The discovery of non-Euclidean geometry on the pseudosphere. Eugenio Beltrami’s letters to Jules Houël (1868-1881). Introduction, notes and commentary by Luciano Boi, Livia Giacardi and Rossana Tazzioli. Preface by Ch. Houzel and E. Knobloch.
(La découverte de la géométrie non euclidienne sur la pseudosphère. Les lettres d’Eugenio Beltrami à Jules Houël (1868-1881).)

*(French)*Zbl 0921.01011
Collection Sciences dans l’Histoire. Paris: Librairie Scientifique et Technique Albert Blanchard. iv, 278 p. (1998).

The 1870’s were a period of great progress in the understanding of geometry, especially non-euclidean geometry. The famous Beltrami model of hyperbolic geometry was introduced in an 1868 memoir of Beltrami entitled “Saggio di interpretazione della geometria non-euclidea,” sent to Jules Houël in 1868. The present volume presents the complete set of 65 letters from Beltrami to Houël between 1868 and 1881, along with many letters to and from various mathematicians (Gennochi, Klein, Lipschitz, Betti, Helmholtz, Bellavitis, Cremona, De Tilly, and Darboux) furnishing related material on the subject. Letters written in Italian are given in the original Italian, preceded by a French translation. The authors provide a 63-page introduction to orient the reader to the mathematical ideas involved in the topic. In addition, the book contains an extensive bibliography and a detailed index. The letters make quite fascinating reading, as one can learn Beltrami’s thoughts on the nature of a variety of subjects, for example Lobachevskij’s horosphere (“One can say only that the horosphere is a surface of curvature zero in non-Euclidean space, which says nothing about its absolute form.”) or the famous pseudosphere (“I have said in the Saggio that the pseudospherical surface, to the extent that it is represented by the variables \(u\) and \(v\) is infinite in every sense and simply connected (einfach zusammenhängende, according to Gauss and Riemann), and that is quite true. But, since the general integral of that surface in the usual coordinates \(x\), \(y\), \(z\), is not known (being an integral that the current state of analysis is probably far from being able to vie), so that the most general form of this surface is unknown, it cannot be proved a priori that it can exist in ordinary space with a twofold infiniteness in every sense and be simply connected.

Reviewer: R.Cooke (Burlington)