Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German

polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose knowledge spans a substantial number of subjects, known to draw on complex bodies of knowledge to solve specific pro ...

known in his day as a

linguist Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure. Linguis ...

and now also as a

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

. He was also a

physicist A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate caus ...

, general scholar, and publisher. His mathematical work was little noted until he was in his sixties.

Biography

Hermann Grassmann was the third of 12 children of Justus Günter Grassmann, an

ordained Ordination is the process by which individuals are consecrated, that is, set apart and elevated from the laity class to the clergy, who are thus then authorized (usually by the denominational hierarchy composed of other clergy) to perform va ...

minister who taught mathematics and physics at the

Stettin Szczecin (, , german: Stettin ; sv, Stettin ; Latin language, Latin: ''Sedinum'' or ''Stetinum'') is the capital city, capital and largest city of the West Pomeranian Voivodeship in northwestern Poland. Located near the Baltic Sea and the Po ...

Gymnasium, where Hermann was educated. Grassmann was an undistinguished student until he obtained a high mark on the examinations for admission to

Prussia Prussia, , Old Prussian: ''Prūsa'' or ''Prūsija'' was a German state on the southeast coast of the Baltic Sea. It formed the German Empire under Prussian rule when it united the German states in 1871. It was ''de facto'' dissolved by an em ...

n universities. Beginning in 1827, he studied theology at the

University of Berlin Humboldt-Universität zu Berlin (german: Humboldt-Universität zu Berlin, abbreviated HU Berlin) is a German public research university in the central borough of Mitte in Berlin. It was established by Frederick William III on the initiative o ...

, also taking classes in

classical languages A classical language is any language with an independent literary tradition and a large and ancient body of written literature. Classical languages are typically dead languages, or show a high degree of diglossia, as the spoken varieties of the ...

, philosophy, and literature. He does not appear to have taken courses in mathematics or

physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...

. 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. 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 ''Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik'', here referred to as A1. 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. Over the next four years, Grassmann passed examinations enabling him to teach mathematics,

chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions ...

, and

mineralogy Mineralogy is a subject of geology specializing in the scientific study of the chemistry, crystal structure, and physical (including optical) properties of minerals and mineralized artifacts. Specific studies within mineralogy include the proces ...

at all secondary school levels. 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

Ernst Kummer Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a '' gymnasium'', the German equivalent of ...

for his opinion of Grassmann. Kummer wrote back saying that Grassmann's 1846 prize essay (see below) 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. Starting during the political turmoil in Germany, 1848–49, Hermann and his brother Robert published a Stettin newspaper, ''

Deutsche Wochenschrift für Staat, Kirche und Volksleben Deutsch or Deutsche may refer to: *''Deutsch'' or ''(das) Deutsche'': the German language, in Germany and other places *''Deutsche'': Germans, as a weak masculine, feminine or plural demonym *Deutsch (word), originally referring to the Germanic ve ...

'', calling for

German unification The unification of Germany (, ) was the process of building the modern German nation state with federal features based on the concept of Lesser Germany (one without multinational Austria), which commenced on 18 August 1866 with adoption of t ...

under a

constitutional monarchy A constitutional monarchy, parliamentary monarchy, or democratic monarchy is a form of monarchy in which the monarch exercises their authority in accordance with a constitution and is not alone in decision making. Constitutional monarchies dif ...

. (This eventuated in 1871.) After writing a series of articles on

constitutional law Constitutional law is a body of law which defines the role, powers, and structure of different entities within a State (polity), state, namely, the executive (government), executive, the parliament or legislature, and the judiciary; as well as th ...

, 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 University of Giessen, official name Justus Liebig University Giessen (german: Justus-Liebig-Universität Gießen), is a large public research university in Giessen, Hesse, Germany. It is named after its most famous faculty member, Justus von L ...

Mathematician

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 Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...

's ''

Traité de mécanique céleste ''Traité de mécanique céleste'' () is a five-volume treatise on celestial mechanics written by Pierre-Simon Laplace and published from 1798 to 1825 with a second edition in 1829. In 1842, the government of Louis Philippe gave a grant of 40,000 ...

'' and from

Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaMécanique analytique'', but expositing this theory making use of the

vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...

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 is now called

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...

and the notion of a

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...

