Lobachevski Illuminated: Content, Methods, and Context of the Theory of Parallels

Abstract

In the 1820\u27s, Nikolai Ivanovich Lobachevski discovered and began to explore the world\u27s first non-Euclidean geometry. This crucial development in the history of mathematics was not recognized as such in his own lifetime. When his work finally found a sympathetic audience in the late 19th century, it was reinterpreted in the light of various intermediate developments (particularly Riemann\u27s conception of geometry), which were foreign to Lobachevski\u27s own way of thinking about the subject. Because our modern understanding of his work derives from these reinterpretations, many of Lobachevski\u27s most striking ideas have been forgotten. To recover them, I have produced an illuminated version of Lobachevski\u27s most accessible work, Geometrische Untersuchungen zur Theorie der Parallellinien (Geometric Investigations on the Theory of Parallels), a book that he published in 1840. I have produced a new English version of this work, together with extensive mathematical, historical, and philosophical commentary. The commentary expands and explains Lobachevski\u27s often cryptic statements and proofs, while linking the individual propositions of his treatise to the related work of his predecessors (including Gerolamo Saccheri, J.H. Lambert, and A.M. Legendre), his contemporaries (including J·nos Bolyai and Karl Friedrich Gauss), and his followers (including Eugenio Beltrami, Henri PoincarÈ, and David Hilbert). This dissertation supplies the contemporary reader with all of the tools necessary to unlock Lobachevski\u27s rich, beautiful, but generally inaccessible world