Hermann Graßmann

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Hermann Graßmann

Hermann Günther Graßmann (born April 15, 1809 in Stettin ; † September 26, 1877 there ) was a German mathematician , physicist and linguist . He is considered to be the actual founder of vector and tensor calculus .

life and work

The father, Justus Günther Graßmann (1779-1852) completed a three-year, actually theological, but also with scientific content at the University of Halle and then taught first as a private tutor, then as vice-principal at the city school in Pyritz and then at the grammar school in Stettin . His subjects were mathematics , physics and drawing . He also wrote some textbooks on elementary mathematics for elementary schools and high schools and founded a physical society in 1835. "The scientific and philosophical approaches laid down in all of these works became a crucial starting point for his son's scientific developments."

Youth and Studies

The young Hermann Günther did not act like z. B. his contemporary and later colleague William Rowan Hamilton emerged as a child prodigy; rather, as he grew up, he was first noticed by his limited mental resilience, forgetfulness and dreaming. Up to the age of 14 one could not deduce his extraordinary talent. But then his interest aroused, and initially - the family was strongly influenced by Pietism - theological studies were also considered, after he had passed the school leaving examination in 1827 with the best grade. In 1827 he began his studies at the Berlin University . There he heard, among other things, the dialectic lecture and the sermons by Friedrich Schleiermacher , which strongly influenced his thinking. "Even during his university days, Hermann Graßmann acquired independent study methods which enabled him to penetrate mathematics as an autodidact later on." During his entire studies, he did not attend a mathematical lecture.

His enormous additional, at first philologically oriented, self-imposed study workload quickly pushed him to the limit of his psychological and physical powers, and he fell ill. He first had to reorient himself in his approach, and he eventually developed a work posture that was more appropriate for him. In a letter exercised Grassmann self-criticism: "The Phlegmatic ... must rather seek to give his reasoning clarity and clarity depth." . The basis for the extraordinary productivity of “this uncanny spirit” (Junghans) in the most varied of scientific fields was thus created. The role of Schleiermacher's dialectics as the key to the regularities of the most diverse areas of science is particularly important. Graßmann writes that one can learn from him "for every science, because he gives fewer positive things than he does skillfully, to attack every investigation from the right side and to continue it independently, and to enable him to find the positive himself." .

On the way to the expansion theory of 1844

In 1830 Graßmann returned to Stettin. He took up his self-study again and dealt with physics and mathematics in "close connection of geometry, arithmetic and combination theory". In 1831 he took a position at the Szczecin teachers' seminar and initially taught German and spatial studies as an assistant teacher and wrote examination papers for the teaching examination committee in Berlin. In these works his early fundamental approach is evident, "in which the mathematical approach is always flanked or even initiated by philosophical considerations" . He received the license to teach mathematics up to secondary school as a senior teacher. In 1834 he passed his first theological exam, but had already decided on a scientific career.

In 1837 he became a scientific teacher at the Otto School in Stettin. In 1839 his first work, conceived for teaching purposes, appeared on the derivations of the crystal figures , for which August Ferdinand Möbius found a certain interest because he had also dealt with this topic on the sidelines. In 1838 he passed his second theological exam, and before that he even signed up for a re-examination in mathematics and physics, certainly to improve his demonstrable mathematical qualification. In 1840 he completed the new examination paper on the theory of ebb and flow , in which he successfully applied newly developed mathematical approaches. Graßmann knew about the importance and efficiency of the vector calculation he had designed and used for the first time in this work. After moving to the Friedrich-Wilhelms-Schule, his main work, expansion theory , appeared in 1844 .

“This book broke the contemporary conception of the treatment of geometry. Extensive philosophical preliminary considerations, presentation of an abstract theory of connections conceived as the basis of all mathematics, sparse use of formulas, rejection of geometry as a mathematical discipline and the development of an n-dimensional, metric-free theory of mathematical diversity "form the theoretical elaboration of his mathematical program. With this work, Graßmann anticipated considerations that are closely related to Bernhard Riemann's later approaches to the theory of n-dimensional manifolds and to Hamilton's concept of quaternions. However, Graßmann was generally not understood, which is why he was ignored by the professional world and the book did not sell at all. One reason for this was certainly the use of terms that he had created himself as an autodidact and not historically sanctioned. Even with his method of representation, which deviated from the Euclidean ideal, he was not yet able to win over the mathematical experts.

