1832年、Jeanette Dreschler と結婚した。二人の間には息子 Eduard と娘 Mathilda が生まれた。Jeanetteは1848年没した。
書籍 Geometrie der Lage （1847）は射影幾何学の代表的な書籍である。Burau (1976) は次のように書いている。
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.[1]
1889年、マリオ・ピエリはシュタウトのこの書籍を翻訳し I Principii della Geometrie di Posizione Composti in un Systema Logico-deduttivo（1898）を著作した。1900年にはブリンマー大学のシャーロット・スコットが、雑誌 Mathematical Gazette へ、シュタウトの多くの作品を英語に翻訳した[2]。1948年のヴィルヘルム・ブラシュケの教科書 Projective Geometry の Vorwortには、若かりし頃のシュタウトの肖像が飾られている。
シュタウトは、1856年 - 1860年に出版された Beiträge zur Geometrie der Lage の3巻で、実射影幾何学を複素射影空間へ拡張した。
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.[3]
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 quadrics is analogous, in three dimensions.[4]
Throw
1857年、Beiträge zur Geometrie der Lage の2巻において、シュタウトはthrows（独: Wurftheorie）と呼ばれる概念を発明した。これは射影調和共役と射影領域（英語版）に深く関連している。ヴェブレンとヤングの射影幾何学の教科書の6章では、点の乗法と加法を通して、 "Algebra of points" （点の代数）を得ている。throw の概念は、複比とも深く関連する。ジュリアン・クーリッジは次のように書いている[5]。
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.
...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.
シュタウトの調和共役の功績の肯定的な評価には次のようなものがある。
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.[8]
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.
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 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 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.[7]
^ abFreudenthal, Hans. “The Impact of Von Staudt's Foundations of Geometry”. In R.S. Cohen. For Dirk Struik. D. Reidel 及び Peter Plaumann & Karl Strambach, ed (1980). “Geometry – von Staudt's Point of View”. Proceedings of NATO Advanced Study Institute (D. Reidel). ISBN90-277-1283-2.
^Dirk Struik (1953). “theorem of von Staudt”. Lectures on Analytic and Projective Geometry. Addison-Wesley. p. 22
John Wesley Young (1930) Projective Geometry, Chapter 8: Algebra of points and the introduction of analytic methods, Open Court for Mathematical Association of America.
