**Karl von Staudt** was a German mathematician who worked on projective geometry.

- When Karl was four years old, in 1802, Rothenburg was annexed to Bavaria and this created a time of major upheaval.
- Johann Christian became the legal council for the city but major changes to the education system after the annexation saw the creation of the four-year Latin school in Rothenburg (now the Reichsstadt Gymnasium) which Karl attended.
- However, at the Carolinum-Alexandrinum Gymnasium he was taught mathematics by Karl Heribert Ignatius Buzengeiger (1771-1835) and from this time on von Staudt knew that mathematics was the subject he loved.
- The best mathematician at this time was Carl Friedrich Gauss in Göttingen and Buzengeiger advised von Staudt that he should go there to Göttingen and study under Gauss.
- This was not a particularly easy thing for von Staudt to do since there were difficulties put in the way of anyone from Bavaria who wished to study in a university outside the Kingdom.
- Von Staudt, however, followed the advice of Buzengeiger and eventually overcoming the administrative problems, matriculated at the University of Göttingen, in the Kingdom of Hanover, on 3 May 1819.
- Gauss was employed in Göttingen as the director of the university observatory so, to work closely with him, von Staudt became involved in work on astronomical calculations.
- Through Buzengeiger he had come to know Karl Feuerbach and in 1820 solved one of his problems by calculating the radius of the first "Feuerbach's circle".
- When he said his final farewell to Gauss, his teacher is reported to have said to him "Staudt, you can now learn nothing more from me".
- Von Staudt then tried the Friedrich-Alexander University of Erlangen where it appears that, based on the work he had done for Gauss together with Gauss's recommendation, he received a doctorate in 1822.
- Von Staudt studied the state examinations to teach mathematics and science at a Gymnasium and on 24 October 1822 he began his teaching practice at the Würzburg Gymnasium.
- Fortunately Prince Ludwig of Bavaria wanted to modernise the University of Würzburg so von Staudt's appointment was approved subject to the condition that he also remained as a teacher at the Würzburg Gymnasium.
- The Director of the Melanchthon-Gymnasium in Nürnberg was aware of von Staudt's fine reputation as a teacher so he offered him a position at his school.
- Unhappy with his situation in Würzburg, von Staudt accepted and began his teaching at the Melanchthon-Gymnasium on 25 October 1827.
- Von Staudt, however, did not wish to leave Bavaria so did not accept Bessel's offer.
- Von Staudt was appointed to the chair of mathematics at the Friedrich-Alexander University of Erlangen on 23 August 1835 and took up the position on 1 October.
- In 1825 von Staudt showed how to construct a regular inscribed polygon of 17 sides using only compasses.
- He turned to Bernoulli numbers and in De numeris Bernoullianis Ⓣ(On the Bernoulli numbers) (1845) gave not only a proof of the von Staudt-Clausen theorem but also provided new significant results about properties of the numerators of Bernoulli numbers, given in form of congruences.
- Von Staudt endeavoured to construct a geometry free from all metrical relations, and exclusively based upon relations of situation.
- In the first part of his work, von Staudt entirely omitted imaginary relations.
- By purely projective methods, von Staudt established a complete method for calculating the anharmonic ratios of the most general imaginary elements.
- Even today, a full proof of von Staudt's theorem takes no less than twenty pages, including a number of unspeakably dull lemmas.
- Garrett Birkhoff, in his treatise on lattice theory, a book purporting to deal precisely with this and related topics, gives the statement of von Staudt's theorem, and then gingerly refers the reader to a proof by Emil Artin that was privately distributed in mimeographed form in the thirties at the University of Notre Dame.
- Theodor Reye, whose lectures on von Staudt's approach to projective geometry were first published in 1866, says in his preface that the austere language, the extreme abstractness of presentation, and the lack of diagrams have hindered the well-deserved recognition of von Staudt's work.
- Perhaps Reye's lectures began to reawaken interest in von Staudt.
- But it seems to me that it was probably Felix Klein, with his interest in the foundations of geometry and the so-called non-Euclidean geometries, who focused attention again on von Staudt.
- Klein in 1873 claimed there was a gap in von Staudt's proof of a key result (what we now call the Fundamental Theorem of Projective Geometry), which could only be filled by an axiom of continuity.
- It remained for later generations to appreciate the impact of von Staudt's work on the foundations of projective geometry.
- Von Staudt aimed to found projective geometry independently of any metric assumptions, an approach closely approximating modern axiomatic form.
- The conclusion is that von Staudt was the first to raise the question of foundations looking for purity of method in projective geometry, using his definition of projectivities by the invariance of harmonicity.
- Von Staudt also gave a nice geometric solution to quadratic equations.
- In 1863, von Staudt was elected a corresponding member of the Bavarian Academy of Sciences.

Born 24 January 1798, Imperial Free City of Rothenburg (now Rothenburg ob der Tauber, Germany). Died 1 June 1867, Erlangen, Bavaria (now Germany).

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Astronomy, Origin Germany

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive