Eduard Study ( ), more properly Christian Hugo Eduard Study (March 23, 1862 – January 6, 1930), was a German

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 ...

known for work on

invariant theory Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descri ...

of ternary forms (1889) and for the study of

spherical trigonometry Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are grea ...

. He is also known for contributions to space geometry, hypercomplex numbers, and criticism of early physical chemistry. Study was born in

Coburg Coburg () is a town located on the Itz river in the Upper Franconia region of Bavaria, Germany. Long part of one of the Thuringian states of the Wettin line, it joined Bavaria by popular vote only in 1920. Until the revolution of 1918, it ...

in the Duchy of

Saxe-Coburg-Gotha Saxe-Coburg and Gotha (german: Sachsen-Coburg und Gotha), or Saxe-Coburg-Gotha (german: Sachsen-Coburg-Gotha, links=no ), was an Ernestine, Thuringian duchy ruled by a branch of the House of Wettin, consisting of territories in the present- ...

Career

Eduard Study began his university career in Jena, Strasbourg, Leipzig, and Munich. He loved to study biology, especially entomology. He was awarded the doctorate in mathematics at the

University of Munich The Ludwig Maximilian University of Munich (simply University of Munich or LMU; german: Ludwig-Maximilians-Universität München) is a public research university in Munich, Germany. It is Germany's sixth-oldest university in continuous operatio ...

in 1884.

Paul Gordan __NOTOC__ Paul Albert Gordan (27 April 1837 – 21 December 1912) was a Jewish-German mathematician, a student of Carl Jacobi at the University of Königsberg before obtaining his PhD at the University of Breslau (1862),. and a professor ...

, an expert in

was at Leipzig, and Study returned there as Privatdozent. In 1888 he moved to Marburg and in 1893 embarked on a speaking tour in the U.S.A. He appeared at a Congress of Mathematicians in Chicago as part of the

World's Columbian Exposition The World's Columbian Exposition (also known as the Chicago World's Fair) was a world's fair held in Chicago in 1893 to celebrate the 400th anniversary of Christopher Columbus's arrival in the New World in 1492. The centerpiece of the Fair, hel ...

and took part in mathematics at

Johns Hopkins University Johns Hopkins University (Johns Hopkins, Hopkins, or JHU) is a private research university in Baltimore, Maryland. Founded in 1876, Johns Hopkins is the oldest research university in the United States and in the western hemisphere. It consi ...

. Back in Germany, in 1894, he was appointed extraordinary professor at Göttingen. Then he gained the rank of full professor in 1897 at Greifswald. In 1904 he was called to the

University of Bonn The Rhenish Friedrich Wilhelm University of Bonn (german: Rheinische Friedrich-Wilhelms-Universität Bonn) is a public research university located in Bonn, North Rhine-Westphalia, Germany. It was founded in its present form as the ( en, Rhine ...

as the position held by

Rudolf Lipschitz Rudolf Otto Sigismund Lipschitz (14 May 1832 – 7 October 1903) was a German mathematician who made contributions to mathematical analysis (where he gave his name to the Lipschitz continuity condition) and differential geometry, as well as numbe ...

was vacant. There he settled until retirement in 1927. Study gave a

plenary address at the International Congress of Mathematicians This is a list of International Congresses of Mathematicians Plenary and Invited Speakers. Being invited to talk at an International Congress of Mathematicians has been called "the equivalent, in this community, of an induction to a hall of fame." ...

in 1904 at Heidelberg and another in 1912 at Cambridge, UK.

Euclidean space group and dual quaternions

In 1891 Eduard Study published "Of Motions and Translations, in two parts". It treats the

Euclidean group In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations ...

E(3). The second part of his article introduces the

associative algebra In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...

of dual quaternions, that is numbers :

q = a + bi + cj + dk \!

where ''a'', ''b'', ''c'', and ''d'' are

dual numbers In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form , where and are real numbers, and is a symbol taken to satisfy \varepsilon^2 = 0 with \varepsilon\neq 0. Du ...

and multiply as in the

quaternion group In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset \ of the quaternions under multiplication. It is given by the group presentation :\mathrm_8 ...

. Actually Study uses notation such that :

e_0 = 1,\  e_1 = i,\  e_2 = j,\  e_3 = k, \!

