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| '''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]]:
| | Hello from United Kingdom. I'm glad to came here. My first name is Penney. <br>I live in a town called Nyadd in western United Kingdom.<br>I was also born in Nyadd 31 years ago. Married in October year 1999. I'm working at the university.<br><br>Visit my web blog :: [http://www.dailystrength.org/people/4282113/journal/11475241 gates] |
| * ''Lectures on [[Differential equation]]s with known [[Infinitesimal transformation]]s'' (1893),
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| * ''Lectures on [[Continuous group]]s'' (1893), and
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| * ''Geometry of [[Contact geometry|Contact Transformations]]'' (1896).
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| All three are now available online through [[archive.org]]; see External links section below.
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| 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]].
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| 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.
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| 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.
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| Scheffers is known for an article on special transcendental curves (including [[W-curve]]s) which appeared in the [[Klein's encyclopedia|''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 [[Niels Henrik Abel|Abel]] and [[Sophus Lie|Lie]] on translation surfaces).
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| 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).
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| Georg Scheffers died August 12, 1945, in [[Berlin]].
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| ==Hypercomplex numbers==
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| {{main|hypercomplex number}}
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| In 1891 Georg Scheffers contributed his article "Zurück-führung komplexer Zahlensysteme auf typische formen" to [[Mathematische Annalen]] (39:293–390).
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| This article addressed a topic of considerable interest in the 1890s and contributed to the development of [[Abstract algebra|modern algebra]]. Scheffers distinguishes between a "Nichtquaternion system" (Nqss) and a Quaternion system (Qss). Scheffers characterizes the Qss as having three elements
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| <math>e_1,\ e_2,\ e_3</math> that satisfy (p 306) | |
| :<math>e_1e_2-e_2e_1 = 2e_3, \quad e_2e_3-e_3e_2 = 2e_1, \quad e_3e_1-e_1e_3=2e_2.</math> | |
| In today's language, Scheffers' Qss has the [[quaternion]] algebra as a [[subalgebra]].
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| Scheffers anticipates the concepts of [[direct product]] of algebras and [[direct sum of modules#Direct sum of algebras|direct sum of algebras]] with his section (p 317) on reducibility, addition, and multiplication of systems.
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| Thus Scheffers pioneered the structural approach to algebra.
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| 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.
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| ==References==
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| * Werner Burau (1975) "Georg Scheffers" [[Dictionary of Scientific Biography]]
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| * {{MacTutor Biography|id=Scheffers}}
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| * [http://genealogy.math.ndsu.nodak.edu/id.php?id=19534 Georg Scheffers] at [[Mathematics Genealogy Project]].
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| ==External links==
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| * 1891: [http://www.archive.org/details/vorlesuberdiff00liesrich ''Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen''] from [[archive.org]]
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| ** (lectures on [[differential equation]]s with known infinitesimal transformations)
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| * 1893: [http://www.archive.org/details/vorlescontingrup00liesrich ''Vorlesungen über continuerliche Gruppen'']
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| ** (lectures on continuous [[group (mathematics)|groups]]), and
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| * 1896: [http://www.archive.org/details/geometriederber00liesuoft ''Geometrie der Berührungstransformationen'']
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| ** (geometry of [[Contact geometry|contact transformations]])
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| {{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
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| | NAME = Scheffers, Georg
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| | ALTERNATIVE NAMES =
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| | SHORT DESCRIPTION = Mathematician
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| | DATE OF BIRTH = November 21, 1866
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| | PLACE OF BIRTH = Altendorf bei Holzminden, [[Germany]]
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| | DATE OF DEATH = August 12, 1945
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| | PLACE OF DEATH = [[Berlin]], [[Germany]]
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| }}
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| {{DEFAULTSORT:Scheffers, Georg}}
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| [[Category:German mathematicians]]
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| [[Category:Differential geometers]]
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| [[Category:1866 births]]
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| [[Category:1945 deaths]]
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Hello from United Kingdom. I'm glad to came here. My first name is Penney.
I live in a town called Nyadd in western United Kingdom.
I was also born in Nyadd 31 years ago. Married in October year 1999. I'm working at the university.
Visit my web blog :: gates