When I was in growing up, I had a copy of Handbook of Mathematical Tables and Formulas by R. S. Burington from 1940. When I first saw the book, much of it was mysterious. Now everything in the book is very familiar, and would be familiar to anyone who has gone through the typical college calculus sequence. But there is one exception: two pages on spherical trigonometry, the study of triangles drawn on a sphere.
Until recently, the only place I’d ever heard of spherical trig was Burington’s old book. I managed to find a book on spherical trig at the Rice University library, Spherical Trigonometry by J. H. D. Donnay, written in 1945. Amazon has several recent books on the spherical trig, but it appears they’re all reprints of much older books. It seems interest in teaching spherical trig died sometime after World War II.
I’m not sure why schools quit teaching spherical trig. It’s a practical subject; after all, we live on a sphere. Surveyors, navigators, and astronomers would find it useful. Somewhere along the way, solid geometry fell out of favor, and I suppose spherical trig fell out of favor with it. The standard math curriculum changed in order to make a bee line for calculus. Presumably this is to meet the needs of science and engineering students who need calculus as a prerequisite for their courses. In the process, subjects like solid geometry were squeezed out. I see the logic in the contemporary sequence, but it is interesting that the sequence was different a generation or two ago.
See Notes on Spherical Trigonometry for a list of some of the elegant identities from this now obscure area of math. For example, the interior angles of a spherical triangle must add up to more than 180°, and the area of the triangle is proportional to the amount by which the sum of these angles exceeds 180°.