One of the many roles of history is to tell a story. The history of the Parallel Postulate is a great story – it spans more than two millennia, stars an impressive cast of characters, and contains some of the most beautiful results in all of mathematics. My immodest goal for this book is to tell this story.

Another role of history is to focus our attention and so to provide a thread of unity through a parade of events, people, and ideas. My goal grows small and quite modest before all of Geometry, especially its recent history. A more modest goal then is to provide a focus in which to view the standard tools of differential geometry, and in so doing offer an exposition, motivated by the history, that prepares the reader for the modern, global foundations of the subject.

In recent years, it has become a luxury to offer a course in differential geometry in an undergraduate curriculum. When such a course exists, its students often arrive with a modern introduction to analysis, but without having seen geometry since high school. In the United States geometry taught in high schools is generally elementary Euclidean geometry based on Hilbert's axiom scheme. The beautiful world of non-Euclidean geometry is relegated to a footnote, enrichment material, or a “cultural” essay. This is also the case in most current introductions to differential geometry.