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I teach a course on non-Euclidean geometry to high schoolers. I'm looking for an article or book that gives a thorough and interesting history of spherical geometry and trigonometry. I'm looking for it for my own learning and possibly to distribute to my students. I'd like something that has some substance to it--not just a couple of facts tossed off for color. A treatment that gives an overview of spherical geometry through the ages would be great, as would a "zoomed in" treatment of some particular episode.


Edit: Some good resources have been suggested below, especially the book suggested by unclejamil. I still haven't found quite what I'm looking for, though. I'd like something that focuses just on the history of the study of the geometry and trigonometry of the sphere, from ancient times to modernity, focusing on major advances and motivations. Any thoughts?

If I can't find such a source ready-made, I'll try to put together a short essay myself.

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up vote 4 down vote accepted

The Mathematics of the Heavens and the Earth: The Early History of Trigonometry will do:

Seems to be what you're looking for. Good luck with the kids. ;)

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+1 for van Brummelen's book. I heard him give a talk on precisely this subject at the 2010 Joint Mathematics Meetings. His presentation is clear and easy to understand, yet amazingly detailed. – Willie Wong Jul 6 '11 at 13:13
+1 This looks like an awesome resource and an interesting read. I've ordered a copy. Thanks! I'm still hoping to find something that focuses more exclusively on the development of spherical geometry--less interlaced with the corresponding planar story and including more information about non-trigonometric approaches and facts about the geometry of the sphere as well. Clearly all of these things are intertwined, but my hope is that there might be something more streamlined (while remaining substantive) for pedagogic reasons. I'd also like to see it reach at least up to the time of Riemann. – Justin Lanier Jul 6 '11 at 22:20

Try Tristan Needham's Visual Complex Analysis. There is a chapter in there on non-euclidean geometry which includes spherical and hyperbolic geometry. An excellent book that shows the links between different geometries, Möbius Transformations, etc.

$\textbf{Edit}$: You don't need to be a genius to see the many beauties of complex analysis in this book; Needham explains many concepts beautifully. However, that being said I know a lecturer who dissed this book asking if it was for physicists. The basis he said that was Needham used things like $ds$ for element of arc-length, this lecturer said that the book was not being rigorous by saying that these are things physicists use without formal definitions.

Note: I may or may not support his view but am just conveying that there are some people who dislike this book.

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+1 for this amazing book – mathmath8128 Jul 6 '11 at 6:11
I, too, once met a lecturer who dissed the book. He was an analyst, and said something along the lines of "well, I'm an analyst, not a geometer, and that's not complex analysis". It is an excellent book, nevertheless. You just have to approach it on its own terms. – Jose L. Lykón Jul 6 '11 at 15:00
I read this book in some depth a couple of summers ago and enjoyed it immensely. The chapter on non-Euclidean geometry is great, but it's not quite what I'm looking for here. – Justin Lanier Jul 7 '11 at 13:51

Today we think of Euler's Polyhedral formula (V+F-E=2) as a result in both geometry, topology, and combinatorics (graph theory) and it is common to give a proof of it using graph theory methods. Euler's "proof" was not correct. The first proof was given by Legendre using methods from spherical geometry! This seems curious but I recently came across a book that makes more sense of what happened in an historical perspective: The book charts how Legendre pioneered a modern view of symmetry and how his work on solid angles was related to his proof of Euler's polyhedral formula. It is also fascinating how without our modern journals and methods to share ideas among scholars Legendre came to do this work about 1794 after Euler's initial work, about 1750.

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