Kästner studied law, philosophy, physics, mathematics and metaphysics in Leipzig from 1731, and was appointed a Notary in 1733.

He gained his habilitation from the University of Leipzig in 1739,[1] and lectured there in mathematics, philosophy, logic and law, becoming an associate professor in 1746.

One of his doctoral students was the physicist and aphorist Georg Christoph Lichtenberg, who became a colleague of his at Göttingen.

Kästner became most well known for his poems, which appeared first in print without his consent in 1781 and were notable for their biting humour and sharp irony on different contemporary personalities.

Geschichte der Mathematik is considered an astute work, but lacks a comprehensive overview of all subsections of mathematics.