# Eduard Study

Eduard Study
Born 23 March 1862
Coburg
Died 6 January 1930 (aged 67)
Bonn
Nationality German
Fields Mathematics
Alma mater Munich
Doctoral students Ernst August Weiß
Known for Geometrie der Dynamen
Invariant theory
Spherical trigonometry

Eduard Study, more properly Christian Hugo Eduard Study (March 23, 1862 – January 6, 1930), was a German mathematician known for work on invariant theory of ternary forms (1889) and for the study of spherical trigonometry. He is also known for contributions to space geometry, hypercomplex numbers, and criticism of early physical chemistry.

Study was born in Coburg in the Duchy of Saxe-Coburg-Gotha. He died in Bonn.

## 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 in 1884. Paul Gordan, an expert in invariant theory 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 the primordial International Congress of Mathematicians in Chicago as part of the World's Columbian Exposition[1] and took part in mathematics at Johns Hopkins University. 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 as the position held by Rudolf Lipschitz was vacant. There he settled until retirement in 1927.

## Euclidean space group and dual quaternions

In 1891 Eduard Study published "Of Motions and Translations, in two parts". It treats the Euclidean group E(3). The second part of his article introduces the associative algebra of dual quaternions, that is numbers

$q = a + bi + cj + dk \!$

where abc, and d are dual numbers and {1, ijk} multiply as in the quaternion group. 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 under the title "Von Bewegungen und Umlegungen, I. und II. Abhandlungen". Eduard Study cites William Kingdon Clifford as an earlier source on these biquaternions. In 1901 Study published Geometrie der Dynamen[2] also using dual quaternions. In 1913 he wrote a review article treating both E(3) and elliptic geometry. This article, "Foundations and goals of analytical kinematics"[3] develops the field of kinematics, in particular exhibiting an element of E(3) as a homography of dual quaternions.

Study's use of abstract algebra was noted in A History of Algebra (1985) by B. L. van der Waerden. On the other hand, Joe Rooney recounts these developments in relation to kinematics.[4]

## Hypercomplex numbers

Main article: Hypercomplex number

Study showed an early interest in systems of complex numbers and their application to transformation groups with his article in 1890.[5] He addressed this popular subject again in 1898 in Klein's encyclopedia. The essay explored quaternions and other hypercomplex number systems.[6] This 34 page article was expanded to 138 pages in 1908 by Élie Cartan, 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:[7]

[Study] 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 numbers, dual quaternions, and split-biquaternions, all being associative algebras over R.

## Ruled surfaces

Study's work with dual numbers and line coordinates was noted by Heinrich Guggenheimer 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. 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 complexen Gebiet" (Shortest path in complex domains) for Mathematische Annalen (60:321–378). Some of its contents were anticipated by Guido Fubini a year before. The distance Study refers to is a Hermitian form on complex projective space. Since then this metric has been called the Fubini–Study metric. 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. Like James Joseph Sylvester, Paul Gordan 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:[8]

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.

## References

1. ^ Case, Bettye Anne, ed. (1996). "Come to the Fair: The Chicago Mathematical Congress of 1893 by David E. Rowe and Karen Hunger Parshall". A Century of Mathematical Meetings. American Mathematical Society. p. 65.
2. ^ E. Study (1903) Geometrie der Dynamen, from Historical Math Monographs at Cornell University
3. ^ E. Study (1913), Delphinich translator, "Foundations and goals of analytical kinematics" from Neo-classical physics
4. ^ Joe Rooney William Kingdon Clifford, Department of Design and Innovation, the Open University, London.
5. ^ E. Study (1890) D.H. Delphenich translator, "On systems of complex numbers and their applications to the theory of transformation groups"
6. ^ E. Study (1898) "Theorie der gemeinen und höhern komplexen Grössen", Encyclopädie der mathematischen Wissenschaften I A 4 147–83
7. ^ M.A. Akivis & B.A. Rosenfeld (1993) Élie Cartan (1869 — 1951), American Mathematical Society, pp. 68–9
8. ^ Paul E.S. Wormer and Josef Paldus (2006) Angular Momentum Diagrams Advances in Quantum Chemistry, v. 51, pp. 51–124
9. ^ Snyder, Virgil (1904). "Review of Geometrie der Dynamen. Die Zusammensetzung von Kräften und verwandte Gegenstände der Geometrie von E. Study" (PDF). Bull. Amer. Math. Soc. 10 (4): 193–200.
10. ^ Study, E. (1904). "Reply to Professor Snyder's review of Geometrie der Dynamen". Bull. Amer. Math. Soc. 10 (9): 468–471. MR 1558146.
11. ^ Emch, Arnold (1912). "Review: Vorlesungen über ausgewählte Gegenstände der Geometrie von E. Study" (PDF). Bull. Amer. Math. Soc. 19 (1): 15–18.
12. ^ Emch, Arnold (1914). "Review: Konforme Abbildung einfach-zusammenhängender Bereiche von E. Study" (PDF). Bull. Amer. Math. Soc. 20 (9): 493–495.
13. ^ Emch, Arnold (1915). "Review: Die realistische Weltansicht und die Lehre vom Raume von E. Study" (PDF). Bull. Amer. Math. Soc. 21 (5): 250–252.
14. ^ Shaw, J. B. (1925). "Review: Einleitung in die Theorie der Invarianten linearer Transformationen auf Grund der Vektorenrechnung von E. Study" (PDF). Bull. Amer. Math. Soc. 31 (1): 77–82.