This is a beautiful introduction to neutral, hyperbolic, and spherical geometry, in which the deductions are synthetic throughout (pp.1–138). It starts with Hilbert’s axioms for neutral geometry (including continuity), moves on to plane neutral geometry, and then to plane hyperbolic geometry, covering for neutral geometry an amount of material similar to that found in [M. J. Greenberg, Euclidean and non-Euclidean geometries. Development and history. 4th ed. New York: W. H. Freeman (2008; Zbl 1127.51001)] (including Propositions 16–19 and 24 from Book I of Euclid’s Elements), and for hyperbolic geometry material comparable to what can be found in [O. Perron, Nichteuklidische Elementargeometrie der Ebene. Stuttgart: B. G. Teubner (1962; Zbl 0101.37403)], with a more in-depth analysis of several functions occurring naturally in plane hyperbolic geometry, and with a proof of the hyperbolic Pythagorean theorem in the form \(\cosh a= \cosh b \cosh c\), where \(a,b,c\) are the sides of a right triangle and \(a\) is the hypotenuse. Area, hypercycles, horocyles, parabolic motions, hyperbolic trigonometry, Lobachevsky’s angle of parallelism are all treated in sufficient depth to deduce the main results of plane hyperbolic geometry. At certain places in the exposition, similarities with spherical geometry are pointed out, such as in the Pythagorean theorem. The last 34 pages of this introduction are devoted to models of plane hyperbolic geometry. First, the authors present the Beltrami-Cayley-Klein model, to then introduce a somewhat unusual algebraic model, in which points are prime ideals of the ring \({\mathbb R}[X]\), that turns out to be the Poincaré upper half-plane model, and then a ‘coherent’ model based on \({\mathbb R}^3\) equipped with a quadratic form \(q_t\), with \(t\in [-1,1]\), in which the connected component \(I_t\) of the identity in the orthogonal group defined by \(q_t\), and maximal abelian subgroups of \(I_t\) play a central role, that allows the transition from hyperbolic to Euclidean and to spherical geometry by varying the value of \(t\in[-1,1]\). Both the algebraic formulation of the Poinacré upper half-plane and the ‘coherent’ model are new in the literature on models of classical geometries.
The authors could have also mentioned the form of the Pythagorean theorem found in the hyperbolic case by M. T. Familiari-Calapso [C. R. Acad. Sci., Paris, Sér. A 263, 668–670 (1966; Zbl 0149.17702)] (see also [C. R. Acad. Sci., Paris, Sér. A 268, 603–604 (1969; Zbl 0184.46302)], M. T. Calapso [Atti Accad. Peloritana Pericolanti Cl. Sci. Fis. Mat. Natur. 50, 99–107 (1970)], and [Rend. Circ. Mat. Palermo, II. Ser. 24, 157–167 (1975; Zbl 0363.50007)] (where the above form of the Pythagorean theorem is also deduced)), and in the hyperbolic and spherical case in [P. Maraner, Math. Intell. 32, No. 3, 46–50 (2010; Zbl 1206.51016)], which is a genuine non-Euclidean version of the area (of circles the radii of which are the sides of a triangle in which the sum of two angles is equal to the third angle) formulation of the Euclidean Pythagorean theorem.
Minor misprints were detected: on page 64, line 2 from above “curve \(AA'\)” should be “curve \(m\)”, and on page 86, line 4 from below, “six” can be lowered to “five”.
For the entire collection see [Zbl 1230.00040].