He was the tutor of Carl Friedrich Gauss in Brunswick and the educator of Lobachevsky at the University of Kazan.

On 23 October 1791 Bartels studied mathematics under Johann Friedrich Pfaff in Helmstedt and Abraham Gotthelf Kästner in Göttingen.

In the winter semester of 1793/1794 he studied Experimental Physics, Astronomy, Meteorology and Geology under Georg Christoph Lichtenberg.

In 1807 he was invited to join the University of Kazan by the founder Stepan Jakowlewitsch Rumowski (1734–1812), and went there in 1808 where he was appointed to the chair of Mathematics.

In 1821 he moved to the University of Dorpat, now Tartu, Estonia, where he founded the Centre for Differential geometry.