By lucky chance the schoolmaster [Büttner] had an assistant, Johann Martin Bartels, a young man with a passion for mathematics, whose duty it was to help the beginners in writing and cut their quill pens for them. Between the assistant of seventeen and the pupil of ten [Gauss] there sprang up a warm friendship which lasted out Bartels' life. They studied together, helping one another over difficulties and amplifying proofs in their common textbook on algebra and the rudiments of analysis. Out of this early work developed one of the dominating interests of Gauss's career [algebra]. ... Bartels did more for Gauss than to induct him into the mysteries of algebra. The young teacher was acquainted with some of the influential men of Brunswick. He now made it his business to interest these men in his find [Gauss].In particular Bartels informed Eberhard August Wilhelm Zimmermann (1743-1815) who had been professor of mathematics, physics and natural history at the Collegium Carolinum in Brunswick since 1766. This was extremely valuable for the young Gauss, but this remarkable meeting of minds between Gauss and his young teacher Bartels led to Bartels becoming determined to pursue his study of mathematics. We should note, however, that in  Dick puts forward a view different from the accepted one which we have given above. He suggests that Bartels had higher tasks to do than to teach calculation at the school, that it is uncertain whether Bartels taught Gauss calculation, and that it is unclear whether Bartels used his influence to help Gauss further his education in other establishments.
Bartels' association with the Collegium Carolinum was formal from 23 August 1788 when he became a visitor there. Then, on 23 October 1791, he entered the University of Helmstedt where he studied under the professor of mathematics Johann Friedrich Pfaff. He then moved to the University of Göttingen where his teachers included Abraham Gotthelf Kästner, the professor of mathematics and physics. However, mathematics was not the only subject that Bartels studied, for in the winter semester of 1793-4 he studied Experimental Physics, Astronomy, Meteorology and Geology.
In 1800, Bartels was appointed to teach mathematics in Reichenau, a Swiss town close to the city of Chur. He met Anna Magdalena Saluz from Chur and they were married in 1803; their daughter Johanna Henriette Francisca Bartels (born 1807) married the astronomer Wilhelm Struve in February 1835. However, in 1801 Bartels had moved to Aarau in the north of Switzerland where he taught at the cantonal school. From 1803 he was back in Germany teaching at the University of Jena and while there, in 1807, he received an invitation from Stepan Rumowski to be the Professor of Mathematics at the Kazan State University. This university had been founded in 1804, the result of one of the many reforms of the Russian emperor Alexander I, and it opened in the following year. Rumowski was responsible for setting up the university and most of the professors he had invited to go there were from Germany. Bartels took up his post at professor of mathematics at Kazan in 1808 and, during the following twelve years, he lectured on the History of Mathematics, Higher Arithmetic, Differential and Integral Calculus, Analytical Geometry and Trigonometry, Spherical Trigonometry, Analytical Mechanics and Astronomy. Vinberg  writes:-
In the first years the atmosphere in the Department was quite favourable. The students were full of enthusiasm. They studied day and night to compensate for lack of knowledge. The professors, mainly invited from Germany, turned out to be excellent teachers, which was not common.In 1808 Nikolai Ivanovich Lobachevsky had the good fortune to study with Bartels at Kazan University not long after he took up his post there. Not only did Bartels assist Lobachevsky with his studies, but he also looked after his young student, supporting him when he got into trouble with the authorities (which happened quite often!) When Lobachevsky was due to graduate it was Bartels who spent three days lobbying the other professors to award him a Master's degree. The university authorities did not want to give Lobachevsky a degree at all because of his poor behaviour. Bartels won the argument and Lobachevsky was awarded a Master's Degree. After graduating in 1811, Lobachevsky remained in Kazan to study with Bartels who guided his reading of Gauss's Disquisitiones Arithmeticae Ⓣ and Laplace's Mécanique Céleste Ⓣ. In 1814 it was mainly due to Bartels that Lobachevsky was appointed as an assistant professor. We should note that Lobachevsky took Bartels' course on the History of Mathematics which, following Montucla, considered in detail Euclid's Elements and his theory of parallel lines. It was this course which made Lobachevsky think about non-euclidean geometry.
When Bartels was about to leave Kazan in 1820 he wrote about his time there, giving a similar impression to that in Vinberg's quote above :-
I was very glad to find there [Kazan], in spite of the small number of students, a lot of enthusiasm for the study of the mathematical sciences. In my lectures on higher analysis I might have at least twenty listeners, so little by little a small mathematical school has arisen.In 1821 Bartels moved to the university in Dorpat (now Tartu in Estonia). Dorpat had been part of Poland, then of Sweden but in 1704 it was annexed to Russia by Peter the Great. The University of Dorpat had been founded in 1632 by Gustavus II Adolphus of Sweden in 1632. However it closed in 1710 and remained empty for almost 100 years before reopening in 1802 as the Kaiserliche Universität zu Dorpat. Bartels founded the Centre for Differential Geometry at Dorpat, and remained there until his death in 1836.
Bartels made most of his contributions to mathematical research after being appointed to the University of Kazan. However, he did not publish his discoveries until after he moved to Dorpat and even then he did not publish them all. Some we only know of because his students included the results in their own work acknowledging that Bartels had given them in his lecture courses. One such result is the famous Frenet-Serret formulas that were discovered first by Bartels. He introduced the method of moving trihedrons. To each point of a space curve Bartels associated a trihedron, which later became called the Frenet trihedron, and Bartels obtained the formulas now known as the Frenet-Serret formulas. We only know of this since they were published in a prize work by his student Carl Eduard Senff in Principal theorems of the theory of curves and surfaces in 1831, with due acknowledgement to Bartels. Frenet gave six of the formulas in 1847 and later Serret gave all nine.
Let us note that Bartels corresponded with Gauss from the time that he was working in Switzerland. The correspondence continued through the years that he worked in Kazan and during the first few years that he was in Dorpat. After Gauss became famous, a joke went round that Bartels was the best mathematician in Germany because Gauss was the best mathematician in the world.
A couple of years after Bartels moved to Dorpat, he became a Privy Councillor in 1823. He was honoured with election to the St Petersburg Academy of Sciences.
Article by: J J O'Connor and E F Robertson