Introduction Classically, the expression “non-Euclidean geometry” refers to the two geometries of non-zero constant curvature, namely, the spherical and the hyperbolic.1 Spheri- cal geometry was developed in Greek antiquity and it attained there a high degree of matureness, especially in the work of Menelaus of Alexandria (1st–2nd centuries A.D.).2 Hyperbolic geometry is a nineteenth century achievement. It was discov- ered by Lobachevsky, Bolyai and Gauss, with a few anticipated results obtained in the eighteenth century, especially in the work of J. H. Lambert who developed a ge- ometry (which, from his point of view, was hypothetical), in which all the Euclidean axioms hold except the parallel postulate and where the latter is replaced by its nega- tion. Although Lambert’s goal was to find a contradiction in the consequences of this system of axioms (like many other prominent geometers, he thought that such a ge- ometry cannot exist), the net result of his investigations is a collection of interesting theorems in hyperbolic geometry.3 In the present volume, the term “non-Euclidean geometry” is used in a broader sense, including geometries such as de Sitter, anti-de Sitter and others which can be developed, in analogy with the three classical geometries, in the framework of projec- tive geometry, using the language of conics, quadrics and quadratic forms. Studying the three classical geometries in the setting of projective geometry is also a nineteenth century achievement; it is due to the visions of Cayley, Klein, Beltrami, Poincaré, and a few others. Today, non-Euclidean geometry in this generalized sense is a very active research field and it seemed to us that editing a book containing self-contained surveys touch- ing several aspects of this domain was desirable for researchers and those who want to learn this subject. This is the reason for which this book is published. 1 The expression “non-Euclidean geometry,” to denote spherical and hyperbolic geometry, was coined by Gauss. 2 The first English edition of Menelaus’ work just appeared in press: R. Rashed and A. Papadopou- los, Menelaus’ Spherics: early translation and al-M¯ah¯an¯ı/al-Haraw¯ı’s version, Scientia Graeco-Arabica 21, Walter de Gruyter, Berlin, 2017. 3 J. H. Lambert, “Theorie der Parallellinien,” in Die Theorie der Parallellinien von Euklid bis auf Gauss, eine Urkundensammlung zur Vorgeschichte der nichteuklidischen Geometrie (P.Stäckel and F. En- gel, eds.), B. G. Teubner, Leipzig 1895. French translation by A. Papadopoulos and G. Théret, La théorie des lignes parallèles de Johann Heinrich Lambert, Sciences dans l’Histoire, Blanchard, Paris, 2014. There is no English translation available of this work. viii V. Alberge and A. Papadopoulos Geometry is one of these mathematical fields (probably the unique one to such a great extent) where the texts of the ancient great authors keep their full value, and going through them is not a matter of vain curiosity nor a matter of being interested in history, but it is the best way to understand present day mathematics. In this sense, the rapid growth of non-Euclidean geometry in the last few decades, after its revival by Thurston in the 1970s, is also a return to the sources of geometry. This is why the names of Menelaus, Pappus, Euler, Lambert, Lagrange, Lexell, Fuss, Schubert, Chasles, Study, Story, and several others are mentioned at several occasions in the various essays that constitute this volume. Going throughout these works also shows how slow is the process of development of geometry, despite the rapid (quantitative) growth of the literature in this field. The slow evolution process of mathematics is marked out by abrupt changes attached to names such as those we just mentioned. Before describing in some detail the content of this volume, we would like to make a few remarks on two topics that are at the heart of the various surveys that constitute it, namely, the notion of area in non-Euclidean geometry, and the geometry of conics. In spherical geometry, the area of a triangle is (up to a constant multiplicative factor) the excess of its angle sum with respect to two right angles. In other words, the area of a triangle with angles A;B;C is equal to A B C , up to a constant C C factor that does not depend on the choice of the triangle. In hyperbolic geometry, the area of a triangle is (again up to a constant factor) the deficiency of the angle sum with respect to two right angles, that is, .A B C/. The fact that in C C spherical (respectively hyperbolic) geometry the angle excess (respectively the angle deficiency) of an arbitrary triangle is positive, is probably the most important feature in that geometry. As a matter of fact, there exist classical proofs of the trigonometric formulae in spherical and hyperbolic geometries