Struggle for recognition

In 1846 Graßmann began a series of publications on the theory of algebraic curves , the elaboration of an approach that had resulted in expansion theory, of course also for the further propagation of his program. However, these were also ignored by his contemporaries.

Then he tried to work on a prize task that had been advertised by the Jablonowskische Gesellschaft since 1844 . It was about the reconstruction and further development of a geometric calculus only sketchily designed by Gottfried Wilhelm Leibniz . Thanks to his new method, only Graßmann was able to tackle this task successfully. He was awarded the award. Inspired by the success, he and his brother Robert Graßmann engaged in further studies and in 1847 applied for a mathematics teaching position at a Prussian university.

He submitted his price paper and the expansion theory , but the report by Ernst Eduard Kummer is particularly devastating. He writes “that this text [the theory of expansion] will continue to be ignored by mathematicians as before; for the effort to familiarize oneself with it seems too great in relation to the real gain in knowledge which one supposes to be able to draw from it ” . The application was rejected.

On the way to the second expansion theory

In 1849 Graßmann married Marie Therese Knappe, who happily gave birth to 11 children over the next few years. He published several articles on the application of expansion theory to the theory of algebraic curves. In 1853 a pioneering essay appeared in the field of color theory, which influences colorimetry to this day. An essay on vowel theory was also published, which is considered to be the forerunner of Helmholtz's theory of resonance . Inspired by the writings of Franz Bopp , he began to deal with languages, mainly Sanskrit , and the then still young historical linguistics .

He worked a lot on arithmetic and on a new elaboration of the expansion theory , which appeared in 1861 and 1862. He had changed the way he presented these books, certainly also stimulated by the criticism of the previous one, and his brother also seems to have exerted a certain influence on them. He now applied the strict , purely formulaic Euclidean representation. With that, however, as Petsche describes, he fell “from one extreme to the other. It is true that the mathematicians could no longer reproach him for the philosophical presentation; In return, however, they were expected to be offered a completely strange mathematical material in the most difficult to access method of representation of that time, without even having an idea of ​​the usefulness of the mathematical development. An echo to the publication of his work [expansion theory] was therefore completely absent ”. With both works, he again applied for a chair at the Ministry of Culture, which was again rejected. The reason given was that there was no better position for him anyway, since Graßmann had taken over his position at the Szczecin Gymnasium in 1852 after the death of his father, which had already been linked to the title of professor.

Finally, disappointed, he turned away from all mathematical studies in order to devote himself entirely to linguistics.

Linguistic work

In this area too, Graßmann produced something new and important; Here he won the broad recognition of his colleagues. In 1863 he published the Hauchdissimilationsgesetz , for which he did not claim authorship. From 1873 his dictionary on Ṛgvedasaṃhitā appeared , which is still in use in Indology today, even if many entries are now outdated. A translation of the same text was then published. The American Oriental Society made him a member in 1876. At the instigation of Rudolf von Roth , the University of Tübingen awarded him an honorary doctorate in the same year; The proposal states: "He is one of the best linguists and Sanskritists ... The translation [of the Ṛgvedasaṃhitā] is far superior to that begun by Alfred Ludwig in Prague ... because of its penetrating understanding and tasteful interpretation" . From today's perspective, Graßmann is one of the most important Veda researchers at the turn of the 19th and 20th centuries.

Late recognition of the mathematical work

Late in Graßmann's life there was general scientific recognition of his mathematical achievements. In his Lectures on Quaternions in 1853, Hamilton had already mentioned Graßmann's expansion theory as groundbreaking, but this had no effect on the whole and, above all, Graßmann himself had not noticed anything. On November 24, 1866 he received a letter from Hermann Hankel in which he expressed his enthusiasm for Graßmann's mathematics. He had noticed Graßmann through reading the Quaternions . “[I] saw to my great joy,” writes Hankel, “that in the same [the two expansion theories] the concept of complex numbers - that's what I call your extensive quantities - [sic!] Is treated and treated in a general way and in such an appropriate manner used when I could only wish for my own clarification ” . Hankel was able to understand Graßmann and a regular correspondence ensued. But Hankel did not yet have decisive scientific weight to ultimately help Graßmann achieve a breakthrough. In 1869 Felix Klein became aware of the name Graßmann through Hankel's theory of complex number systems . This in turn referred his colleague Alfred Clebsch to him. At the instigation of Clebsch, Graßmann was finally elected as a corresponding member by the Göttingen Society of Sciences in 1871 . In 1872 Victor Schlegel , a colleague of Graßmann's at the Stettiner Gymnasium, published the first attempt at a closed external representation of Graßmann's views, the system of spatial theory . The growing recognition was unstoppable. Sophus Lie even came to Stettin to ask Graßmann to explain how he handled Pfaff's problem . The vector calculation and vector analysis then prevailed against the quaternions by the end of the 19th century, in particular through the work of Josiah Willard Gibbs and Oliver Heaviside . Gibbs came across the vector concept independently of Grassmann, but recognized its priority (correspondence with Victor Schlegel).

