Georg Scheffers
Encyclopedia
Georg Scheffers was a German mathematician
specializing in differential geometry. He was born on November 21, 1899 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
:
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 curve
s and the second on surface
s. 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 derivative
s and integral
s. In 1958 this book was republished for the fourteenth time.
Scheffers is known for an article on special transcendental curves (including W-curve
s) 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
.
(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 sytem" (Nqss) and a Quaternion system (Qss). Scheffers characterizes the Qss as having three elements
that satisfy (p 306)
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 Herman 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.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
specializing in differential geometry. He was born on November 21, 1899 in the village of Altendorf near Holzminden
Holzminden
Holzminden is a town in southern Lower Saxony, Germany. It is the capital of the district of Holzminden. It is located directly on the river Weser, which here is the border to North Rhine-Westphalia.-History:...
(today incorporated into Holzminden). Scheffers began his university career at the University of Leipzig
University of Leipzig
The University of Leipzig , located in Leipzig in the Free State of Saxony, Germany, is one of the oldest universities in the world and the second-oldest university in Germany...
where he studied with Felix Klein
Felix Klein
Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
and Sophus Lie
Sophus Lie
Marius Sophus Lie was a Norwegian mathematician. He largely created the theory of continuous symmetry, and applied it to the study of geometry and differential equations.- Biography :...
. Scheffers was a coauthor with Lie for three of the earliest expressions of Lie theory
Lie theory
Lie theory is an area of mathematics, developed initially by Sophus Lie.Early expressions of Lie theory are found in books composed by Lie with Friedrich Engel and Georg Scheffers from 1888 to 1896....
:
- Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen (1891)
- (lectures on differential equationDifferential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s with known infinitesimal transformations)
- (lectures on differential equation
- Vorlesungen über continuerliche Gruppen (1893)
- (lectures on continuous groupsGroup (mathematics)In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
), and
- (lectures on continuous groups
- Geometrie der Berührungstransformationen (1896)
- (geometry of contact transformationsContact geometryIn mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle and specified by a one-form, both of which satisfy a 'maximum non-degeneracy' condition called 'complete non-integrability'...
)
- (geometry of contact transformations
In 1896 Scheffers became docent
Docent
Docent is a title at some European universities to denote a specific academic appointment within a set structure of academic ranks below professor . Docent is also used at some universities generically for a person who has the right to teach...
at the Technical University of Darmstadt, where he was raised to professor
Professor
A professor is a scholarly teacher; the precise meaning of the term varies by country. Literally, professor derives from Latin as a "person who professes" being usually an expert in arts or sciences; a teacher of high rank...
in 1900. From 1907 to 1935, when he retired, Scheffers was a professor at the Technical University of Berlin
Technical University of Berlin
The Technische Universität Berlin is a research university located in Berlin, Germany. Translating the name into English is discouraged by the university, however paraphrasing as Berlin Institute of Technology is recommended by the university if necessary .The TU Berlin was founded...
.
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
Textbook
A textbook or coursebook is a manual of instruction in any branch of study. Textbooks are produced according to the demands of educational institutions...
consisted of two volumes, one on curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
s and the second on surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
s. 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
Analytic geometry
Analytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...
as well as calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
of derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
s and integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
s. In 1958 this book was republished for the fourteenth time.
Scheffers is known for an article on special transcendental curves (including W-curve
W-curve
In geometry, a W-curve is a curve in projective n-space that is invariant under a 1-parameter group of projective transformations. W-curves were first investigated by Felix Klein and Sophus Lie in 1871, who also named them. W-curves in the real projective plane can be constructed with straightedge...
s) which appeared in the Enzyklopädie der mathematischen Wissenschaften
Klein's encyclopedia
In mathematics, Klein’s encyclopedia refers to a German mathematical encyclopedia published in six volumes from 1898 to 1933. Felix Klein and Wilhelm Meyer were organizers of the encyclopedia. Its title in English is "Encyclopedia of mathematical sciences including their applications", which is...
in 1903: "Besondere transzendenten Kurven" (special transcendental curves). He wrote on translation surfaces for Acta Mathematica
Acta Mathematica
Acta Mathematica is a journal publishing original research papers in all fields of mathematics. The journal was founded by Gösta Mittag-Leffler in 1882 and is published by Institut Mittag-Leffler, a research institute for mathematics belonging to the Royal Swedish Academy of Sciences...
in 1904: "Das Abelsche und das Liesche Theorem über Translationsflächen" (the theorem of Abel
Niels Henrik Abel
Niels Henrik Abel was a Norwegian mathematician who proved the impossibility of solving the quintic equation in radicals.-Early life:...
and Lie
Sophus Lie
Marius Sophus Lie was a Norwegian mathematician. He largely created the theory of continuous symmetry, and applied it to the study of geometry and differential equations.- Biography :...
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
Berlin
Berlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...
.
Hypercomplex numbers
In 1891 Georg Scheffers contributed his article "Zurück-führung komplexer Zahlensysteme auf typische formen" to Mathematische AnnalenMathematische Annalen
Mathematische Annalen is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann...
(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 sytem" (Nqss) and a Quaternion system (Qss). Scheffers characterizes the Qss as having three elements
that satisfy (p 306)In today's language, Scheffers' Qss has the quaternion
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
algebra as a subalgebra
Subalgebra
In mathematics, the word "algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operation. Algebras in universal algebra are far more general: they are a common generalisation of all algebraic structures...
.
Scheffers anticipates the concepts of direct product
Direct product
In mathematics, one can often define a direct product of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets, together with a suitably defined structure on the product set....
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
Literature review
A literature review is a body of text that aims to review the critical points of current knowledge including substantive findings as well as theoretical and methodological contributions to a particular topic...
going back to the work of Herman Hankel. In §14 (p 386) Scheffers reviews both German and English authors on hypercomplex numbers. In particular, he cites Eduard Study
Eduard Study
Eduard Study was a German mathematician known for work on invariant theory of ternary forms and for the study of spherical trigonometry. He is also known for contributions to space geometry, hypercomplex numbers, and criticism of early physical chemistry.Study was born in Coburg in the Duchy of...
’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
Edmond Laguerre
Edmond Nicolas Laguerre was a French mathematician, a member of the Académie française . His main works were in the areas of geometry and complex analysis. He also investigated orthogonal polynomials...
on linear systems, a rich source of hypercomplex numbers.