Paul Feuerbach had been sixteen years of age when he left home, against his father's wishes, to study philosophy and history at the University of Jena. He became highly successful as a lawyer and eventually was able to make peace with his father who forgave him, after much pleading, for running away and disobeying him. On 16 November 1802, Paul wrote to his father describing his young son Karl (the subject of this biography) who was approaching the age of 30 months at this time (see , also quoted in ):-
... my youngest son Karl is a healthy, red cheeked, fat youngster, who runs around happily and has little thought for anything except food.The family moved around as Paul Feuerbach's fame took him from one city to another. They were in Jena when Karl was born but soon moved to Kiel. In the winter of 1804 Paul writes in a letter (see ) that he has just taken his wife and three boys from Kiel to Landshut in Bavaria. They made the journey in an open-topped wagon in freezing weather. Later they moved from Landshut to Munich (about 65 km) and, in 1814, from Munich to Bamberg (over 200 km north of Munich). Karl and his elder brother Anselm were attending a Gymnasium in Munich when their father moved to Bamberg and they were left in Munich to complete their schooling. Both boys entered the University of Erlangen in 1817 where they were brilliant students. In fact Paul Feuerbach's fame had been recognised by King Maximilian Joseph who had ennobled him in 1808 and awarded him free university education for all his sons. In 1819 Paul Feuerbach was appointed as President of the Court of Appeals in Ansbach. He wrote a letter in December of that year in which he proudly describes his son Karl's outstanding abilities in mathematics and physics and his achievements at the University of Erlangen. Clearly Paul, coming from a family with a tradition in law, would have liked his sons to follow the 'family profession' but he writes that although Karl has given thought to the legal profession, nevertheless, his intention at the time was to become an engineer in the army.
After spending three years studying at the University of Erlangen, Karl Feuerbach moved to the University of Freiburg im Breisgau in 1820. This was a move designed to allow him to study under Karl Heribert Ignatius Buzengeiger (1771-1835) who had taught in Ansbach before been appointed to the University of Freiburg in the previous year; indeed this move was very fruitful as Feuerbach had hoped. Feuerbach's father, however, although as one might imagine very proud of his son's accomplishments, nevertheless was worried that Karl might find problems in Bavaria since he was not a Bavarian by birth. Moreover, as a Protestant he was likely to suffer discrimination in the predominantly Roman Catholic Bavaria. These fears seemed groundless, however, when by the age of twenty-two, Karl Feuerbach had been awarded his doctorate, been appointed to a professorship in mathematics at the Gymnasium at Erlangen and had published an extremely important mathematical work Eigenschaft einiger merkwürdigen Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten Linien und Figuren Ⓣ.
His life, however, did not go well and it is unclear whether the misfortunes that followed were simply that or whether they were motivated by political enemies of his father. There seems little doubt that as a student Feuerbach had been, as many young men, somewhat rebellious. He had joined an organisation with political aims (again many students do this) but he had also behaved in an irresponsible way incurring debts despite the wealthy family from which he came. One day as he walked to the Gymnasium at Erlangen where he was teaching, he was arrested. Around twenty young men, all members of the same student organisation, had been rounded up and arrested at this time. Feuerbach and the other nineteen were imprisoned in the New Tower in Munich. A description of what happened after this is contained in a letter his father wrote on 1 July 1827  and is recounted in :-
... the weeks and months were allowed to drag on without action and that Karl was not allowed to receive letters or visitors. Karl became obsessed by the idea that only his death could free his 19 companions. He made two separate attempts at suicide. One evening he was found unconscious from loss of blood after having cut the veins in his feet. He was transferred to a hospital and then made a second attempt at suicide by jumping out a window. He was saved from death by landing in a deep snow bank but was permanently crippled as a result of the accident. The father writes bitterly of the fact that he was not more closely guarded, particularly since he had been declared on the verge of a mental breakdown when he was arrested.It was not long after his attempted suicide that Karl Feuerbach was released on condition that he was looked after by Friedrich Wilhelm Thiersch (1784-1860), a classical scholar and educationalist and friend of the Feuerbach family, who had taught Karl when he had attended the Gymnasium in Munich. This release allowed Feuerbach to await trial out of prison and indeed the trial eventually went ahead only to result in the release and total vindication of all concerned (one of the men had died in prison so there were only 19 to be declared 'not guilty'). Feuerbach had been undertaking mathematical work while in prison and once released, all he wanted to do was to be with his brothers, sisters and father in Ansbach while he continued with this research. He was not in a sufficiently strong mental state to resume teaching for about a year and when he did it was at the Gymnasium at Hof. This appointment as professor of mathematics had been arranged by King Maximilian Joseph who made strenuous efforts to help all the young men who had been imprisoned. However, Feuerbach was not happy at Hof, missing the quiet life he had with his family at Ansbach. His mental state deteriorated and two of his brothers went to Hof to help him to go to Erlangen for medical treatment. Part of the reason for choosing Erlangen was the knowledge that Feuerbach had been happy there. He made a steady recovery and soon he appeared to be sufficient well to resume teaching, which he did in 1828 at the Gymnasium at Erlangen. However, he was not really in a good enough mental state as events were to show :-
... one day he appeared in class with a drawn sword and threatened to cut off the head of every student in the class who could not solve the equations he had written on the blackboard. After this episode he was permanently retired. Gradually he withdrew more and more from reality. He allowed his hair, beard, and nails to grow long; he would stare at occasional visitors without any sign of emotion; and his conversation consisted only of low mumbled tones without meaning or expression.He only lived for a further six years and these he spent in Erlangen living as a recluse.
Feuerbach's fame is as a geometer who discovered the nine point circle of a triangle. This is sometimes called the Euler circle but this incorrectly attributes the result. Feuerbach also proved that the nine point circle touches the inscribed and three escribed circles of the triangle. These results appear in his 1822 paper, and it is on the strength of this one paper that Feuerbach's fame is based. He wrote in that paper:-
The circle which passes through the feet of the altitudes of a triangle touches all four of the circles which are tangent to the three sides of the triangle; it is internally tangent to the inscribed circle and externally tangent to each of the circles which touch the sides of the triangle externally.The nine point circle which is described here had also been described in work of Brianchon and Poncelet the year before Feuerbach's paper appeared. However John Sturgeon Mackay notes in  that Feuerbach gave:-
... the first enunciation of that interesting property of the nine point circlenamely that "it is internally tangent to the inscribed circle and externally tangent to each of the circles which touch the sides of the triangle externally." The point where the incircle and the nine point circle touch is now called the Feuerbach point.
You can see a diagram showing the Feuerbach point at THIS LINK.
Feuerbach did undertake further mathematical research. He sent a note from Ansbach to the journal Isis (dated 22 October 1826) entitled Einleitung zu dem Werke Analysis der dreyeckigen Pyramide durch die Methode der Coordinaten und Projectionen. Ein Beytrag zu der analytischen Geometrie von Dr. Karl Wilhelm Feuerbach, Prof. d. Math. Ⓣ. This note announced results which were to appear in full in a later publication and indeed they did in a 48-page booklet Grundriss zu analytischen Untersuchungen der dreyeckigen Pyramide Ⓣ published in 1827. This is a second major work by Feuerbach and it has been studied carefully by Moritz Cantor who discovered that in it Feuerbach introduces homogeneous coordinates. He must therefore be considered as the joint inventor of homogeneous coordinates since Möbius, in his work Der barycentrische Calcul Ⓣ also published in 1827, introduced homogeneous coordinates into analytic geometry.
Article by: J J O'Connor and E F Robertson