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Karl Feuerbach's father Paul J A Ritter von Feuerbach who was a professor of law and wrote the Bavarian criminal code. Of his 8 children 5 sons were to be awarded a doctorate, 3 of them becoming a professor, the most famous being the philosopher (without ever being professor) Ludwig A Feuerbach (1804-72) who was one of the very influential critics of religion and thus of great importance for Marx and marxism. Karl was a brilliant student. By the age of 22 he had been awarded his doctorate, been appointed to a professorship at the Gymnasium at Erlangen and had published an extremely important mathematics paper.
His life, however, did not go well. His career as a teacher only lasted six years and even these were years of great difficulty due to ill health. In 1828 Feuerbach retired from teaching, unable to cope any longer with teaching given his state of health. He only lived for a further six years and these he spent in Erlangen living as a recluse.
Feuerbach was 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. 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.
Feuerbach did publish a further work in 1827. This is a second major work and it has been studied carefully by Moritz Cantor. In this work, Moritz Cantor has discovered, 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
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