Johann Pfaff (1765-1825) was a famed German mathematician who studied calculus and differential equations. His greatest contribution to the study of mathematics was the first general method of integrating partial differential equations of the first order, which revolutionized the study of partial differential equations.

Johann Pfaff was born in Stuttgart, Germany, on December 22, 1765. He was the son of a financial counselor in Wurttemberg (a small duchy that would later become part of Germany) and his mother was the daughter of a member of the exchequer of Wurttemberg. Thus, he was born into a prominent place among the civil servants of the area.

He was the second of seven sons and, although he was the most famous of the children, he wasn't the most successful. His youngest brother, J. Wilhelm Pfaff, was also a mathematician, as was his younger brother Christoph, who studied chemistry, medicine, and pharmacy. Interestingly, Johann attended a school in Stuttgart that was heavy on discipline but light on education, and although he attended there until the age of 19, he was exposed to very little mathematics. He left in 1785 having studied law, which was to serve in his career as a civil servant.

This didn't stop Johann from studying mathematics on his own, however, and his independent study was noticed by the Duke of Wurttemberg, who encouraged Johann to attend the University of Gottingen where he studied mathematics and physics for two years. After this, in 1787, Pfaff moved on to Berlin, where he studied astronomy for a year and wrote his first paper, focusing on an astronomical problem.

In 1788, Pfaff was elected mathematical chair at the university at Helmstedt on the recommendation of his physics professor at Gottingen and his own writings, chief of which was one that investigated unusual differential equations. Seeing his promise, he was chosen for the chair at Helmstedt and stayed there until 1810.

During his time there, he helped mentor Gauss and built up a strong mathematics department, but there simply wasn't enough money to sustain the school, and it closed in 1810. Distraught, Pfaff moved on to chair the mathematics and astronomy departments at Halle.

The period from 1810 to 1815 proved to be Pfaff's peak period as a mathematician. He published numerous papers on partial differential equations and other special functions and contributed greatly to Gauss's work on ellipses inside quadrilaterals. In 1815, he published his masterwork outlining the method for integrating partial differential equations of the first order, but the mathematics world was not ready for it, and it would remain largely unnoticed until after his death.

Pfaff remained at his post until he passed on in 1825. His greatest work wouldn't be rediscovered until 1827, when a number of papers appeared utilizing his method, so he wouldn't live to see his greatest success. His method for integrating partial differential equations is still in use today.