Title: カール・フォン・シュタウト (Karl Georg Christian von Staudt)
Section: Throw

1857年、『Beiträge zur Geometrie der Lage』の2巻において、シュタウトはthrows（ドイツ語: 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"と呼ばれる点を得ている。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。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:                    (        C        A        ,        B        D        )        +        (        C        A        ,        B                  D          ′                )        =        (        C        A        ,        B        S        )        .                       {\displaystyle (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                    (        C        A        ,        B        D        )        ⋅        (        C        A        ,        D                  D          ′                )        =        (        C        A        ,        B                  D          ′                )        .              {\displaystyle (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.ヴェブレンとヤングの定理10には次のような要約がある。The set of points on a line, with                               P                      ∞                                {\displaystyle P_{\infty }}   removed, forms a field with respect to the operations previously defined. 更には次のような記述もある。...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.throwsはジョン・スティルウェル（2005）によって"projective arithmetic"と表現されている。また、次のセクション"Projective arithmetic"には以下の記述がある。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.シュタウトの実数の構成法に関する功績は不完全であった。その一つの問題は、有界な数列が密集点を持たなければならないという点である。 ハンス・フロイデンタールは次のように言及した。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.実射影平面の輪環の順を研究した数学者の一人に、イタリアの数学者ジョバンニ・ヴァイラーティがいる。この順の科学には、分離関係と呼ばれる四元関係が要求される。この関係を用いて、単調数列と極限の概念が循環的な"line"で処理できる。すべての単調数列は極限値を持ち、"line"は完備空間になる。これらの発展は、射影幾何学の                              R                      {\displaystyle \mathbb {R} }  の公理の性質を取り出す動きとして、シュタウトの可換体の公理の演繹は触発された。