,

Geometrie der Lage, was not widely read as it was difficult to understand; primarily

because there are no illustrations, applications, sketches or references to other work,

nor is a foundation laid down on axioms; and many new technical terms are not used,

not always with explanation [3].

When he was twenty-one (in 1819) von Staudt graduated from the University of

Gottingen in Germany. He then began studying astronomy under Gauss. Three years

later he received his doctorate degree in astronomy from the University of Erlangen,

after publishing some astronomical articles. He then taught and gave lectures for

thirteen years, until in 1835 he was appointed professor of mathematics at the

University of Erlangen. Twelve years later, at age 49 he published Geometrie der Lage.

Von Staudt spent his life in central Germany. The University of Gottingen is only about

300 kilometers north of his birth place, Rothenburg ob der Tauber, and the University of

Erlagen is only about 75 kilometers east.

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1 In older literature, projective geometry is sometimes called "higher geometry," "geometry of position," or "descriptive geometry" (Cremona 1960, pp. v-vi) [4].

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Although he wasn’t adventurous when it came to where he lived, von Staudt was

adventurous with his mathematics. “Von Stuadt's notions usually drew upon preceding

ones in interesting and often novel ways. He was 'the most original and profound of the

projective geometers of the German school his passion was for unity of

method' (Coolidge, 1945, "A history of the conic sections and quadric surfaces" Oxford

61)” [3].

“In geometry, which after a long sleep since antiquity had just been flourishing

again for half a century, von Staudt’s approach was a striking new example of that old

strategy, an example that was bound more and more to determine the spirit of geometry

in the 19th century [1].” According to Reich he is a founder, along with Poncelet,

Gergonne, and Steiner, of projective and synthetic geometry. He was first in developing

a theories of projective geometry that were completely non-metrical. Like Poncelet and

Steiner von Staudt was fond of the ‘law of reciprocity’ (the ‘principal of duality’) and

presented many theorems and exercises in two parallel columns where the words

‘point’ and ‘plane’ and the words ‘connection’ and ‘intersection’ were interchangeable to

make new theorems.Like Euclid, von Staudt did not sufficiently axiomatize his work, and

“around 1897 Mario Pieri axiomatized von Staudt's geometry [3],” which has led to it

being less accessible to readers than the work of other mathematicians of his time.

Some of the key ideas that von Stuadt worked with were: that the locus of all

vanishing points in a plane was a vanishing straight line, that all vanishing points and

lines were located on an infinitely distant plane, Euclid’s parallel axiom, ‘harmonic

configurations’ of collinear points, colllineations in spatial systems, and the

correspondences between elements of configurations. “He defined a collineation as a

point-point relationship, transforming a straight line into a straight line while preserving

harmonic configurations [3].” He did this by inverting previous definitions in an attempt

to “avoid any appeal to metric relations [1].” An example of this is given by Freudenthal

when he explains von Staudt’s redefinition of harmonicity of a point quadruple (which,

since antiquity, was defined as the cross ratio found in the figure of the complete

quadrilateral) started from this incidence figure to define by it what von Staudt called a

harmonic quadruple, he then showed that the fourth member of a harmonic quadruple is

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determined by three of them, as it follows from weakened versions of Desargus’

theorem or the quadrilateral theorem [1].

Although there is little information to be found on von Staudt and although his

work is not well known, the importance of his theories and definitions should be

recognized, as they helped other mathematicians to forward the areas of projective and

synthetic geometry. Von Staudt's work influenced David Hilbert, who played in

influential role in the axiomatization of geometry, in the early 1890's. Von Staudt’s work

also influenced geometry through his use of collineation, which gave a possible new

basis for projective geometry. “The figure upon whom von Staudt's influence was most

marked was Theodor Reye, who's book (much easier to read than von Staudt's)

became so well known that it had five editions, up to 1923. In his foreword Reye

emphasized the high quality, importance, and elegance of von Staudt's contributions to

synthetic geometry [3].”

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Bibliography

[1] Freudenthal, Hans. “The Impact of von Staudt’s Foundations of Geometry.”

Geometry - von Staudt’s Point of View. Ed. Peter Plaumann & Karl Strambach.

Dordrecht, Holland: D. Reidel Publishing Company, 1981. 401-425.

[2] "Karl Georg Christian von Staudt." Encyclopædia Britannica. Encyclopædia

Britannica Online Academic Edition. Encyclopædia Britannica Inc., 2011. Web. 22

Nov. 2011. <http://www.britannica.com.turing.library.northwestern.edu/EBchecked/

topic/564309/Karl-Georg-Christian-von-Staudt>.

[3] Reich, Karin. “Karl Georg Christian von Staudt, Book on Projective Geometry

(1847).” Landmark Writings in Western Mathematics 1640 - 1940. Ed. I. Grattan-

Guinness. Amsterdam: Elsevier, 2005. 441-447.

[4] Stillwell, John. The Four Pillars of Geometry. New York: Springer, 2005. 140.

[5] Weisstein, Eric W. "Projective Geometry." From MathWorld--A Wolfram Web

Resource. http://mathworld.wolfram.com/ProjectiveGeometry.html

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