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Triangles before Logarithms: Trigonometry in the Lost Century

Glen Van Brummelen (Quest University) began his lecture in the MAA Carriage House on March 10 with a quick test: “Name for me a 16th-century mathematician,” he challenged.

Although the audience did come up with one, most people can’t. Van Brummelen noted that while 17th-century luminaries such as Newton and Leibniz and Descartes trip off the tongue, even math enthusiasts are hard-pressed to name a mathematician active between 1500 and 1599.

Glen Van Brummelen in the Carriage House

“Why is that?” the historian of mathematics asked. “Did nothing happen in the 16th century?”

Far from it, as Van Brummelen convinced listeners with his talk, “Triangles before Logarithms: Trigonometry in the Lost Century.” The lecture was cosponsored by HOM SIGMAA, the history of mathematics special interest group of the MAA.

It turns out that the 16th century was a pivotal one for trigonometry, a trigonometry Van Brummelen says is much richer and more vast than what is taught today. The ancient Greeks did trigonometry more often on the surface of a sphere than in the plane, but spherical trigonometry has vanished from school curricula.

“It disappeared in the year 1955,” Van Brummelen said, “which is a year I mark as a tragic moment in the history of mathematics.”

The pinnacle of medieval trigonometry, according to Van Brummelen, was Johannes Regiomontanus, author of what, in the early days of printing, became the standard trigonometry text in Europe.

“In the year 1533 everybody thought the way Regiomontanus thought,” Van Brummelen said. In terms of only the sine and the versed sine, that is, and without any symbolic algebra to lighten the linguistic load of mathematical propositions.

Regiomontanus may have penned the go-to treatise on trig, but it was Georg Rheticus whose book—16 pages of trigonometric tables—Van Brummelen deems “one of the strangest books ever to appear on the Catholic list of banned books.” Was the Church threatened by Rheticus’s tidy geometric formulation of six trigonometric functions? Probably not. More likely it was Rheticus’s connection to Copernicus—he convinced Copernicus to publish his work and wrote the introduction to it—that got the trig tables blacklisted.

Most of Rheticus’s contemporaries ignored his work, but not François Viète, whom Van Brummelen calls the most famous mathematician of the 16th century and the father of symbolic algebra. Viète was first and foremost a trigonometer, Van Brummelen said, and devised notation to facilitate his work with triangles.

But since it was his notation, all but Viète had a hard time fathoming it.

“Some of you professional mathematicians know what happens when somebody works in his room for 20 years, never leaves the room, invents his own notation, gets this whole wonderful structure, and nobody knows what the heck he’s talking about,” said Van Brummelen. “That was François Viète. Nobody had a clue what he was up to—and it was brilliant.”

Near the end of his talk, Van Brummelen reiterated the importance of the so-called lost century. And it wasn’t just the transformation of trigonometry that made the 1500s crucial to the history of mathematics and science more broadly.

“This is the century where the barrier between mathematics and the physical world broke,” Van Brummelen said. “This is the century where mathematics became the engine of science. It’s not calculus that did it. It’s trigonometry.”

At the beginning of the 16th century, the “practical geometry” used to erect a cathedral, for example, relied on similar triangles and the Pythagorean theorem. And as late as 1581, Maurice Bressieu relegated his real-world application of trigonometry to an appendix. Metrices astronomicae contains, as an afterthought almost, what Van Brummelen described as “the first story problem in trigonometry,” in which the author finds the height of a castle. And how did Bressieu begin his discussion of the problem?

With an apology, a testament to the disconnect between cutting edge and applied mathematics: “Hoping that this will not be offensive to the reader. . . .”  

By the early 17th century, however, after the first use of the word “trigonometry” and the application of trigonometric techniques to geodesy and geography, gnomonics and architecture, what had been known as the “doctrine of triangles” had come into its own.

Van Brummelen doubts it’s a coincidence that Galileo declared mathematics “the language in which God has written the universe” in 1623, soon after the curtain fell on the 16th century.  

“I’d like to think it was trigonometry that allowed him to make this statement,” Van Brummelen said. —Katharine Merow