# Eduard Study

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 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 biquaternions

In 1891 Eduard Study published "Of Motions and Translations, in two parts". It treats Euclidean space through the space group. The second part of his article constructs a seven-dimensional space out of "dual biquaternions", that is numbers

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

where abc, and d are dual numbers and {1, ijk} multiply as in the quaternion group. He uses these conventions:

$e_0 = 1,\ e_1 = i,\ e_2 = j,\ e_3 = k, \!$
$\varepsilon _0 = \varepsilon ,\ \varepsilon _1 = \varepsilon i,\ \varepsilon _2 = \varepsilon j,\ \varepsilon _3 = \varepsilon k. \!$

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 to highlight the applications of this algebra. Due to Eduard Study's profound and early exploitation of this eight-dimensional associative algebra, it is frequently referred to as Study Biquaternions. Study's achievement is celebrated, for example, in A History of Algebra (1985) by B. L. van der Waerden, who also cites Clifford's earlier note.

Since the space group is important in robotics, the Study biquaternions are a technical tool, now sometimes referred to as dual quaternions. For example, Joe Rooney has profiled the use of this algebra by several modelers of mechanics (see external link).

## Hypercomplex numbers

In 1898 Eduard Study was the author of an article on hypercomplex numbers in the Klein's encyclopedia. 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 priority in his title with the words "after Eduard Study".

In the 1993 biography of Cartan by Akivis and Rosenfeld, one reads:

[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.

Thus in the study of classical associative algebras over R there are two special ones: Study's quaternions (4D) and Study's biquaternions (8D).

## 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 Weg 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:

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.