Presents the axiom systems of non-Euclidean geometry. This book states and proves theorem after theorem of (hyperbolic) non-Euclidean geometry.

'In this book Girolamo Saccheri set forth in 1733, for the first time ever, what amounts to the axiom systems of non-Euclidean geometry. It is in this book that, for the first time in history, theorem after theorem of (hyperbolic) non-Euclidean geometry is stated and proved. Why - one cannot help asking - did Saccheri not evaluate correctly what he had achieved, why did he not claim credit for the discovery of non-Euclidean geometry?...The main content of the Notes are a series of annotations to Saccheri's text by the eminent scholars Paul Stackel and Friedrich Engel. These annotations originally appeared as footnotes to these authors' German translation of Euclides Vindicatus and were, with some modifications, translated into English by F. Steinhardt' - from the Preface to the second English edition.