. He went on to develop those methods in his A1 and A2. In 1844, Grassmann published his masterpiece (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 Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

is put into the algebraic form he advocated, the number three has no privileged role as the number of spatial

dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...

s; the number of possible dimensions is in fact unbounded. Fearnley-Sander describes Grassmann's foundation of linear algebra as follows: Following an idea of Grassmann's father, A1 also defined the

exterior product In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of the ...

, also called "combinatorial product" (in German: ''kombinatorisches Produkt'' or ''äußeres Produkt'' “outer product”), the key operation of an algebra now called

exterior algebra In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...

. (One should keep in mind that in Grassmann's day, the only

axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...

atic theory was

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...

, and the general notion of an

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...

had yet to be defined.) In 1878,

William Kingdon Clifford William Kingdon Clifford (4 May 18453 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his ...

joined this exterior algebra to

William Rowan Hamilton Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...

quaternions In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...

by replacing Grassmann's rule ''e_pe_p'' = 0 by the rule ''e_pe_p'' = 1. (For

, we have the rule ''i''² = ''j''² = ''k''² = −1.) For more details, see

Exterior algebra In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...

. A1 was a revolutionary text, too far ahead of its time to be appreciated. When Grassmann submitted it to apply for a professorship in 1847, the ministry asked

for a report. Kummer assured that there were good ideas in it, but found the exposition deficient and advised against giving Grassmann a university position. 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 curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...

s and

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...

s, 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 Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...

: 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). 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; his theory's four color laws are still taught, as Grassmann's laws. Grassmann's work on this subject was inconsistent with that of

Helmholtz Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Association, ...

. Grassmann also wrote on

crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wor ...

electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...

, and

mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...

. In 1861, Grassmann laid the groundwork for Peano's axiomatization of arithmetic in his ''Lehrbuch der Arithmetik''. 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

. The result, ''Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet'' (A2), fared no better than A1, even though A2 manner of exposition anticipates the textbooks of the 20th century.

Response

In the 1840s, mathematicians were generally unprepared to understand Grassmann's ideas. In the 1860s and 1870s various mathematicians came to ideas similar to that of Grassmann's, but Grassmann himself was not interested in mathematics anymore.

Adhémar Jean Claude Barré de Saint-Venant Adhémar Jean Claude Barré de Saint-Venant (23 August 1797 – 6 January 1886) was a mechanician and mathematician who contributed to early stress analysis and also developed the unsteady open channel flow shallow water equations, also known as ...

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. One of the first mathematicians to appreciate Grassmann's ideas during his lifetime was

Hermann Hankel Hermann Hankel (14 February 1839 – 29 August 1873) was a German mathematician. Having worked on mathematical analysis during his career, he is best known for introducing the Hankel transform and the Hankel matrix. Biography Hankel was born on 1 ...

, whose 1867 ''Theorie der complexen Zahlensysteme''. In 1872

Victor Schlegel Victor Schlegel (4 March 1843 – 22 November 1905) was a German mathematician. He is remembered for promoting the geometric algebra of Hermann Grassmann and for a method of visualizing polytopes called Schlegel diagrams. In the nineteenth centur ...

published the first part of his ''System der Raumlehre'', which used Grassmann's approach to derive ancient and modern results in

plane geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...

Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...

wrote a negative review of Schlegel's book citing its incompleteness and lack of perspective on Grassmann. Schlegel followed in 1875 with a second part of his ''System'' according to Grassmann, this time developing higher-dimensional geometry. Meanwhile, Klein was advancing his

Erlangen program In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' It is nam ...

, which also expanded the scope of geometry. Comprehension of Grassmann awaited the concept of

s, which then could express the

multilinear algebra Multilinear algebra is a subfield of mathematics that extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of ''p' ...

of his extension theory. To establish the priority of Grassmann over Hamilton,

Josiah Willard Gibbs Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in t ...

urged Grassmann's heirs to have the 1840 essay on tides published. A. N. Whitehead's first monograph, the ''Universal Algebra'' (1898), included the first systematic exposition in English of the theory of extension and the

. With the rise of

differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

the exterior algebra was applied to

differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...

s. In 1995 Lloyd C. Kannenberg published an English translation of The Ausdehnungslehre and Other works. For an introduction to the role of Grassmann's work in contemporary

mathematical physics Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...

see ''

The Road to Reality ''The Road to Reality: A Complete Guide to the Laws of the Universe'' is a book on modern physics by the British mathematical physicist Roger Penrose, published in 2004. It covers the basics of the Standard Model of particle physics, discussing ...

