"The mark of a really great idea," says mathematics faculty member Glen Van Brummelen, "is its usefulness in different, unexpected contexts."

Like, for example, the world of sailors navigating courses through uncharted oceans. Or medieval Muslims finding the direction of Mecca for daily prayer. Or astronomers yearning to understand the movement of the stars, and geographers calculating the dimensions of planet Earth.

The really great idea at the center of all these? Trigonometry.

Despite its centrality in a vast variety of human endeavors, a new history of trigonometry hasn't been published in over 100 years. That will soon change, as Van Brummelen is working on just such a book through a fellowship with the prestigious Dibner Institute at MIT.

The Dibner Institute, which supports advanced research in the history of the natural sciences, mathematics, and technology, created the fellowships in order to "bring together a group of outstanding scholars, offering them a focused and hospitable setting where the priority is untrammeled study and informal exchange." Scholars from around the world apply, and competition is always fierce--particularly last year, because the program will not be offered beyond this year.

Van Brummelen is fascinated by the way trigonometry finds its roots in a number of different locations and circumstances. That very depth and breadth, however, is also what makes approaching the subject so difficult. "To come to terms with so many different influences on a single mathematical idea is a huge adventure," says Van Brummelen. Naturally, questions of culture and perspective come into play. "The biggest challenge is coming to a sophisticated understanding of several cultures and their interactions: Babylon, Greece, India, Islam, and the early West. None of these cultures uses the same concepts that we do, and it is oh-so-easy to think you understand a text, when you're missing some crucial concept."

Those challenges are compounded by the number of controversies surrounding ancient trigonometry--one of the biggest surprises of his research, says Van Brummelen. For instance, just where and how was trigonometry born? "Some claim that the measurements of slopes that were made by ancient Egyptian pyramid builders counts as a sort of trigonometry. Others say that certain Babylonian tablets talk about measurements of triangles, and so they count too. I reject both of those, since I don't really see a moving back and forth between angles and lengths there. So, there are actually four or five possible starting points for the subject, separated by almost 2000 years and three cultures.

"It's also rather curious," Van Brummelen continues, "that the story of trigonometry in ancient Greece has almost completely been rewritten in the last 35 years. You would think, since the Greek texts have been around so long, that the storyline would be more or less in place and only quibbles over details remain, but that's not the case. The last time a history of trigonometry was written was in 1900/1903, in German-and I find that I cannot use that book as a resource at all."

Fortunately, the Dibner fellowship opened the door to a wealth of many other research materials, including Harvard's library resources, with documents like a paper on Indian trigonometry written by John Playfair in 1798. Van Brummelen was also granted access to the Burndy Library on the MIT campus, owned by the family that runs the Institute. "It has first editions of every scientific work you can think of from the last 500 years, including a copy of Newton's Principia Mathematica that was annotated by Newton himself, in his own handwriting."

Van Brummelen hopes his book will afford a view into a part of human history and a pillar of mathematics that is too often given short shrift. "I've always been interested in trigonometry," he says, "partly because it's so poorly taught in schools-perhaps the worst of any aspect of the math curriculum." That's unfortunate, he says, because it's also "a beautiful synthesis of the two polar opposites of mathematics, geometry and number. And it's a perfect illustration of the maxim that mathematics needs the real world, particularly the sciences, to grow."

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