One of the many examinations for which Grassmann sat, required that he submit an essay on the theory of the tides. In 1840, he did so, taking the basic theory from Laplace's Mécanique céleste and from Lagrange's Mécanique analytique, but expositing this theory making use of the vector methods he had been mulling over since 1832. This essay, first published in the Collected Works of 1894–1911, contains the first known appearance of what are now called linear algebra and the notion of a vector space. He went on to develop those methods in his A1 and A2 (see references).
In 1844, Grassmann published his masterpiece, his Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik, hereinafter denoted A1 and commonly referred to as the Ausdehnungslehre, which translates as "theory of extension" or "theory of extensive magnitudes." Since A1 proposed a new foundation for all of mathematics, the work began with quite general definitions of a philosophical nature. Grassmann then showed that once geometry is put into the algebraic form he advocated, the number three has no privileged role as the number of spatial dimensions; the number of possible dimensions is in fact unbounded.
Fearnley-Sander (1979) describes Grassmann's foundation of linear algebra as follows:
|“||The definition of a linear space (vector space)... became widely known around 1920, when Hermann Weyl and others published formal definitions. In fact, such a definition had been given thirty years previously by Peano, who was thoroughly acquainted with Grassmann's mathematical work. Grassmann did not put down a formal definition --- the language was not available --- but there is no doubt that he had the concept.
Beginning with a collection of 'units' e1, e2, e3, ..., he effectively defines the free linear space which they generate; that is to say, he considers formal linear combinations a1e1 + a2e2 + a3e3 + ... where the aj are real numbers, defines addition and multiplication by real numbers and formally proves the linear space properties for these operations. ... He then develops the theory of linear independence in a way which is astonishingly similar to the presentation one finds in modern linear algebra texts. He defines the notions of subspace, linear independence, span, dimension, join and meet of subspaces, and projections of elements onto subspaces.
...few have come closer than Hermann Grassmann to creating, single-handedly, a new subject.
Following an idea of Grassmann's father, A1 also defined the exterior product, also called "combinatorial product" (In German: äußeres Produkt or kombinatorisches Produkt), the key operation of an algebra now called exterior algebra. (One should keep in mind that in Grassmann's day, the only axiomatic theory was Euclidean geometry, and the general notion of an abstract algebra had yet to be defined.) In 1878, William Kingdon Clifford joined this exterior algebra to William Rowan Hamilton's quaternions by replacing Grassmann's rule epep = 0 by the rule epep = 1. (For quaternions, we have the rule i2 = j2 = k2 = −1.) For more details, see exterior algebra.
A1 was a revolutionary text, too far ahead of its time to be appreciated. Grassmann submitted it as a Ph. D. thesis, but Möbius said he was unable to evaluate it and forwarded it to Ernst Kummer, who rejected it without giving it a careful reading. Over the next 10-odd years, Grassmann wrote a variety of work applying his theory of extension, including his 1845 Neue Theorie der Elektrodynamik and several papers on algebraic curves and surfaces, in the hope that these applications would lead others to take his theory seriously.
In 1846, Möbius invited Grassmann to enter a competition to solve a problem first proposed by Leibniz: to devise a geometric calculus devoid of coordinates and metric properties (what Leibniz termed analysis situs). Grassmann's Geometrische Analyse geknüpft an die von Leibniz erfundene geometrische Charakteristik, was the winning entry (also the only entry). Moreover, Möbius, as one of the judges, criticized the way Grassmann introduced abstract notions without giving the reader any intuition as to why those notions were of value.
In 1853, Grassmann published a theory of how colors mix; it and its three color laws are still taught, as Grassmann's law. Grassmann's work on this subject was inconsistent with that of Helmholtz. Grassmann also wrote on crystallography, electromagnetism, and mechanics.
Grassmann (1861) set out the first axiomatic presentation of arithmetic, making free use of the principle of induction. Peano and his followers cited this work freely starting around 1890. Lloyd C. Kannenberg published an English translation of The Ausdehnungslehre and Other works in 1995 (ISBN 0-8126-9275-6. -- ISBN 0-8126-9276-4).
In 1862, Grassmann published a thoroughly rewritten second edition of A1, hoping to earn belated recognition for his theory of extension, and containing the definitive exposition of his linear algebra. The result, Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet, hereinafter denoted A2, fared no better than A1, even though A2's manner of exposition anticipates the textbooks of the 20th century.
Read more about this topic: Hermann Grassmann
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