CTK Insights

20 Dec

The Forgotten Art of Spherical Trigonomtery

A new book Heavenly Mathematics from the Princeton University Press comes with a subtitle, The Forgotten Art of Spherical Trigonometry. Glen Van Brummelen, the author of the book, writes in the Preface:

Today almost no trigonometry texts even mention the existence of spherical counterpart. The only book on the subject continuously in print (Clough-Smith 1966) is difficult to obtain and available only from nautical booksellers. This paucity comes strangely at a time when new applications of spherical trigonometry are being found. GPS devices have some of its formulas built in. It's amusing to see bibliographies of research papers in computer graphics and animation (for use in movies like those made by Pixar) referring to nothing older than last week, except for that stodgy old spherical trig text.

In all fairness, the author's declaration is only partially true. A quick search on amazon.com for "spherical trigonometry" revealed a dozen books, some old (among which a 2005 reprint of J. Casey's 1889 classic) but some pretty new. However, there is no denying it that, notwithstanding numerous and important applications, the subject of spherical trigonometry has been excluded from high school and teacher education curricula many decades ago. I, for one, do not have a book dedicated to the subject, although some of the books in my library have chapters covering the essentials.

The present book is very well written; it leaves a clear impression that the author intended to endear - not merely present and teach - spherical trigonometry to the reader. Although not a history book, there are separate chapters shedding light on the approaches to the subject in the ancient, medieval, and modern times. There are also chapters on spherical geometry, polyhedra, stereographic projection and the art of navigation. The book is thoroughly illustrated and is a pleasant read. Chapters end with exercises; the appendices contain a long list of available and not so available textbooks and recommendations for further reading organized by individual chapters. The book made a valuable addition to my library. I freely recommend it to math teachers and curious high schoolers.

That said, I'd like to digress to a pet peeve of mine. My first (practically a knee jerk) reaction on receiving the book was to the subtitle The Forgotten Art of Spherical Trigonometry. The author makes it clear that, although forgotten as a curricular subject, spherical trigonometry has been successfully used in various applications, including modern technology - I am just back from a trip during which I've been heavily relying on a GPS device. What do you make of that? My reading is that there was a small (relative to the world population) number of scientist and engineers that, when the need arose, had been able to locate the tools originally lacking from their knowledge base; they've been able to locate the tools and then successfully employ them for development and exploitation of novel technology. Those people may have been missing in specific knowledge but not in intelligence necessary for acquiring new knowledge on the go. The majority of the population is happily using the results of the effort by a few people without realizing nor caring to learn how much effort went into the new development, or what knowledge was required.

Now, do you really care how much mathematics went into building bridges, refrigerators, airplanes, or gasoline refinery? Mathematics is being used in every conceivable piece of technology, every branch of science, while some of it even proves useful to practically everyone in everyday life. So, my question is, Is it truly necessary to burden the students with a study of the subject which most of them won't be using in their work on the pretext of its extremely universal usefulness? I believe that past and present educational reforms keep beating a dead horse.

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