\varepsilon _0 = \varepsilon ,\  \varepsilon _1 = \varepsilon i,\  \varepsilon _2 = \varepsilon j,\  \varepsilon _3 = \varepsilon k. \!

The multiplication table is found on page 520 of volume 39 (1891) in

Mathematische Annalen ''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, ...

under the title "Von Bewegungen und Umlegungen, I. und II. Abhandlungen". Eduard Study cites

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 hi ...

as an earlier source on these

biquaternion In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions co ...

s. In 1901 Study published ''Geometrie der Dynamen'' also using dual quaternions. In 1913 he wrote a review article treating both E(3) and

elliptic geometry Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines ...

. This article, "Foundations and goals of analytical kinematics" develops the field of

kinematics Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fiel ...

, in particular exhibiting an element of E(3) as a homography of dual quaternions. Study's use of

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 ter ...

was noted in ''A History of Algebra'' (1985) by

B. L. van der Waerden Bartel Leendert van der Waerden (; 2 February 1903 – 12 January 1996) was a Dutch mathematician and historian of mathematics. Biography Education and early career Van der Waerden learned advanced mathematics at the University of Amst ...

. On the other hand, Joe Rooney recounts these developments in relation to kinematics.

Hypercomplex numbers

Study showed an early interest in systems of complex numbers and their application to transformation groups with his article in 1890. He addressed this popular subject again in 1898 in ''

Klein's encyclopedia Felix Klein's ''Encyclopedia of Mathematical Sciences'' is a German mathematical encyclopedia published in six volumes from 1898 to 1933. Klein and Wilhelm Franz Meyer were organizers of the encyclopedia. Its full title in English is ''Encyclope ...

''. The essay explored

quaternion 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 quater ...

s and other hypercomplex number systems. This 34 page article was expanded to 138 pages in 1908 by

Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometr ...

, who surveyed the hypercomplex systems in ''Encyclopédie des sciences mathématiques pures et appliqueés''. Cartan acknowledged Eduard Study's guidance, in his title, with the words "after Eduard Study". In the 1993 biography of Cartan by Akivis and Rosenfeld, one reads: :

tudy Tudy may refer to: People * Tudy of Landevennec, Breton saint Places * Île-Tudy, France * St Tudy St Tudy ( kw, Eglostudi) is a civil parish and village in north Cornwall, England, United Kingdom. The village is situated in the River Ca ...

defined the algebra °H of ' semiquaternions' with the units 1, ''i'', ''ε'', ''η'' having the properties

i^2 = -1, \ \varepsilon ^2 = 0, \ i \varepsilon = - \varepsilon i = \eta. \!

: Semiquaternions are often called 'Study's quaternions'. In 1985 Helmut Karzel and Günter Kist developed "Study's quaternions" as the kinematic algebra corresponding to the group of motions of the Euclidean plane. These quaternions arise in "Kinematic algebras and their geometries" alongside ordinary quaternions and the ring of 2×2 real matrices which Karzel and Kist cast as the kinematic algebras of the elliptic plane and hyperbolic plane respectively. See the "Motivation and Historical Review" at page 437 of ''Rings and Geometry'', R. Kaya editor. Some of the other hypercomplex systems that Study worked with are

, dual quaternions, and split-biquaternions, all being

s over R.

Ruled surfaces

Study's work with

dual number In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form , where and are real numbers, and is a symbol taken to satisfy \varepsilon^2 = 0 with \varepsilon\neq 0. Du ...

s and line coordinates was noted by

Heinrich Guggenheimer Heinrich Walter Guggenheimer (July 21, 1924 – March 4, 2021) was a German-born Swiss-American mathematician who has contributed to knowledge in differential geometry, topology, algebraic geometry, and convexity. He has also contributed volume ...

in 1963 in his book ''Differential Geometry'' (see pages 162–5). He cites and proves the following theorem of Study: The oriented lines in R³ are in one-to-one correspondence with the points of the dual unit sphere in D³. Later he says "A differentiable curve A(''u'') on the dual unit sphere, depending on a ''real'' parameter ''u'', represents a differentiable family of straight lines in R³: a

ruled surface In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, t ...

. The lines A(''u'') are the ''generators'' or ''rulings'' of the surface." Guggenheimer also shows the representation of the Euclidean motions in R³ by orthogonal dual matrices.

