Johann Friedrich Pfaff

Johann Friedrich Pfaff (born December 22, 1765 in Stuttgart ; † April 21, 1825 in Halle (Saale) ) was a German mathematician and member of a dynasty of university professors (see below). He taught at the Universities of Helmstedt and Halle , where he mainly dealt with analysis and partial differential equations .

Life

Pfaff was the second oldest of twelve children of the Stuttgart tax councilor Friedrich Burkhard Pfaff and his wife Maria Magdalena , née Brand . Two of his brothers also became scientists: Christoph Heinrich Pfaff (1773–1852) and Wilhelm Andreas Pfaff (1774–1835).

From 1774 to 1785 he attended the Hohe Karlsschule , a military school in the ducal Solitude Palace near Stuttgart, where he completed legal studies in 1785. In addition, he did self-study in mathematics and was sent on a multi-year educational trip by Carl Eugen von Württemberg , sovereign and founder of the school. From 1785 he studied mathematics and physics at the instigation of the Duke in Göttingen with Abraham Gotthelf Kästner and Georg Christoph Lichtenberg . In 1787 he went to the Berlin observatory to Johann Elert Bode and in the following year on an educational trip a. a. to Jena, Gotha, Prague and finally to Vienna . At Lichtenberg's instigation, he was appointed professor of mathematics at the University of Helmstedt in 1788 .

A historic achievement of the committed university professor was to have recognized the genius of the young Carl Friedrich Gauß ; In 1799 he was the reviewer for his dissertation and supported his application to do his doctorate in absentia at the University of Helmstedt . Other important students were Johann Christian Martin Bartels (1769–1836), Christian Ludwig Gerling (1788–1864), Carl Brandan Mollweide (1774–1825) and the later Tübingen law professor Heinrich Eduard Siegfried von Schrader (1779–1860). Pfaff also proved to be a talent promoter through a letter of recommendation to Alexander von Humboldt in Göttingen .

In terms of university policy, Pfaff was committed to maintaining the University of Helmstedt as a Braunschweig State University. When it was closed in 1810 because of anti-French activities during the Napoleonic Wars , Pfaff went to the University of Halle, which had just reopened, and in 1812 after Prof. Klüngel's death also became the director of the observatory there . At that time, both places belonged to the Kingdom of Westphalia founded by Napoleon .

He was also a respected teacher in Halle, among others from August Ferdinand Möbius (1790–1868), Johann August Grunert , Johann Joseph Schön and Karl Friedrich Wex .

Pfaff left behind extensive mathematical manuscripts that are kept in the university library in Halle. In 1812 he was admitted to the Prussian Academy of Sciences as a corresponding member and in 1817 as a foreign member . From 1821 he was a corresponding member of the Académie des Sciences in Paris . In 1793 he became a corresponding and in 1798 honorary member of the Russian Academy of Sciences in Saint Petersburg . In 1793 he was elected a corresponding member of the Göttingen Academy of Sciences .

relationship

In 1803 Pfaff married his cousin Caroline Brand , daughter of the pastor Christoph Brand . Christoph Brand was the brother of Johann Pfaff's mother and son of the church and chamber councilor Gottfried Brand . With Caroline, Pfaff had the sons Carl (later historian) and Ludwig .

His brothers were Christoph Heinrich Pfaff (1773–1852), professor of chemistry in Kiel, and Johann Wilhelm Andreas Pfaff (1774–1835), professor of mathematics and astronomy in Dorpat, Würzburg and Erlangen. The sons of Johann Wilhelm Andreas, in turn, were professors Hans Ulrich Vitalis Pfaff (mathematics) and Alexius Burkhardt Immanuel Friedrich Pfaff (mineralogy), his daughter was the writer Pauline Damajanti , who later married the liberal journalist Carl Ludwig Theodor Brater and shared with him Daughter Agnes Sapper , also a writer, had.

Pfaff as a mathematician

Pfaff dealt intensively with the theory of partial differential equations and around 1810 with the simplification of certain differential expressions ( "Pfaff's problem"), the general solutions of which were given by HGGraßmann (1809-77). The " Pfaffian forms " became part of the basic training in mathematical analysis and are still a research topic today. Despite a very positive review by Gauss, Pfaff's work was initially hardly noticed until Jakobi emphasized its importance in 1827.

Other mathematically significant works are a Latin textbook on analysis in the spirit of Leonhard Euler in 1797 , or the solution of a publicly set problem for the largest ellipse inscribable in a convex square (1810), which Gauß and Mollweide also dealt with.

Important publications are:

1788 Attempt of a new method of summation along with other related analytical remarks

1794 Analysis of an important task of Mr. La Grange , archive of pure and applied mathematics

1796 On the advantages which a university grants a country , (Häberlins) Staats-Archiv

1797 Disquisitiones analyticae maxime ad calculum integralem et doctrinam serierum pertinentes

1810 Determination of the largest in a square, as well as in a triangle, ellipse to be described , in FXZach , monthly correspondence for the promotion of geography and celestial science
1814/15 Methodus generalis, aequationes differentiarum partialium, necnon aequationes differentiales vulgares, utrasque primi ordinis, inter quotcunque variabiles complete integrandi , treatises of the kings. Academy of Sciences in Berlin, p. 76-135. German by G. Kowalewski:
General method, partial differential equations and ordinary differential equations, both of the first order, to be completely integrated in any number of variables , Ostwald's classic of exact sciences, Volume 129, 1902
Collection of letters exchanged between Johann Friedrich Pfaff and Duke Carl von Württemberg, F. Bouterwek, A. v. Humboldt, AG Kästner, and others (Ed. Carl Pfaff 1853, with biography).