"So the life of a great scientist who had been misunderstood for a long time and who struggled in intellectual isolation for the progress of mathematics came to an end." Shortly before his death, he experienced a new edition of the expansion theory from 1844 after it had been found that almost the entire first edition had been crushed due to lack of sales.

On the occasion of Hermann Graßmann's 200th birthday, an international scientific conference took place in Potsdam and Stettin in September 2009, which examined the contexts and the history of his work as well as the further development of his ideas in the present in an interdisciplinary manner.

His son Hermann Graßmann the Younger was also a mathematician who worked on the edition of the Collected Works, continued Graßmann's theory of expansion and became a professor in Giessen.

Graßmann's vector calculation

Some basic ideas of Graßmann's vector calculation:

  • Relationships between spatial quantities can be described with the aid of algebraic linking laws
  • Conception of the lines AB and BA as opposite quantities (consideration of the negative in the geometry), in addition to the length of a line, its direction is important
  • In contrast to Hamilton, Graßmann is interested in extending his thoughts to n dimensions
  • AB + BC = AC applies even if A, B, C are not in a straight line
  • if all elements of a line are subjected to the same changes (today: parallel displacements), the resulting line is the same as the original.
  • the geometric product ( wedge product ) of two segments is the area of the parallelogram formed from them

Graßmann already used the terms linear dependency and independence, basis and dimension , albeit under different names. Graßmann speaks of lengths and sizes, not of vectors.

See also

Publications (selection)

  1. Derivation of the crystal shapes from the general law of crystal formation. In: Program discussion of the Stettiner Ottoschule 1839 [= GW 2.2, pp. 115–146]
  2. Theory of the ebb and flow. Examination paper from 1840. In: GW 3,1, pp. 8–203
  3. Outline of German language teaching. In: Program discussion of the Stettiner Ottoschule 1842, pp. 2-56
  4. The science of extensive size, or expansion theory, a new mathematical discipline. 1st part: The linear expansion theory. Leipzig 1844 [Reprint: 1878] [= GW 1,1, pp. 4–312]
  5. Geometrical analysis linked to the geometrical characteristic invented by Leibniz. Award-winning typeface. Leipzig 1847 [= GW 1,1, pp. 321–398]
  6. On the theory of color mixing. In: Poggendorfs Annalen der Physik und Chemie 89 (1853), pp. 69–84 [= GW 2.2, pp. 161–173]
  7. Overview of the acoustics and the lower optics. Vowel theory. In: Programmabhandlung des Stettines Gymnasium 1854 [= GW 2,2, S. 174–202]
  8. Textbook of mathematics for higher education institutions. Part 1: arithmetic. (PDF; 9.6 MB) Berlin 1861
  9. The expansion theory. Completely and strictly justified. Berlin 1862 [= GW 1,2, pp. 1–383]
  10. Dictionary for the Rigveda. Leipzig 1873–1875 [6. Edition Wiesbaden 1996, ISBN 3-447-03223-5 ]
  11. Rig Veda. Translated and provided with critical and explanatory notes. 2 vols. Leipzig 1876–1877
  12. [GW:] Collected mathematical and physical works. 3 vols. Leipzig 1894–1911 [reprint: New York 1972]