Hermitian form metric

In 1905 Study wrote "Kürzeste Wege im komplexen Gebiet" (Shortest paths in the complex domain) for

(60:321–378). Some of its contents were anticipated by

Guido Fubini Guido Fubini (19 January 1879 – 6 June 1943) was an Italian mathematician, known for Fubini's theorem and the Fubini–Study metric. Life Born in Venice, he was steered towards mathematics at an early age by his teachers and his father, wh ...

a year before. The distance Study refers to is a

Hermitian form In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allow ...

complex projective space In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...

. Since then this

metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathe ...

has been called the

Fubini–Study metric In mathematics, the Fubini–Study metric is a Kähler metric on projective Hilbert space, that is, on a complex projective space CP''n'' endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and ...

. Study was careful in 1905 to distinguish the hyperbolic and elliptic cases in Hermitian geometry.

Valence theory

Somewhat surprisingly Eduard Study is known by practitioners of

quantum chemistry Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions ...

. Like

James Joseph Sylvester James Joseph Sylvester (3 September 1814 – 15 March 1897) was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory, and combinatorics. He played a leadership ...

believed that invariant theory could contribute to the understanding of chemical valence. In 1900 Gordan and his student G. Alexejeff contributed an article on an analogy between the coupling problem for angular momenta and their work on invariant theory to the '' Zeitschrift für Physikalische Chemie'' (v. 35, p. 610). In 2006 Wormer and Paldus summarized Study's role as follows: : The analogy, lacking a physical basis at the time, was criticised heavily by the mathematician E. Study and ignored completely by the chemistry community of the 1890s. After the advent of quantum mechanics it became clear, however, that chemical valences arise from electron–spin couplings ... and that electron spin functions are, in fact, binary forms of the type studied by Gordan and Clebsch.

Cited publications

''Über die Geometrie der Kegelschnitte insbesondere deren Charakteristikenproblem.''
Teubner, Leipzig 1885.
''Methoden zur Theorie der ternaeren Formen.''
Teubner, Leipzig 1889.
''Sphärische Trigonometrie, orthogonale Substitutionen, und elliptische Functionen: Eine analytisch-geometrische Untersuchung.''
S. Hirzel, Leipzig 1893. * ''Aeltere und neuere Untersuchungen über Systeme complexer Zahlen''
Mathematical Papers Chicago Congress

''Die Hauptsätze der Quaternionentheorie.''
Gaertner, Berlin 1900.
''Geometrie der Dynamen. Die Zusammensetzung von Kräften und verwandte Gegenstände der Geometrie.''
Teubner, Leipzig 1903.
''Vorlesungen über ausgewählte Gegenstände der Geometrie.''
Teubner, Leipzig 1911 * ''Konforme Abbildung einfach-zusammenhängender Bereiche''. Teubner, Leipzig 1913.
''Die realistische Weltansicht und die Lehre vom Raume.''
Friedr. Vieweg und Sohn, Braunschweig 1914.
''Einleitung in die Theorie der Invarianten linearer Transformationen auf Grund der Vektorenrechnung.''
Friedr. Vieweg und Sohn, Braunschweig 1923.
''Mathematik und Physik - Eine erkenntnistheoretische Untersuchung.''
Friedr. Vieweg und Sohn, Braunschweig 1923.
Theorie der allgemeinen und höheren komplexen Grossen
in ''Encyklopädie der mathematischen Wissenschaften'', weblink to

University of Göttingen The University of Göttingen, officially the Georg August University of Göttingen, (german: Georg-August-Universität Göttingen, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany. Founded ...

References

* Werner Burau (1970) "Eduard Study" in

Dictionary of Scientific Biography The ''Dictionary of Scientific Biography'' is a scholarly reference work that was published from 1970 through 1980 by publisher Charles Scribner's Sons, with main editor the science historian Charles Gillispie, from Princeton University. It consi ...

. *

External links

* *
Appendix to Geometrie der Dynamen on the foundations of kinematics
(English translation)
"Foundations and goals of analytical kinematics"
(English translation)
"A New Branch of Geometry"
English translation)
"On non-Euclidian and line geometry"
(English translation) {{DEFAULTSORT:Study, Eduard 1862 births 1930 deaths 19th-century German mathematicians 20th-century German mathematicians People from Coburg University of Bonn faculty Geometers