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Hermann Günther Grassmann (April 15, 1809, Stettin (Szczecin) – September 26, 1877, Stettin) was a German polymath, renowned in his day as a linguist and now admired as a mathematician. He was also a physicist, neohumanist, general scholar, and publisher. His mathematical work was not recognized in his lifetime. Grassmann was the third of 12 children of Justus Günter Grassmann, an ordained minister who taught mathematics and physics at the Stettin Gymnasium, where Hermann was educated. Hermann often collaborated with his brother Robert. Grassmann was an undistinguished student until he obtained a high mark on the examinations for admission to Prussian universities. Beginning in 1827, he studied theology at the University of Berlin, also taking classes in classical languages, philosophy, and literature. He does not appear to have taken courses in mathematics or physics. Although lacking university training in mathematics, it was the field that most interested him when he returned to Stettin in 1830 after completing his studies in Berlin. After a year of preparation, he sat the examinations needed to teach mathematics in a gymnasium, but achieved a result good enough to allow him to teach only at the lower levels. In the spring of 1832, he was made an assistant at the Stettin Gymnasium. Around this time, he made his first significant mathematical discoveries, ones that led him to the important ideas he set out in his 1844 paper referred to as A1 below. In 1834 Grassmann began teaching mathematics at the Gewerbeschule in Berlin. A year later, he returned to Stettin to teach mathematics, physics, German, Latin, and religious studies at a new school, the Otto Schule. This wide range of topics reveals again that he was qualified to teach only at a low level. Over the next four years, Grassmann passed examinations enabling him to teach mathematics, physics, chemistry, and mineralogy at all secondary school levels. Grassmann felt somewhat aggrieved that he was writing innovative mathematics, but taught only in secondary schools. Yet he did rise in rank, even while never leaving Stettin. In 1847, he was made an "Oberlehrer" or head teacher. In 1852, he was appointed to his late father's position at the Stettin Gymnasium, thereby acquiring the title of Professor. In 1847, he asked the Prussian Ministry of Education to be considered for a university position, whereupon that Ministry asked Kummer for his opinion of Grassmann. Kummer wrote back saying that Grassmann's 1846 prize essay contained "... commendably good material expressed in a deficient form." Kummer's report ended any chance that Grassmann might obtain a university post. This episode proved the norm; time and again, leading figures of Grassmann's day failed to recognize the value of his mathematics. During the political turmoil in Germany, 1848 - 49, Hermann and Robert Grassmann published a Stettin newspaper calling for German unification under a constitutional monarchy. (This eventuated in 1871.) After writing a series of articles on constitutional law, Hermann parted company with the newspaper, finding himself increasingly at odds with its political direction. Grassmann
had eleven children, seven of whom reached adulthood. A son, Hermann
Ernst Grassmann, became a professor of mathematics at the University
of
Giessen. 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 papers, referred to below as A1 and A2. In 1844,
Grassmann published his masterpiece, his Die Lineare
Ausdehnungslehre, ein neuer Zweig der Mathematik [The Theory of Linear
Extension, a New Branch of Mathematics], 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, then 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: 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 [in what is now the usual
way] 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,
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. 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. Grassman'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. Curiously, Grassmann (1861) has
never been translated into English. NOTE: Lloyd C. Kannenberg published
an English translation of The Ausdehnungslehre and Other works in 1995. In 1862,
Grassman 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 [The
Theory of Extension,
Thoroughly and Rigorously Treated], hereinafter denoted A2, fared no better
than A1, even
though A2's
manner of exposition anticipates the textbooks of the 20th century. The only
mathematician to appreciate Grassmann's ideas during his lifetime was Hermann
Hankel, whose 1867 Theorie
der
complexen
Zahlensysteme helped
make
Grassmann's
ideas better known. Grassmann's
mathematical
methods
were slow to be adopted but they directly
influenced Felix
Klein and Élie
Cartan. A.N.
Whitehead's first monograph, the Universal
Algebra (1898),
included the first
systematic exposition in English of the theory of extension and the exterior
algebra. The theory of extension led to the development of differential
forms and to the
application of such forms to analysis and geometry. Differential
geometry makes
use
of the exterior
algebra. Adhémar
Jean
Claude Barré de Saint-Venant developed a vector calculus
similar to that of Grassmann which he published in 1845. He then
entered into a dispute with Grassmann about which of the two had
thought of the ideas first. Grassmann had published his results in
1844, but Saint-Venant claimed that he had first developed these ideas
in 1832. Disappointed
at
his
inability to be recognized as a mathematician, Grassmann turned
to historical linguistics.
He
wrote
books on German grammar, collected folk songs, and learned Sanskrit.
His
dictionary
and his translation of the Rigveda (still in print) were
recognized among philologists. He devised a sound law of Indo-European
languages, named Grassmann's
Law in his
honor.
These philological accomplishments were honored during his lifetime; he
was elected to the American
Oriental
Society and
in 1876, he received an honorary doctorate from the University
of
Tübingen. |