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Georg Scheffers (21 November 1866 – 12 August 1945) was a German
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
specializing in
differential geometry.
Life
Scheffers was born on 21 November 1866 in the village of Altendorf near
Holzminden
Holzminden (; nds, Holtsminne) is a town in southern Lower Saxony, Germany. It is the capital of the district of Holzminden. It is located on the river Weser, which at this point forms the border with the state of North Rhine-Westphalia.
Histor ...
(today incorporated into Holzminden). Scheffers began his university career at the
University of Leipzig
Leipzig University (german: Universität Leipzig), in Leipzig in Saxony, Germany, is one of the world's oldest universities and the second-oldest university (by consecutive years of existence) in Germany. The university was founded on 2 Decemb ...
where he studied with
Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
and
Sophus Lie
Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations.
Life and career
Marius Soph ...
. Scheffers was a coauthor with Lie for three of the earliest expressions of
Lie theory
In mathematics, the mathematician Sophus Lie ( ) initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory. For instance, the latter subject is ...
:
* ''Lectures on
Differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s with known
Infinitesimal transformation
In mathematics, an infinitesimal transformation is a limiting form of ''small'' transformation. For example one may talk about an infinitesimal rotation of a rigid body, in three-dimensional space. This is conventionally represented by a 3×3 s ...
s'' (1893),
* ''Lectures on
Continuous group
In mathematics, topological groups are logically the combination of Group (mathematics), groups and Topological space, topological spaces, i.e. they are groups and topological spaces at the same time, such that the Continuous function, continui ...
s'' (1893), and
* ''Geometry of
Contact Transformations'' (1896).
All three are now available online through
archive.org
The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, ...
.
In 1896 Scheffers became
docent
The title of docent is conferred by some European universities to denote a specific academic appointment within a set structure of academic ranks at or below the full professor rank, similar to a British readership, a French " ''maître de con ...
at the
Technical University of Darmstadt
Technical may refer to:
* Technical (vehicle), an improvised fighting vehicle
* Technical analysis, a discipline for forecasting the future direction of prices through the study of past market data
* Technical drawing, showing how something is co ...
, where he was raised to
professor
Professor (commonly abbreviated as Prof.) is an academic rank at universities and other post-secondary education and research institutions in most countries. Literally, ''professor'' derives from Latin as a "person who professes". Professors ...
in 1900. From 1907 to 1935, when he retired, Scheffers was a professor at the
Technical University of Berlin.
In 1901–1902 he published a famous two-volume textbook entitled ''Anwendung der Differential- und Integralrechnung auf die Geometrie'' (application of differential and integral calculus to geometry). The first volume subtitled ''Einführung in die Theorie der Curven in der Ebene und in Raum'' was published in 1901 and dealt with
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
s. The second volume subtitled ''Einführung in die Theorie der Flächen'' (introduction to the theory of
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
s) was published in 1902. A second edition was published in 1910 (volume 2, 1913), and a third edition in 1922.
In 1907 Scheffers published the first two volumes of his extensive revision and rewriting of
Georg Bohlmann's 1897–1899 revision of
Harnack Harnack is the surname of a German family of intellectuals, artists, mathematicians, scientists, theologians and those in other fields. Several family members were executed by the Nazis during the last years of the Third Reich.
* Theodosius Harnac ...
's 1884 German translation of
Serret
Serret is a surname. Notable people with the surname include:
*Dulce María Serret (1898–1989), Cuban pianist and music teacher
* Joseph Alfred Serret (1819–1885), French mathematician
*Marie-Ernestine Serret
Marie-Ernestine Serret (1812– ...
's famous two-volume ''Cours de calcul différentiel et intégral'', which was first published by Gauthier-Villars in 1868. In 1909 Scheffers published the third and last volume of his rewriting of Bohlman's version of Serret's two-volume work. For a new edition, Scheffers added an appendix with 46 pages of historical notes for the first and second volumes.
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
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
of
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s and
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
s. In 1958 this book was republished for the fourteenth time.
Scheffers is known for an article on special transcendental curves (including
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 r ...
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 Abel'sche und das Lie'sche Theorem über Translationsflächen"(the theorem of
Abel
Abel ''Hábel''; ar, هابيل, Hābīl is a Biblical figure in the Book of Genesis within Abrahamic religions. He was the younger brother of Cain, and the younger son of Adam and Eve, the first couple in Biblical history. He was a shepherd ...
and
Lie on translation surfaces).
Other books written by Scheffers ar
''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 12 August 1945, in
Berlin
Berlin ( , ) is the capital and List of cities in Germany by population, largest city of Germany by both area and population. Its 3.7 million inhabitants make it the European Union's List of cities in the European Union by population within ci ...
.
Hypercomplex numbers
In 1891 Georg Scheffers contributed his articl
"Zurückführung 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
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ...
. Scheffers distinguishes between a "Nichtquaternion system" (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 In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations.
"Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operat ...
.
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
A literature review is an overview of the previously published works on a topic. The term can refer to a full scholarly paper or a section of a scholarly work such as a book, or an article. Either way, a literature review is supposed to provid ...
going back to the work of
Hermann Hankel
Hermann Hankel (14 February 1839 – 29 August 1873) was a German mathematician. Having worked on mathematical analysis during his career, he is best known for introducing the Hankel transform and the Hankel matrix.
Biography
Hankel was born on ...
. In §14 (p 386) Scheffers reviews both German and English authors on hypercomplex numbers. In particular, he cites
Eduard Study
Eduard Study ( ), more properly Christian Hugo Eduard Study (March 23, 1862 – January 6, 1930), was a German mathematician known for work on invariant theory of ternary forms (1889) and for the study of spherical trigonometry. He is also known f ...
’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 Nicolas Laguerre (9 April 1834, Bar-le-Duc – 14 August 1886, Bar-le-Duc) was a French mathematician and a member of the Académie des sciences (1885). His main works were in the areas of geometry and complex analysis. He also investigate ...
on linear systems, a rich source of hypercomplex numbers.
See also
*
Scheffers conditions
References
* Werner Burau (1975) "Georg Scheffers"
Dictionary of Scientific Biography
The ''Dictionary of Scientific Biography'' is a scholarly reference work that was published from 1970 through 1980 by publisher Charles Scribner's Sons, with main editor the science historian Charles Gillispie, from Princeton University. It consi ...
*
Georg Scheffersat
Mathematics Genealogy Project
The Mathematics Genealogy Project (MGP) is a web-based database for the academic genealogy of mathematicians.. By 31 December 2021, it contained information on 274,575 mathematical scientists who contributed to research-level mathematics. For a ty ...
.
External links
* 1891
''Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen''from
archive.org
The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, ...
** (lectures on
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s with known infinitesimal transformations)
* 1893
''Vorlesungen über continuerliche Gruppen''** (lectures on continuous
groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
), and
* 1896
''Geometrie der Berührungstransformationen''** (geometry of
contact transformations)
{{DEFAULTSORT:Scheffers, Georg
19th-century German mathematicians
Differential geometers
1866 births
1945 deaths
20th-century German mathematicians
People from Holzminden
Academic staff of the Technical University of Berlin
