**Hermann Günther Grassmann** was a

GermanThe Germans are a Germanic ethnic group native to Central Europe. The English term Germans has referred to the German-speaking population of the Holy Roman Empire since the Late Middle Ages....

polymathA polymath is a person whose expertise spans a significant number of different subject areas. In less formal terms, a polymath may simply be someone who is very knowledgeable...

, renowned in his day as a

linguistLinguistics is the scientific study of human language. Linguistics can be broadly broken into three categories or subfields of study: language form, language meaning, and language in context....

and now also admired as a

mathematicianMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

. He was also a

physicistPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

,

neohumanistHumanism is an approach in study, philosophy, world view or practice that focuses on human values and concerns. In philosophy and social science, humanism is a perspective which affirms some notion of human nature, and is contrasted with anti-humanism....

, general scholar, and publisher. His mathematical work was not recognized in his lifetime.

## Biography

Grassmann was the third of 12 children of Justus Günter Grassmann, an ordained minister who taught mathematics and physics at the Stettin

GymnasiumA gymnasium is a type of school providing secondary education in some parts of Europe, comparable to English grammar schools or sixth form colleges and U.S. college preparatory high schools. The word γυμνάσιον was used in Ancient Greece, meaning a locality for both physical and intellectual...

, 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

PrussiaPrussia was a German kingdom and historic state originating out of the Duchy of Prussia and the Margraviate of Brandenburg. For centuries, the House of Hohenzollern ruled Prussia, successfully expanding its size by way of an unusually well-organized and effective army. Prussia shaped the history...

n universities. Beginning in 1827, he studied

theologyTheology is the systematic and rational study of religion and its influences and of the nature of religious truths, or the learned profession acquired by completing specialized training in religious studies, usually at a university or school of divinity or seminary.-Definition:Augustine of Hippo...

at the University of Berlin, also taking classes in

classical languagesClassics is the branch of the Humanities comprising the languages, literature, philosophy, history, art, archaeology and other culture of the ancient Mediterranean world ; especially Ancient Greece and Ancient Rome during Classical Antiquity Classics (sometimes encompassing Classical Studies or...

,

philosophyPhilosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...

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

mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

or

physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

.

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** (see references).

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

mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

,

physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

,

chemistryChemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....

, and

mineralogyMineralogy is the study of chemistry, crystal structure, and physical properties of minerals. Specific studies within mineralogy include the processes of mineral origin and formation, classification of minerals, their geographical distribution, as well as their utilization.-History:Early writing...

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

KummerErnst Eduard Kummer 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 high school, where he inspired the mathematical career of Leopold Kronecker.-Life:Kummer...

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.

During the political turmoil in Germany, 1848-49, Hermann and Robert Grassmann published a Stettin newspaper calling for German unification under a

constitutional monarchyConstitutional monarchy is a form of government in which a monarch acts as head of state within the parameters of a constitution, whether it be a written, uncodified or blended constitution...

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

constitutional lawConstitutional law is the body of law which defines the relationship of different entities within a state, namely, the executive, the legislature and the judiciary....

, 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 GiessenThe University of Giessen is officially called the Justus Liebig University Giessen after its most famous faculty member, Justus von Liebig, the founder of modern agricultural chemistry and inventor of artificial fertiliser.-History:The University of Gießen is among the oldest institutions of...

.

## 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's

*Mécanique céleste* and from

LagrangeJoseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...

'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 algebraLinear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

and the notion of a

vector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

. 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* [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

geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

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

dimensionIn physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

s; the number of possible dimensions is in fact unbounded.

Fearnley-Sander (1979) describes Grassmann's foundation of linear algebra as follows:

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 algebraIn mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs...

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

axiomIn traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...

atic theory was

Euclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

, and the general notion of an

abstract algebraAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

had yet to be defined.) In 1878,

William Kingdon CliffordWilliam Kingdon Clifford FRS 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 honour, with interesting applications in contemporary mathematical physics...

joined this exterior algebra to

William Rowan HamiltonSir William Rowan Hamilton was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques...

's quaternions by replacing Grassmann's rule

*e*_{p}e_{p} = 0 by the rule

*e*_{p}e_{p} = 1.

*(For quaternions, we have the rule *i

^{2}* = *j

^{2}* = *k

^{2}* = -1.)* For more details, see

exterior algebraIn mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs...

.

