Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century (Springer Undergraduate Mathematics Series)

Based on the latest historical research, Worlds Out of Nothing is the first book to provide a course on the history of geometry in the 19th century. Topics covered in the first part of the book are projective geometry, especially the concept of duality, and non-Euclidean geometry. The book then moves on to the study of the singular points of algebraic curves (Plücker’s equations) and their role in resolving a paradox in the theory of duality; to Riemann’s work on differential geometry; and to Beltrami’s role in successfully establishing non-Euclidean geometry as a rigorous mathematical subject. The final part of the book considers how projective geometry rose to prominence, and looks at Poincaré’s ideas about non-Euclidean geometry and their physical and philosophical significance. Three chapters are devoted to writing and assessing work in the history of mathematics, with examples of sample questions in the subject, advice on how to write essays, and comments on what instructors should be looking for. Preface Contents List of Figures Mathematics in the French Revolution The French Revolution Some mathematicians Monge Descriptive geometry Poncelet (and Pole and Polar) Poncelet reminisces Poncelet's mathematics Poncelet, Traité des propriétés projectives des figures, 1822 [202, pp. xix--xxvii] Commentary Pole, polar and duality Theorems in Projective Geometry The theorems of Pappus, Desargues and Pascal Some properties of some transformations Alternative treatment of cross-ratio and the fourth harmonic point Porismata Poncelet's Traité Poncelet's singular claims Meeting Cauchy responds Other responses Poncelet's more conventional methods Duality and the Duality Controversy Pole and polar Gergonne versus Poncelet Curves of higher degree Gergonne on the principle of duality Poncelet, Chasles, and the Early Years of Projective Geometry What was done -- differing opinions Institutions and careers Chasles What was done? Chasles, Steiner and cross-ratio Extracts from Chasles' Aperçu historique Chasles on descriptive geometry Chasles on Monge and his school Chasles on Monge's work A quick introduction to modern projective geometry The real projective plane Projective spaces Euclidean Geometry, the Parallel Postulate, and the Work of Lambert and Legendre Saccheri Lambert Legendre Lambert on the consequences of a non-Euclidean parallel postulate Gauss (Schweikart and Taurinus) and Gauss's Differential Geometry Gauss Schweikart and Taurinus What Gauss knew Gaussian curvature János Bolyai János and Wolfgang Bolyai János Bolyai's new geometry János Bolyai's section 32 Lobachevskii Lobachevskii and Kasan Lobachevskii's new geometry Lobachevskii's first foundations of geometry Astronomical evidence From Lobachevskii's Geometrische Untersuchungen, 1840 [153] Opening remarks Concluding remarks Publication and Non-Reception up to 1855 Minding's surface The Bolyais read Lobachevskii Final years of János Bolyai Final years of Lobachevskii Gauss's death, Gauss's Nachlass On Writing the History of Geometry -- 1 Assessment and advice Reading and writing the history of mathematics Practice questions References and footnotes Appendices Names Your essays An assignment on the first 12 chapters Advice Across the Rhine -- Möbius's Algebraic Version of Projective Geometry Möbius's Barycentric calculus Barycentric coordinates Projective transformations Duality Central projection from one plane to another A note on duality Möbius's introduction of projective coordinates Plücker, Hesse, Higher Plane Curves, and the Resolution of the Duality Paradox Higher plane curves Cubic curves Plücker's resolution of the duality paradox Confirmation by others Plücker Hesse The Plücker Formulae Singular points The non-singular cubic curve in the plane The non-singular quartic curve in the plane 28 real bitangents The Mathematical Theory of Plane Curves Non-singular points and tangents Double points Homogeneous coordinates First and subsequent polars The first polars of a circle Inflection points Hessians Finding tangents with homogeneous coordinates References Complex Curves Complex by necessity Complex numbers in geometry The introduction of complex curves The introduction of complex points -- the example of elliptic functions Riemann: Geometry and Physics Riemann