After the award of his doctorate, Stolz habilitated at the University of Vienna and worked there as a Privatdozent until 1869. He obtained a scholarship to allow him to study in Germany from 1869 to 1871. He went first to Berlin where he attended lectures by the three great mathematicians Weierstrass, Kummer and Kronecker. He then studied at Göttingen, attending lectures by Klein and Clebsch before returning to Austria. His period of study in Germany marked the addition of a new topic to his research interests. He was charmed by Weierstrass's approach to analysis and began to investigate problems in that area; these have proved to be his most significant contributions. He also struck up a friendship with Klein at this time and after returning to Innsbruck the two began to correspond. We briefly mention some topics in their correspondence below.
In July 1872 Stolz was appointed an extraordinary professor in his home town of Innsbruck. He was appointed to an ordinary professorship at the University of Innsbruck in July 1876 and in the same year he married. Despite many attempts to get him to accept a chair in Vienna, he remained in his home town building up a strong reputation despite low funding and only a few students but attracting colleagues like Gegenbauer.
We mentioned the Stolz-Klein correspondence above. In , 32 from Stolz to Klein, found in the archives at Göttingen, and 25 from Klein to Stolz from the archives at Innsbruck, are described and a summary of the contents of the mathematically interesting letters is given. Topics covered are those of interest to Klein and to Stolz and include Klein's Erlangen program, nowhere differentiable continuous functions, geometry, and the topology of the line.
His topics of research were wide ranging as Robinson points out in :-
Stolz's earliest papers were concerned with analytic or algebraic geometry, including spherical trigonometry. He later dedicated an increasing part of his research to real analysis, in particular to convergence problems in the theory of series, including double series; to the discussion of the limits of indeterminate ratios; and to integration.Examples of his early papers are Die geometrische Bedeutung der complexen Elemente in der analytischen Geometrie Ⓣ (1871), Beweis einiger Sätze über Potenzeihen Ⓣ (1875), Über die singulären Punkte der algebraischen Functionen und Curven Ⓣ (1875), Allgemeine Theorie der Asymptoten der algebraischen Curven Ⓣ (1877), and Die Multiplicität der Schnittpunkte zweier algebraischen Curven Ⓣ (1879). In the 1870s Weierstrass's ε, δ approach to analysis became to standard approach and, in B Bolzano's Bedeutung in der Geschichte der Infinitesimalrechnung Ⓣ (1881), Stolz pointed out that Bolzano had suggested a similar approach even before Cauchy had attempted his own way of making analysis more rigorous.
Archimedes' axiom states that given two magnitudes a and b one can find a multiple of a which exceed b and also a multiple of b which exceeds a. This can actually be traced back to Eudoxus but the term 'Archimedes' axiom' used today was coined by Stolz in 1882 in Zur Geometrie der Alten, insbesondere über ein Axiom des Archimedes Ⓣ. On the same topic he also wrote Zur Geometrie der Alten, insbesondere über ein Axiom des Archimedes Ⓣ (1883) and Über das Axiom des Archimedes Ⓣ (1891). He also returned to infinitesimals of the sort considered by Newton and attempted to make this approach rigorous. Balanzat points out in  that the first correct definition of differential in two variables to Stolz in Grundzüge der Differential und Integralrechnung, I Ⓣ (1893).
In 1885 the first part of Vorlesungen über Allgemeine Arithmetik Ⓣ was published on Allgemeines und Arithhmetik der Reelen Zahlen Ⓣ, with the second part on Arithhmetik der Complexen Zahlen Ⓣ appearing two years later. This two volume work, like all Stolz's books, was very well written; it became a classic. Another book, published in 1891 was Grössen und Zahlen Ⓣ. He published a series of books under the title Leçons nouvelles sur l'analyse infinitésimale et ses applications géométriques Ⓣ. The first part Principes généraux Ⓣ appeared in 1894 followed by étude monographique des principales fonctions d'une seule variable Ⓣ (1895), and two further volumes. In 1902, in collaboration with his student J A Gmeiner, he published the two volume work Theoretische Arithmetik Ⓣ. A second edition was published in 1915, ten years after Stolz's death.
Article by: J J O'Connor and E F Robertson