literature

  • Joachim Buhrow: Hermann Graßmann - late recognition of an original mathematician . In: Mathematics Lessons. Volume 6. 1993, pp. 14-24.
  • Joachim Buhrow: Hermann Günther Graßmann (1809–1877) . In: Pomerania. Journal of Culture and History. Issue 1/2010, ISSN  0032-4167 , pp. 41-42.
  • Moritz Cantor , August LeskienGraßmann, Hermann . In: Allgemeine Deutsche Biographie (ADB). Volume 9, Duncker & Humblot, Leipzig 1879, pp. 595-598.
  • Kurt Elfering: About linguistic research and the aspirant law . In: Schreiber, work and effect , pp. 33–35.
  • Friedrich Engel : Graßmann's life. Along with an index of the writings published by Graßmann and an overview of the handwritten estate. In: GW 3.2, pp. 1-400.
  • Friedrich Engel Hermann Graßmann , DMV annual report 1910
  • Friedrich Engel : Hermann Graßmann (1809-1877) . In: Martin Wehrmann , Adolf Hofmeister and Wilhelm Braun (eds.): Pommersche Lebensbilder . Volume 2: Pomerania of the 19th and 20th centuries . Leon Sauniers Verlag, Stettin, 1936, pp. 74-84.
  • Desmond Fearnley-Sander Hermann Grassmann and the creation of linear algebra , American Mathematical Monthly, Volume 86, 1979, pp. 809–817, Online (received the Lester R. Ford Award in 1980)
  • F. Junghans: Hermann Graßmann. In: Journal for mathematical and natural science teaching 9 (1978), pp. 167-169, 250-253.
  • Gottlob Kirschmer:  Graßmann, Hermann. In: New German Biography (NDB). Volume 7, Duncker & Humblot, Berlin 1966, ISBN 3-428-00188-5 , p. 5 f. ( Digitized version ).
  • Hans-Joachim Petsche: Graßmann . Birkhäuser, Basel [etc.] 2006 (Vita Mathematica 13), ISBN 3-7643-7257-5 .
  • Hans-Joachim Petsche, Lloyd Kannenberg, Gottfried Keßler and Jolanta Liskowacka (eds.): Hermann Graßmann - Roots and Traces. Autographs and Unknown Documents. Text in German and English . Birkhäuser, Basel [etc.] 2009, ISBN 978-3-0346-0154-2 .
  • Hans-Joachim Petsche, Albert C. Lewis, Jörg Liesen and Steve Russ (eds.): From Past to Future: Graßmann's Work in Context. The Graßmann Bicentennial Conference, September 2009 . Springer Basel AG, Basel 2010, ISBN 978-3-0346-0404-8 .
  • Hans-Joachim Petsche and Peter Lenke (Eds.): International Grassmann Conference. Hermann Grassmann Bicentennial: Potsdam and Szczecin, September 16-19, 2009; Video recording of the conference . 4 DVDs, 16:59:25. Universitätsverlag Potsdam, Potsdam 2010, ISBN 978-3-86956-093-9
  • Victor Schlegel: Hermann Graßmann. His life and his works. Leipzig 1878.
  • Peter Schreiber (Ed.): Hermann Graßmann. Work and effect. International symposium on the occasion of the 150th anniversary of the first publication of the 'linear expansion theory' (Lieschow / Rügen, 23-28 May 1994) . Greifswald: Ernst Moritz Arndt University, subject areas mathematics / computer science 1995.
  • G. Schubring (Ed.) Hermann Günther Graßmann (1809-1877): Visionary Mathematician, Scientist and Neohumanist Scholar , Dordrecht 1996.
  • Arno Zaddach: Graßmann's algebra in geometry, with side glances at related structures. Mannheim, Leipzig, Vienna, Zurich 1994, ISBN 386025474X .

Web links

Individual evidence

  1. Graßmann wrote himself with "ß", see: Petsche, Graßmann , p. 103, note 2
  2. ^ The Pomeranian Newspaper . No. 3/2008, p. 6.
  3. Petsche, op.cit., P. 15
  4. Petsche, op.cit., P. 23
  5. Petsche, op.cit., P. 27
  6. Petsche, op.cit., P. 146
  7. Petsche, op.cit., P. 30
  8. Petsche, op.cit., P. 31
  9. Petsche, op.cit., P. 42
  10. See Junghans, Graßmann , p. 168
  11. Petsche, op.cit., P. 53
  12. Petsche, op.cit., P. 89
  13. ^ Friedrich Wilhelm: Alfred Ludwig. In: Volume XV. Neue Deutsche Biographie, 1987, accessed on August 31, 2017 .
  14. See Karin Reich : About the honorary doctorate at the University of Tübingen in 1876 . In: Schreiber, work and effect , pp. 59–61
  15. Petsche, op.cit., P. 94
  16. ^ Michael Crowe, A history of vector analysis, University of Notre Dame Press 1967, Dover Reprint 1985
  17. Petsche, op.cit., P. 102
  18. Graßmann Bicentennial Conference ( Memento of the original from March 2, 2009 in the Internet Archive ) Info: The archive link has been inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.  @1@ 2Template: Webachiv / IABot / www.uni-potsdam.de