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Pangeometry. Edited and translated by Athanase Papadopoulos. (English) Zbl 1208.51013
Heritage of European Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-087-6/hbk). xii, 310 p. (2010).
This is the first integral English translation of Lobachevsky’s Pangeometry of 1855. The translation of this classic, which is printed together with its Russian and French original and translated from the latter, is not only for the use of historians of mathematics, but for offering present-day students of hyperbolic geometry a “fresh point of view on the subject, namely, a model-free point of view which is not to be found in most of the more modern textbooks on hyperbolic geometry”. The translation is made significantly more readable than its French original first and foremost by adding figures (the original had none), as well as by shortening long sentences and adding an index.
The trilingual text of the Pangeometry is followed by a biography of Lobachevsky taken from his collected works, and an enlightening commentary on the Pangeometry – discussing its content, the reception of hyperbolic geometry, models and model-free hyperbolic geometry, a look at all of Lobachevsky’s work in geometry, and a short list of references.
The commentaries are meticulous and help both the historically inclined and the geometrically inclined reader, in particular whenever the latter intends to study hyperbolic geometry over the real numbers. The reviewer found in the historical part only two places where relevant literature could have been added: (i) in footnote 153 on page 258, where E. Breitenberger [Arch. Hist. Exact Sci. 31, 273–289 (1984; Zbl 0554.01020)] would have belonged, and (ii) on page 269, where the author mentions the work of de Tilly – as well as that it “probably has not received the attention it deserves” – referring to the in-depth historical survey of 19th century non-Euclidean geometry, J.-D. Voelke [Renaissance de la géométrie non euclidienne entre 1860 et 1900. Bern: Peter Lang (2005; Zbl 1078.01014)], in which de Tilly’s work receives a whole chapter, would have been appropriate.
In the geometrical part itself, in particular in “On models, and on model-free hyperbolic geometry” (pp. 280-284), in which the author has words of praise for the synthetic (model-free) approach in hyperbolic geometry, his enthusiasm being tempered only by the admonition that “one should not fall in the opposite trap of giving a course based on axioms, where geometry becomes an abstract exercise in logic”, mentioning a split that occurred in the 20th century history of hyperbolic geometry would have been enlightening. The split referred to above is one between viewing the strongest form of continuity as being an integral part of hyperbolic geometry, and thus viewing the real numbers as central to geometry (hyperbolic or otherwise), and one starting with D. Hilbert [Math. Ann. 57, 137–150 (1903; JFM 34.0525.01)] – where via the calculus of ends, hyperbolic geometry, including hyperbolic trigonometry, was developed without continuity, over any Euclidean field – and leading to the astonishing results collected and systematized in the arguably most important book on metric geometry since Euclid’s Elements, [F. Bachmann, Aufbau der Geometrie aus dem Spiegelungsbegriff. 2. ergänzte Aufl. Berlin-Heidelberg-New York: Springer-Verlag. (1973; Zbl 0254.50001)]. Unlike the continuous hyperbolic geometry tradition (encompassing differential geometry, low-dimensional topology, hyperbolic manifolds, complex analysis), in the latter tradition all reasoning has remained model-free. This more axiomatic approach (which, however, is very far from being “an abstract exercise in logic”) has never managed to cross the French-German language frontier, having been rejected from the very beginning by Poincaré, which explains why students of hyperbolic geometry in France (or in the U.S.A. or in Russia, for that matter) are not likely to be exposed to synthetical reasoning in hyperbolic geometry.
The elementary introduction to synthetic plane hyperbolic geometry [O. Perron, Nichteuklidische Elementargeometrie der Ebene. Stuttgart: B. G. Teubner (1962; Zbl 0101.37403)] may also be added to the author’s short list of references (pp. 285–289).

MSC:
51M10 Hyperbolic and elliptic geometries (general) and generalizations
51-03 History of geometry
01A75 Collected or selected works; reprintings or translations of classics
01A55 History of mathematics in the 19th century
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