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<DIV><FONT face=Arial size=2>Dear Histling:</FONT></DIV>
<DIV><FONT face=Arial size=2> Winfred
Lehmann's <U>Reader in Nineteenth Century Historical Indo-European
Linguistics</U> (Indiana U.P., 1967) is no doubt familiar to
many as a standard resource for courses in historical linguistics or the
history of linguistics, for which a reading knowledge of German on the part of
students cannot be assumed. The present writer has assigned parts of it in this
capacity, and on the second-hand markets of the Internet the book is priced
at a level which makes clear that it is still a valued work. Lehmann's
brief introductions to the works translated are often the main sources of
information easily available to students on the authors included. For this
reason, the present writer was startled recently when he came across a
quotation from one introduction included in Lehmann's
collection, in which the biographical information was
inaccurate and puzzling. It is discussed here because the actual facts are both
interesting and perhaps suggestive in several more general
senses.</FONT></DIV>
<DIV><FONT face=Arial size=2> The
section in question is devoted to the work of Hermann Grassmann. Lehmann
states on p.110 that "Hermann Grassmann (1809-1877) was a banker who was
compelled to retire because of tuberculosis." Unfortunately, the only accurate
facts in this sentence are Grassmann's name and dates. Next it is
stated that "In his leisure he occupied himself with mathematics and
linguistics", which is merely but regrettably misleading as to the place of
these occupations in his life.</FONT></DIV>
<DIV><FONT face=Arial size=2> Who was
Hermann Guenther Grassmann? Born on April 15 of 1809, he came from a scholarly
German family resident in Pomeranian Stettin, now the Polish city of
Szczecin. Starting the University of Berlin at the age of 18, he studied
theology and classics there for three years, but in 1830 he returned to Stettin,
where he undertook </FONT><FONT face=Arial size=2>private study of physics
and mathematics, natural history, theology, and philology, to prepare himself
for a career of secondary teaching. In the spring of 1832, he entered his
lifetime career of gymnasium (advanced high school) teaching when he began as
assistant teacher at the Stettin gymnasium. During the academic year 1834-1835,
he taught at the technical high school in Berlin, succeeding the geometer
Jakob Steiner when Steiner moved to a professorship at the University of
Berlin. In 1835, he returned to </FONT><FONT face=Arial size=2>Stettin, where he
joined the faculty of the Otto Schule gymnasium. He spent his life as a
gymnasium-level teacher, never working in a bank and never able to
obtain a university position. He did not retire, but remained a
gymnasium instructor (with the nominal rank of professor) until he died on
September 26, 1877 from cardiac problems rather than
from tuberculosis. </FONT></DIV>
<DIV><FONT face=Arial size=2>
Grassmann's life and mathematical contributions are recounted in a number
</FONT></DIV>
<DIV><FONT face=Arial size=2>of standard references. Particularly valuable is
the eight-page survey by W.Burau and C.J.Scriba, pp.192-199 in vol. XV of
the <U>Dictionary of Scientific Biography</U> ed. C.C. Gillispie
(Scribner's, 1978), which is drawn on in the
present discussion. His mathematical work on vector analysis
(the first version published in 1844 is usually referred to as
<U>Ausdehnungslehre</U>) was so far ahead of its time that Grassmann's
limited professional advancement is attributed as much to the lack of
specialists qualified to assess his work as to the notorious difficulty of
its exposition. </FONT><FONT face=Arial size=2>This work is the main focus of
the third chapter of Michael J. Crowe's solid <U>History of Vector
Analysis </U> (1967,1985, Dover). To give a notion of the
significance of Grassmann's mathematical work, it is useful to quote Crowe
(1985, pp.54-55): "What Grassmann created was above all a mathematical
system, not just a new mathematical idea or theorem. His creative act cannot be
compared with such mathematical discoveries as the Pythagorean theorem or
Newton's version of the calculus. Rather it is best thought of as comparable to
such creations as non-Euclidean geometry or Boolean algebra." Because of its
breadth and generality, it is difficult to summarize, and those interested
may well refer to the discussions of Burau and Scriba (1978), of Crowe
(1985) cited above, and of Klein (1926) cited
below. Grassmann's </FONT><FONT face=Arial size=2>original work in
physics was better recognized than his mathematics during most of his lifetime.
