The origins of the tangent

The tangent and the secant -------------------------

The study of the properties of curves took off in the 17^th century with the introduction of Cartesian coordinates. Determining the tangent at a point was regarded as a fundamental problem. Since scientific works were written in Latin, this line was denoted by tangens, the present participle of a verb meaning "to touch." At the same time, French mathematicians such as Descartes used the French equivalent, "touchante," in their correspondence. The Latin term had prevailed in France by the following century.

Unlike the tangent, which merely brushes the curve, the secant cuts straight through it without a second thought. The former is often regarded as "the limit" of the latter. The adjective "secant" was formed during the Renaissance from the present participle of the Latin verb secare, meaning "to cut." The use of "secant" to denote a line can already be found in the work of Simon Stevin in the early 17^th century. The same root appears in several mathematical terms, including "segment," a piece of a line with "both ends cut off." Originally, the term segment of a circle was also used for what we now call an arc of a circle.

Naming the trigonometric functions --------------------------------

The word "sine" arose from a confusion between two Arabic words. The term "half-chord," used by the Indian astronomer and mathematician Aryabhata (476–550), was transformed into a word denoting a fold in a garment or veil; it was then translated into Latin as sinus, which has the same meaning.