# Georg Scheffers

Georg Scheffers was a German mathematician specializing in differential geometry. He was born on November 21, 1866 in the village of Altendorf near Holzminden (today incorporated into Holzminden). Scheffers began his university career at the University of Leipzig where he studied with Felix Klein and Sophus Lie. Scheffers was a coauthor with Lie for three of the earliest expressions of Lie theory:

All three are now available online through archive.org; see External links section below.

In 1896 Scheffers became docent at the Technical University of Darmstadt, where he was raised to professor in 1900. From 1907 to 1935, when he retired, Scheffers was a professor at the Technical University of Berlin.

In 1901 he published a German translation of the French textbook on analysis by Joseph Serret. The title was Anwendung der Differential- und Integralrechnung auf die Geometrie (application of differential and integral calculus to geometry). This textbook consisted of two volumes, one on curves and the second on surfaces. A second edition was published in 1910 (volume 2, 1913), and a third edition in 1922.

Another very successful book was prepared for students of science and technology: Lehrbuch der Mathematik (textbook of mathematics). It provided an introduction to analytic geometry as well as calculus of derivatives and integrals. In 1958 this book was republished for the fourteenth time.

Scheffers is known for an article on special transcendental curves (including W-curves) which appeared in the Enzyklopädie der mathematischen Wissenschaften in 1903: "Besondere transzendenten Kurven" (special transcendental curves). He wrote on translation surfaces for Acta Mathematica in 1904: "Das Abelsche und das Liesche Theorem über Translationsflächen" (the theorem of Abel and Lie on translation surfaces).

Other books written by Scheffers are Lehrbuch der Darstellenden Geometrie (textbook on descriptive geometry) (1919), Allerhand aus der zeichnenden Geometrie (1930), and Wie findet und zeichnet man Gradnetze von Land- und Sternkarten? (1934).

Georg Scheffers died August 12, 1945, in Berlin.

## Hypercomplex numbers

{{#invoke:main|main}} In 1891 Georg Scheffers contributed his article "Zurück-führung komplexer Zahlensysteme auf typische formen" to Mathematische Annalen (39:293–390). This article addressed a topic of considerable interest in the 1890s and contributed to the development of modern algebra. Scheffers distinguishes between a "Nichtquaternion system" (Nqss) and a Quaternion system (Qss). Scheffers characterizes the Qss as having three elements ${\displaystyle e_{1},\ e_{2},\ e_{3}}$ that satisfy (p 306)

${\displaystyle e_{1}e_{2}-e_{2}e_{1}=2e_{3},\quad e_{2}e_{3}-e_{3}e_{2}=2e_{1},\quad e_{3}e_{1}-e_{1}e_{3}=2e_{2}.}$

In today's language, Scheffers' Qss has the quaternion algebra as a subalgebra.

Scheffers anticipates the concepts of direct product of algebras and direct sum of algebras with his section (p 317) on reducibility, addition, and multiplication of systems. Thus Scheffers pioneered the structural approach to algebra.

Though the article covers new ground with its exploration of Nqss, it is also a literature review going back to the work of Hermann Hankel. In §14 (p 386) Scheffers reviews both German and English authors on hypercomplex numbers. In particular, he cites Eduard Study’s work of 1889. For volume 41 of Mathematische Annalen Scheffers contributed a further short note, this time including reference to 1867 work by Edmond Laguerre on linear systems, a rich source of hypercomplex numbers.