Crowe (1985)notes that by 1860, only five mathematicians in Europe
are known to have come to appreciate Grassmann's work to some extent. When
he was elected in 1864 to membership in the Leopoldina, founded in 1652 as
the oldest scientific society of Germany, it was for his contributions to
physics, not his mathematics.</FONT></DIV>
<DIV><FONT face=Arial size=2> Building
on his classical training, after the political storms of 1848-1849
Grassmann began the study of ancient Indo-European languages and comparative
linguistics for which he is known to linguists. In 1854, his musical gifts
contributed to his development of a theory of vowel
acoustics in research which was limited by the available acoustic
equipment, but won the respect of Helmholtz, the contemporary physicist most
qualified to judge it, according to Klein (1926). In fact, Grassmann's
theory of vowels is considered a substantial and independent anticipation
of that of Helmholtz (1859). It stimulated an experimental investigation by
J. Lahr in a doctoral dissertation of 1885 (<U>Die Grassmansche
Vocaltheorie im Lichte des Experiments</U> Leipzig 1885; p.94 in <U>
Wiedemanns Annalen</U> Bd. 26, 1886; cf. P.Gruetzner p.468 in <U>Ergebnissse der
Physiologie</U> I, 1902). In the early 1860s came Grassmann's studies of
comparative Indo-European phonology, which are the main reference of his
fame in linguistics for Grassmann's Law, although Lehmann justly points out
partial predecessors in this area. </FONT></DIV>
<DIV><FONT face=Arial size=2> Disappointed
by the poor reception of his second exposition of his main
mathematical work in 1862, Grassmann changed the main focus of
</FONT></DIV>
<DIV><FONT face=Arial size=2>his original investigations, turning to the study
of the Rig Veda, the most archaic document of Vedic Sanskrit. Although his
metric translation of the Rig Veda into German (Berlin 1876-77) is now not well
known, his <U>Woerterbuch zum Rig-Veda</U> (Leipzig 1873-1875), still
available in two different editions published on opposite sides of the
earth, is a monument to his </FONT><FONT face=Arial size=2>labor and analytic
penetration of the text.</FONT></DIV>
<DIV><FONT face=Arial
size=2> Publication of
Grassmann's works on Sanskrit was followed by membership in the
American Oriental Society and an honorary doctorate from the University of
Tuebingen. His death was recognized in obituaries whose authors included the
linguists B. Delbrueck (<U>Augsburger Allg. Zeit.</U> 1877, No.291,
supp.) and August Leskien (jointly with the mathematician Moritz
Kantor in <U>Allg. Deutsche Biographie</U> IX, pp.595-598, Leipzig 1879).
Delbrueck would recall Grassmann later </FONT><FONT face=Arial size=2>as "this
remarkable scholar, who in a certain respect stands unique among us"
(<U>Einleitung in das Studium der indogermanischen Sprachen</U> 6th ed.
Leipzig 1919, p.124). </FONT><FONT face=Arial size=2>During the years
just before his death, increasing recognition of his mathematical work must have
brought some satisfaction, and in the year of his death (1877) he revised the
1844 version of his work on vector analysis, in an edition which
appeared in 1878. </FONT></DIV>
<DIV><FONT face=Arial
size=2> Those who wish to
assess Grassmann's position in mathematics have only to turn to standard
histories, such as E.T.Bell's <U>The Development of Mathematics</U>
</FONT><FONT size=2><FONT face=Arial>(2nd ed. McGraw-Hill 1945 pp.198-206),
C.B.Boyer and U.Merzbach's modern <U>History of
Mathematics</U></FONT></FONT><FONT face=Arial size=2> </FONT><FONT face=Arial
size=2>(2nd ed. Wiley 1991, pp.584-586), and Felix Klein's
respectful 9-page treatment in his <U>Vorlesungen ueber die Entwicklung der
Mathematik im 19. Jahrhundert</U></FONT><FONT face=Arial size=2> (Goettingen,
1926, pp.173-182). Klein admitted the significance of Grassmann's influence on
his own Erlangen Program in geometry of 1872, his inaugural program
pronounced when he assumed his professorship in Erlangen. However, it is fair to
say that the nature and significance of Grassmann's linguistic work
have not always been clearly appraised from the mathematical side: Boyer
and Merzbach describe his linguistic work as "being a specialist in Sanskrit
literature" (1991, p.585).</FONT></DIV>
<DIV><FONT face=Arial size=2> From a
linguistic point of view, it is intriguing to comment on a relevant parallel to
Grassmann's work. </FONT><FONT face=Arial size=2>The closest contemporary
parallel to the primary mathematical work of Grassmann is the work of the Irish
mathematician William Rowan Hamilton (1805-1865) known under the label
of quaternions. Klein (1926), Bell (1945), and Crowe (1967) provide more or
less detailed expositions of this work </FONT><FONT face=Arial size=2>and
its relation to that of Grassmann. Hamilton, like Grassmann, was