**A1** was a revolutionary text, too far ahead of its time to be appreciated. Grassmann submitted it as a

Ph. D.Doctor of Philosophy, abbreviated as Ph.D., PhD, D.Phil., or DPhil , in English-speaking countries, is a postgraduate academic degree awarded by universities...

thesis, but

MöbiusAugust Ferdinand Möbius was a German mathematician and theoretical astronomer.He is best known for his discovery of the Möbius strip, a non-orientable two-dimensional surface with only one side when embedded in three-dimensional Euclidean space. It was independently discovered by Johann Benedict...

said he was unable to evaluate it and forwarded it to

Ernst KummerErnst Eduard Kummer 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 high school, where he inspired the mathematical career of Leopold Kronecker.-Life: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,

MöbiusAugust Ferdinand Möbius was a German mathematician and theoretical astronomer.He is best known for his discovery of the Möbius strip, a non-orientable two-dimensional surface with only one side when embedded in three-dimensional Euclidean space. It was independently discovered by Johann Benedict...

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 lawIn optics, Grassmann's law is an empirical result about human color perception: that chromatic sensation can be described in terms of an effective stimulus consisting of linear combinations of different light colors...

. Grassmann's work on this subject was inconsistent with that of Helmholtz. Grassmann also wrote on

crystallographyCrystallography is the experimental science of the arrangement of atoms in solids. The word "crystallography" derives from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and grapho = write.Before the development of...

,

electromagnetismElectromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...

, and

mechanicsMechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....

.

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 (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 algebraLinear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

. 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 HankelHermann Hankel was a German mathematician who was born in Halle, Germany and died in Schramberg , Imperial Germany....

, whose 1867

*Theorie der complexen Zahlensysteme* helped make Grassmann's ideas better known. This work

Grassmann's mathematical methods were slow to be adopted but they directly influenced

Felix KleinChristian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...

and

Élie CartanÉlie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...

. 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 algebraIn mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs...

. The theory of extension led to the development of

differential formIn the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...

s and to the application of such forms to

analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

and

geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

. Differential geometry makes use of the

exterior algebraIn mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs...

. For an introduction to the role of Grassmann's work in contemporary

mathematical physicsMathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...

, see Penrose (2004: chpts. 11, 12).

Adhémar Jean Claude Barré de Saint-VenantAdhémar Jean Claude Barré de Saint-Venant was a mechanician and mathematician who contributed to early stress analysis and also developed the one-dimensional unsteady open channel flow shallow water equations or Saint-Venant equations that are a fundamental set of equations used in modern...

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.

## Linguist

Disappointed at the inability of others to recognize the importance of his mathematics, Grassmann turned to historical

linguisticsLinguistics is the scientific study of human language. Linguistics can be broadly broken into three categories or subfields of study: language form, language meaning, and language in context....

. He wrote books on German grammar, collected folk songs, and learned

SanskritSanskrit , is a historical Indo-Aryan language and the primary liturgical language of Hinduism, Jainism and Buddhism.Buddhism: besides Pali, see Buddhist Hybrid Sanskrit Today, it is listed as one of the 22 scheduled languages of India and is an official language of the state of Uttarakhand...

. His dictionary and his translation of the

RigvedaThe Rigveda is an ancient Indian sacred collection of Vedic Sanskrit hymns...

(still in print) were recognized among philologists. He devised a sound law of

Indo-European languagesThe Indo-European languages are a family of several hundred related languages and dialects, including most major current languages of Europe, the Iranian plateau, and South Asia and also historically predominant in Anatolia...

, named

Grassmann's LawGrassmann's law, named after its discoverer Hermann Grassmann, is a dissimilatory phonological process in Ancient Greek and Sanskrit which states that if an aspirated consonant is followed by another aspirated consonant in the next syllable, the first one loses the aspiration...

in his honor.

These philological accomplishments were honored during his lifetime; he was elected to the

American Oriental SocietyThe 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....

and in 1876, he received an honorary doctorate from the University of Tübingen.

## See also

- Exterior algebra
In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs...

- Grassmann number
In mathematical physics, a Grassmann number, named after Hermann Grassmann, is a mathematical construction which allows a path integral representation for Fermionic fields...

- Grassmannian
In mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological...

- Grassmann's law
Grassmann's law, named after its discoverer Hermann Grassmann, is a dissimilatory phonological process in Ancient Greek and Sanskrit which states that if an aspirated consonant is followed by another aspirated consonant in the next syllable, the first one loses the aspiration...

(phonology)
- Grassmann's law (optics)
In optics, Grassmann's law is an empirical result about human color perception: that chromatic sensation can be described in terms of an effective stimulus consisting of linear combinations of different light colors...

## External links