“A line that cuts through a circle gave trigonometry one of its six fundamental functions. The Latin word for "cutting" now describes a ratio that approaches infinity.”
Latin secare means "to cut." Its present participle, secans, means "cutting." In geometry, a secant line is one that cuts through a curve at two points, as opposed to a tangent, which merely touches. The concept is ancient—Euclid's Elements discusses lines intersecting circles—but the trigonometric function came later.
In the 1590s, Georg Joachim Rheticus, a student of Copernicus, began compiling massive trigonometric tables. Thomas Fincke, a Danish mathematician, coined the term "secans" in his 1583 work Geometriae rotundi for the ratio of the hypotenuse to the adjacent side in a right triangle. The secant function was the reciprocal of cosine.
The secant has a peculiar property: it is undefined at 90 degrees and 270 degrees, where it shoots toward positive or negative infinity. In graphical terms, the secant curve has vertical asymptotes—gaps where the function ceases to exist. A line that cuts becomes a function that breaks.
Navigation, physics, and engineering all relied on secant tables before electronic calculators. Ships crossing the Atlantic used secant values for course corrections. The Mercator projection, which made straight-line navigation possible on flat maps, depends on the integral of the secant function—a formula that took over a century to solve after Mercator published his map in 1569.
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The secant is the least intuitive of the six trigonometric functions. Students learn sine and cosine first; the secant arrives later, feeling redundant. Why not just write 1/cos? But for centuries, when calculations were done by hand with printed tables, the secant had its own column, its own identity.
A line that cuts through a circle. A ratio that lunges toward infinity twice per revolution. The secant names both a geometric act and its algebraic consequence—the cut, and what happens when cutting becomes impossible.
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