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Continuing his studies at Heidelberg, Mayer submitted his habilitation thesis to that university and gained the right to teach at universities in 1866. He taught at Heidelberg for the rest of his life, becoming an extraordinary professor in 1871 and marrying Margerete Weigel in the following year. He was promoted to an ordinary professor in 1890.

Wussing writes in that:

As a professor, Mayer enjoyed great respect from his colleagues and students. His activities as a researcher ... earned him membership in numerous learned societies ...

Mayer worked on differential equations , the calculus of variations and mechanics. He emphasised the principle of least action in all his work which followed the path of Lagrange and Jacobi . His work on the integration of partial differential equations and a search to determine maxima and minima using variational methods brought him close to the investigations which Lie was carrying out around the same time.

Engel received his doctorate from Leipzig in 1883 after studying under Mayer and writing a thesis on contact transformations. Engel became a valuable assistant to Lie for several years but towards the end of the 1880s the relationship between Engel and Lie broke down. In 1892 the lifelong friendship between Lie and Klein broke down and the following year Lie publicly attacked Klein . Mayer was connected with the whole episode through his friendship with Klein , being an editor of Mathematische Annalen, and perhaps most significantly since his work was closely related to that of Lie .

The book contains a collection of 186 letters exchanged between Klein and Mayer over the years from 1871 to 1907. The letters provide insights into the scientific and personal relations among Klein , Mayer and Lie over the period.

Wussing writes in that:

through the subsequent works of Mayer, Lie 's achievements became famous relatively quickly.

Source:School of Mathematics and Statistics University of St Andrews, Scotland