In the autumn of 1862, Lüroth passed the matriculation examination for the University of Bonn and he began to study astronomy with Friedrich Wilhelm Argelander (1799-1875). Argelander had been a student of Wilhelm Bessel, who had set up an observatory which had begun operating in Bonn in 1845. He had worked on a star catalogue in collaboration with Eduard Schönfeld who had advised Lüroth that Bonn was an excellent place to study astronomy. Lüroth began to make astronomical observations but he soon realised that his poor eyesight was not going to give him a future in that subject so his interest turned to mathematics. He had already published a number of works in astronomy in addition to the paper mentioned above, namely Ephemeride der Calypso Ⓣ (1862) Drei Ringmikrometer-Beobachtungen der Urania am 5-füssigen Fraunhofer Ⓣ (1862) and Fünf ebensolche Beobachtungen der Asträa Ⓣ (1863). Having to give up his ambition to study astronomy was certainly not an easy one for the young man and in later life he described this as one of the most painful disappointments of his life.
He studied mathematics at a number of universities as was the usual practice of German students at that time. Between the years 1863 and 1865 he was a student at the University of Heidelberg where he attended lectures by Otto Hesse and Gustav Kirchhoff and was advised by them on topics for research. He received a doctorate in 1865 for a thesis Zur Theorie des Pascalschen Sechsecks Ⓣ on the Pascal configuration. After graduating from Heidelberg, Lüroth went to Berlin where he spent 1865-66 attending lectures at the University from several leading mathematicians including Karl Weierstrass. In 1866 he went to the University of Giessen where he was strongly influenced by Alfred Clebsch. Now Clebsch had arrived in Giessen in 1863 and at that time had changed his research topic from applied mathematics to pure mathematics. He had collaborated with Paul Gordan at Giessen on the theory of abelian functions and their joint monograph on that topic was published around the time Lüroth arrived in Giessen. It was working with Clebsch in Giessen that directed Lüroth's research towards geometry and function theory.
Lüroth habilitated at the University of Heidelberg in the summer of 1867 after submitting the thesis Zur Theorie der windschiefen Flächen and he was appointed to the teaching staff. He taught there for a year before being appointed to the Technische Hochschule in Karlsruhe in 1868. Joseph Dienger (1818-1894) had been a secondary school teacher before being appointed to the Technische Hochschule in Karlsruhe in 1850. He worked there until 1868 and initially Lüroth was appointed to help Dienger out but, after Dienger left, he became a candidate to fill the vacant professorship and in January 1869 he was appointed as a full professor of mathematics at the Technische Hochschule in Karlsruhe. He was only 25 years old at the time, remarkably young to receive such a prestigious appointment, but he had impressed the appointing panel with his outstanding teaching ability and his likeably personality. Lüroth married Karoline Antonie Schepp (1850-1920) (known as Antonie) in 1875 and their only child, a daughter Emilie, was born in Karlsruhe in 1876. Antonie was the sister of Lieutenant Adolf Schepp (1837-1905) who collaborated with Lüroth on translations of English and Italian mathematical texts into German. In the year that his daughter was born, Lüroth was director of the Technische Hochschule in Karlsruhe, a position he held in 1876-77. Lüroth spent twelve years at Karlsruhe, then in 1880 he moved to Munich, teaching at the Technische Hochschule there for three years. In 1883 Lüroth moved again, this time being appointed to the University of Freiburg. He was to spend the rest of his career in Freiburg im Breisgau. He was vice-rector of the university in 1889-90.
The authors of  write about his abilities:-
From his youth he was distinguished by the ability to grasp new ideas quickly and clearly, and to recognize which are important. Because he had a remarkable memory, Lüroth was effortlessly at home in almost all areas of mathematics, including applied mathematics, many branches of astronomy, geodesy and even more remote areas of knowledge. ... It was the treatises that were most difficult to understand that attracted him the most. From out of this capacity, which he maintained throughout his life, his actual productive activity has been developed, which stretches, with rare versatility, to geometry and mechanics, to astronomy and geodesy, to probability theory, set theory and the logical foundations of mathematics, on function theory and algebra.As we have mentioned above, Lüroth was taught by Hesse and Clebsch and because of their influence he continued to develop their work on geometry and invariants. He published results in the areas of analytic geometry, linear geometry and continued the directions of his teachers in his publications on invariant theory. In 1869 Lüroth discovered the "Lüroth quartic" and it appears in his paper Einige Eigenschaften einer gewissen Gattung von Kurven vierter Ordnung Ⓣ which was published in the first volume of Mathematische Annalen in 1869. The Lüroth quartic is a nonsingular quartic plane curve which contains the ten vertices of a complete pentalateral. Lüroth's work on this curve came out of an investigation he was carrying out into when a ternary quartic form could be represented as the sum of five fourth powers of linear forms.
