Ana Historia Mathematica A history of non-euclidean geometry: Evolution of the concept of a geometric space: By B. A....
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HM 18 REVIEWS 373 A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space. By B. A. Rosenfeld. Translated by Abe Shenitzer. With the editorial assistance of Hardy Grant. New York (Springer-Verlag). 1988. xi + 471 pp. Reviewed by Jeremy Gray Faculty of Mathematics, Open University, Milton Keynes, MK7 6AA, England This comprehensive book goes from the early history of spherical geometry to work by Chevalley and Weyl among others. The most valuable chapters are undoubtedly the opening ones. I know of no history of spherical geometry; here we find a good account of Greek work, including that of Menelaus and Ptolemy, and also of Indian and, especially, Arabic or Islamic work. From al-Biriini and Nasir al-Din, we pass to Regiomontanus, Copernicus, and Vi&e before finishing our journey with Euler. Chapter 2 treats one of the author’s favourite topics, the theory of parallels. The treatment of Islamic sources is again most noteworthy. Together with K. Jaouiche’s work [ 19861, it makes possible a study of the topic for those denied Arabic. It differs from Jaouiche’s book by being cast in the form of extensive summaries of various manuscripts, whereas Jaouiche provides translations into French. This chapter concludes with an account of work by Legendre, Gurev (a Russian critic of Legendre), and Farkas Bolyai; Lobachevskii is deferred to a later chapter. Chapter 3 considers various projections, notably the stereographic projection, and its use in the analemma and the construction of astrolabes. Here the works of Commandino and Benedetti might have been mentioned. Rosenfeld proceeds from ibn Sinan to Kepler, and then to Desargues and Newton. The by-nowcustomary run through more modern treatments at greater speed takes us to conformal and projective transformations in the work of Euler and Lagrange, Poncelet and Mobius. Chapter 4 on geometric algebra is more diversely presented. The longest treatment is reserved for Vi&e, and Descartes’ achievements are but summarized. The chapte; r ends with some hints about vector geometry and spaces of higher dimensions. Chapter 5, like its predecessors, is better when it is fresher. It deals well with many aspects of Islamic debates on the philosophy of space; Mach stands in for many modern writers in an account that selects only positivism for discussion. With Chapter 6, we get for the first time in English a current Russian view of Lobachevskii. Rosenfeld describes what he did and what its implications were. He is particularly good on describing the actual reception of Lobachevskii’s ideas in Russia. Then comes his account of the work of Janos Bolyai, Gauss, Schweikart, and others. Rosenfeld does not enter an opinion on the vexed question of whether Gauss really discovered non-Euclidean geometry, but the implication of his account is that the honor must go to Lobachevskii (and the reviewer agrees). The chapter ends with an account of the discovery of various models of REVIEWS 374 HM 18 non-Euclidean geometry: the Beltrami model, the Klein model, and the Poincare model, in particular. Chapter 8 deals with higher dimensional spaces, and the more substantial Chapter 9 focuses on curvature. Not just Gauss and Riemann are discussed; Rosenfeld goes on to discuss Einstein and the impact of general relativity, affine connections, fibrations, and Betti numbers. Topological properties of spaces are also considered. Chapter 9 looks at groups of transformations. The influence of Rosenfeld’s late friend, Isaac Yaglom, to whom the book is dedicated, is apparent. Most of the account is devoted to Lie groups and symmetric spaces. The book concludes in chapter 10 with a look at geometrical interpretations and applications of the theory of algebras, a topic on which Rosenfeld has himself done original mathematical work. The book seems to fall into two halves. The second half may be dispatched briefly. It comes at the end of most chapters, occupies the final two, and looks at modern topics. Here the treatment is rushed and, from an historical standpoint, superficial. Often very little more is provided than a summary of the mathematics with little attention to the historical context. The real difficulties in attempting to do better than this are known to everyone who has tried, but they are not solved here, and the result may prove indigestible, although accurate, mathematically speaking. The first half, coming at the start of most chapters, is much more worthwhile. The author commands an astonishing range of languages and has been well-placed to deal with a variety of topics. The result is a series of thorough discussions, often with new material, on topics that have not been written about usefully before or about which Rosenfeld finds new things to say. It is to be hoped that others will expand upon the insights in this half of the book. There are the usual problems with lack of references to Western literature, but this is a problem the other way round too, except when translations as good as this one are available. In the difficult times for Glasnost that one must foresee at the end of 1990, it is more charitable to express the hope that increased contact between East and West will lead to more accounts of the history of mathematics as well-informed as this one. REFERENCES Jaouiche, K. 1986. La Thtorie A des Paralkles en Pays d’lslam. Paris: Vrin. History of Algebraic and Differential Topology, 1900-1960. Dieudonne. Boston and Base1 (Birkhtiuser). 1989. xxi + 648 pp. By Jean Reviewed by Jeremy Gray Faculty of Mathematics, Open University, Milton Keynes, MK7 &IA, England The origins of algebraic topology lie in almost every branch of 19th-century mathematics. As one example, the central theorem in Riemannian complex analy-