‎"LÜROTH, JACOB.‎
‎Beweis eines Satzes tiber rationale Kurven. - [LÜROTH'S THEOREM]‎

‎Leipzig, B. G. Teubner, 1876. 8vo. Bound in recent full black cloth with gilt lettering to spine. In ""Mathematische Annalen"", Volume 9., 1876. Entire volume offered. Library label pasted on to pasted down front free end-paper. Small library stamp to lower part of title title page and verso of title page. Very fine and clean. Pp.163-165 [Entire volume: IV, 575 pp.].‎

Reference : 47130

‎First printing of Lüroth's Theorem, a celebrated case of rational variety within algebraic variety. ""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"" (DSB).Lüroth's problem concerns subextensions L of K(X), the rational functions in the single indeterminate X. Any such field is either equal to K or is also rational, i.e. L = K(F) for some rational function F. In geometrical terms this states that a non-constant rational map from the projective line to a curve C can only occur when C also has genus 0. That fact can be read off geometrically from the Riemann-Hurwitz formula.""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."" (DSB)‎