linguistically gifted; by the age of thirteen, he had
acquired one language for every year of his age, under the tutelage of his
uncle, the Reverend James Hamilton. Unlike Grassmann, however,
Hamilton's linguistic talents were never a focus of his mature
original intellectual activity, except for his composition of mediocre
poetry. </FONT><FONT face=Arial size=2>Also unlike Grassmann, who
published a number of books for use only as high-school
textbooks, turned his genius too often to committee-work and
political, social, and church issues, and was the center of a productive
family, the reclusive Hamilton, proclaimed a professor before he
finished his university program, left behind at his death many unpublished
papers and about 60 large
manuscript books of unpublished mathematical research
(E.T.Bell <U>Men of Mathematics</U> Simon and Schuster 1937,
pp.340-361).</FONT></DIV>
<DIV><FONT face=Arial size=2> It
is tempting to wonder whether this parallelism of achievement in a particular
mathematical field and linguistic gifts
is founded on cognitive parallels between mathematics
and logic on the one hand and the formal aspect of linguistic structure on
the other hand. A counter-argument might be that a third
scientist who contributed much to vector analysis, the Yale
mathematical physicist Josiah Willard Gibbs (1839-1903), who correctly
valued Grassmann's work as much more general than the corresponding work of
Hamilton (Burau and Scriba, 1978; Bell, 1945; L.P.Wheeler <U>Josiah Willard
Gibbs</U>, 2nd ed. Yale U.P. 1952, ch. 7), is not known for such outstanding
linguistic achievements, although his studies in Paris and Heidelberg
(1866-1869) surely demanded practical linguistic competence. Neither were
such linguistic gifts displayed by the British physicist Oliver
Heaviside (1850-1925), a more modern exponent of vector methods, who
did not in any case have the advantages either of university education or
of a learned family background (pp.211-212 in <U>Dictionary of Scientific
Biography</U> VI, ed. C.C.Gillispie, Scribner 1972). </FONT></DIV>
<DIV><FONT face=Arial size=2> Now
it is useful to recur to the
apparently disparate combination of</FONT></DIV>
<DIV><FONT face=Arial size=2>academic areas in which Grassmann was active.
In terms of his home environment, Grassmann's father </FONT><FONT face=Arial
size=2>Justus Guenther Grassmann taught physics and mathematics in the
Stettin gymnasium, and was an original researcher in crystallography
and combinatorial mathematics (M. Cantor pp.598-599 in <U>Allg. deutsche
Biographie</U> IX, Leipzig 1879). Thus it is reasonable to infer that
although he was not at all regarded by his father as the prodigy which Hamilton
was, Hermann Grassmann received a substantial early orientation in these
scientific fields, from which his study of theology and classics might
seem to be an intellectual deviation, if he had not come from a long line
of Protestant pastors whose traditions included both
scholarly and artistic interests (Klein, 1926). </FONT></DIV>
<DIV><FONT face=Arial
size=2> When Grassmann went to
the still young University of Berlin in 1827, he studied partly
under August Neander (1789-1850), who devoted himself to early Christian
church history. Grassmann also studied under Neander's own teacher
Friedrich Schleiermacher (1768-1834), who was still quite active and must have
offered much broader intellectual perspectives as a philosopher and theologian
who is regarded as a foundational thinker in the history of Protestant
Christianity (K. Barth <U>Protestant Theology in the Nineteenth
Century</U> 1952, trans. 1972 SCM Press). The influence of both Grassmann's
father and of Schleiermacher on Grassmann's mathematics has been analyzed
(Crowe, 1985). It has also been argued that the philosophical aspect of
Schleiermacher's training made itself felt later in the abstract
philosophical character of Grassmann's mathematical exposition, much to the
detriment of its acceptance. </FONT></DIV>
<DIV><FONT face=Arial size=2>
Grassmann's study of classics in Berlin took place under the aegis of
a scholar who would be quite distinctive among classicists of any
time. August Boeckh (1785-1867) was in his bloom as a scholar
who defined philology as the </FONT><FONT face=Arial
size=2>reconstruction of all aspects of a past
culture. Boeckh focussed in much of his most brilliant work on
quantitative phenomena, on weights, measures, and coinage standards in
antiquity, on the economics and chronology of ancient Greece, and on Greek
music and poetic metrics. Boeckh liked to think of both
physical and metaphysical systems as founded on astronomy. For
this reason, he dedicated a series of investigations to Greek astronomy
(cf. U. von Wilamowitz-Moellendorff <U>History of Classical Scholarship</U>