that are exclusively based on these properties. The oldest such proofs that we are aware of are due to L. Gérard,4 a student of Poincaré. Since the trigonometric formulae contain in essence all the geo- metric information on a space, spherical and hyperbolic geometry are essentially the geometries where angle excess or, respectively, angle deficiency of all triangles are positive. Thus, it is not surprising that several chapters in the present volume concern the notion of area and its use in non-Euclidean geometry. After area, one naturally considers volume in higher-dimensional spherical and Euclidean geometries, and here we enter into the realm of difficult problems. One might recall in this respect that Gauss, in a letter to Wolfgang Bolyai (the father of János Bolyai, the co-discoverer of hyperbolic geometry), dated March 6, 1832,5 after he gave an outline of a proof of the area formula for a hyperbolic triangle as angle defi- ciency, asked his friend to suggest to his son to work on the determination of volumes 4 L. Gérard, Sur la géométrie non euclidienne, Thèse № 768, Faculté des Sciences de l’Université de Paris, Gauthier-Villars, Paris, 1892. 5 See Gauss’s Collected works, Vol. 6, p. 221, and the article by P. Stäckel and F. Engel “Gauss, Die beiden Bolyai und die nichteuklidische Geometrie,” Math. Annalen 2 (1897), 149–167. Introduction ix of tetrahedra in three-dimensional hyperbolic space. It is conceivable that Gauss was unaware of the fact that this subject was extensively studied by Lobachevsky dur- ing the same period, but he probably knew from his own experience that obtaining formulae for volumes of tetrahedra in hyperbolic and spherical spaces is a difficult problem. It turns out indeed that there is no simple formula for the volume of a hyperbolic (or spherical) tetrahedron in terms of the dihedral angles that is compa- rable to the formula for the area of a triangle. The existing formulae (first obtained by Lobachevsky) give expressions of volume in terms of an integral function which bears the name Lobachevsky function and which is closely connected to the Euler dilogarithm function. These formulae involve the value of the Lobachevsky function at the dihedral angles of the tetrahedron. Several computations of volumes of tetrahedra were conducted after Lobache- vsky’s work but the progress was slow. Some of the most important conjectures that are open in three-dimensional geometry of hyperbolic (or spherical) space concern volume. We mention incidentally that the computation of volumes of hyperbolic tetrahedra is the subject of Chapter 8 of Thurston’s notes on three-dimensional ge- ometry.6 Talking about volume, let us also mention that there are interesting formulae for volumes of Euclidean tetrahedra that are due to Euler. The latter, in a letter to his friend Christian Goldbach, dated November 14, 1750,7 announced a formula for the volume of a simplex in terms of its side lengths, a three-dimensional analogue of Heron’s formula for the area of a triangle.8 Later on, he gave the proof of that formula, in a paper published in 1758.9 In the same paper, he provided several other formulae for volumes of Euclidean simplices, including a formula in terms of side lengths, and another one in terms of lengths of the three edges abutting on a solid angle along with the three plane angles that these sides form. Finally, let us note that computing area and volume, beyond those of polygons and polyhedra, is another subject that has a long history whose origin can be traced back to the work of Hippocrates of Chios (5th century B.C.) who studied areas of lunes (intersections of two discs) and other figures, and which was brought to a high degree of sophistication in the work of Archimedes (3rd century B.C.). The reader can imagine that computing areas and volumes without the use of the mechanical methods of modern calculus is always a challenging problem. 6 W. P. Thurston, Geometry and topology of three-manifolds, Princeton Lectures Notes, Princeton, N.J., 1979. 7 Letter № 149 in Euler’s volume of his Opera omnia containing his correspondence with Goldbach, Series quarta A, Vol. IV, Birkhäuser, Basel, 2014. 8 Heron’s formula is contained in the Codex costantinopolitanus Palatii Veteris, ed. and transl. by E. M. Bruin, Leiden, 1964. 9 L. Euler, Demonstratio nonnullarum insignium proprietatum, quibus solida hedris planis inclusa sunt praedita, Novi Commentarii academiae scientiarum petropolitanae 4 (1752), 140–160, in Opera om- nia, Series 1, Vol. XXVI, 94–108. x V. Alberge and A. Papadopoulos Let us pass to our second topic, projective geometry and conics. We note right away that the study of conics dates back to Greek antiquity, where the most important treatise on the subject is the multi-volume work of Apollonius of Perga (3rd–2nd c.