Nagel attended classes at the Gymnasium in Stuttgart and then after taking the county examinations in 1817 he was accepted into the Evangelical Seminaries of Maulbronn and Blaubeuren in Baden-Württemburg. These two Protestant Gymnasiums had been founded in the middle of the 16th century to provide a broad education to gifted pupils. They were boarding schools with the younger boys going to Maulbronn and then transferring to Blaubeuren. Nagel studied in these Gymnasiums for four years and, although the aim was an education based on theology, he also studied science and mathematics at Blaubeuren. He soon realised that he had a passion for mathematics.
In 1821 he entered the Evangelical Monastery in Tübingen and began his studies of theology at the Eberhard-Karls University of Tübingen. However he also attended lectures on mathematics and physics at the university given by the astronomer Johann Gottlieb Friedrich von Bohnenberger (1765-1831), who was the professor of mathematics and astronomy at the University of Tübingen, and Friedrich Joseph Pythagoras Riecke (1794-1876) who was a lecturer in Tübingen until 1823 when he left to become Professor of Mathematics and Physics in Hohenheim. Nagel had hoped to study under Christoph Friedrich von Pfleiderer, professor of mathematics at Tübingen, but Pfleiderer died in September 1821 just when Nagel was starting his studies.
Nagel completed his studies in 1825 and, having qualified as a clergyman, entered the Church. His first church was in Kirchentellinsfurt, about 7 km east of Tübingen and later he moved to the church in Hengen about 30 km south east of Tübingen. While in Hengen he applied for a teaching position in Oldenburg but he was not successful. In December 1826 he applied for a teaching position at the Lyceum in Tübingen. He was accepted and began teaching both there and at the Realschule in Tübingen in 1827. He taught both mathematics and biology at these schools. Being in Tübingen, he took the opportunity to continue his study of mathematics at the university. Advised by Bohnenberger, he submitted his thesis De triangulis rectangulis ex algebraica aequatione construendis Ⓣ to the Faculty of Arts of the University of Tübingen in October 1827 and was appointed as a privatdocent. At the University he lectured on Euclidean geometry and on mathematical and physical geography.
When Nagel realised that his chances of becoming a professor at Tübingen were very slight, he decided to accept the offer of an appointment as Professor of Mathematics and Natural Science at the Gymnasium in Ulm. He took up this position on 1 November 1830 and at the same time he taught at the Realschule in Ulm. In September 1844 he became Rector of the Real-Schule in Ulm and he continued to hold this position for 25 years. He retired from teaching in 1875. As to Nagel's approach to teaching mathematics we note that he writes in his book Die Idee Der Realschule, Nach Ihrer Theoretischen Begründung Und Praktischen Ausführung Dargestellt Ⓣ (1840) that an education in mathematics is necessary:-
... to expose the ideas laid down in nature: the simple pure utterances of the deity.Nagel published six mathematical articles, the most important of which is Untersuchungen über die Eigenschaften der wichtigsten mit dem Dreieck in Verbindung stehenden Kreise Ⓣ (1835) and Untersuchungen über die wichtigsten zum Dreiecke gehörigen Kreise. Eine Abhandlung aus dem Gebiete der reinen Geometrie Ⓣ (1836). In these papers he studied various points in a triangle which occur as the intersection of concurrent lines. He is most famous for one of these points of intersection that is today called the Nagel point.
There is another way to construct a, b and c since they are the points where the three excircles touch the triangle as shown in the diagram.
Particularly interesting is Nagel's idiosyncratic, purely elementary geometrical method of proof. In his papers he proved the existence of the Nagel point and other points such at the Gergonne point and the Middle point. The Middle point M of a triangle is the point where the three lines, each formed by joining the centre of an excircle to the midpoint of the side of the triangle that it touches, concur. The Gergonne point G is the point where the three lines, each formed by joining the point the incircle touches a side to the opposite vertex, concur. Other interesting properties of the Nagel point is that it, the centroid and the incentre of a triangle are collinear.
Other mathematical publications by Nagel include Theorie der periodischen Decimalbrüche nebst Tabellen zur leichteren Verwandlung gewöhnlicher Brüche in Decimalbrüche Ⓣ (1845).
Nagel made other contributions to Ulm in addition to serving as a teacher and rector there. In particular he acted as an advisor to the city magistrates and it was Nagel who was instrumental in setting up gas lighting in Ulm. He was also responsible for setting up a technical training school to provide commercial training to pupils. Another of his achievements was founding a mathematical club in Ulm.
Article by: J J O'Connor and E F Robertson