From Euclid to Leibniz: the long hunt for the line that just touches
A tangent is the straight line that just touches a curve at one point, heading the same way. Pinning that down took roughly two millennia, from Euclid's circles around 300 BC through Fermat and Descartes to Leibniz, whose idea of two infinitely close points still underlies the modern version.
The word comes from the Latin tangens, touching. Euclid discussed tangents to circles in book III of the Elements, and Apollonius, in his Conics around 225 BC, described a tangent as a line leaving no room for another straight line between it and the curve. Archimedes found the tangent to his spiral by imagining a point travelling along it.
In the 1630s Fermat devised adequality to compute tangents, including those of the parabola, while Descartes independently used normals, relying on the fact that a circle's radius always meets it at right angles. Roberval treated a curve as the path of a point combining simpler motions, de Sluse and Hudde produced algebraic recipes, and work by Wallis and Barrow fed into the calculus of Newton and Leibniz.
Older definitions caused trouble. One from 1828 described a line that touches a curve without cutting it, which would deny any tangent at an inflection point, where the tangent does cross over. The modern view follows Leibniz: draw secant lines through a point A and a second point B on the curve, and the tangent is their limit as B slides toward A. That limit requires differentiability, so where two circular arcs meet at a sharp vertex, no single tangent exists.
Formally, the tangent to y = f(x) at x = c passes through (c, f(c)) with slope equal to the derivative there, making it the best straight-line approximation to the function at that point. Circles, parabolas, ellipses and hyperbolas have no inflection points, a cubic has exactly one, and a sine wave has two per period. The same notion extends to tangent planes on surfaces and is one of the basic ideas of differential geometry.
Source: Tangent