1921, trans. Johns Hopkins 1982). There is a probable continuity here in
that Grassmann's first major physical work, carried out in 1839-40, was on
the theory of the tides, which result from the gravitational attraction of the
moon and sun, and was carried out partly in relation to his study of the
analytic mechanics of Lagrange and of the celestial mechanics
of Laplace. Another point of continuity is that according to Cantor
and Leskien (1879) Grassmann had continued to study mathematics privately
in Berlin, as well as attending to his formal studies.</FONT></DIV>
<DIV><FONT face=Arial size=2> In
a more general sense, the nature of Grassmann's work in linguistics and
Sanskrit in relation to mathematics can be appreciated by anyone who has
ever been immersed in the formalisms of logical derivation (e.g. W. Quine
<U>Methods of Logic</U> , Holt 1959) or computer programming, which
are closely related to the logical substrate of stemmatics in
textual criticism and specification of sequences of linguistic changes in
historical linguistics. Given his exposure to technical aspects of
philology under Boeckh, it is quite understandable that Grassmann
should turn to making useable at least a nucleus of the data which were
indispensable for comparative linguistics, the linguistic forms of the Rig
Veda. Thus his linguistic and mathematical studies were not
entirely unrelated, but overlapped to a considerable extent in
that they represented different aspects of a single fundamental mode of
thought. </FONT></DIV>
<DIV><FONT face=Arial size=2> </FONT><FONT
face=Arial size=2> It is likely that many linguists have
acquired from varied sources a more precise appreciation of </FONT><FONT
face=Arial size=2>Grassmann's background and contributions than is
available in Lehmann's book. This was certainly true for
the scholar from whom the present writer first heard of
Grassmann's achievements both in mathematics and comparative linguistics,
the </FONT><FONT face=Arial size=2>Indo-Europeanist Joshua Whatmough.
Whatmough's own interest in mathematics sensitized him to the duality of
Grassmann's work. </FONT></DIV>
<DIV><FONT face=Arial size=2> Sadly,
Grassman is recorded not to have been a great teacher in the secondary
school context, although he eventually advanced to a senior position in the
Stettin gymnasium (Klein, 1926; Burau and Scriba, 1978).
Whether the match of genius and such routine work is productive is
uncertain. In the older German educational system, it </FONT><FONT
face=Arial size=2>was sometimes possible for such distinguished figures as
the mathematicians Jakob Steiner and Karl Weierstrass </FONT><FONT
face=Arial size=2>to advance from secondary school teaching to a university
post. As another example of someone who achieved greatly while remaining within
the secondary educational system, later it was possible for
the outstanding Hellenist Edwin Mayser to issue his six
well-nourished volumes on the language of the Ptolemaic Greek papyri mainly from
a Stuttgart gymnasium. It is instructive to reflect on whether such things
would be likely to happen in modern American schools.</FONT></DIV>
<DIV><FONT face=Arial size=2>
Grassmann's limited professional advancement may ring a strong chord for
those present-day linguists who must make their way outside of university
circles. </FONT></DIV>
<DIV><FONT face=Arial size=2>Felix Klein (1926, p.174), distinguished in his
later years as a statesman of science, made a different point in this regard in
discussing Grassman's life: </FONT></DIV>
<DIV><FONT face=Arial size=2>"We academics grow up in acute competition
with those striving in the same direction, like a tree in the middle of a
forest, which must grow high and thin in order to be able to exist at all and
capture its place in the light and air; on the other hand, he who stands
alone, like Grassmann, can grow out fully in all directions, bring his
essence and work to a harmonious perfection and rounding
out." </FONT><FONT face=Arial size=2>Although he lacked the collegial
stimulation of the immediate university environment, Grassmann did not
lack a drive towards creative intellectual discovery. This
follows from words of Klein stated elsewhere in answer to a request
for an explanation of the nature of mathematical discovery, words which have
broader application and must have applied well to Grassmann: "You must have a
problem", said Klein. </FONT><FONT face=Arial size=2>"Choose one definite
objective and drive ahead toward it.You may never reach your goal, but you
will find something of interest on the way." (Bell, 1937, p.419).
</FONT></DIV>
<DIV> </DIV>
<DIV><FONT face=Arial
size=2>
Donald S. Cooper, Ph.D. </FONT></DIV>
<DIV><FONT face=Arial
size=2> </FONT></DIV>
<DIV><FONT face=Arial size=2>
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