Friedrich Ludwig Wachter

Friedrich Ludwig Wachter (born July 20, 1792 in Cleve , † probably April 3, 1817 in Danzig ) was a German mathematician and astronomer , a student of Carl Friedrich Gauß , who wrote contributions to the early history of non-Euclidean geometry .

Life

Wachter was the son of Christian Friedrich Wachter , who worked 1803-1823 as a grammar school director at the Hammonense grammar school in Hamm (Westphalia) . There, because of his recognized giftedness, Wachter completed only two of the five lower grades of the six school years, although his father allowed him to attend classes for four years. In his Abitur certificate from 1809 it says: "All the teachers give the most praiseworthy testimony to the set behavior and moral character of this young man; his diligence in studying was tireless and often had to be withheld for fear that he might harm his physical health. He lived only in the sciences. "

From 1809 Wachter studied mathematics and astronomy at the University of Göttingen under Carl Friedrich Gauß , who, like Johann Franz Encke and other students, included him in the astronomical calculations of the orbit of asteroids (especially Juno ) and on whose influence he published as a student ( Astronomical Yearbooks 1814, 1815). Before submitting his doctoral thesis, he became a grammar school teacher in Altenburg in 1813 on the recommendation of Gauss, who had a high opinion of von Wachter . From November 1813 he did a year of military service in the Wars of Liberation against Napoleon and then submitted his doctoral thesis in astronomy (originally he had planned a topic of differential geometry), which was published in 1815. From 1816 he was a high school professor in Danzig.

Wachter was interested in non-Euclidean geometry and, like Gauss before, discussed a book about attempts to prove the parallel postulate. He visited Gauss in April 1816, and Gauss encouraged him to further study non-Euclidean geometries (for which the parallel postulate, Euclid's 11th postulate, does not apply); Gauss himself had already dealt with these at that time, as he informed Wachter, without, however, ever publishing anything about them (except in letters and hinted at in a book review). But Gauss had already developed a non-Euclidean trigonometry and seems to have asked ( Paul Stäckel ) Wachter to develop it for himself. In 1817 Wachter published the 16-page text Demonstratio axiomatis in Euclideis undecimi in Danzig, which was intended as a preliminary study for a planned book. In it he draws some conclusions from the invalidity of the parallel axiom (similar to Giovanni Girolamo Saccheri in 1733 , whose work he was not familiar with). Nevertheless, he also tries to prove the postulate of parallels, in an attempt which, according to Stäckel, far surpassed all other such attempts in originality. Wachter also reports on the content of this treatise in a letter to Gauß dated February 25, 1817, announces to him that his treatise will be sent and sends the treatise to Friedrich Wilhelm Bessel . In the letter he also expresses his regret that Gauss did not reply to the first letter of December 1816. What Gauss really thought of Wachter's efforts is conveyed in a letter of April 28, 1817 to Heinrich Wilhelm Olbers (at that time he did not yet know about Wachter's disappearance): Although Wachter penetrated the essence of the matter more than his predecessors, his is Evidence no more succinctly than any other.

Wachter had already considered the spherical geometry on a sphere before (in the letter to Gauss in 1816) and understood Euclidean geometry as the limit case of a sphere of infinite radius. According to Stäckel, this idea was not new at the time and had already been published in 1806 (Theses spherologiae, Berlin) by the Cologne high school teacher Grashof (1770–1841).

Wachter was last seen on April 3 (Maundy Thursday) 1817 on his usual evening stroll, during which he was unaccompanied. Despite an intensive search, his body was never found. In 1827 he was pronounced dead. Gauss was deeply shocked by the news of his death. The fact that the body had not been found and no suicide note spoke against a suicide. The authorities looked for him intensively and a substantial reward (200 thalers) for information about him was offered. One still remembered the disappearance of the Austrian officer and mathematician Georg Freiherr von Vega , whose body was pulled from the Danube in 1802, where at first suicide was suspected, but later a robbery and murder could be proven. But there was no evidence of a murder either, and no similar incidents were known in Danzig at the time. Some contemporaries suspected a suicide due to the unstable character of Wachter. In the letter from Wachter's father to Gauss of May 10, 1817, he also quotes in detail from a letter from his son in which he reports his high expectations in his work, which, according to Wachter, only Gauss could understand of contemporary mathematicians and with what he could understand achieved something that mathematicians had tried in vain since the time of Euclid. Kurt Biermann suspects that on April 3rd a letter from Gauß reached Wachter, which made this Gauss known negative judgment and that Wachter committed suicide as a result. In addition, he was blackmailed by a woman with whom he had a casual relationship.