Some of his work on rational curves, in particular his contributions in the paper Beweis eines Satzes über rationale Kurven published in Mathematische Annalen in 1876, was extended to surfaces by Guido Castelnuovo in 1895. In 1883 Lüroth published his method on constructing a Riemann surface for a given algebraic curve. Lüroth also worked on the big problem of the topological invariance of dimension. Albert C Lewis, reviewing , writes:-
After Cantor's discovery, published early in 1878, of the "paradox" of the one-to-one correspondence of the unit square with the unit line segment, "the next problem" was "the invariance of dimension through a consideration of continuity", which Dedekind had formulated in a letter to Cantor. Jacob Lüroth, Johannes Thomae, Enno Jürgens, Eugen Netto, as well as Cantor himself, addressed this problem .... Lüroth assumed a one-to-one correspondence between two coordinate manifolds of different dimension and arrived at a contradiction but only for certain special cases of particular choices of dimension.Although Lüroth made some useful progress, this difficult problem was not completely solved until the work of L E J Brouwer in 1911.
Among his other work, Lüroth undertook editing. He was an editor of the complete works of Ludwig Otto Hesse (published in 1897) and of Hermann Grassmann (published in 1893 and 1902). We mentioned above that he collaborated with his brother-in-law, Adolf Schepp, in translating texts from English and Italian into German. Their most famous translation was of Ulisse Dini's 1878 treatise which they published under the title Grundlagen für eine Theorie der Funktionen einer veränderlichen reellen Grösse Ⓣ (1892). He also has some fine results on logic, a topic he worked on in collaboration with his friend Ernst Schröder. Karl von Staudt's ideas of geometry interested Lüroth and he further developed von Staudt's complex geometry in papers such as Über das Rechnen mit Würfen Ⓣ (1873) and Das Imaginäre in der Geometrie und das Rechnen mit Würfen. Darstellung und Erweiterung der v. Staudt'schen Theorie Ⓣ (1875). He also wrote two important books. He published the first of these, Grundriss der Mechanik Ⓣ, in 1881. This mechanics book makes heavy use of the vector calculus, being the first one to do so. A second book, Vorlesungen über numerisches Rechnen Ⓣ, was published in 1900.
Another of his contributions, for which he has failed to be given the credit that he deserves, was his discovery of the t-distribution in 1876. The estimation of the parameters of the normal distribution by inverse probability had been treated by Carl Friedrich Gauss. Lüroth carried on with Gauss's analysis of the linear normal model by considering the joint distribution of the parameters. He published his work on the t-distribution in Vergleichung von zwei Werten des wahrscheinlichen Fehlers Ⓣ (1876). This statistical test was rediscovered by William Gosset, who coincidently was born in the year that Lüroth made his discovery, and its discovery is now attributed to Gosset who published it under the name 'Student' in 1908.
Lüroth received many honours for his contributions. For example, he was elected to the Royal Bavarian Academy of Sciences in 1882, elected to the German Academy of Scientists Leopoldina in 1883, and elected to the Heidelberg Academy of Sciences 1909.
Lüroth continued to work at Freiburg despite his health deteriorating due to heart problems. He showed remarkable courage and, despite being in severe pain, tried to appear cheerful to all those round him. Eventually he was forced to stop work and, with his wife and daughter, he went to Munich where he hoped to recover. However, he died in Munich following a heart attack. Alexander von Brill and Max Noether give this final tribute :-
Lüroth most carefully maintained and preserved with unswerving loyalty an intercourse with his friends at all times. Absolutely reliable, he could show towards them a spirit of sacrifice, which went far beyond the expected level. What efforts Lüroth has spent on encouraging an appreciation of the work of his late friend, Ernst Schröder who, for a long time, stood alone with his efforts to introduce a conceptual notation in Germany. And to Julius Weingarten, who was still brought in his final years to Freiburg im Breisgau, where a position as visiting professor, was given at the university essentially due to Lüroth, which gave the lonely scholar in his last years a desired sphere.
Article by: J J O'Connor and E F Robertson
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