# Lueroth (or Lüroth), Jakob

# Lueroth (or Lüroth), Jakob

(*b*. Mannheim, Germany, 18 February 1844; *d*. Munich, Germany, 14 Septemper 1910)

*mathematics*.

Lueroth’ first scientific interests lay in astronomy. He began making observations while he was still in secondary school, but since he was hampered by bad eyesight he took up the study of mathematics instead, He attended the universities of Heidelberg, Berlin, and Giessen from 1863 until 1866; he had already, in 1865, written his doctoral dissertation on the Pascal configuration. In 1867 he became *Privatdozent* at the University of Heidelberg, and two years later, when he was still only twenty-five years old, he was appointed professor ordinaius at the Technische Hochschule in Karsruhe. From 1880 until 1883 he taught at the Technische Hochschule of Munich, and from the latter year until his death, at the University of Freiburg.

Lueroth’ first mathematical publicationa were concerned with questions in analytical geometry, linear geometry. and theory of invariants, a develpment of the work of his teachers Hesse and Clebsch. His ame is associated with three specific contributions to science. The first of these, a covariand of a given ternary form of fourth degree, is called the “Lueroth quartic,” and Lueroth discovered it when he examined, following Clebsch, the conditin under which a ternary quartic form may be represented as a sum of five fourth-powers of linear forms. In 1876 he demonstrated the “Lueroth theorem,” whereby each uni-rational curve in rational— Castelnuovo in 1895 proved the analogous but more difficul theorem for surfaces. Finally, the “Clebsch-Lueroth method” may be employed in the construction of a Riiemann surface for a given algebraic cure in the complex plane.

In addition, Lueroth worked in other areas of mathematics far removed from algebraic geometry. He obtained partial proof of the topological in variance of dimension (proved in 1911 by L. Brouwer) and, following the work of Staudt, did research in complex geometry. He was also involved in the logical researches of his friend Schroder and published two books in applied mathematics and mechanics. These were *Grundriss der Meclumik,* in which he used the vector calculus for the first time* and *Vorlesungen Uber immerisches Rechnen.* Lueroth collaborated in editing the collected works of Hesse and Grassmann.

## BIBLIOGRAPHY

I. Original Works. A complete bibliography of Lueroth’s works may be found in the Brill and Noether obituary (see below). The papers containing his main discoveries are “Einige Eigenschaften einer gewissen Gattung von Kurven 4. Ordnung,” in *Mathematische Annalen,***2** (1869), 37-53; “Das Imaginäre in der Geometrie und das Rechnen mit Würfen,” ibid., **8** (1875), 145-214; “Beweis eines Satzes tiber rationale Kurven,” ibid., **9** (1876), 163-165; and” Ober die kanonischen Querschnitte einer Rie-mannschen Fliehe,”in *Sitzungsberichte der physikalisch-medizinischen Sozietät in Erlangen*, **15** (1883), 24-30. He also published Grundriss der Mechanik (Munich, 1881), and Vorlesungen über numerisches Rechnen (Leipzig, 1900).

II. Secondary Literature. An obituary is A. Brill and M. Noether, “Jacob Lueroth,” in*Jahresberichte der Deutschen Mathematikervereinigung,***20** (1911), 279-299.

Werner Burau

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**Lueroth (or Lüroth), Jakob**