'' by

Roger Penrose Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, philosopher of science and Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics in the University of Oxford, an emeritus fello ...

Linguist

Grassmann's mathematical ideas began to spread only towards the end of his life. Thirty years after the publication of A1 the publisher wrote to Grassmann: “Your book ''Die Ausdehnungslehre'' has been out of print for some time. Since your work hardly sold at all, roughly 600 copies were used in 1864 as waste paper and the remaining few odd copies have now been sold out, with the exception of the one copy in our library”. Disappointed by the reception of his work in mathematical circles, Grassmann lost his contacts with mathematicians as well as his interest in geometry. The last years of his life he turned to historical

linguistics Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure. Linguis ...

and the study of

Sanskrit Sanskrit (; attributively , ; nominally , , ) is a classical language belonging to the Indo-Aryan branch of the Indo-European languages. It arose in South Asia after its predecessor languages had diffused there from the northwest in the late ...

. He wrote books on

German grammar The grammar of the German language is quite similar to that of the other Germanic languages. Although some features of German grammar, such as the formation of some of the verb forms, resemble those of English, German grammar differs from that of ...

, collected folk songs, and learned Sanskrit. He wrote a 2,000-page dictionary and a translation of the ''

Rigveda The ''Rigveda'' or ''Rig Veda'' ( ', from ' "praise" and ' "knowledge") is an ancient Indian collection of Vedic Sanskrit hymns (''sūktas''). It is one of the four sacred canonical Hindu texts (''śruti'') known as the Vedas. Only one Sh ...

'' (more than 1,000 pages), which earned him a membership of the American Orientalists' Society. In modern Rigvedic studies, Grassmann's work is often cited. In 1955 the third edition of his dictionary to ''Rigveda'' was issued. Grassmann also noticed and presented a

phonological rule A phonological rule is a formal way of expressing a systematic phonological or morphophonological process or diachronic sound change in language. Phonological rules are commonly used in generative phonology as a notation to capture sound-related o ...

that exists in both Sankskrit and Greek. In his honor, this phonological rule is known as Grassmann's law. These philological accomplishments were honored during his lifetime; he was elected to the

American Oriental Society The American Oriental Society was chartered under the laws of Massachusetts on September 7, 1842. It is one of the oldest learned societies in America, and is the oldest devoted to a particular field of scholarship. The Society encourages basic ...

and in 1876, he received an honorary doctorate from the

University of Tübingen The University of Tübingen, officially the Eberhard Karl University of Tübingen (german: Eberhard Karls Universität Tübingen; la, Universitas Eberhardina Carolina), is a public research university located in the city of Tübingen, Baden-Wü ...

Publications

* A1: ** ** * * * A2: **1862.
Die Ausdehnungslehre. Vollständig und in strenger Form begründet.
'. Berlin: Enslin. ** English translation, 2000, by Lloyd Kannenberg, ''Extension Theory'',

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

, * 1873.
Wörterbuch zum Rig-Veda
'. Leipzig: Brockhaus. * 1876–1877. ''Rig-Veda''. Leipzig: Brockhaus. Translation in two vols.
vol. 1
published 1876, vol. 2 published 1877. * 1894–1911.
Gesammelte mathematische und physikalische Werke
'' in 3 vols. Friedrich Engel ed. Leipzig: B.G. Teubner. Reprinted 1972, New York: Johnson.

Citations

References

* * * * * * * * * * * * * * Note: Extensiv
online bibliography
revealing substantial contemporary interest in Grassmann's life and work. References each chapter in Schubring.

External links

* The MacTutor History of Mathematics archive: ** *

Discusses the role of Grassmann and other 19th century figures in the invention of linear algebra and vector spaces.

s home page.

From Past to Future: Grassmann's Work in Context
"The Grassmann method in projective geometry"
– A compilation of English translations of three notes by Cesare Burali-Forti on the application of Grassmann's exterior algebra to projective geometry
C. Burali-Forti, "Introduction to Differential Geometry, following the method of H. Grassmann"
(English translation of book by an early disciple of Grassmann)
"Mechanics, according to the principles of the theory of extension"
– An English translation of one Grassmann's papers on the applications of exterior algebra {{DEFAULTSORT:Grassmann, Hermann 1809 births 1877 deaths Scientists from Szczecin 19th-century German mathematicians Linear algebraists Linguists from Germany 19th-century German physicists People from the Province of Pomerania Color scientists Humboldt University of Berlin alumni Translators from Sanskrit 19th-century translators