Tangent

Lore & Background

Euclid referenced the tangent to a circle in Book III of his Elements around 300 BC. Apollonius, in his work Conics around 225 BC, defined a tangent as a line such that no other straight line could fall between it and the curve. Archimedes found the tangent to an Archimedean spiral using a geometric method that related the spiral's properties to the tangent of a circle, not by considering the path of a moving point. In the 1630s, Fermat developed the technique of adequality to calculate tangents, similar to taking the difference between f(x+h) and f(x) and dividing by a power of h. Independently, Descartes used his method of normals, based on the observation that the radius of a circle is always normal to the circle itself. These methods led to the development of differential calculus in the 17th century. Roberval discovered a general method of drawing tangents by considering a curve as described by a moving point whose motion is the resultant of several simpler motions. René-François de Sluse and Johannes Hudde found algebraic algorithms for finding tangents. Further developments included those of John Wallis and Isaac Barrow, leading to the theory of Isaac Newton and Gottfried Leibniz. An 1828 definition of a tangent as 'a right line which touches a curve, but which when produced, does not cut it' was later dismissed, as it prevented inflection points from having any tangent. Modern definitions are equivalent to those of Leibniz, who defined the tangent line as the line through a pair of infinitely close points on the curve.

Reader's Guide

The concept of the tangent line is one of the most fundamental notions in differential geometry and has been extensively generalized, notably to the tangent space. The geometrical idea of the tangent line as the limit of secant lines serves as the motivation for analytical methods used to find tangent lines explicitly. The question of finding the tangent line to a graph, known as the tangent line problem, was one of the central questions leading to the development of calculus in the 17th century. Descartes stated that constructing the tangent to a curve was 'the most useful and most general problem in geometry that I know.' The precise mathematical formulation of the limit was given by Cauchy in the 19th century. Calculus provides rules for computing derivatives of functions, allowing equations of tangents to be found for power functions, trigonometric functions, exponential functions, logarithms, and their combinations. However, calculus also demonstrates that for some functions and points, the limit determining the slope of the tangent line does not exist, meaning the function is non-differentiable. This can occur if the geometric tangent exists but is a vertical line, which cannot be given in point-slope form since it has no slope. The tangent line is also related to the tangent plane to a surface at a given point, which is the plane that 'just touches' the surface at that point.

Did You Know?

• Euclid made several references to the tangent to a circle in Book III of his Elements around 300 BC.
• Apollonius defined a tangent as a line such that no other straight line could fall between it and the curve.
• An 1828 definition of a tangent as a line that 'touches a curve, but which when produced, does not cut it' was later dismissed because it prevented inflection points from having any tangent.
• The word 'tangent' comes from the Latin 'tangens,' meaning 'touching.'