Pfaff's problem

In Helmstedt in 1788 Pfaff published works on a new type of derivation of differentiation rules , in 1788 and later on the summation of certain series , and in 1793, following on from Euler , on the series expansion for integrals of certain powers.

In 1815 Pfaff published his most important work "Methodus generalis aequationes differentiarum particularum ... complete intigrandi" . It is about Pfaff's problem of integrating partial differential equations of the first order of shape with ${\ displaystyle \ omega = dF}$

{\ displaystyle \ qquad \ omega = \ sum _ {k = 1} ^ {n} f_ {k} (x_ {1}, \ ldots, x_ {n}) \, \ mathrm {d} x_ {k}. }

Today one calls a Pfaffian form in the variables . The problem was to represent the solution as a total differential , which Pfaff achieved through variable transformation. ${\ displaystyle \ omega}$ ${\ displaystyle n}$ ${\ displaystyle x_ {1}, \ ldots, x_ {n}}$

The method, initially only recognized by Gauß as significant, was expanded in 1827 by Jacobi (1804-1851). Pfaff had considered the case of an even number of variables , Jacobi was able to extend the procedure to an odd number of variables.

literature

Moritz Cantor : Johann Friedrich Pfaff . In: Allgemeine Deutsche Biographie (ADB). Volume 25, Duncker & Humblot, Leipzig 1887, p. 592 f.
Karin Reich : Mishaps on the life and work of the mathematician Johann Friedrich Pfaff (December 22, 1765– April 21, 1825). In: Amphora. Festschrift for Hans Wussing on his 65th birthday. , ed. by Sergei S. Demidov… Basel: Birkhäuser 1992, ISBN 3-7643-2815-0 , pp. 551-596.
Gerhard Betsch: Johann Friedrich PFAFF - Analysis and History of Science between EULER and CAUCHY. In: Jürgen Maaß (Ed.): Kepler Symposium Philosophy and History of Mathematics, Univ.-Verl. Trauner, Linz 2005 (History of Natural Science and Technology Series), Volume 5, pp. 108–120, ISBN 3-85487-840-0 .
Gerd Leibrock (ed.): The correspondence between Carl Friedrich Gauß and Johann Friedrich Pfaff. Rauner, Augsburg 2008, ISBN 978-3-936905-27-4 .

Web links

Literature by and about Johann Friedrich Pfaff in the catalog of the German National Library
Biography and appreciation of his work from the University of Halle
John J. O'Connor, Edmund F. Robertson : Johann Friedrich Pfaff. In: MacTutor History of Mathematics archive .
Spektrum .de: Johann Friedrich Pfaff (1765–1825) April 1, 2019

Individual evidence

^ Members of the previous academies. Johann Friedrich Pfaff. Berlin-Brandenburg Academy of Sciences , accessed on May 22, 2015 .
^ Foreign members of the Russian Academy of Sciences since 1724. Johann Friedrich Pfaff. Russian Academy of Sciences, accessed October 17, 2015 .
↑ Holger Krahnke: The members of the Academy of Sciences in Göttingen 1751-2001 (= Treatises of the Academy of Sciences in Göttingen, Philological-Historical Class. Volume 3, Vol. 246 = Treatises of the Academy of Sciences in Göttingen, Mathematical-Physical Class. Episode 3, vol. 50). Vandenhoeck & Ruprecht, Göttingen 2001, ISBN 3-525-82516-1 , p. 188.
^ Cantor : Pfaff, Johann Wilhelm Friedrich . In: Allgemeine Deutsche Biographie (ADB). Volume 25, Duncker & Humblot, Leipzig 1887, p. 593 f .; Günther Oestmann: Pfaff, Wilhelm. In: New German Biography (NDB). Volume 20, Duncker & Humblot, Berlin 2001, ISBN 3-428-00201-6 , p. 292 ( digitized version ).

personal data
SURNAME	Pfaff, Johann Friedrich
ALTERNATIVE NAMES	Pfaff, Friedrich
BRIEF DESCRIPTION	German mathematician
DATE OF BIRTH	December 22, 1765
PLACE OF BIRTH	Stuttgart
DATE OF DEATH	April 21, 1825
Place of death	Halle (Saale)

[1] Members of the previous academies. Johann Friedrich Pfaff. Berlin-Brandenburg Academy of Sciences , accessed on May 22, 2015 .

[2] Foreign members of the Russian Academy of Sciences since 1724. Johann Friedrich Pfaff. Russian Academy of Sciences, accessed October 17, 2015 .

[3] Holger Krahnke: The members of the Academy of Sciences in Göttingen 1751-2001 (= Treatises of the Academy of Sciences in Göttingen, Philological-Historical Class. Volume 3, Vol. 246 = Treatises of the Academy of Sciences in Göttingen, Mathematical-Physical Class. Episode 3, vol. 50). Vandenhoeck & Ruprecht, Göttingen 2001, ISBN 3-525-82516-1 , p. 188.

[4] Cantor : Pfaff, Johann Wilhelm Friedrich . In: Allgemeine Deutsche Biographie (ADB). Volume 25, Duncker & Humblot, Leipzig 1887, p. 593 f .; Günther Oestmann: Pfaff, Wilhelm. In: New German Biography (NDB). Volume 20, Duncker & Humblot, Berlin 2001, ISBN 3-428-00201-6 , p. 292 ( digitized version ).