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. We can then see a thread of unity through a parade of events, people, and ideas. My more modest goal is to provide a focus with which to view the standard tools of elementary differential geometry, and discover how their history emerges out of Geometry writ large, and how they developed into the modern, global edifice of today.

In recent years, to offer a course in differential geometry to undergraduates has become a luxury. 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 high school geometry is generally elementary Euclidean geometry based on Hilbert's axiom scheme. Such an approach is a welcome introduction to the rigors of axiomatic thinking, but the beauty of Euclidean geometry can get lost in the carefully wrought two-column proof. If mentioned at all, the marvels of non-Euclidean geometry are relegated to a footnote, enrichment material, or a “cultural” essay. This situation is also the case in most current introductions to differential geometry.