literature

• Waldo Dunnington : Gauss. , American Mathematical Society, 2004 (first 1955), pp. 179, 267-268.
• Paul Stäckel : Friedrich Ludwig Wachter - a contribution to the history of non-Euclidean geometry. In: Mathematical Annals. Vol. 54, 1901, p. 49, online , with a translation of Wachter's demonstration from 1817 and two letters from Wachter to Gauß from December 12, 1816, February 25, 1817, a letter from Wachter to Bessel (March 17, 1817) and a letter from Wachter's father to Gauß dated May 18, 1817.
• Kurt-Reinhard Biermann : I am deeply shaken - new attempt to clear up Wachter's death. In: Communications Gauss Society. No. 35, 1998. bibcode : 1998GGMit..35 ... 41B
• Kurt-Reinhard Biermann: Disappeared and Lost. Friedrich L. Wachter: A criminal case from the history of mathematics. In: Culture and Technology. Issue 2, 1998, p. 26.
• Wilhelm Hinke: Friedrich Ludwig Wachter - an unsolved criminal case. In: Messages from the Friends of the Hammonense Gymnasium in Hamm. Issue 58 (1998), p. 8f.
• Martha Küssner: Johann Friedrich Posselt - Friedrich Ludwig Wachter - Johann Carl Eduard Schmidt - three hopeful Gauss students who died young. In: Communications of the Gaußgesellschaft. No. 17, 1980, p. 48.

References

1. ↑ Quoted in: Walter Siegmund: Das Gymnasium Hammonense from 1657–1957. In: Festschrift for the 300th anniversary of the state high school in Hamm. 1657-1957. Hamm 1957, pp. (9-127) 99; reprinted in: Festschrift for the 325th anniversary of the Hammonense grammar school. 1657-1982. Hamm 1982, there p. (31-156) 123.
2. ^ Göttingen learned advertisements, April 20, 1816, anonymous
3. ↑ Matthias Metternich Complete Theory of Parallels 1815.
4. ↑ Stäckel, Mathematische Annalen 1901. According to Stäckel, Gauss himself was not sure about the nature of non-Euclidean geometry at the time, but increasingly came to believe that the postulate of parallels could not be proven.
5. ↑ Stäckel, loc. Cit. Nor did he know Johann Heinrich Lambert's work on it
6. ↑ printed by Stäckel
7. ^ Letter from Wachter to Gauß, December 12, 1816. Printed in Gauß Collected Works, Vol. 8 and also in Stäckel.
8. ↑ Dunnington, Gauß, p. 179. In a letter to Gerling dated May 15, 1817, Gauß wrote that he was deeply shaken . He also attested that he had a good character , excellent talents , and a pure passion for science . According to Gauss, his metaphysical enthusiasm , which would have led him astray, would have been corrected with increasing age. Gauss suspected an accident (drowning) rather than a murder. There is no question of a suspicion of suicide.
9. ↑ At that time the wind still came three days after the disappearance of the sea, so that a corpse would not have been driven out to sea.
10. ^ Bernhard von Lindemann , an astronomer and friend of Gauss. Quoted from Biermann Disappeared and Lost
11. ↑ Biermann, Disappeared and Lost . Such a letter was not found in Wachter's estate and there is no evidence of it, for example, in the estate of Gauss or his letters. Biermann suspects, however, that Gauss had previously sent Wachter the negative judgment he had communicated to Olbers.
12. ↑ Martha Küssner, quoted from Biermann disappeared and missing