Veronese’s Non-Archimedean Linear Continuum

In 1907 Hans Hahn of Vienna published an article on non-archimedean systems of quantities [1]. The study of such systems, according to Hahn, goes back to Paul du Bois-Reymond and Otto Stolz. (It actually goes back further — consider horn angles in ancient Greece, for example.) The work of du Bois-Reymond was published between 1870 and 1882 (Hahn cites only two articles, 1875 and 1877). That of Stolz appeared from 1879 to 1896 (Hahn cites articles of 1881, 1883 and 1891). Hahn also observes that Rodolfo Bettazzi handles some questions of this kind in his Teoria delle grandezze of 1890 [2], and that Giuseppe Veronese built a geometry without use of the Archimedean axiom in his “mathematically and philosophically significant” Fondamenti di geometria of 1891, which was translated into German by Adolf Schepp with some changes by Veronese in 1894 [3, 4]. Veronese subsequently answered various objections to his work — Hahn cites articles of 1896, 1897 and 1898. Tullio Levi-Civita, as Hahn says, gave an arithmetical representation of Veronese’s continuum in 1892/1893 and 1898 [5, 6]. Finally, Hahn cites Arthur Schoenflies’ article of 1906 [7].