Who was Hermann Grassmann?

[Donald Cooper]

Dear Histling:
        Winfred Lehmann's Reader in Nineteenth Century Historical Indo-European Linguistics  (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.
        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.
       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 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 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.  
        Grassmann's life and mathematical contributions are recounted in a number 
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 Dictionary of Scientific Biography 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 Ausdehnungslehre) 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. This work is the main focus of the third chapter of Michael J. Crowe's solid History of Vector Analysis  (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 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.
        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 (Die Grassmansche Vocaltheorie im Lichte des Experiments Leipzig 1885; p.94 in  Wiedemanns Annalen Bd. 26, 1886; cf. P.Gruetzner p.468 in Ergebnissse der Physiologie 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.   
       Disappointed by the poor reception of his second exposition of his main mathematical work in 1862, Grassmann changed the main focus of 
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 Woerterbuch zum Rig-Veda (Leipzig 1873-1875), still available in two different editions published on opposite sides of the earth, is a monument to his labor and analytic penetration of the text.
        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 (Augsburger Allg. Zeit.  1877, No.291, supp.) and August Leskien (jointly with the mathematician Moritz Kantor in Allg. Deutsche Biographie IX, pp.595-598, Leipzig 1879). Delbrueck would recall Grassmann later as "this remarkable scholar, who in a certain respect stands unique among us" (Einleitung in das Studium der indogermanischen Sprachen 6th ed. Leipzig 1919, p.124).  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. 
        Those who wish to assess Grassmann's position in mathematics have only to turn to standard histories, such as E.T.Bell's The Development of Mathematics (2nd ed. McGraw-Hill 1945 pp.198-206), C.B.Boyer and U.Merzbach's modern History of Mathematics (2nd ed. Wiley 1991, pp.584-586), and Felix Klein's respectful 9-page treatment in his Vorlesungen ueber die Entwicklung der Mathematik im 19. Jahrhundert (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).
        From a linguistic point of view, it is intriguing to comment on a relevant parallel to Grassmann's work. 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 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. 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 Men of Mathematics Simon and Schuster 1937, pp.340-361).
         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 Josiah Willard Gibbs, 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 Dictionary of Scientific Biography VI, ed. C.C.Gillispie, Scribner 1972). 
        Now it is useful to recur to the apparently disparate combination of
academic areas in which Grassmann was active. In terms of his home environment, Grassmann's father 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 Allg. deutsche Biographie 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). 
        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 Protestant  Theology in the Nineteenth Century 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.  
        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 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 History of Classical Scholarship 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.
        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 Methods of Logic , 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.   
        It is likely that many linguists have acquired from varied sources a more precise appreciation of 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 Indo-Europeanist Joshua Whatmough. Whatmough's own interest in mathematics sensitized him to the duality of Grassmann's work.  
        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 was sometimes possible for such distinguished  figures as the mathematicians Jakob Steiner and Karl Weierstrass 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.
        Grassmann's limited professional advancement may ring a strong chord for those present-day linguists who must make their way outside of university circles. 
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: 
"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." 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. "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).  

                                                       Donald S. Cooper, Ph.D. 
          
         
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