Riemann's publications Riemann on geometry Surfaces Riemannian geometry From Riemann's Habilitationsvortrag Differential Geometry of Surfaces Basic techniques Geodetic projection Introducing Beltrami's Saggio Beltrami's Teoria of 1868 The Saggio From Beltrami's Saggio (Essay) of 1868 [13] Legendre's error References Beltrami, Klein, and the Acceptance of Non-Euclidean Geometry Beltrami's version Gauss's posthumous contribution Kant? Felix Klein Klein at Erlangen ... ... and beyond Klein's Cayley metric Klein's unification of geometry The Erlangen Program in the 1890s Weierstrass and Killing On Writing the History of Geometry -- 2 Assessment questions Advice Cremona Salmon Lobachevskii's account in 1840 Projective Geometry as the Fundamental Geometry The rise of projective geometry Cremona Cremona's projective geometry Salmon Anxiety -- Pasch Helmholtz Free mobility Hilbert and his Grundlagen der Geometrie Hilbert Hilbert and geometry The Grundlagen der Geometrie Desargues' theorem Impact References The Foundations of Projective Geometry in Italy Peano and Segre Enriques Pieri Conclusions Veronese's theory of projection and section in higher dimensions Henri Poincaré and the Disc Model of non-Euclidean Geometry Poincaré A prize competition Poincaré's discovery of non-Euclidean geometry The Poincaré and Beltrami discs Poincaré and Klein Circumcircles Inversion and the Poincaré disc Inversion References Is the Geometry of Space Euclidean or Non-Euclidean? How to decide? Poincaré's conventionalism Enriques disputes Poincaré on the subjective experience of a non-Euclidean space Poincaré's arguments Summary: Geometry to 1900 References What is Geometry? The Formal Side Nagel's thesis From Hilbert's Grundlagen der Geometrie What is Geometry? The Physical Side Geometry and physics Einstein The special theory of relativity The paradoxes of special relativity Minkowski Einstein, gravity and the rotating disc From Einstein's Relativity: The special and general theory [62] What is Geometry? Is it True? Why is it Important? Truth Mathematical truths Proof Frege versus Hilbert Relative consistency Poincaré on the relative consistency of Euclidean and non-Euclidean geometry On Writing the History of Geometry -- 3 Assessment questions Advice on writing such essays How the essays will be graded Von Staudt and his Influence Von Staudt Von Staudt's Geometrie der Lage Klein's response to von Staudt Non-orientability Axiomatics -- independence References Bibliography Some Geometers Index Worlds Out of Nothing is the first book to provide a course on the history of geometry in the 19th century. Based on the latest historical research, the book is aimed primarily at undergraduate and graduate students in mathematics but will also appeal to the reader with a general interest in the history of mathematics. Emphasis is placed on understanding the historical significance of the new mathematics: Why was it done? How - if at all - was it appreciated? What new questions did it generate? Topics covered in the first part of the book are projective geometry, especially the concept of duality, and non-Euclidean geometry. The book then moves on to the study of the singular points of algebraic curves (Plücker's equations) and their role in resolving a paradox in the theory of duality; to Riemann's work on differential geometry; and to Beltrami's role in successfully establishing non-Euclidean geometry as a rigorous mathematical subject. The final part of the book considers how projective geometry, as exemplified by Klein's Erlangen Program, rose to prominence, and looks at Poincaré's ideas about non-Euclidean geometry and their physical and philosophical significance. It then concludes with discussions on geometry and formalism, examining the Italian contribution and Hilbert's Foundations of Geometry; geometry and physics, with a look at some of Einstein's ideas; and geometry and truth. Three chapters are devoted to writing and assessing work in the history of mathematics, with examples of sample questions in the subject, advice on how to write essays, and comments on what instructors should be looking for Based on the latest historical research, Worlds Out of Nothing is the first book to provide a course on the history of geometry in the 19th century. Topics include projective geometry, especially the concept of duality, non-Euclidean geometry, and more