To open one of his public lectures, before a large audience of high-school students, the famous Fields medalist Cédric Villani took up the recurring question of what maths is good for. His answer, in substance, was that mathematics is a truly remarkable tool—one that is not natural, but was invented and developed to describe and understand the world, and to act upon it. It is, above all, a tool for solving problems, as is the case for the other sciences. Moreover, mathematicians work according to well-defined rules (of logic) and precise criteria (aesthetic, artistic…).
Mathematical theories are built by solving problems that may be theoretical or practical, internal to mathematics or raised by concrete situations. One emblematic "internal" problem, which arose very early in the history of humankind and was taken up by many great scholars of earlier centuries, concerns the concept of the tangent to a curve. Let's look at one particular aspect of this question, restricting ourselves to the case of plane curves assumed to be "regular". The aim is to study how such a curve "deviates" from one of its tangents in the neighborhood of the point of tangency under consideration.
Let's start by drawing a circle
In the plane, consider a "smooth" curve (C) and one of its points P, from which the tangent (T) to (C) is drawn. In the "immediate vicinity" of P, (C) very nearly coincides with the line (T); the curve then gradually moves away from (T) (except, of course, in the particular—and not very interesting—case where (C) is itself a line). How does the deviation of (C) from (T) vary?
The answer obviously depends on how much (C) curves near P. If the curve (C) represents a road with a bend at point P, then, depending on whether the turn is "wide" or "tight", (C) moves "a little" or "sharply" away from (T).
